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15.
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Lesson CXLVIII.
THEORY OF HABMONIC INTERVALS
(Intrpduction to Counterpoint)
A sequence of two pitch-units produces
a melodic interval.
A simultaneous combination of
two pitch-units produces a harmonic interval.
The
technique of correlation of simultaneous melod.ies •
de_pends er1tirely upon the composition of harmo11ic
intervals.
Any number of simultaneous parts (voices)
in counterpoint are formed by the pai'Fs.
These
pairs may be conceived as voices immediately
'
.
adjacent in pitch, as well as in any other form of
vertical arrangement (i.e. over 1, over 2, etc.).
The suc cess of harmonic versatility of
counterpoint depends upon the manifold of harmonic
intervals used in a certain style.
Limited quantity
of harmonic intervals results in limited forms of
the harmonic versatility of co unterpoint.
Thus, t:t1e
study of harmonic intervals becomes one of the
important prerequisites of counterpoint.
Harmonic intervals have dual origin:
1. Physical
•
•
2. Musical
Tl1e physical origin of harmonic intervals
leads back to the simplest ratios.
•
The musical origin
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16.
of intervals i s based on selective and combinatory
processes.
All semitones, i.e. units of the equal
temperament of twelve, are the structural units of
all other harmonic intervals available in such
equal temperament.
As they appear in our hearing,
they amount to the following forms:
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i = 1,
i
=
5,
1 = 9,
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i = 2,
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1
=
=
6,
i = 3,
i
10, 1
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=
7,
i = 4,
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11, 1 =.12
This completes the entire selection
within one actave range.
An addition of intervals
to an octave produces musically identical intervals
over one octave, as the similarity of different
pitch-units within the ratio of 2 t o 1 is so great
that they even have identical musical names .
The
system of musical notation introduces, among other
forms of confusion, tl1e dual system of the interval
nomen
. clature .,
Thus, an interval containing three
semitones may b e called either a minor third or an
augmented second.
Simple ratios of acoustical int.ervals are
merely approximate equivalents of the harmoni c intervals of equal temperament.
It is not scientifically
'
correct to think the way the majority of acousticians
d o, that a 5 to 4 ratio is an equivalent of a major
U·
third or a� to 5, of a minor third, or a 7 to 4, of
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17.
a minor seventh, etc., as these intervals deviate
considerably from their equivalents in equal
•
temperament.
It is utterly impossible to follow the
methods established by some acousticians in studying
the type and quality of intervals in the equal
••
•
temperament of twelve as compared to their equiva
lents in the simple acoustical ratios*
The so-called
consonance is a totally different type of intervals
musically or acoustically.
If music had
to use
'.,
acoustical consonances only, yet being confined to an
equal temperament of twelve, the only real consonance
would be an octave, i.e. no two pitch-units bearing
different names would ever be used, and we would
never have either any harmony or counterpoint.
The
reason for this is th.at no other intervals than an
octave or a perfect fifth, with a certain allowance,
are consonances within the equal temperament..
All
other intervals are quite complicated ratios.
Thus,
the art of music has its own possibilities based on
the limitations within a given manifold of our tuning
system.
Acoustical consonances produce a so
called natural harmonic scaj..e, which consists of a
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fundamental with all its partials appearing in the
sequence of a natural harmonic series (i.e. 1, 2, 3,
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18.
4, 5, 6, 7, a, 9, etc.) .
The ratios of acoustical
consonances are equivalent to the ratios of
vibrations producing pitches.
For example, a .£
2
ratio means that if the actual quantities representing
both the numerator and the denominator were
multiplied by a considerable number value, they
would actually sound as pitches.
,
•
3 , as suc,h,
While
2
sounds to our ear as the resultant of interference
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of 3 to 2,
cycles per second sounds to our ear
as a perfect fifth.
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Our ear accepts pitch-units and their
ratios as they reach said ear and the auditory
consciousness and not as they are induced upon us
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in the traditional musical schooling.
For example,
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19.
a melody played simultaneously in the key of c and
in the key of b next to it, or a seventh above,
sounds decidedly disturbing to musicians of our
time.
Yet an interval that is musically identical
is acoustically so different that being placed
three octaves apart it produces a musically consonant
•
•
The reason for this is that in such
impression.
absolute intervals as seventh three octaves apart
•
approximates the the 15 to 1 ratio, 1.e� the sound
of a 15th harmonic in relation to its fundamental.
And when the pitches are so far apart the deviation
from equal temperament becomes less obvious for our
pitch discrimination.
The following tables offer a
group of examples illustrating musically consonant
intervals which are usually classified as dissonances,
and with their correspondence to the proper location
of harmonics.
In all these cases no octave
substitution can be made without affecting the
actual state of consonance.
Figure II •
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(please see next page)
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Likewise, musical consonances being
placed into a wrong pitch register, such as low
register, produce upon our ear an effect of musical
dissonances.
The reason for this is that being an
approximation of simple ratios they require the
placement of their fundamentals at such low
frequencies that they are below the range of
audibility.
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For example, a major third being associated
ratio would require that the fundamental be
located two octa�es below the fourth h�rmonic.
Music
being played in major thirds in the contra-octave
simply would not permit the physical existence of
such fundamental.
The following tables offer three
•
examples of the low setting of intervalsr
Figure III.
a
(please see, next page)
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With this understanding in mind we can
see that no serious theory of resolution of
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dissonant intervals may be devised without specifica-
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23.
tions to exact octave. locati on of the i nterval.
Thus, when we come to the theory of resolution of
i ntervals it wi ll merely be offered for the purpose
of the versati le treatment of the prog ressi ons of
harmonic i ntervals-, and not for the purpose of
exterminati on of dissonances.
••
Esthetically as well
as physiologi cally we desire sequences of tensi on
and release, and as different harmonic i ntervals
produce different degrees o f tension the versatility
of the sequence of i ntervals wi ll satisfy such
requirements.
C
It has often beeB-.1. the case that music
written according to the rules and regulations of
the dogmati c counterpoi nt does not sound esthetically
as convincing as its counterpart in the XVI or XVII
Century.
This inferi or quality is due to the
limited quantity of harmonic intervals and the forms
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treatment of the latter.
A. Classificatio n of Harmonic Intervals
within the Equal Temperament of Twelye
All harmonic intervals may be classified
into two groups:
•
1. With regard t o their density. i.e.. the
fullness of sonority, and
,
2. With regard to their tensi on, i.e.. their
dissonant quality.
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24•
Classif ication of density evolves from
the intervals producing the emptiest effect upon
our ear up to the intervals producing the fullest
effect�
The following table is only an approximate
one; nevertheless, it serves the purpose with a
certain degree of approximation, 1. e� the first few
••
•
intervals sound decidedly empty and the last few
sound decidedly full, while in the center there are
a f ew intermediate ones•
•
Figure IV.
•
••
•
Classif ication of tension is based upon
the separation of consonances f rom the dissonances
and the separation of the consonances and dissonances
by nam.e f rom the consonances and dissonances E,Y.
sonority.
All cases when consonances and dissonances
correspond respectively by name and sonority imply
the diatonic intervals�
And all cases when
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25.
consonances and dissonances do not correspond t o
their original names produce chromatic intervals.
The group of diatonic cor1sonances includes perfect
unisons, perfect octaves, perfect fifths, perfect
•
fourths, major thirds, minor thirds, major sixths,
minor sixths.
The group of diatonic dissonances
includes major and minor seconds, major and minor
sevenths, major and minor ninths.
All the chromatic
intervals are classified into augmented and diminished.
•
The Augmented Intervals:
Unison, 2nd
3rd, 4th, 5th, 6th.
The Diminished Intervals:
Octave, 7th, 6th, 5th, 4th, 3rd.
;Figure V.
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2nd
n
The augmented unison
"
n
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n
n
3rd
4th
equivalent to minor 2nd by sonority.
"
n
n
n
n
n
n
n
n
"
n
major 3rd
"
perfect 4th n
no diatonic interval.
n
"
minor 6th by sonority.
minor 7th n
"
The diminished octave "
n
n
major 7th "
n
n
n
n
n
n
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n
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n
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n
5th
6th
7th
6th
5th
4th
3rd
n
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n
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major 6th "
ferfect 5th"
n
no diat-0nic interval.
maJor 3rd by sonority.
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major 2nd n
Thus, the following intervals are
•
n
n
n
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•
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n
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consonanc-es by sonori ty. The augmented 2nd, 3rd, 5th;
the diminished 7th, 6th, 4th.
All other chromatic
intervals will be treated as dissonances with the
•
resoluti ons corresponding either to diatonic or to
chromatic dissonances.
r
•
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27.
Lesson CXLIX.
B. Resoluti on of Harmonic Intervals
The necessity of varying tension implies
the procedl.ll'e known as resoluti on of intervals.
It
is important to realize that the variation of tension
may be g radual as well as sudden, i . e. the transition
•
from a more dissonant harmonic interval to a less
dissonant one and finally i nto a fully consonant one
i s as desirable as a di rect transit ion from e.xtreme
tension to full consonance .
In the followi ng tables i ntervals such
as perfect 4th and 5th are included as well, not
for the purpose of relievi ng them from tensi on, but
for the pur�ose of devising different useful manipula
tions forming contrapuntal sequences.
The quantity of
resolutions known to a composer has a definite effect
upon the harmonic versatility of his counterpoint.
For e:xaniple, if one knows only four re.solutia>ns • of a
major 2nd (which is the �sual case) as compared to
the twelve possible resoluti ons, the amount of
musical possibi lities is considerably less.
Thinking
in terms of vari ati ons one can see that the number
of permutati ons available from four or from twelve
. .
'
elements is so diffebent i n quantity that they cannot
twentyeven be compared (the first giving/four variations
•
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28.
and the s eco nd givi ng 479, 001, 600 vari ations) .
i s easy to see that having such losses on the.
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quantity of resoluti ons of each harmoni c interval,
the loss on the total of v.ersatili ty of counterpo int
i s incalculable..
There is no need i n memorizing all
the details of the reso luti on o f intervals , as there
are general underlyi ng pri nciples evolved thr oug h
the
tradi ti on of centuries•
1.
All diatonic i ntervals resolve
thr oug h either outward or i nward or o�lique motion
of each voice on a semi tone or a wh ole tone. *
2. When a resoluti on is ob tained
through oblique moti on the sustained voice may
produc e a leap on a melodic i nt erval of a perfect
4th, either up or dow n.
3. All i ntervals known as 2nds have a
tendency to expand.
All i ntervals kn1.,,wn as 7ths
have a tendency to contract.
All 7ths are the
exact equivale nts of 2nds i n the oc tave i nversi on
(i.e. all pi t ch-uni ts are identi cal with those of
the 2nds) . All the 9ths have a tendency to contract..
All the 4ths and 5ths are neutral, i .e. they either
expand or contract. ·
Thus,. the enti re range of permutations of
semi tones and whole tones, wi th their respective
* An 1 = 3 is also correct when such an i nterval
- represents two adjacent musical nan1es (c - dff,
for example) .
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directions, constitutes the entire manifold of
resolutions ..
Refer to Resolution of Diatonic Intervals
chart be.low.
Resolution of Diatonic Intervals
•
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Seconds
Ninths
Sevenths
---
Fourths
- --
and Fifths
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--
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(enharmonic)
The following is a complete table of
resolutions of diatonic intervals ..
The i11tervals in
parentheses are the secondary resolutions.
They are
used in all cases when the first resolution produces
a dissonance.
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All chromati c i ntervals which are
augmented have a tendency of expansion•
.And all
chromatic int ervals which are dimi nished have a
tendency of contraction.
The method of reasoni ng
in resolving augmented or diminished intervals is
d-f
is a 2nd derived throug h augmentation
as follows:
C
of a major second, either through alteri ng of d t o'd�
or of c� to c�.
d
d�
a 2nd
or ,J.,�
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C
Thus, ori gi nally it mi g ht have been
Consi dering the dual ori gin of such
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further in the same di recti•..,n, i . e .. to e; or if c t,
is the alterati on of c� it has the inertia of
moving to b�.
Such two steps taken individually or
simultaneo usly constitute the fundamental resolutions.
An
analogoos procedure must be·applied to the
diminished intervals where the diminutions are
produced throug h inward alteration.
The following is a complete table of
resol uti ons of chromatic i ntervals.
When a chromatic
interval resolves into a consonance by sonority, the
si g n "enh. " is placed above it (enharmonic).
When
the interval of resolution is surrounded by paren
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Figure
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(please see next page)
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In the old counterpoint we often find a
different type of resolutions from the ones described
above.
They were known as kambiata resolutions,
which are conceived as a melo<:lic step of 'a 3rd instead
of a 2nd.
No good reason bas ever been given why such
resolutions would be used.
••
I offer an hypothesis for
the explanation . of these resolutions which I believe is
the only one to be correct •
As the tradition of old counterpoint
•
was developed, while the pentatonic (5 units) scales
'
.
were in use, some of the pitch-units o'r full diatonic
(heptatonic = 7 units) scales were absent. Thus, if
d
we find that in an interval
, d moves to e, while
C
•
c moves to a (instead of b) , a kambiata takes place
merely because such scale may be a pentatonic scale
and the unit b does not exist.
This approach offers us a definite
principle of resolution of int ervals in the scales
which have not been in use in the classical
traditional music confining all the resolutions
merely to the next step with the f'o.llow�g musical
name ,,,
For example, in harmonic a minor, the interval
a
1f may be resolved through moveme· nt of th� lower
g
u
voice only to f,, as no other pitch-unit with the name
f exists in such scale,,,
· This concludes the Theory of Harmonic
Intervals ,,,
0
0
•
•
S C H I L L I N G E R
J O S E P H
C O R R E S P O N D E N C E
C O U R S E
With: Dr . Jerome Gross
Subject : Music
Lesson CL.
Theory of Correlated Melodies .
(Counterpoint)
As counterpoint represents a system of
correlation of melodies in simultaneity and continuity,
it is absolutely essential to be thoroughly familiar
with the constitution of melody.
.
Only by being
familiar with the ma terial of the Theory of Mel ody is
the successful accomplishment of such task possible.
Correlation of melodi es is usually considered to be
one of the most difficult procedures.
As the structural
constitution of one melody is unknown theoretically, the
combination of two unknown Quantities is an entirely fan
•
tastic task to undertake.
It is not only a problem of putting two
voices together, but a problem of either combining two
melodies already made, or a composition of two melodies
•
with distinct individual characteristics .
As each
melody consists of several components, such as the
rhythm of duratior1s, attacks, melodic forms, the forms
of tra jectorial mo tion, etc., the correlation of two
·C
melodies in addition to the above described components
•
0
0
•
•
•
2.
adds one more : harmonic correlation.
Thus, counterpoint
can briefly be defined as a system of correlation of
rhythmic, melodic and harmonic forms in two or more
conjugated melodies.
As the' forms concerning one individual melody
,
are known thro ugh the previous material , we will first
cover the field of harmonic correlation which is based .
.•
on tJ:1e Theory of Harmonic Intervals .
After covering this
particular branch we shall return to the other forms of
correlation for the purpose of achieving t�e final
results offered by the contrapuntal technique.
A. Two-Part Counterpoint
L
The fundamental technique in writing two-part
counterpoint is based on writing one new melody to a
given melody.
A given melody is usually abstracted
from its rhythm of durations, thus producing a purely
melodic form which may be taken from a choral as well as
from a popular song.
The usual way of presenting such
an abstracted melodic form is in whole notes.
Such a
melodic form is usually known as Cantus Firmus (firm
cl1ar¢: = canonic or established chant) .
Our abbrevia
tions for Cantus Firmus will be C .F. and for the . melody
written to it, counterpoint or C. P.
The first forms of
counterpoint will be classified through the quantity of
attacks in C.P. as against one attack in C.F.
Thus,
0
0
•
•
0
all the fundamental forms of counterpoint will be as
follows:
CP
-
CP
CF
- a
. . . . . . n
2,
3
1,
er- •
This form of counterpoint, through inter
national agreement for a number of centuries, implies ·
the usage of co nsonances only.
As we shall have four
fundamental forms of harmonic correlation and some
of these forms will be polytonal (i.e., there will be
two different keys used simultaneously), we will have
to use consonances by name and by sonority.
The
positive requirements for harmonic correlation in
2-part C.P. are:
a. The variety of types of intervals (i. e.,
intervals expressed by different numbers).
b. The variety of density.
c. Well defined cadences expressed through the
leading tones moving into the ir axes.
d. Crossing of C.P. and C.F. is permissible
when necessary.
The negative requirements are:
a. The elimination of co nsecutive intervals whioh
are perfect unisons, octaves, 4ths and 5ths.
C
0
0
•
No consecutive dissonances.
Thus, the only
intervals to be used in parallel motion are
thirds and sixths.
b. Motion toward such intervals only through
contrary (outward or i nward) directions.
c. No repetiti on of the same pitch-unit in CP
unless it i s i n a di fferent octave.
The forms of harm onic relations previously
••
used in time continuity (see Theory of Pitch Scales)
will be used in counterpoint as the forms- of
'
simultaneous harm onic correlation.
•
1.
u. - u.
2.
u. -
3. P. -
Forms of Harmonic Correlation
Unitonal - Unimodal (identical scale
structure and key si gnat ure).
P. Unitonal - PolyJjlodal (a family scale
u.
wit h identical key signature).
Polyt onal - Unimodal { i dentical scale
structure, dif ferent key signature) .
4. P. - P. Polyt onal - Polymodal (di fferent scale
structure, different key signature).
::( n the XIV Century, . in the ca.se of
Guillaume de .Machault* we find a fully developed type
2, and in some cases an undeveloped type 3.
Only the
* The phonograph records of a Mass written by this
composer for the coronation of Charles V are
available. (Les Paraphonistes de St. Jean des Matines
and Brass E nsemb le conducted by Van) . The reconstruction
of Machault ' s 2 and 3-Part Madrigals in our musical
not ation is publi shed by the Hi·storic Musi cological
Society of Leipzig in 1926. Not aYailable in U.S.A •
'--"'
•
•
•
0
0
•
L.
ignorance and vanity of the contemporary composers
make them believe that they are the discoverers of
polytonal counterpoint.
The greatest joke is on
the modern French composers who make the claim of
priority, not being aware that their direct musical
ancestors were the originato rs o f this style centuries
ago.
••
It is also unfortunate that the idea of poly
tonality goes hand in hand with the so-called
"dissonant counterpoint", i.e., the counterpoint of
•
continuous tension without release.
Music based on
polytonality with resolutions is a v ery fruitful,
highly promising and almost undiscovered field .
The usual length of C. F . is about 5, 7, 9
L
or more bars, p referably in odd numbers (this require
ment is traditional) .
The selecti on of different key
signatures for the types 3 and 4 is entirely optional.
Any two scales, the root tones o f which produce a
conso nance, may be used for this type of counterpoint.
The best way of cor1struc ting exercises is the placement
of C.F. on a central staff surrounded by two staves
below, and two staves above, assigning each staff for
•
a different type of counterpoint.
In the followi ng group o� exercises each
part mus t, be played individually with C.F.
Thus, each
example produ.ces four types o f counterpoint with a
I
•
•
0
0
6•
•
historical emphasis of eight centuries, as the first
and second types were consid erably developed during
the middle ages, and the third and the fourth types
are mostly used in the music of today •
•
It is important to realize that all forms
of traditional contrapuntal writing were based on the
••
conception of each melody being in a different mode,
and w e can even trace the polytonal forms (though in
their embryonic form) as far back as the XIII Century.
•
•
•
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7.
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Lesson CLI.
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accompaniment , doub le pedal point may be used in
.
additi on t o 2-part cou nterpoint . The root tones of
both cont rapuntal part s become the axes which must
be assig ned as chordal functions of a double pedal
For exam ple, cou nterp oint type l (giving t he
same pitch-units for poth voi ces) may be consi dered
as a root t one or a 3rd or a 5t h, etc., of a simple
c hord structure.
Then, havi ng c as a axi s for bot h
cont rapu ntal parts, the pedal point will become
...._,,
.
,_
'
As a temporary devic e for harmonic
point.
•,
.
-
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.
0
0
a.
C
g or
C
, c,
a f
e
e tc.
This device is applicable to all four
types of counterpoint.
For example, in type 2, if one
contrapuntal part is ionian c and the other ae olian a,
the y may represent a root and a 3rd, or a 3rd and a
•
The pedal point in such case
5th, etc., respectively.
will be
e
C
or
f'
a
two axes as c and
•
pedal points.
In the types 3 and 4 with such
e tc.
a P,
w e may use
e ),
al,
or
C
f'
e tc.
as
Each double pedal point must last
through the entir e co1:1trapuntal continu:i:ty.
More £lexible forms o f harmo�1zation of
the 2-part counterpoint will be offered later.
CP
CF
2 a
In devising two attacks of a counterpoint
against one attack of the C.F . , the following combina
tions of barillonic intervals ar e possible :
(c - consonanc e ;
C
- C
d - dissonance )
c - d
d -
C
d - d*
In the old counterpoint all the se cas e s were used in
both strict and free style, with the
*In scal ewise contrary motion only.
•
exc eption
of a
0
0
dissonance being on the first beat.
Thus, each bar may start with either a
consonance or a dissonance.
And, in the case of
�� == 2, all dissonances require immed iate resolutions .
•
Here are a few examples of such contrapuntal exercises.
Figure II.
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12..
Lesson CLII.
CP
CF
= 3 a
Three attacks of CP against one attack
of CF offer the following combinations of harmonic
intervals:
C - C - C
•
C
-d
-· C
d
- C
C
C - C
-
- d � resolution
d - C - d
d - d*-
�
resolution
C
c - d - d*
The d - c - c combination offers a new
device which becomes possible with three and more
attacks .;
\Ale shall call it a del.ayed (or indirect)
resolution.
Instead of resolving a tense interval we
move it to another cot1sonance, after which we resolve
the dissonance.
This device accomplishes two things:
(1) it produces a psychologica l suspense, thus
making music more intriguing;
* I n scalewise contrary motion only
L
•
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13 .
(2) it produces ipso facto a more expressive
melodic form.
Examples of Delayed Resolut�ons
Figu,re III,.
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CP
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4 a
•
Four attacks of C.P. against one attac-k of
C.F . offer the following combinations of harmonic
intervals:
C - C - C - C
c - c - c - d � resolution
C - C C -
d -
d -
C
C - C
·d - c - c - c
C - C C -
d - d*
d - d*-
d - d**
V
•
C
- C
-
d
C--
d
d
- C - C
C
- d -
d - C - d - C
+
-t
resolution
resolution
There are wider possibilities in the field
CP
of delayed resolution for CF = 4 •
Parallel axes, centrifugal and centripetal
forms become more prominent.
*In scalewise co11trary motion only.
�Either as * or two independent dmssonances, both of
which are resolved by the follo wing c - c in any
order.
0
0
15 ..
Exawples of Delayed Resolutions
Figure V.
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It is also useful to know all the
advantageous starting points for the scalewise
passages ending with a consonance.
Examples of Passages Ending with a
Consonance.
Figure VI.
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,
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16.
CP = 4
a
CF
Figure VII.
Examples of
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17 .
•
Lesson CLIII.
CP
CF
5 a
It is no longer necessary to tabulate all
the possible combinations of c and d.
Best melodic qual ity of CP results from
••
an extensive use of delayed resolutions .
The latter ,
being combined with the variety of intervals and v-,ith
the scalewise passages produce most versatile forms
of melody.
The devices for delayed resolution,
impossible for less attacks than five, are as follows :
•
d, d
c
..!,_;,i
d , c , i.e. : the first dissonance is
"�
followed by the second dissonance with its resolution,
then by the repetition of the first dissonance with
its resolution;
d...,
, d2 c
71
d 2 c , i .e.: the first dissonance is
�
followed by the second dissonance without resolution,
followed by the resolution of the first dissonance,
then by the repetition of the second dissonance
followed by its resolution.
Examples of Delayed Resolutions.
Figure VIII.
(please see next page)
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19 .
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CF
CP
6 a
The new devices for delayed resolutions
possible with six attacks:
d , d2 d , c
d 2 c , i. e . : the first dissonance, the
...__::,,
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second dissonance, the repetition of the first
.
.
dissonanc-e with its resolution, the repetition of the.
•
•
second dissonance with its resolution;
d1 d
o
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d
c
c , i.e. : the first dissonance, the
�
second dissonance, the resolution of the 'first
dissonance, the repetition of the second dissonance,
the delay, the resolution of the second dissonance;
•
d , d2 c
'--=="
d, c
.....
c , i.e. : the first dissonance, the
:,,
second dis sonance with its resolution, the repetition
of the first dissonance, a delay, resolution of the
•
first dis sonance;
d1 c
....,_
c
_,,
d
�
c
c , i . e . a combination of two groups
�
by three, each consisting of a dissonance, a delay and
a resolution.
Other combinations can be devised in a
similar way.
For example: d 1 c
..,
a combination of 2 + 4 .
"'
'
d2 c
d 2 c , which is
;,,
While using six attacks agai nst CF, it i s
easy to devise a great variety of melodic forms and
interference pattern (see: Melodi:za.tion of Harmony) .
.,
0
0
20.
•
Exam ples of De layed Resolutions.
Fip;ure XI .
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m
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.
Examples of Scalewise Passages Ending wit h
a C onsonance.
Figure XII.
•
· Examples of
CP
CF
= 6 a
Figur e XIII.
•
(please see next page)
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0
0
21.
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22.
•
CP - 7 a
C, Seven attacks of CP against one of CF offer
new forms of delayed resolutions.
•
The number of new
comb inations grows, and it becomes quite easy to
develop vari ous melodic forms, built on parallel,
converging and diverging axes.
Examples of Delayed Resolutions.
Figure XIV,
•
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.....
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j
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=L::. ·-
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Examples of Scalewise Passage,s Ending
aI
with a Consonance.
Figure XV.
•
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23.
u
- CP
Examples of CF
= 7 a
Figure XVI .
(4)
I•
•
CP =
CF
8
a
Eight attac k s of CP against one of CF offer
a great variety of melodic form s.
The latter can be
obtained through the technique of delayed resolutions.
It is equally fruitful t o devise melodic forms by
mea n s of attac k-groups.
For example, thinking of 8 as
0
0
24.
i
series represented through its binomials a.nd
Interference groups can be carried out
trino�ials.
in counterpoint in the same way �s in the Melodization
of Harmony, where such groups were used against the
attacks of H.
Examples of Delayed Resolutions.
Figure XVII .
•
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,
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All scalewis e passages ending w ith a
1�
•
•
•
•
consonance must start and end with the same pitch
unit, as such is the property of our seven-name
musical system.
Example s of Scalewise Passages
Ending with a Consonance,
Figyre XVIII.
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XIX.
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.- ,.., r
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Fig ure
= 8 a
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CP = 8 a gives sufficient technical
CF
equipment for any greater quantity of att acl<:s. It is
CP - 12 a an d
desirable to devise such cases as CF
CP = 16 a, as they provide very usable material for
CF
the animated forms of passage-like obligato. Under
usual · (traditional) treatment, such groups with many
attacks of CP agains t CF remain uniform or nearly
•
-
0
0
26.
uniform in durations.
The most important conditions f or
obtai ning an expressive counterpoint:
(1) abundance of dissonances;
(2) delayed resolutions;
(3) interference attack- g roups•
••
•
•
•
•
•
0
0
. .
J O S E P H
S C H I L L I N G E R
C O R R E S P O N D EN C E
With: Dr,• Jerome Gross
C O U R S E
Subject: Music
•
Lesson CLIV.
Composi tion of the Attack-Groups in
Two-Part Counterpoint.
In all the previous forms of counterpoint the
attack-group of CP against each attack of CY was
P =
A const.
constant: G
CF
The monomial attack group cons�sted of any
•
desirable number of attacks: A = a, 2a, 3a, . . . ma .
•
CP.
Now we arrive at binomial attack-groups for
This can be expressed as
counterpoint written to two successive attacks of the
ca11tus firmus consists of two . differ.ent attack-groups.
For instance :
'
(1 )
(3)
CP ,
CF, +
CP2
CF 2
_
-
a •
2a + a
a '
(2 )
(4)
CP 2
+ CF
2
_
-
3 a + 2a .
a
a '
-- a
a ' •••
8a
The selection of number values for the attacks
of CP against the attacks of CF depends on the amount of
contrast desired in the two successive attack-groups of
CP.
All further details pertaining tothis matter
0
0
2.
•
are in the respective chapter of the Theory of
Melodization .
•
Binomial attack-groups are subject to permut.ations.
For example :
CP, + C.P 2- = 4a + 2a
CF ,
a
a "
CF 2
This
•
binomial attack group can be varied further through the
Suppose CF has 8a.
permutations of the higher order.
Then the whole contrapuntal continuity will acquire the
following distribution of the att ack-groups:
CP, + CP,
CFa
CF,
CP 1-t
CF , -r
-
+
_CP, +
CF,-
4a + 2a + 2a + 4a +
a
a
a
a
2a+
a
4a +
a
4a +
a
2a
a
•
Polynomial attack-groups of CP against CF can
be devi.sed in a similar fashion.
The resultants of interference , their variations,
involution groups and series of variable velocities can be
used as material for this purpose.,
Examples of polynomial att ack-groups of
(1)
OP4-b
CF , - t,
_
-
3a + � + 2a + 2a + �
a
a
a,
a
a
(2 )
CP ,-8
=
2a + � + ,! + � + 2a + � + ,! + ,!
a
a
a
a
a
a
a
a
(3)
CP &->
CF ,-5'
=
(4)
CF ,_,.
+ 3a •
a '
+ 2a •
a
.! + 2a + 3a + 5a + 8a .
a
a
a '
a
a
= 9a
a
+ 6a + 6a + 4a
a
a
a
CP .
CF .
•
Simplest duration-equivalents of attacks
will be used in the following examples.·
'
•
0
0
,
- ·..'\·
Figµre XX •
•
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4.
.
At this stage it shou ld not be difficu lt to
develop the t echn ique of writing one att ack of CP t o a
g rou p of attacks of CF.
In an exercise CF must be so
constructed as to permit the matching of one attack
agai nst a g iven attack- group.
I n a g iven melody, when
compos:l.ng a counterpa.rt, it is necessary to compose the
.
attack�g roups first • This shou ld be accom plished with
.
a view upon the possibilities of the t reat ment of
harm onic int ervals.
,
Whenever the assumed g rou p does
not permit t o carry out the resolut ion• requirements
'
(such as expanding of the second, contracting of the
seventh or the ninth, et c. ), the attack-group itself
must be reconst ructed •
•
As it was mentioned before, it is qu ite
pract ical to re-wri.te the given melody into u niform
durations first, and then t o assig� the advant ageous
attack- g rou ps.
After the cou nterpoint is written,. the
original scheme of durations ean be reeonstruc�ed.
Vf ith the pr.esent equipment , only such
melodies can be used as cantus f irmu s which are built
on one scale at a time, and the scale itself must belong
t o the Fir st Grou p (see Theory of P itch Scales) .
The procedure it self of dist ributing the
att ack- g roups of a given melody is a nalogous t o that
used in the branch of Harmo.nization of Melody, where
•
0
0
5.
attacks of a given melody were distributed in
relation to the quantity of chords accompanying
..
them.
The follov,ing is a melody subjected to
different attack treatments for the purpose of
writing a counterpart to it.
••
•
Figure XXI,
(please see next page)
•
•
0
0
Fig . XXI.
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0
0
7.
Lesson CLV.
In writing a counterpart to a given melody
(but without any considerations of the given harmonic
accompaniment) it is important to consider :
(1) the composition of attacks, and
(2 ) the composition of duratioas.
Composition of attacks depends upon the
'
degree of animation of the given melody.
If a lively
melody is to be compensated, the countermelody should
•
'
be devised on the basis of reciprocation of' attacks and,
finally, durations.
•
•
All the techniques pertaining to
variations of two elements serve as material for the
two part compensation (counterbalancing) •
If a lively melody is to be contra.sted, the
countermelody should be devised by summing up groups
of attacks together with their durations.
The sums of
durations of the given melody, with the specified number
of attacks ag.ainst eaeh attack of the countermelody,
define the durations of the counterpart.
If a slow melody is to be compensated
(counterbalanced) by a slow co unterpart, the technique
. of reciprocation of attacks and durations should take
place.
Variations of two elements provide such a
technique.
If a slow melody is to be contrasted, the
countermelody should be devised fir st by defining the
•
0
0
8.
number of attacks in the countermelody against each
indiv idual attack of the given melody, after which
the sum of the attacks of the counterpart will
represent the duration, equivalent to the duration of
one attack of the given melody.
Melodies where animated portions alternate
••
with the slow ones, or with cadences, are particularly
suited for the compensation method.
In such a case
when one melody stops, the other moves and vice versa.
We shaJ 1 analyze now th.e prob·lem of writing
the counterpart to a given melody.
Let us take Ben Jonson rs "Drink to Me Only
With Thine Eyes".
The melody reads as follows:
,
•
,''
"
'
..
�
�
•
-,
I
•
•.
....
...
-c
-
�
Reconstruction of this melody into a CF
gives it the following appearance :
)I
•
�
•
'
j
....
'
•
.....
This is a fairly animated type of melody.
0
0
•
Let us devise the scheme of durations for CP.
One
be to make each attack of CP correspond to T.
Thus
•
of the •simplest solutions for a contrasting CP wo� ld
we would obtain CP
=
4a and a
=
6t.
For a less
moderate contrast we could assign CP = Sa and a = 3t.
To obtain CP of the counterbalancing ty pe wou ld require
the assignment of two contrasting elenents, if such
can be found in CF.
As T, = 2a and T2
=
6a, and as T3 =
= 5a and T� = a, this CF provides suf f ic ient material
•
for assig ning two elements an d f or compensating them in
CP. There is of course no way to counterbalance the
original v·ersion of this melody.
•
Thus, we have obtained the following three
solutions, each different but equally acceptable.
Figure XXII,
(please see next page)
•
0
0
•
If �
Q
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;,.�
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(Fig. XXII, cont . )
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0
0
12.
•
Lesson CLVI.
Direct Composition of Durations in
0
Two-Part Counterpoint,
In composing an original two-part counter
point it is often desirable to compose the two counter
parts rhythmically first.
;
The entire technique
concerning binomials and their variations (see Theory '
of Rhythm) is applicable in this case.
Counterbalancing (compensation) is achieved
tbro-ugh the permutation of binomia1s. ·, and ',this may
follow through the higher orders .,
For example:
•
3
J.
,,,
J ,I J
q.
iJJ
q.
d.
i11
jJJ J.
4,
J.
JJJ
11l i1l �-
•
Which part is written first (thus becoming
CF) is not essential in such a case.
It is essential,
however, to write one part completely, and not section
by section.
GP must be written after CF is completed.
For more diversified rhythmic continuity,
C
•
0
0
•
resultant s with an even number of terms can be us ed.
T he binomials cons tantly reci procate i n such a case.
8
J.
For exam ple: T = r
8t ) .
8+7(+
I J I' J . J J
,,r] J
J . PJ
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u
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In all such cases (c ont inuous· reciprocat ion
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,
vary .
Still more homogeneous effects of rhythm
in both counterparts may be achieved through t he use
of variations of rests or s plit-unit groups. The
groups thems elv es do not have to be binomials; the
two best of any poly nomial groups take place .
F or exam ple: (a) rests
4 .L. 1 J J
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All the above described d.evices permit
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used simultaneously .
as a group.
Any number of terms can be used
The limitation of two parts corresponds
to the two power-groups (adjacent or non-ad jacent
0
0
16.
povrers).
In all such cases the number of attacks
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equals the quantity of terms in the polynomial.
CP = 2a,
Thus a binomial squared gives CF
a trinomial squared gives
gt
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Still greater cont�asts can be acl1ieved
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non-adjacent powers...
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Nothing prevents the composer from using
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•
fourth power groups against e ubes, etc.
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9
8
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Figure XXIII,
Exampa.es of Two-Part C ounterpoint wit h Pre
Compos ed Duration-Groups.
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Fig . XXI I I .
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20.
Lesson CLVII,
Chromatization of the Diatonic Counterpoint
It seems to be easy to write a chroma tic
counterpart to any diatonic melody, as any suitable
pitch-units can be chosen from the entire chromatic
scale.
Such countermelodies, however, contain one
general defect: the neutral character which comes with
a uniform soale.
To an average listener it sounds as
if any pitch-unit would be equally as acceptable in
place of the ones already set.
This peculiarity of
musical perce1,Jtion is due to the inl1eri ted and
0
cultivated diatonic orientation.
•
An average listener hears chromatic units
as an ornamental supplement to a diatonic scale.
Such
chromat.ic units are commonly used as auxiliary tones
moving into the diatonic units of a given scale, thus
forming directional unit$.
Diatonic uni ts are
perceived as independent pitches (though in a certain
grouping in sequence) .
Chromatic units are perceived
as dependent pitches leading il1to diatonic pitches.
Music constructed entirely chromatically, i.e., without
diatonic dependence usually belongs to a different
cat�gory than the dia· tonic music with directional
units ..
It is known under the name of
• "twelve-tone" music.
•
11
atonal'', or the
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0
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21.
For this reason, we shall use chromatic
counterpoint with diatonic dependence only.
Such a
counterpoint can be devised at its best by means of
inserting the passing or the auxiliary chromatic units
I
post factum.
This technique is applicable to all four
••
types of harmonic relations .
It is important that the
conversion of a diatonic counterpoint into chromatic
does not affect the est-ablished forms of resolutions.
The remodeling of durations can be
accomplished by means of split-unit groups.
This
device allo-vi,s to preserve the character of rhythm
•
which was originally set •
Figure XXIV,
Examples of Chromatic Variati o ns of
the Diatonic Coun�erpoint.
(please see next page)
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J O S E P H
S C H I L L I N G E R
C O R R E S P O N D E N C E
C O U R S E
With; pr. Jerome Gross
Lesson CLVIII.
Subject: Music
•
Composition of C ontrapuntal C on tinuity.
Extension of any given contrapuntal continuity
is based on geometrical .mutations.
The fundamental technique of geometrical
•
muta.tions for the two-part counterpoint is the inter
change of music assigned to CF and CP .
Assuming that
CF represents the actual part and CP -- the actual
\.._,/
•
counterpart , we obtain the two variants for each voice:
CP
CF
+
CP , where both CF and both CP are identical, but
CF
appear in a different octave .
In the old systems of counterpoint it was
known as "vertical convertibility in octave" .
We shall
look upon it merely as two variants of exposition for any
co�nterpoint and consider such a convertibility to be an
inherent property of counterpoint as such.
By applying the princi ple of variation of
two elements ad infinitum, i. e . , through permutations of
the higher orders, we can compose an entire piece of
music from one contrapuntal exposition.
Figure XXV .
Example of Contrapuntal Cont inuity of the Thir§
Order Produced Through the Permutation of Parts
2_f t!le Original EtcPosi tion •
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geometrically, becomes sub j ect to gua�.ra11t •rotation (see:
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Geometrical Pro j ections of Music ) , we obtain the four
variations of t:t1e georlletrical positi ons :
(i) , @ , © ,
@ .
Through t:t1e verti cal permutation of parts
•
As each variant
two-part exposi tion yields two variants.
has four rotatior1al posi ti o11s, the total number of variants
fo r 011e two-par t contr_apuntal exposition is ei ght :
CF @
a '
GP
GP @
a '
CF
CF @
b
CP
CF In"'\
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Wl:1en making a transit ion from one form into
another in the same part, place the respective pi tch-unit
in its nearest pitch position.
This is true of both : the
octave and the geometrical inversion.
for
©
The axis of inversion
and @) is the axis of CF (or the part assumed to
bear its meaning) .
Figure µvr.
Examples of the Variants of One Exposition .
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(Fig. XXVI, cont.)
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•
The eight variants of contrapuntal exposition
p;n y
combination of the selected variants produces a complete
form of coatinuity, i.e. , a whole composition.
The sele ction of various geometrical inversions
•
must be guided by a definite tendency with reg.ard to the
amount and distribution of corlt rasts..
All the considera
tions pertaining to this matter were discussed in the
•
Geometrical Pro jections of Music.
The most importa11t principle to remember is;
(1 )
@ a nd @ are identical in intonation and converse
in temporal structure;
(2)
•
© and
®
,�
... ...
;
�
are ide11tical in intonation and converse
--
I... •t. '�
..., !
•
can be selected in any desir able co mbinati on .
J
i.=..�
I
L
•
I-
.
;;:�,.-
-
�
'
..
--
..
0
0
in temporal structure;
•
•
•
(3)
G) and @ are converse in intonation and identical
(4)
@
(5)
@ and © are converse in intonation
(6)
in temporal structure;
•
.
and
©
are converse in intonation and converse
in temporal structure;
and ide11.tical
in temporal structure;
•
a11d
@
are converse J.Il iritonation and converse
•
in temporal structure.
There is a way to obtain identical temporal
structures for all geometrical inversions: any
symmetrical group is identical with its converse .
For ·
instance:
(1)
C
3
)
r5+4
= 4 + 1 + 3 + 2 + 2 + 3 + 1 + 4
u , , , 1 u , , ,, u, ,, , u
•
•
0
0
10.
There is also a way to obtain an identical
\....J
pitch-scale for all geometrical inversions, when
desirable.
The original seale must be symmetrically
constructed (whi.ch does not necessarily place it into
the Third or the Fourth Group).
pitch units in
©
In such a case the
and @ are not idei.-itical but the
scale structur e (that is, the set of intervals) is •
•
•
For instance:
•
C - e �- f - g -
®
@
b v _ g - f - e t;, -
@
C -
,
b V (3 + 2 + 2 + 3)
C
d - f -· g - a - C
a - g - f - d
(3 + 2 + 2 + 3)
(3 + 2 + 2 +
1'
J
3) 1'
( 3 + 2 + 2 + 3)
i
Examples of complete forms of contrapu·ntal
•
continuity based on geometrical inversions:
CP /4:" + CF @ + CF @ + CP @ + CF /4.' + CF 'c" + CP @ •
CP
CP
CF
CP�
CP �
CF · '
(1) CF \!:/
CF -'a' + CP 'b' + CF 0 + CP ' (2) cp
'
lS-'
CF \V
CP �
CF�
CP
(3) CP© + CF 'b' + . C P 'a', + CF /:;"\ + CF
CP @ '
cp '-!V
CF �
CP �
CF
•
We shall apply t.t1e first of the above schemes
of continuity to the theme based on the exposition in
type II of Fig. XX.VI.
The theme will be used in its
original ST version (i .e. , without the added balance).
u
0
0
11.
;Figure XXVII •
-'
-
,
;
II -e
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EA)�:
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0
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V
12.
As we have seen before, the interchangeability
of CF and CP produces two forms for each geometri cal
p osition.
This property can be utilized for the purpose
of producing conti nui ty based on imitati on.
The two
reciprocal expositi ons follow i ng one another are planned
i11 su ch a man ner, that the fir st one cor1 sists of an
unaccompanied CF only, whi le the second has both parts.
••
When CF exchanges its positions, the resulting effect is
im itation •
•
In• the fol lowing example, Fig,. XXVI type III,
'
will serve as a theme.
The com plete continuity will follow this
v
scheme: CF @ + gf@ +
I
gi@ + g;(g) + g�@ •
Figure XXV III .
•
(please see next page)
'
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14.
Lesson CLIX.
Correlation of Melodic For�s in Two-Part Counterpoint
We have achieved s0 far the harmonic and the
temporal correlation of .t�o me:,lodic parts.
•
Melodic fo1-- ms
have been plam1ed in some gene ral way, and many details
were merely the outcome of the harmonic treatment of intervals.
•
•
Now we arrive at the point where systematic
treatment in correlating melodic f9rms becomes necessary .
As melody is expre. ssed fundamentally by means of an axial
co mbination, the correlation of two melodies becomes
essentially tile problem of coordination between the twQ
axial groups.
We shall start this analytical survey with
monomial axes for both CF and CP.
Under such conditi ons the following 25 forms
become possible.
·' -0a ·' -Oa ·' -bo ·' -Ob '· -c0 '· -Oc ·' -d0 '· -0d '·
-
- - - - - - - -
a . b . a . C . a . d . a . b . C • b .
a ' .a , b ' a ' c ' a , d ' b ' b ' c '
-,- -,- - -.
d . b . C • d . C • d
o , d , d
b
d , c
It is important to note that the various
f orms of balancing and unbalancing are inherent with the
above combinations.
The analysis of two parts being
parallel or .contrary is not suf' ficient, as, under either
conditions, one voice may be balancing and the other may
0
0
15.
be unbalanci ng, or both voices may be balancing as
well as unbalancing.
For example:
CF _ b . d . b . a
CP - b ' b '
' ct •
c
In the first case both voiaes are parallel
and balancing ; in the seco11d case both voices are
parallel, out CF is unbalancing and CP is balancing;
in the third case
both voices are contrary, but both
•
•
are balancing; in the fourth case both voices are
contrary, but both are unbalancing.
It · follows from the above consi._derations,
that in order to achieve continuous motion in two-part
counterpoint, it is necessary to introduce ah unbalancing
axis in one of the parts when the othe r part is moving
toward balance, unless a c.adence is desired.
J.s. Bach
Aad more of parallel motion than it is usually believed
to be, but he always managed to avo id cadencing, except
where it is obviousl y intended.
•
On the other hand,
many academic theoreticians advocate an abundance of
contrary motion as being· essentially contrapuntal.
This
in itself is of little importance, and beeorues a source
of monotony, unless coupled with the composition of
balance relations between CF and CP.
Thus, the selection of axial combinations
for the two counterparts (or for one · counterpart to a
u
given part). depends upon t.he form of expression.
0
0
16.
Axial relations with regard to their
directions are: (1) parallel ; (2) contrary; ( 3 ) oblique.
Axial relations with regard to their balancing
tendencies are:
(1) � ; (2)
i;
( 3)
( )
;
4
�
i .
In addition to this, the zero-axis expresses
a coritinuous state of balance ..
•
All further development of oorrelating axial
•
combinati ons of two melodies follows the ratio developIDent
•
of the quantities of axes in one part
111
r�la tion to
another.
•
Under such cond iti ons, all the above described
· cp
cases refer to one category only: CF = ax, i.e. , one
secondary axis of counterpoint corresponds to one
secondary axis of ca11tus firmus; ax is an abbreviation of
the word axis •
Nov; we arrive at the binomial relations of
axial groups of the counterpoint in rela tion to the
C &"l.tus firmus :
CP
CF
2ax and
ax
ax
2ax
Under such conditio11s, a monomial axis of one
part corresponds to · a binomial axial combination of
another.
For instance:
0
0
•
17.
CP
-
CP
CF
-
O+a • a+b • c+d • b+O • d+a
0 , b , a , C ' 0 ' • • •
0
O+a
•
,
b • a • C
a+b , c+d , b+O
,
•
0
d+a
'
• • •
It is easy to see that the re are 200 such
simultaneous combinati ons, as the re are 10 origi nal
binomial axial combinations, each havi ng 2 permutations.
•
20 combinations are now combi ned verti cally with 5
monomials (O, a, b, c, d).
This produces 20 • 5 = 100.
Finally, · 100 must be multiplied by 2, as each simultaneous combination can be inverted.
·.
The period of duration of or1e axis equals to
the sum of durations of the two axes constituting the
binomial.
Thus, in a combination:
axnt - T C� = 2
ax
mt
+
1, the time period
Y
�
C
axpt
for both parts is the same.
Time rati os for the binomial axes must be
selected in accordance with the seri e s which the
•
monomial axis represent.
For instance, the duration of ax of CF is 8T;
then, CP can be matched as any binoroial of 8 series .
Let us select the 5+3 binomial of this �ries..
Now we
can define the simultaneous temporal relations as follows :
CP _ ax5T + ax3T
CF
ax8T
In a simultaneo us combination of a binomial
0
0
18 ..
versus monomial axial combina tion it acquires the
following significanc e : during the period of duration
of a monomial axis (balanced, balancing or unbalancing)
its counterpart has two phases which may be: :U+u; U+B;
B+U; B+B.
If we sing le out a continuous balance
(O-axis) as an independent form, we obtain 12 forms of
••
balance relations between CP and CF, when one of them
is a binomial and the other a monomial.
CF _ ax _ O . . 0 .
CP - 2ax - U+U ' U+B '
•
u .
u . ...._;.,.,..
U+U ' U+B '
B
.
B
.
U+U ' U+B '
•
0
.
B+U '
u
0
.
B+B '
B+U '
B+B '
B '
B+u
B
B+B •
The same quantity is available for
ax
GP - -28x •
CF
..;;__.
If 0-axis partic ipates in a binomial , there are 15 more
combina.�ions, as O+U, O+B, B+O, O+O would have to be
multiplied by 3.
Let us select one of the possible combinations,
U+U
CP
_
_
d+a
_
2ax
and let it be:
•
C
CF
ax
Suppose CF = BT and we matc h the previously
selected time-ratio1 for CP.
CP
CF
appears as follows :
Then the correlation of
CP _ d5T + a3T
c8T
CF -
In this case CP unbalances for 5T in the
direction below its P.A., and unbalan� es still furt her
•
•
0
0
19 .,
in the di rection above its P . A. for 3T .,
While this
happens, CF moves. steadily toward its own P.A. in the
upward direction, during the course of 8T.
FifillE e XXIX .
•
CP
-·---- - · - - - -
- - - - - · ··
CF -- - - - - - - - - - - - - - - - - •
•
In the same fashion, trinomial axial combina
•
tions of one part can be correlated with a monomial axis
of another.
The quantities of simultaneous combinations
equal the number of trinomials times 5 .,
There are 60 trinomials with two identical
terms (see Theory of Melody) and 60 trinomials with all
This yields: 120 • 5 = 600 for CP and
CF
CF
the same quantity for CP •
terms dif" ferent.
As t.h e number of axes in one part is three
and in the other part -- one, we can write:
CP -- 3ax
CF
ax or
CP
CF
ax
- .......
3ax
T
•
In each case� the tri nomial requires three
temporal coefficients, the sum of which equals to that
of monomial.
0
0
20 •
•
g� = 3� = axmt + a�� + axpt ,
+ pt =
�"lhere mt + nt +
T.
Let T equal 5.
Then, by selecting 2+2+1,
which is one of the trinomials of
ax2T + ax2T + axT •
ax5T
CP
CF
i
series, we obtain :
The trinomial distribution of the o, U arid .B
•.
gives the following number of the forms of balance.
O+O+U;
O+O+B;
U+U+O;
U+U+B ; B+B+O ;
B+B+U.
Each of the above 6 combinatio�s has 3
permutations, giving the total o f 6 • 3 = 18.
When each
of these variations is placed against O, U or B in the
counterpart, the number of forms becomes tri pled : 18•3 =
•
= 54.
g� and g� have 54 forms each.
Thus both
But the above forms contain trinomials with
The addition of trinomials without
two identical terms.
identical terms produces one combination: O+U+B, which
has 6 permutations�
•
These 6 forms, being placed ag ainst
the three poss-ible forms of the counterpart, produce
CP and CF
CF
CP
•
balance of
for
CF
CP
•
have 18 forms each.
The total of trinomial combinations of
CP is 54 + 18 = 72, and the same number
CF
0
0
21 .,
When secondaI"'"f axes are su bstituted for
the forms of
each case gives more than one
For example: CP = U+O+B •
CF
U
solution.
balance,
CF
CP
- a '· u =
(1 ) u (2) 0
(3) B
CP
CF
-
-
0
u
d;
= a ,·
u-
•
d.
B = c;
b;
Then
the following solutions are available:
a+O+b • a+O+b • a+O+c • a+O+c •
' a ' d ,
'
a
d
d+O+b
a
d+O+ b
, d
•
,
•
d+O+c • d+O+c
•
,
a
d
5
5
We yield the following
Let us assign the pr eviously discussed
series trinomial time ratio.
solutions:
T_.;.........;
+ 0;::.;2=..:T::;..,...;+�b:.=..
T • a2T + 02T +
a�2-=CP = =
'
d5T
CF
a5T
a2T + 02T + cT . d2T + 02T +
'
a5T
d5T
bt
a2T + 02T + cT
'
'
a5T
bT
. d2T + 02T +
'
d5T
•
bT
d2T + 02T + cT • d2T + 02T + oT •
'
a5T
d5T
Figure XXX.
(please see next page)
•
•
.
•'
0
0
22.
•
1
t
I
T
1
CP
I
T
OF
.
•
GP
•
CF
•
•
I
I
-
.
� � "I:
t
J,f
..
•
,r
l
•
l
I
;,-
•j
t
· · · · · · --··
..
�
•
t"I
•
:r
I
i
1
i
�
-.;;,
- • -, • - • ·_! _ __
'
l-
•
- - • - • j••
1
t
•
1
-
. . . - - .. - . •- •··
�
1 .•
•
• I.
.t
,
lr -
•
I
l
1
[
I
•
\._)
•
•
0
0
23.
Lesson CLX.
Ultim ately a polynomial axial combination
Cfu"'l
serve as the counterpart to a monomial axis.
The
effect of such � correlation is instability (poly
nomial) versus stability (monomial).
•
The selection of
forms of O, U and B depends upon the effects of balance
necessary in each particular case.
The abundance of
unbalancing axes results in restless, disquieting,
unstable melodies.
Such melodies are termed as
dramatic, passionate, ecstatic, etc .
The abundance of
balancing and the 0-axes produces the restful, quiet,
stable melodies.
They are usually termed as contem
plative, epical, serene.
Examples of composition of
Let m
U+B+U+B+U.
= 5; then:
g� = 5axax •
CP
CF
_ max
ax
•
balance-group:
Let us consider the following
..
Let us assume that the two extreme terms
are identical, but different from the middle one.
Then
. the possibilities for the u , s are:
(1) a+d+a
and (2) d+a+d
Let us select the fir st combination .
Let us
assume that both B ' s are identical but on the opposite
side of P.A. from the two identical urs.
c+c for the B+B.
Then we get:
The entire axial combination for the
CP appears as follows:
0
0
•
24 •
CP = a+c+-d+c+a
Let CF be represent ed by B, and let it be
b, in order to achieve greater variety of balancing
forms of CP in relation to CF .
CP _ a+c+d+c+a
CF
b
..
•
Let the duration of the entire group be 16T.
!
Let the temporal coe fficients correspond to
series
on the basis of t = 2T. Then, by selecting a quinti
nomial ( for the five axes of CP) , we obtain the
following temporal scheme:
CP
a4T + c2T + d4T + c2T + a4T
CF = _____b_,,,,1_
6=
T------•
Figur e XXXI .
r + L • r-- �i-,;"; ;
CP
1
•
CF
I
1- - 1
.i
..
!
. • .i • • • • __.• • • • -· _. • ....
�•._., -1•-•
-
1
• • ,. • .. • e;
I
1
l
The temporal ratios, discussed so far,
CP = 1, 2, 3, ... m.
referred to the form CF
.
Such axial relations can be further developed
into polynomial groups in both CF and CP: .
•
/
0
0
25.
(1) Through the technique prev1ously applied to the
. composition of attack-groups (see Melodization
of Harmony) ;
(2) By the direct application of ratios producing
interference.
The first technique makes it possible to
.
•
match any desirable number of axes of the CP against each
axis of the
cf.
Let us take CF with 4 axes.
Vve can match
2, 3 or more axes of CP against each axis of CF and in
any desirable sequence�
For example:
•
CP = 2ax + 2ax + 2ax + 2ax
ax
ax
ax
CF
ax
By assigning temporal coefficients in such
a way that the sum of durations in each 2ax of CP
corresponds to the duration of ax of CF, we acquire a
With the temporal coefficients based
synchroni zed CP
•
CF
on r5+4 , for instance, we obtain the following correlation :
CP = ax4T + axT + ax3T + ax2T + ax2T + ax3T + axT + ax4T
ax5T
ax5T
ax5T
CF
ax5T
Let O+b+c+a be th e axial combination of CF,
and (O+a) + (O+b) + (b+O) + (a+O) -- the axial combination
acquires the f olloYv"ing appearance.
of CP. Then CP
CF
CP = 04T + aT + 03T + b2T + b2T + 03T + aT + 04T
CF
05T
b5T
c5T
a5T
•
•
0
0
26.
Figure XXXII.
•
I
I
. .. ... _... .J . ... .. . -�
•
.
•
- ......___.I,__..t�_
'I, .]__..,._..'· ___
L
l
I
When proportionate relations of the temporal
coefficients of
g�
are desirable and a 'constant number
of the axes of CP is assigned against each axis of CF,
the technique of distributive involution solves the
problem •
•
For example :
CP
3ax
3ax
3ax
.,.. = 9ax ..,.
CF
3ax - ax + ax + ax •
To carry out this form of correlation in
proportions, we shall select the square o f 2+1+1 of the
series.
CP _ ax4T + ax2T + ax2T + ax2T + axT + axT + ax2T + axT + axT
axl:T
CF
ax8T
ax4T
Let the axial combination for both CP and CF
be the trinomial a+b+c.
Then:
+___
T_
+_
T + __
c_
b_
T_
a__
2__
+__
T_
T_
+_
CP _ a4T + b2T + c2T + __
c_
T •
a__
b_
2_
a8T
b4T
CF
c4T
•
•
0
0
27 •
•
r1ggre XXXIII.
i· • .. f-{ 'f.l
...
I
.t -
CP
•
i
I
4
t
i . ...... . -'• ._
I,
•r
-
----......
I
'
t •�
.
"·
•. 1I
"
j
·I
J.
•
•
� ·J
'f.
•
i
1
-•
I 1
I I
t
l
,..
..
i
l
Most complex temporal relations result from
the quantities of axes in CP and CF, whi ch' produce
'interference ratios .,
We shall discuss here only the
simplest f orms of such interference, which require
•
uniform temporal coefficients for both CP and CF, only
different in value.
This corresponds to Binary SY!}
chronization as described in th e Theory of Rhythm .
th is sense an
%
In
ratio represents the number of
secondary axes in the two counterparts.
2 ratio.
2
Let us take
CP _ 3ax
CF - 2ax
CP = 2ax
or �
3ax •
Under such conditions
After synch ronization, the
first expression appears as follovvs:
•
CP _ -----=---=-ax2T + ax2T + ax2T
ax3T + ax3T
CF Let CF consist of O+d and CP -- of a+d+O.
•
•
Then :
CP _ a2T + d2T + 02T
CF
03T + d3T
•
•
0
0
V
28.
Figure XXXIV.
r
� :t
r
•
CP
l
--
r
1
•
• ..
;
.
Series of acceler·ations used in• their
•
•
reciprocal directions serve as another material for
the temporal coefficients of CP
CF • This technique
produces two counterparts in the form of growth versus
decline.
An example:
CP - axT + ax2T + ax3T + ax5T
CF - ax5T + ax3T + ax2T + axT
Axial combinations:
CP _ a+b+c+d
CF - a+b+c+d •
Hence:
- ---------------
CP - aT + b2T + c3T + d5T
CF
a5T + b3T + c2T + dT
Figure x:t.rv .
CP '
- . . ..
....... ,.
r
l
. ··- . . ·· ···
CF
I
r
l
L
0
0
This case illustrates the f.act that even
identical axial combinations in both counterparts can
be made contrasting by the reciprocation of temporal
coefficients.
An obvious contrast of some axial combinations
against their ovm magnified versions can be achieved by
means of the coefficients of duration applied to the
•
original group of temporal coefficients�
•
An example :
•
CP = 2(ax3T + axT + ax2T + ax2T)
- ax6T + ax2T + ax4T + ax4T
CF
CP _ a+b+c+d
CF
a+b+c+d •
Axial combination:
•
Hence:
CP = a3T + bT + c2T + d2T + a3T + bT + c2T + d2T
abT + b2T + c4T + d4T
CF
�
r+
Figure XXXVI.
t
I
'
'
1
-1
II
l
I
i
•
I
....
·
.·
,.
.
GP ,
',
Cf
i
..
j
�
..•.
+-
L
�
'I-
--
I
I
�
i �4 f 1
.
. . . . ' .. . ..' . I
- . .. . .
1
.
l -.
..,, ---- . . .
t
t
I
�
.I
t
1
•
I
0
0
30.
Lesson CLXI,
After the correlation of temporal coefficients
has been established, the cqrrelation of pitch ranges of
both counterparts must follow.
Identi cal secondary axes may have a different
rate of speed.
In terms of pitch ranges it means that a
greater range may be covered in the same period of time
as the smaller range.
Identi cal axes having dif ferent pi tch-ranges
produce noticeable amount of contrast.
CP _ axT2P
axTP •
CF
Then:
Let a be the axis in both parts.
CP _ - aT2P
aTP •
CF
Figµre XXXVI
.- I .
CP
CF
IiI 'I '
• • ---- - ...
1 · • ., - - - -
i - • - ---�·
•
•
·-· ·+ . ......
•
·r· ··
I
.l . . . . � . .
•
When the two counterparts are repr.esented by
•
the axes identical with respect to balance, but non
identical in structure, the contrast becomes still more
obvious.
,
J
0
0
V
31 .,
(1 )
CP _ B
CF - B •
CP _ b2P . c2P . b3P . c3P . b3P . c3P . • • •
CF - cP ' bP ' cP ' pP ' c2P ' b2P '
•
Figur e XXXVIII.
'
I
r r
t
I
1
•
I.
,
l
· .r :_
(2)
l
f
'
I
CP _ U
CF - U •
CP = a2P . d�P . a3P . d3P . a3P . d3P . • • •
CF
dP ' aP ' dP ' aP ' d2P ' a2P '
Figure XXXIX .
•
•
. . . .. . . . ...., . . . .
'
\.J
-
1
0
0
32.
Still greater contrasts result from juxta
position of pitch ranges of the two counterparts,, when
the axial structures differ with respect to balance.
CP - U
CF B .
CP -_ -a2P . a2P . -d2P . d2P
CF
bP ' cP ' bP ' cP '
•
• •
•
Figure XJ, .
•
I
•
•
•
0-axis is not to be concerned with, when
correlating pitch-ranges of the two counterparts.
As pitch-ratios may be in direct, oblique or
inverse relatior1s with the time-ratios in each part,
correlation of the tv10 counterparts offers the following
fundamental possibilities:
CP = T+P direct .
CF
T+P direct '
T+P oblique .
T+P oblique '
T+P opl�que
T-;-P direct '
T+P inverse
T+P inverse .
T+P inverse
T+P inverse ,,
T+P oblique '
T+P direct
'
The second, the third and the fifth forms
u
have another varia nt each (by inversion).
Thus, the
•
0
0
33.
total number of the above relati ons is 6+3 = 9.
E xamples:
CP _ T+P direct
T+P direct
CF
(1)
CP _ bTP + o2T2P + a4T4P '.
d4T4P + b 3T3P
CF -
(2)
CP _ aTP + b2T2P + a 3T3P + d4T4P
•
CF
04T + a3T3P + c2T2P + bTP
Fi ure XLI.
•
(•)
T
I
•
(�)
C p . . . . ..�.. . . . . . . .. . . . . . . .. . . . . .
...
•
.•
CF
•
t
I
L
t
CP
T+P direct
CF - T+P oblique
�
-
-----=;.._..,;._;;;.
CP _ a4T4P + c2T2P
( l) CF
- dT3P + c2T2P + d2TlP
;
CP = b 3T3P + dTP + c2T2P + a2T2P
(2) CF
dT4P + b3T3P + c4TlP
•
l
I
•
0
0
Figure XLI I .
I
CP .. • • • • • • • • • • • •
l
I -
(p . . . . . .
•
..
•
•
!
Cf
•
Cf" ..... ... . . . . . . . . . . . .. . . .
•
CP _ T+P inverse
CF - T+P direct
(1)
CP
CF
(2 )
dT
P_+
T�
T=
P
2�
+ a=T=P-=CP = =a
2T
�d=2�T�2�
�2�P;,---+_.;.d�2=
�
2P�___
+ =
��a=2=
=P
;.__,,+
•
=
T
c
+
CF
c4 l P + c3T2P + 2T3P
cT4P
=
a6T2P + b3T4P
b4T4P + d2T2P + c2T 2P + dTP
Figur e ,XLI I I .
-,.------:------- __,..-----
(1)
.,
CP
CF
CP
.... • • • .. • • • .. • • •
•
• • •• • • • • •
c;:, F . - . . . . . . . . . . . .. ... . . . . •·
+
+
0
0
35.
CP _ T+P obligue
CF - T+P oblique
(1 )
CP _ a3TlP + a2T2P + bT3P + b3TlP + b2T2P + aT3P
c3T5P + d4T4P + c5T3P
CF
(2)
_
CP �
CF
bT5P + a2T4P + d3T3P + b4T2P + a5TlP
a7T3P + b5T5P + c3T7P
Figure
• XLIV .
••
•
(t)
(l)
. ..
r•
'
.
;,
C p -----.
. .:. . . . . . T .. ..
•
'
Cp .. . . -.
- • • . .....
j
..
•
....
•
•
•
CF · · · · ·- · · · · · · · · · · · • • ..J. ... . . . . .
·
CF
t
. . . .... . -. . . __. . . . . . . . . . . . .... . . . . . . ......
•
•
f
•
CP == T-:-P oplig!}e
T+P i nverse
CF
•
(1)
CP _ b3T2P + · c3T3P + b2T3P
CF - aT2P + b2TlP + c2TlP + d3TlP '
(2 )
CP _ a4T3P + d3T3P + a3T4P
CF - cT4P + b2T3P + b3T2P + c4TlP
0
0
r
36.
Figure XLV .
(l.)
..
CP
. . . . . . .. ... . . . . ..... ... .. . . . . .
p
C
CF
C F ···· · · · · · · · · · · : · . .. . . . ...
•
•
•
CP _ T+P inv erse
CF - T+P inverse
(1 )
CP _ a3T lP + cT3P + c3T lP + aT3P
a5T3P + b3T5P
CF -
(2 )
CP -_ cT2P + c2TlP + b2TlP + b4T2P
CF
d6T3P + d3T6P
•
'
•
0
0
37 .
Fig ure XLVI_.
•
-
-
·CP
f"
,.
•
•
• • • •
+C F
I
•
•
•
•
(2.)
;
•
t
t- T -+
i
•
..+-'"·
,
CF
._
�t�,t
__,. T
---;---1--.--;--,-
..,.-.....-1i_
..-f ...,_I_____
I --1-[-\
.
.
.
--
I
..
I
J
I '
•
-.,1'
••
,.l
•
--l--!..
r
•
...
•
-
-
_,....,..
• •
-. ,, ..
. . .. ... . . .'.. •.. . . - • . .. .
....
_.
_
..
••
.....
•
Example
of App lication
'
CP _
T+P direct
�
CF T+P inv erse
CP - a4T4P + b3T3P + a3T3P + b2T2P
b8TlP + d4T2P
CF
T(CF) - (4+3+3+2) 2
-
( 16+12+12+8) + (12+9+9+6) +
+ ( 12+9+9+6) + (8+6+6+4 ).
T(CP )
-
•
( [I] +l+l+ l+ ltl+l+ l+ l+l+ l+l)
•
r ➔
1
0
0
•
38 ..
A�ial combination of
'
CP
CF
i n its general form:
f
•
C p . . .... . .� · . ... . . . . .
t
..
' . ..,
•
•
'
Let CF be constructed from C-ma j . na t. d0
scale
a11d
CP -- from A�- maj . nat . d� s cale.
P = 5p wit h approxim ation.
•
Let
Under such conditions, the
range of CF will be abo ut an octave an d a half, and
the range of CP -- about two octaves . .
Figure XLVII.
'
( please see next page)
0
0
39 ..
(Fig ., XLVII)
• •
II
,'-:�
,
.,..
-
•
__ �
I
.
c;::
·
.
I..'flt ..-� .
••
•
I t.t.
.. • •
'9
.
,_
,_
l
-
•
•
z
,
: �= .
,,_.
.. . '•.
I
r
:•.:
'
•
��
�
.
�
-
...,..
.... ,,:---..., ...
•
..
•
�I•►
�
.
¾
.:;:!
I
J..
• . Pj.,:
� �
�
....
... . ,.
-
2·
�
�-··
�
'�
•••
•
•
•
�
'
.
'
.
l
.•
•
•
J
r
,,.�
•
-·
�
•
...
.
•'-- ,
..--..
-� b'.
�,
�.
•
•
•
,.
,_.
•
- t»:
�-
.
I
•
..
'
, ::•• •·:
�
.
'
-�
:•
,,
��- -
-I
,, . .
�
I�
.- · �
••
:
'I: ·
I
-'
!
•
'
.
•
-
-
I �-
--
- ta..�r,.J�
•
)
,
•>
•
a
•�T
-
-I -
�
�-
-
l
•
•
,.-
Z:!•
-
[
•
-�
•'
.
Composition of the counterp�rt to a given melody
by means of axial correlation.
In order to accomplish the process o f
correlation of counterparts by means of axial
correlati on, it is necessary to reconstruct the axial
group of the given melody first.
After the analysis
of TP ratios of CF has been accomplished, it is
I
'ti •
•
,
•
0
0
•
40.
I
important to detect whether the T+P is of direct,
�
After this, the general
oblique or inverse form.
planning of the CP axial combination must follow.
Fir st -- with respect to T+P correlation, and second -
with respect to the axial combination itself and its
T+P ratios.
•
•
The following graph is a transcription of
Ben Jon son ' s "Dr ink to Me Only With Thine Eyes" .
Figyr e XLVIII.
•
.
'-.__,)
C
,
••
•
.
,,
..
#
,
•
\
\
'
�
,.
\
•
\
\
I_
• • • •
I•
-
••
•
•
•
\
I
,, ,.
I
I
,
•
'
'
'
..
•
I
�
•
• ••
•
•- -
.,
•
' '•
-
-•
-'-'if
'... ,
''
�
-.
,.J.
•
'
• • • • •
•
--- .
•
•
-....:......:.;.
.i
._...
•
==
This melody contains a modal modulation.
P.A. 1 is .. Phrygian (d 2 ) ar1d P .A . 2 is Ionian (d0 ) .
The entire axial gr.oup gradually gravitates toward
.
P .P. • 2 , where it forms its absolute balance.
If vve take
into account all the minute crossings, analysis of the
axial group appears as follows.
0
0
P . A . , = a6t + b2t + dt + ct + a2t + b3t + d3t
P.A. 2 = b3t + 05t + llJ .
The modulation is performe d by establishing
the correspondence between d3t (P.A. , ) and b3t (P.A. 2 ) .
We can say that: d3t (P.A. , ): b3t (P.A. 2 ).
As pitch
ranges are approximately equal, the P- ratio may be
regarded as constant .
Let us devise a counterpart in 1+4 time•
ratio • . This would mean that CP v,ould have only. one
secondary axis.
As the general tendency, of CF is
gradual gravitation toward balance in the course of
two oscillations (which correspond to four directions
•
and eight ;individual axes), we shall introduce b-axis
for the counterpart..
Then CP will consist of one
gravitating to,vard balance.
direction, consistently
'
such conditions
development .
gi
Under
represents a complete cycle of
This counterpart corresponds to the case
(2) in group (a) of Fig. XXII , where CP has an Aeolian
P.A . (ds ) .
figure XLIX.
(please see next page)
0
0
•
42.
(Fig . XLIX)
•
••
•
•
• •
••
..
•
'•
'
•
•
, .__
\
\
\
•
•
CF P. A . : C
• • • • • • •
•,
••
,
..
•,
•
. '•
••
•
•
. ,,__
•
•
•
•
•
••
• • • • • • • • • • • • • •
•
,.__..;.._�---- • •••
•
•
r
•
P.A . • .
CP
- ---
•
C
•
•
--
•
•
A
•
0
A
o
•
J
--
-
• • • • • • • • • •
.
-. . . .
--
. . _______
....;::;..-
•
�
,ti
t
•
lfll
-
-
�
1
-
j
...
::i
•
•
�
•
j
�·
-
!§;;
t
.; � ".
r
•
1
•
0
0
S C H I L L I N G E R
J O S E P H
C O R R E S P O N D E N C E
C O U R S E
Subject : Music
With: Dr. Jerome Gross
Lesson CLXII.
The Use of Symm�tric Scales
in Two-Part Coupterpoint
•
The unity of style requires th.at both counter
•
•
parts are based on symmetric scales, if one of them is •
•
Scales of the Third Group and scales of the
'
. Fourth Group, mostly in contracted form, s�rve as
material for counterpoint.
It is acceptable to have one
counterpart in the Third Group and another either in the
u
•
Third or in the Fourth Group.
When the two counterparts
belong to the different groups, two cases can be observed :
of pitche�;
(1) both scales have identical set
"
set' .of pitches.
(2) both scales have different
'
•
Example :
�
(1)
�
�
(2)
't'
\.__/
�
-
Tr
-
T,
-
T,
-
T,
C
C
C
T,
Ta
T
a;_ d�- e - a4 - C
-
f
-
T2
dl,- e - f -
-
C
L.....
-
..._
�
d
-
·e � - f
d - f - g
!•- a L..-
T,
C
-
T2
- g4F- a - b
f'f'
L
T,
-
!�- b
Ta
\
V-
C
d' - ef- e� '--
r*-
a - b
-
Ti
C
0
0
2.
Relations of the harmonic axes of the two
counterparts can be carried out in all four forms
previously used.
Their meaning with regard to symmetric
scales appears as follows:
Type I (U.u.)
: both scales have the same T , , the same
number of tonics and an identical set
of pitch-units;
•
•
Type II (U. P.)
•
: both scales have the same number of
tonics, their sets of pitch-units are
identical, but their harmonic axes are
on different tonics;
Type III (P .U.) : both scales have an identical form of
symmetry (the quantity of tonics) and
•
an identical set of pitch-units; none
of the tonics of one scale have common
pitches with the tonics of the other,
. .
i . e . , the two sets of tonics belor1g to
the mutually exclusive sets of pitches;
Type IV (P.P.)
: the two scales belong to either identical
or non-identical forms of symmetry;
their sectional scales are of nonidentical structure, yet belonging to
•
one family (according to the classifica
•
u
tion offered in the First Group of
Scales) ; the two sets of · tonics belong to
the mutually exclus,ive sets of pitches.
•
0
0
3.
E xampJ�s of two-par t counterpoint executed
in the scales of the Thir d and the Fourth Gr oup
F igure L .
,. .
IIU,.
+
..
'-"
-
,_
L
TYPE. I.
I
Ii•
U C.f
.,
,.
.•)
r�;
I
I . ,_
eF.
,,
•
�
.
,
-
r-
'
,I
..
�
-.
•
r
-
�
•
,.
•
•
.•
G 4:'tlf
'
-
I r' t'
�
i
-
�
�-
.,
•
.
-
-
•
.
--
,•
_J
I ••
• ;j,
. . ll •
..
•
--
s
•
,I
�
�
,
,,
�
,.
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5•
•
Lesson CLXIII .
Continuous (Canonic) Imitation .
•
The source of Continuous Imi t_ation, usually
known as Canoni c , is a well known phenomenon of
acoustical resonance, bearing the name of Hellenic
Before any composer existed on this planet,
nymph Echo.
•
•
•
•
nature created by chance a quintuple echo "Lorelei"
(which can be justly called five-part canon) dis covered
on the Rhein.
Admiral Wrangel (Russian) describes a
- a canyon
place in Siberia, where the river Lena enters
.,
about 600 feet high and where a pistol shot rapidly
repeats itself more than a hundred times.
u
•
How would you like that for a canon?
Music theorists, which is typical of their
species, think canon to be a purely esthetic development.
VVhatever they think, it is a natural phenomenon and tl1e
most ancient form of musical continui ty.
Th e
. re is a common belief that it requires a
great skill to write a canon.
In reality, the real cause
of any dif.ficulty in writing in this form is methodo
logical incompetence.
Both music theorists and composers
are guilty, because they have not been able to forroulate
the principles of continuous imitation.
· I will not discuss the case of Sergei Ivano
vic h Taneiev, as his interpretation of the canon
'
0
0
6.
•
requires the knowledge of his "Con�ertible Counter
point of Strict Style " , which is a highly complicated
system and deals with the Strict Style only.
Besides,
it does not bring the solutiun to melodic and rhyth mic
forms, being mostly preoccupied with the vertical and
horizontal convertibility of intervals in the harmonic
•
serise.
••
Canon is a complete composition written in
the form of continuous imitation.
•
The usual academic approach to this form is
such that the student is taught first how to write an
11
ordinaryn imitation (scientifically: discontinuous
imitation ) .
•
After not getting anywhere with this form
of imitation, he begins to struggle with the canon .
As from the start the principles of any imitation are
not disclosed . to him, it doe s not make any difference
whether the imitation is discont inuous or continuous.
Once such principles are defined and the technique is
specified, it becomes obvious that the discontinuous
imitation is merely a special case of continuous
0
imitation.
With this in view, we shall establish the
principles of continuous imitation .
Continuous imitation consists of one melody,
coexisting in two different parts in its different
and at a constant velocity.
phases
.
-
•
,
)
0
0
'
• 7.
•
This melody, being of identical structure
in both parts, may vary in intonation.
The latter
condition takes place only when the scale-structure
itself varies.
The temporal organization of cont inuous
imitation has no direct influe11ce on the duration of a
canon.
Longer rhythmic groups are preferable, however,
as continuous recurrence of the same rhythmic structure
.
becowes, eventually, monotonous.
The main sour ce of con tinuous - self-stimula-tion
'\.
in a canon is its melodic form, i.e. , the axial group.
With the devices offered in the Theory of Melody (see
•
Chapter II) it is possible to evolve an axial group of
great extension and, if necessary, wit.bout any repe t.itions.
Thus, the continuance of melodic flow becomes completely
protected.
The correlation of harmonic types and the
treatment of harmonic intervals remains the same as for
all· other forms of contrapuntal technique.
This permits
to compose canons in unitonal as well as in polytonal types.
..
Temporal Strucj:;ure of Cop;tinuous Imitatiop.
A complete composition based on continuous
imitation is known as canon.
The duration of continuous imitation or of a
canon is the multiple of its temporal structure.
The
temporal structure of a two-part canon is related to the
0
0
8•
theme of the canon as 371.
The first third is the
•
aru,.ouncement, the second third is the imitation of
announcement in the first voice and the counterpoint in
the seco:t1d voice, and the last third is the imitation of
the first portion ·of counterpoint in the second voice and
the second portion of counterpoint in the first voice .
..
After the temporal scheme is exh austed, it begins to
repeat itself with new iritonations.
If we designate the first entering voice as
Pi
(whether upper or lower) , the second , entering voice
as "P! r , the first ann ouncement as CPj , the first portion
of counterpoint as CP a, the second portion of counter
point as CP, etc. , the temporal structure of a canon
•
ap�ears as follows :
The continuation of the
temporal structure does not alter the process, merely
increasing the subnumerals of CP in the original
relation :
+ CP,
+
+ CP-.
CP�
• • •
The temporal structure of any two-part canon
is based on two elements, which appear as reciprocal
permutations.
All forms of variation of two elements
are applicable therefore to two part canons (see Theory
,
I
0
0
Let a and b be two elements representing
of Rhythm).
any kind of duration-groups.
a + b + a
Then,
, and the continuation of tbe
a + b
temporal structure assumes the following appearance :
a + b + � + b + � + ·b + � +
.
•
•
a +
a +
a +
=
•
The duration of a temporal structure is t�e
real factor controlling the flow of the canon .
The
longer the structure (not by speed, b y the quantity of
attacks) , the greater the fluidity of the canon.
Duration-groups of all kinds are acceptable as temporal
struct�res for continuous, . imitation and for the canon.
,
A. Temporal structures
compose� from the parts
•
of resultants.
•
(1)
(2 )
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10.
B. Temporal s�ructures c9mppsed from
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I
0
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12.
D. Temporal structures composed from
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•
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'
15.
Lesson CLXIV.
Composition of canons in. all four typ��
of harmonic correlation.
As canon is a duplication of melody at a
certain
tj.me interval, the differences o f intonation
in the two counterparts are due to scale-structures.
•
•
•
Thus, type I produces identical intonations, type II -
non-identical ir1tonations, type III -- identical
intonations and type IV -- non-identical intonations.
The choice of axes in all four forms of correlation
remains based on the . original principle: cor1sonance
between the axes of two counterparts.
In types II and
IV the starting P.A. can be in a dissonant relation with
•
the P.A. of the first voice, but it must end on a
•
consonance.
As continuous imitation can go on indefinitely,
•
we have to know the exact technique of bringing it to a
close .
Cadences are produced by ·the leading tones moving
into their primary axis.
As the first moving voice
defines what happens to the second voice, all that
l.S
necessary is to produ ce a leading tone in the first .
.
.
moving voice.
•
When this portion of melody
•
l.S
transferred
second voice, the first voice produces it•s own
to the
•
leading tone, after which both voices close on their
primary axes.
•
0
0
16.
The use of symmetric pitch-scales is
applicable to cano11s as well.
Exa,oples of two-part canons in all four tYJ?es
•
of harmonic correlation.
Figure LI.
•
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Compositi on of Canonip Cont�nuity by means
of Geometrical Inversions .
•
•
The original canon can be considerably extended
by means of geowetrical inversion.
Tl1e voice entering first produces the axis of
inversio11 for the positio11s
©
and @ .
The final
balance of the last cadence must not participate in the
sequence of inversions, as this would disrupt the
It must be used only at
continuous flov, of the canon .
the very end of the coroposition, if the canon er1ds in
position @ or @ .
Otherwise a new balance must be
added.
Under sue� conditions, the canon consists of
several cor1trasti11g and indevendent sections of
continuous imitation.
Example of a canon developed thrOU£h the
use of geometrica� inversions.
-
•
•
I
0
0
21 .
•
Figure LII .
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23 .
Lesson CLXV.
Fugue
A complete -compositi o n based on discontinuous
imitation constitutes a Fugue.
,
A fragmentary (incomplete) composition based
on discontinuous imitati on constitutes a Fugato.
All other names established - in the past, like
Sinfonia, Invention , Praeludium, Fughetta refer to the
same fundamental form , i . e . , Fugue.
The difference is
mostly in the magnitude of the composition\. (Fugue,
Fughetta) or in the type of harmonic correlation of the
counterparts.
Thus a Fugue which is unitonal-unimodal is
called Invention, Praeludium or Sinfonia.
Praeludium
'0eing the loo sest term o f all, as in many cases it has
not.hing in common with the Fugue.
A Fugue which is
.
unitonal-polymodal (and of a specified polymodal ity) is
called Fugue.
As in my opinion the presence or absence of
polymodality · as well as or polytonality is a matter of
harmonic specificati ons, which vary with time and place,
any complete compositi on based on disc9ntinuous imitation
can rightly be cal�ed fugue.
Fugato usually appears as a polyphonic
episode in a homophonic composition.
•
•
•
0
0
The Form of a Fugue
The temporal struc ture of a fugue depends
on the quantity of themes (sub je cts).
It is customary
to call the fugue with one theme a nsingle fugue n and
the fugue with two themes a "double fugue".
Triple
fugues are very rare, and a real triple fugue requires
many parts (voices), otherwise th e idea that each part
is a theme becomes nonsensical.
For this reason it is expedient to confine
the two-part counterpoint to. fugues with one theme only.
Tr1e general characteristic of all fugues is
the appearance of the theme in all parts in sequence.
The complete thematic cycle is known as an exposition.
•
In two-part counterpoint the first entering voice
announces the theme (we shall call it CF, for the sake
of unity in terminology) , after which the second voice
enters w ith the imitation.
called
necho".
11
This imitatio11 is usually
reply" and might as well have been called
In fact, it is tr1e same theme, with the possible
difference caused by the form of harmonic correlation.
Thus, reply in the types I and III is identical v1ith the
theme, whereas in the types I I and IV it is non-identical,
insofar as the scale-structure is modified.
At the time the second entering voice makes
its announcement (CF ) , the first entering voice evolves
a counterpart (CP) to it .
Thus the form of the first
0
0
25.
exposition (E , ) is as follows:
Pr - ---"'+---CFCP
-CF
Prr -
and the form of any other
--
exposition (En ) is : En =
-
CF + CP
•
CP + CF
In both cases the definition of the first
entering voice (P1 ) and second er1tering voice (P 11 ) ca�
be inverted.
•
In a fugue where CF and CP represent the only
thematic material and no int erludes are used, the ent ire
\.
composition acquires the following form :
F = E , + E 2 + E 3 + • • • + En •
•
In homophonic music this corresponds to a
theme with variations.
In the fugue the variation
technique consists of geometrical inversions of the
•
original exposition.
The counterpoint to the theme can be either
thru
constant (i.e . , the CP is carried out/the a1tire fugue),
or variable (i.e. , a new CP is composed for each
exposition).
Statistically, the use o f constant or
variable CP is about fifty-fift y.
In the XVII and XVIII
Centuries constant CP was somewhat o f a luxury , as the
counterpoint which we consider to be general technique ,
at that time was known as vertically convertible
counterpoint, which was believed to be more difficult to
'-J
0
0
26.
execute.
On the other hand, old composers did not lmow
the technique of geome trical inversi ons, but used tonal
inversions instead and merely as a trick, on some
special occasi ons .. , With the systematic use of geome trical
inversions, fugue becomes greatly diversified.
Under
such a conditi on, constant CP becoroes merely a practical
convenience.
••
Once the theme and the count erpoint are
composed (preparati on of one expositi on) , you get the
entire fugue by means of quadrant rotation arranged in
any desirable sequence ..
If rotatior1s refer to the entire
E, the fugue assumes the follo,ving appearance:
F = E
+ . • • , where m, n and
' \!!!.I
t::'\
p ar e any of the geometric al inversi ons.
F or example:
•
•
Such scheme s are subject t o composers r
i.J.1ventiveness.
Quadrant rotation may affect each appearance.
of the theme, then theme and reply appear in the different
geome trical positions.
. .
0
0
27•
•
For example :
(1 ) E =
(2) E =
=
CF @ + CP
CP
+ C�
(3) E
••
I t is important to note that position is
_always identical for two Si1DUltaneous parts.
''-
Thus,
CF @ means that CP set against it �s also in position @ .
Quadrant rotation applied to theme and
reply pr9duc es al together 16 geometrical forms of
exposition.
Forms o f Imitation Evolved
Through Four_ Quadrants
Figure LIII.
b
•
d
b
C
•
d
0
0
28 •
All cases referring to one geometrical
position for the ·entire . E form the diagonal arrange
ment (heavily outlined) on the above table and appear
to be special cases of the general rotary system.
It is easy to see that with this technique a
fugue of any length can be composed without any effort.
An example of fugal scheme employing
•
•
•
guadrant rotation •
,
CF @ + CP @)
CP @ + CF@
+
•
+
+
c� + CP @
+
E
3
CP@) + CF @
CF @ + GP @
CP @ + CF G)
E 41
+
CF G) + cp·
CF G) + �P G)
CP @ + CF G)
Ee
+
(Q)
CP @ + CF ©
E.r +
+
CP G) + CF @
CF � + CP @
+
CF @) + CP �
E,
CP @ + CF @
CP {li) + CF @
CF @ + · CP @
As this example shows, CF may appear in
the same voice successively, when its geometrical
position alters.
The form of fugue where counterpoint is
varied vdth some or with each of the expositions can
•
0
0
29.
r
'--'
also be subjected to quadrant rotation.
The general scheme of such a fugue appears
as follows :
+
••
CF
CP a
+
+
+ CP 1)
CF
CF + CP ,
F =
....- - �F
V
CPa
CF s
+ CF
E2
+
CF
+
+
CP
CF
E3 +
E 'I + • • •
µi example with application of the quadrant rotation
F =
•
CF
CF (E) + CP 3 ®
E3
CF
CP
2@+
I
CF + CP 3
+
CP 2 + CF
@
{g) E.(
+
+
CF
+ CP 2
CP 1 + CF
E &.4 +
CP� + CF �
CF (ii) + CP , (g)
E6
In· the old fugue the· elimination of monot ony
was usually achieved by means of Interludes .
An inter
lude (we shall term it : I ) is a contrapuntal sequence
of the imitation or of the general type.
Statistics
show that about 50 out of 100 interludes are thema tic
•
(i.e., based on elements of CF or CP) and the rest
neu tral (i.e. , using themat ic element.s of its own) .
As in th e case of coun terpoint itself, I
may be COHJ,Posed once and rotated af'tervfards..
other cases a new I may be composed each time.
In
In the
0
0
30 .
old classical fugues i nterludes served mostly as a
bridge between ��e E • s , and leading into new key.
In our fugues of �fpes I and II they can serve the
Sallle purp ose, whereas in types I I I and IV the interludes
are hardly necessary, as the key variety i s already
inherent with the group of different symmetric tonics.
.
•
•
As Vfe shall see late1"' , the fact that we h·ave two parts
does not limit the quantity of tonics .
The general scheme of a fugue with interludes
appears as follows:
This form is equivalent to the First Rondo
of the homophonic music.
I f , I 2 , I 3 , . . . may be either identical
(though in different geometrical posi tions) or totally
different.
I n , i . e . , the last interlude is quite a
common featur e iri the old fugues and has the rnea11ing
of a conclusion (coda) .
By rotating the same interlude
we acquire new modulaton directi ons.
The method of composing a fugue by this
system consists of the foll o,vi ng stages :
(1) Com_posi tior1 of the theme;
(2) Composition of the counterpoint (one or more)
to the theme; this is equi,ralent to the
u
,,,,
preparation of an exposition;
0
0
31.
(3) Prepa rati on of the exposition (or of all
expositions if there is more than one
count erpoint) in fo·ur geomet rical posi t1ons:
CF @
'
CP
CF @ •
'
CP
CF @ •
'
CP
(4) Composition of the interlude ( s) ;
CF @ •
,
CP
(5) Preparation of the four geometrical positions
••
'
of the interlude( s) ;
(6) C om position of the scheme of F ;
(7) Assembling the fugue.
•
•
•
•
(_J
•
0
0
•
J O S E P H
S C H I L L I N G E R
C O RR E S P O N DEN C E
•
C O U R S E
-
With: Dr. Jerome Gro� s
pubject: :U.u si c
Lessoµ CLXVI,
C omposi tion of the, Theme
Theme in a fugue i s of utmost importance, as
it constitutes at least one half of the entire composition.
Nobody yet has defined clearly
•
fug al theme.
the requirements for a
A good fugal theme is usually ascribed to
the composer' s genius, and this is neither help nor
consola tion to the student of tl1is subject.
We want to
know precipelY_, what makes the_ melody a sµi tabl� fugal
•
theme, as experience shows that: (1 ) not every g ood or
•
expressive melody makes a sui table fugal theme, and (2)
•
not every sui table fugal theme is a good melody for any
other purpose. C ompos.ers, who were. outstandi n g melodists,
fai led to show important achieveme nts as contrapuntalists
(Chopi n, · sohuman n� Liszt, Chaikovsky and others) .
-The
first regu1r�ment for a fugal theme i s.
that it must be an ipcomplet� m elodic form .
In the best
and most typical fugues by J.S. Bach we find that su ch
incomplete melodic forms follov1 their com pletion
as
=
*•
■
e
a
a
•e
=
•
counterpoint evolvin g during the ann ouncement of the
theme i n the second voice.
An i ncomplete melodic form i n this case means
that at the moment the second voice starts the theme� the
1
•$(
0
0
•
2.
first voi ce does not arriv� at its primary axis,
For an illustrati on, let us take Fugue II,
Vol. I, Well Tempered Clavi chord (later to be referred to
as w.T.C.) by J.s. Bach.
Figure LIV.
Pl:
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The theme ends on the first sixteenth of the
third bar, while the melodic form com pletes itself on the
third quarter of the same bar.
It is interesting to note
that the theme (and the melodi c f orm) is constructed on
0
In order to present his announcethe binary axis:
a.
ment clearly, Bach uses
½(
= � ) at the point where the
theme might have stopped otherwise, reserving the eig hth
until the reply i s far on its way of developing.
Thus
Bach eliminates the danger of stopping, which, indeed, i f
reali zed, would have sppiled the en tire fugue.
Another
important detail i s the j uxtaposition of db- axis i n CP
f
versus 0-axis in CF.
0
0
•
All other requirements for a fugal theme
really derive from the first one: all such resources
of tempo ral rhythm and axial. forms ca n be used which
•
•
demonstrate an unfinished melodic structure in the
process of its fo rmation,
The presence of any one of the f ollowing
..
structural characteristics, as well as of !AY combinatio ns
of the latter, produces a suitable fugal theme.
$
(
(l) The p resence of rests; particularly a decreasing
series of rests, combined with an inc�easing number
of attacks; stop-and- go effects; gaining momentum
effects.
•
(2) The sequence of decreasing durati on-values: rhythmic
•
acceleration in the broadest sense.
(3) Dialo gue effects achieved by means of binary axes,
and by means o f atta ck-groups contrasting in f orm,
like legato-staccato.
(4) Effects of g rowth, ach ieved by mea ns of binary and
ternary diverging axes.
•
(5) The p resence of resistance forms (including
repetitio n, phasic and periodic rotation), particularly
leadi ng to climaxes.
Coµibinati ons of the above techniques applied to
p
. ne theme make the latter more saturated and tense, which
increases the fugal characteristic •
I
0
0
•
4.
Figure LV,
Fugal theme:5 b)'.' . J, s_. Bach and by just J. s.
(Numbers in musical examples refer to the
preceding classifications ) .
w. ,: e .
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8.
As it follows from the abov� examp les, the
total duration of a theme (i n terms of quantities of
attacks, or in terms of bars) largely depends up on
the eomposer t s deci sion.
However, the following
generalization is true for most classical fugues: the
duration of the fugal theme i s i n 1nvers� p�oportion
to the number of parts.
Indeed, the first theme of Fugue IV, Vol. I,
W.T.C. has only five attacks; the theme in Fugue XXII,
•
Vol. I, W. T.c. has six attacks.
•
written i n five parts.
Both of � thes . e fugues are
On the other hand, Fugue X of the
sam e volume, wri tten in two parts, has a theme of
twenty-six attacks.
It is not important that the reply enters
on the same time-unit of the measure as the theme.
Quite
• to the contrary, difference in the starting moments (in
relati on to the bar) adds interest . to the whole composi
tion, as it produces an element of surprise.
unsuitable for fugues can be subjected
Themes
,
to some alterati ons , which will make them sui table.
•
It can be demonstrated, by reversing the
procedure, that the mere addi ti on of 0-axis to any
melodic form cap render it sui table as a fugal theme.
J.S. Bach t s theme from the "Toccata and Fugue" i n
D- minor for Organ, being deprived of its 0- axis, loses
..
all its fugal quali ty. When 0-axis is taken out, the
0
0
9.
axial combinati on becomes: b+a+c+a.
Thi s theme seems
t o have nothing but rotati on in relatively narrow range.
The inclusion of 0-axis produces a n effect . ·or- growing
resi stance, and the axial combination becomes:
- 0
d+c+c
•
Figure
•
• LVI.
•
•
The number of bars i n a fugal t heme is an
optional qua ntity.
It may be pair or odd.
integral or f ractional.
It may be
Both odd and fractional are
preferable to pai r and integral, becau se t he latter two
suggest a cadence at the end of the t heme.
I believe
one of the factors that influ enced Buxtehude and all
the Bachs is the awareness of cantus firmus (in a
strict sense) as a theme.
odd number of attacks.
Canti firmi usually had an
0
0
.-
•
10.
Lesson CLXVII,
•
Prepa ration of, the Expo�ition
After se·lecti ng the theme, th.e com.f)os'er must
dev9 te himself to the preparati on of fugal exposition.
As it is easy, with this method, to write
· f our types of fugues on one trieme, i t becomes desirable
to prepare f our expositi ons for the future f ugues.
•
In
a two-part fugue, the enti re preparati on of E consists
merely of wri ti ng CP to CF.
It is advisable that the
expositi on prepared f or each ty pe would be written out
i n all geometrical positi9ns.
This saves time during
the period of assembling the fugue.
Fugues of type IV
often require preparation of two exposi tions, as when
CP
the axes exchange in cf , CP may not fit, and a new
G
counterpoint must be written (CP11 ) .
To make the demonstrati on of all techniques
pertaining to fugue concise, we shall use
theme •
•
&
very brief
Figyre LVII,
•
( please see followi ng pages)
•
•
0
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■ 4·.
[41
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Composition of th� E2rn osi tions
Compo sition of the expositions in ty pe I
does not require any special co nsiderations, as bo th
parts have an identical P.A.
In ty pe II, the modal modulations o f CF,
and its respectively related
of modal sequence.
CP,
must be in one system
For example, if P.A. of CF , is £
and P .A. of CP , is !_, the axis of CF2 (reply ) must be �
and CP2 (counterpoint to rep ly ) must have P.A. o n .Q.,
in order to retain the axial unity in the first part
for the course of one exposition, an d in order to preserve
•
0
0
•
g� as it was origi nally
the vertical relation of
conceived.
The entire structure of the fugue ( from the
above relations) appears as follows:
F =
•
(CF3 + CPa)�
CP 2£ + CFlf !
(CF, + CP , )£
CF:a �
(CF,r + CP.r) !
CI!, � + CF6 �
where .Q., .!., f, .!!, • • • are the primary axes of the
respective parts.
-
Likewise i f
becomes:
C
+
A
A
-c+d-
-
, the sequence of P.A. • s
C
+ • . •
•
I n type III, the tonal ( key) modulations of
CF, and its respectively related CP, must be in one
•
This sequence preserves
system of symmetric sequence.
CP
its constant CF relation only when CP2 (the reply )
forms its P.A. in symmetric inversion to the original
setting.
Let us take the symmetry of '!m,.
For example:
CP
_
§. y
--c •
CF - --
-
In order to
preserve the axial relation where CP is 3 semitones
above CF, the reply must appear from the opposite
•
•
equidista nt point,
from a.
Th is allows the
relativ e stability of both parts, as CP, being three
semitones abov e CF requires th e �-axis.
The structure of such a fugue, evolved on
(_)
four points of symmetry (tonics) , appears as follows:
•
)
0
0
•
15.
A simi lar case evolved from t h ree points of
symm.etry (3J2 ), where
§� =
sequence of- P.A. ' s:
C
-a
••
•
a► +
+
-c+e-
� , g ives t he following
-
-aV+c-
In type IV, in order to carry out t he sequence
of P.A. ' s in symmetric in versi on of t he original setting,
it often becomes necessary t o prepare two 'independent
expositions:
•
E = �:I
and E' = �;��- , as CP may be 1n a
different intervallic relation to CF2 tha n it is t o CF, •
The differen ce usually appears i n vari ations on semit one
or whole tone, which results in most disturbing relat ions,
such as a seco nd instead of a thi rd.
For t his · reason,
example in Fig. LVII offers t wo expositions.
It is easy t o see t he unfitness of CP I as a
counterpoint to reply, by exchangi ng it wit h P.A. of CF.
The sequence of symmet ric P .A. • s• in type IV
of Fig. LVII would develop on t he basis of its pre-set
expositi ons:
E =
CP1
CF
-
and E• =
CPII
CF'
•
0
0
16.
Consi deri ng the enharmonic equality of e4F
and f, a:#f and bp e-tc . � and the fac t that CF i s ev olv ed
in natural major d0 and CF• in natural major <4 , we
obtain the following structure for the fugue:
•
F ::
•
t
I
'
I n the old classical fugues reply appears on
•
•
(CF + CP11 )t
(CF + CP1 1 )£. E + (CF + CP 11) f. E
+
,
1
(CPr + CF )!,if. a
CF � '
(CP1
CF•)g_#
the dominant (i.e., seven semi tones abov e or fiv e semi
tones below the theme) . If there was a sequence of
expositions before the interlude took place, the theme
would usually have returned to the tonic . Acc ording to
our type II, if CF , =
•
�
and CF a = t, CF3 should have been
CF'f should have been a etc . However, this was not the
C
c ase in the fugues of the classi c al period, and there was a
g ood reason for it. As the tQning of mean temperament
! and ! )
(the two-c oordinate system:
developed abberation,
while deviati ng from the tuning center ( =· 1 ) , i t was not
possible to get satisfactory intona ti on in the co urse of
travelin g throug h
Cs
or
C-s-
P.A.• s. And thoug h equal
temperament has overcome this defec t, the habit remained
with the c omposers till the end of XIX Century .
•
•
•
0
0
17 •
.
Le s s on CLXVIII .
,Prepareation ¢' �he Inter lud e s
•
Inter ludes (I, , I 2 , • • • Im) s erve as bridges
between the expos itions .
The las t interlude, if the
fugue end s with one, i s a pos tlu�� (coda) .
s erve two pur pos es :
Interludes
(1) to diver t the li s tener • s attenticn from the
per s i s tenc e of theme ;
•
(2) to produc e a modulatory transi tion fr om one
•
key-axis to another.
•
T he fir s t form i s confi ned to one key, but may
have any number of s ucces s ive P.A. • s , thus produc ing
modal modulati ons (U .-P.) between the two adjacent
expositions having the s ame key-axis (U. -U. and U.-P. ) .
The second form contains different key-�es (P .-U. and
P.-P . ) and c onnec ts the two adjacent expos itions having
different key-axes (P. -U. and P.-P. ).
•
Both form s of
interludes may be either neutral or thematic. Neutral
'
interludes are bas ed on the material of r hythm, or
intonation, or both, not appeari ng i n any of the
expos i tion.
T hematic inter ludes borr ow their material
of rhytbm, or i ntonati on, or both from either CF or CP of
the expos ition. Fur thermore, any of the above des cribed
types of interludes can be executed either i n general or in
imitati ve c ounter point.
•
•
•
0
0
18.
The dur ation of an interlude depends on the
duration of the exposition and the quantity of interlude�.
The form of an in terlude itself has an influence upon its
duration.
In order to construc t a pe,rfect fugue, the
duration of int erludes must be pu t into some def inite
correspondence with the duration of exp ositions..
••
Assuming
one exposition as a temporal unit, we arrive at the
following fu ndam ental schemes for the temporal organization
of in terlu des:
•
(1) T (E) = T (I) , i.e., the duration of � interlude
equ als to that of an exposition.
This presupposes
an equal duration for each of the- interludes;
•
(2 ) T (E) ) T (I), i.e., the dur ation of an exposition
is longer than that of an interlude.
An exact
ratio must be established in ea.ch case;
(3) T (E) < T (I), i.e. , the duration of an interlude
is lo nger than that of an exposition.
ratio mu st be established in each case. •
An exact
(4) r-' = I,T + I2 2T + 1 3 3T + . .• , i. e. , each successive
in terlude becomes longer.
T he du rations of
consecutive interludes may evolve in any desirable
type of progression (natural, arithmetic, geometric,
involu tion, summation etc. ).
The resulting effect
of su ch fugue- structures is tha t the interludes in
cour se of time, begin to dominate the theme.
the persistence of the theme diminishes.
Thu s
0
0
19.
(5)
\..._J
P
= I,nT + I 2 (n-l) T + I8 (n-2) T + • • • , i . e. ,
each successive in terlude becomes shorter.
The
res ulting effect is opposite to that of . (4) : the
domination of th eme oVcer in terludes grows i n the
cours e of time.
(6)
•
•
r ,
i.e., the s equence of interlud& s o.evelops
along s ome form of rhythmic grouping.
•
As convertibi lity an d quadrant rotation are
general properties , the same interlude ma y be used
several times , during
the cou rse of a fugtle.
•
This, being
combined wi th key-transpositi ons, offers a n enormou s
variety of resou rces, at the same time cons ervi ng the
•
compos erts energy •
Non-Modu lating Inte� ludes
(Types I and II)
N on-modulating interludes can be either neu tral
or thematic and they can be evolved i n general or imitative
counterpoint.
Figure LVIII,
(1) An example of Interlude type II executed in general
counterpoint.
Non- th�matic (Neu traJ. ) .
(2) An example of interlude type II executed in
imitative counterpoint.
This one is thematic wi th
reference to CF of Fig. LVII.,
(pleas e see next page)
0
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20.
(Fi g. LVIII)
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Modulating Interludes
I. •Modulating Counterpoint Evolved throug h Harmonic
Technique.
Contrary to the general notion, J.S. Bac h ' s
c ounterpoi nt is less "contrapuntal" than it i s believed
to be.
And especially so when it comes to tonal (key
to-key) modulati ons.
It i s obv.ious that Bac h as well as
many other important contrapunta1 is ts thoug ht of key-to
key transitions i n terms of modulating c h ords.
See, for
example, J.S. Bac h • s W.T.c. , Vol. I, Fugue No. X (a two
part fugue) in E- minor.
The harmonic background of
this fugue is very distinct, and this fugue is rather
typical and not excepti onal.
(
I
•
j=
0
0
21 •
..
I t is easy to convert any modu lating chord
progre.ssion wri tten in four-part harmony into two-part
harmony.
Chord structures of two-part harmony have the
followi ng functions:
(1) 8 (3)
=
1, 3 ; used instead of S (5) of the three-part
structure;
(2) 8 (5) = 1, 5 ; used instead of S(5) of the three-part
structure;
.
(3) 8(7) = 1, 7; used ins tead of 8(7) of the four-part
structure.
G
•
s (,)
Figur e LI
• X1
s
...
In order to obtain an in terlude from a four
pau- t chord-progression i t is necessary to select the
corresponding chor dal fu ncti ons which wou ld translate
the fou r-part structures into. two-part structures.
The
voice-leading pertai ning to two-part harmony will not
be discussed here, as any posi tion of two functions is
equally as acceptable for the present purpose.
.,,.
Both
•
•
parts are more or less i n the vicinity of the fou r-part
0
0
22.
harmony range.
The fin al step consists of developing
melodio figurati on in both parts, but with somewhat
contrasting rhythms of durations and attacks.
Modulating interludes can be either neutral
(general counterpoint) or thematic (imitative counter
In the latter case, thematic material is either
point).
borrowed from CF or CP of the expositions, or is enti�ely
indepen dent.
•
Examples of Modulating Interlu�es
r1gure LX.
(1) Neutral and (2 ) Them atic.
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An
interlude can be � ed i n the same fugue
more than once, appearing i n the different geomet rical
positi ons.
It also can be t ra nsposed t o any desi rable
key-axis, in any of the four qua drants.
II. Modulating Counterpoint Ev olved throug h Me lodic
Technique.
•
This new technique is bei ng offered i n order
t o enable t h e c omposer to ca rry out the pure c ontrapuntal
st y le, even when a key-t9- key tra nsition is desi rable.
•
Modulating counterpoint consists of two
independently modulati ng melodies (see modulation in the
Theory of Pit ch Scales), whose prima ry axes are i n a
•
c onstant simu_ltaneous relationship ax an y given key-poi nt
of the sequence.
After t he vertica l dependence has been
established (the harm onic interval between CP and CF),
it becomes necessary to assign to the primary axis of CP
the meaning of t he tonic which is near�st t o CF th roug h
th� scale of key-signatures.
Let t h e exposition end in t he key of c, an d
-
-
let CF end on c an d CP end on a . Then a becomes a - minor
-
(as the key nearest t o t he key of C t hru t he scale of
key signatures; A- major would be far more rem ote).
Thus
we have established a constant dependence where CP is the
minor key t h ree semi tones. below CF.
The next step c onsists of planning the
modulation of P I ( originally: CF). Let t he modulation be
0
0
u
to the key of f- minor •
Then :
IT
= C + d + G + f
Now we assume that i n ord&r to retain the
original vertical dependence between Pr and P11 , each
axis of a major key must be reciprocated by a minor • key,
and vice versa. Then:
•
pf
II
•
_
C + d + G
a + F + e
, i .e., while P r
modulates
from Q to .5!, P11 modulates from .!. to E, ax:1d when P
I
modul� tes from A to Q, P11 modulate·s f rom E, to ,!:!;
finally both parts arrive at CF havi ng an A►-axis and
CP having an f-axis.
The period of modulation f rom key to key in
both parts i s approximateli the same.
Examples of Modulati ng Interludes
Figure LXI .
(1) Neutral and (2) Thematic
.... .
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25.
The easiest way to a ompo.s e modulating
interludes by the contrapuntal technique is through a
sequence of procedures:
(1 ) PI modulates to the first in termediate key;
( 2 ) Prr
(3) PI
(4) PII
n
"
"
"
"
"
"
second
fl
"
"
"
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fl
"
.fl
and so on, until the entire modulation i s completed •
•
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Lesson C;LXIX,
26.
Composition of the Fugue
The p rocess of assembling a fugu e c onsists of
planning the general sequence of expos itio ns, interludes,
the ir geometrical p ositions and their p rimary axes (key-axes) .
In the following group of fugues only such
materia ls were used, wh ich were p repared in advance (see
Fig. LVII, LVIII, LX and LXI).
The f irst three fugues have inte rlude s ( of both
harmonic and melodic type), while the fourth has none, as
key-variety is suf f ic iently great without it. The la st
'
fugue has indepe ndent c ounterpoints for the theme a nd the
\.
G
•
rep ly.
The la tter. �re interchanged in E s.
E , (i)
It + E 2 @
+
The form of Fugue I (F. ig. LXII):
+
Ea @
+
I a€)• + E i# @@
+
I a G) + E.r@
The f orm of Fugue II (Fig.· LXII) :
6
•F
C
E , @+ E 2 @ + I , + E3 @ + E t1 @ + E.r@) + 1 2 + E6 �
©
The form of Fugue III (Fig. LXII):
0
AP
The form of Fugue IV (Fig. LXII):
0
(E, + E2 + E 3 + E'I + E_, )@ + E& @ + E 7 @ + E8 G)
Figure LXII.
(please see following pages)
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Lesson CXLVIII.
THEORY OF HABMONIC INTERVALS
(Intrpduction to Counterpoint)
A sequence of two pitch-units produces
a melodic interval.
A simultaneous combination of
two pitch-units produces a harmonic interval.
The
technique of correlation of simultaneous melod.ies •
de_pends er1tirely upon the composition of harmo11ic
intervals.
Any number of simultaneous parts (voices)
in counterpoint are formed by the pai'Fs.
These
pairs may be conceived as voices immediately
'
.
adjacent in pitch, as well as in any other form of
vertical arrangement (i.e. over 1, over 2, etc.).
The suc cess of harmonic versatility of
counterpoint depends upon the manifold of harmonic
intervals used in a certain style.
Limited quantity
of harmonic intervals results in limited forms of
the harmonic versatility of co unterpoint.
Thus, t:t1e
study of harmonic intervals becomes one of the
important prerequisites of counterpoint.
Harmonic intervals have dual origin:
1. Physical
•
•
2. Musical
Tl1e physical origin of harmonic intervals
leads back to the simplest ratios.
•
The musical origin
•
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•
16.
of intervals i s based on selective and combinatory
processes.
All semitones, i.e. units of the equal
temperament of twelve, are the structural units of
all other harmonic intervals available in such
equal temperament.
As they appear in our hearing,
they amount to the following forms:
••
i = 1,
i
=
5,
1 = 9,
•
•
i = 2,
i
1
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11, 1 =.12
This completes the entire selection
within one actave range.
An addition of intervals
to an octave produces musically identical intervals
over one octave, as the similarity of different
pitch-units within the ratio of 2 t o 1 is so great
that they even have identical musical names .
The
system of musical notation introduces, among other
forms of confusion, tl1e dual system of the interval
nomen
. clature .,
Thus, an interval containing three
semitones may b e called either a minor third or an
augmented second.
Simple ratios of acoustical int.ervals are
merely approximate equivalents of the harmoni c intervals of equal temperament.
It is not scientifically
'
correct to think the way the majority of acousticians
d o, that a 5 to 4 ratio is an equivalent of a major
U·
third or a� to 5, of a minor third, or a 7 to 4, of
•
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17.
a minor seventh, etc., as these intervals deviate
considerably from their equivalents in equal
•
temperament.
It is utterly impossible to follow the
methods established by some acousticians in studying
the type and quality of intervals in the equal
••
•
temperament of twelve as compared to their equiva
lents in the simple acoustical ratios*
The so-called
consonance is a totally different type of intervals
musically or acoustically.
If music had
to use
'.,
acoustical consonances only, yet being confined to an
equal temperament of twelve, the only real consonance
would be an octave, i.e. no two pitch-units bearing
different names would ever be used, and we would
never have either any harmony or counterpoint.
The
reason for this is th.at no other intervals than an
octave or a perfect fifth, with a certain allowance,
are consonances within the equal temperament..
All
other intervals are quite complicated ratios.
Thus,
the art of music has its own possibilities based on
the limitations within a given manifold of our tuning
system.
Acoustical consonances produce a so
called natural harmonic scaj..e, which consists of a
0
fundamental with all its partials appearing in the
sequence of a natural harmonic series (i.e. 1, 2, 3,
0
0
18.
4, 5, 6, 7, a, 9, etc.) .
The ratios of acoustical
consonances are equivalent to the ratios of
vibrations producing pitches.
For example, a .£
2
ratio means that if the actual quantities representing
both the numerator and the denominator were
multiplied by a considerable number value, they
would actually sound as pitches.
,
•
3 , as suc,h,
While
2
sounds to our ear as the resultant of interference
•
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of 3 to 2,
cycles per second sounds to our ear
as a perfect fifth.
Figure I_.
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Our ear accepts pitch-units and their
ratios as they reach said ear and the auditory
consciousness and not as they are induced upon us
0
in the traditional musical schooling.
For example,
•
0
0
19.
a melody played simultaneously in the key of c and
in the key of b next to it, or a seventh above,
sounds decidedly disturbing to musicians of our
time.
Yet an interval that is musically identical
is acoustically so different that being placed
three octaves apart it produces a musically consonant
•
•
The reason for this is that in such
impression.
absolute intervals as seventh three octaves apart
•
approximates the the 15 to 1 ratio, 1.e� the sound
of a 15th harmonic in relation to its fundamental.
And when the pitches are so far apart the deviation
from equal temperament becomes less obvious for our
pitch discrimination.
The following tables offer a
group of examples illustrating musically consonant
intervals which are usually classified as dissonances,
and with their correspondence to the proper location
of harmonics.
In all these cases no octave
substitution can be made without affecting the
actual state of consonance.
Figure II •
•
(please see next page)
l
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20.
C
(Fig. II)
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Likewise, musical consonances being
placed into a wrong pitch register, such as low
register, produce upon our ear an effect of musical
dissonances.
The reason for this is that being an
approximation of simple ratios they require the
placement of their fundamentals at such low
frequencies that they are below the range of
audibility.
•
with
•
i
For example, a major third being associated
ratio would require that the fundamental be
located two octa�es below the fourth h�rmonic.
Music
being played in major thirds in the contra-octave
simply would not permit the physical existence of
such fundamental.
The following tables offer three
•
examples of the low setting of intervalsr
Figure III.
a
(please see, next page)
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C
(Fig. III)
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With this understanding in mind we can
see that no serious theory of resolution of
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dissonant intervals may be devised without specifica-
0
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23.
tions to exact octave. locati on of the i nterval.
Thus, when we come to the theory of resolution of
i ntervals it wi ll merely be offered for the purpose
of the versati le treatment of the prog ressi ons of
harmonic i ntervals-, and not for the purpose of
exterminati on of dissonances.
••
Esthetically as well
as physiologi cally we desire sequences of tensi on
and release, and as different harmonic i ntervals
produce different degrees o f tension the versatility
of the sequence of i ntervals wi ll satisfy such
requirements.
C
It has often beeB-.1. the case that music
written according to the rules and regulations of
the dogmati c counterpoi nt does not sound esthetically
as convincing as its counterpart in the XVI or XVII
Century.
This inferi or quality is due to the
limited quantity of harmonic intervals and the forms
0£
treatment of the latter.
A. Classificatio n of Harmonic Intervals
within the Equal Temperament of Twelye
All harmonic intervals may be classified
into two groups:
•
1. With regard t o their density. i.e.. the
fullness of sonority, and
,
2. With regard to their tensi on, i.e.. their
dissonant quality.
0
0
24•
Classif ication of density evolves from
the intervals producing the emptiest effect upon
our ear up to the intervals producing the fullest
effect�
The following table is only an approximate
one; nevertheless, it serves the purpose with a
certain degree of approximation, 1. e� the first few
••
•
intervals sound decidedly empty and the last few
sound decidedly full, while in the center there are
a f ew intermediate ones•
•
Figure IV.
•
••
•
Classif ication of tension is based upon
the separation of consonances f rom the dissonances
and the separation of the consonances and dissonances
by nam.e f rom the consonances and dissonances E,Y.
sonority.
All cases when consonances and dissonances
correspond respectively by name and sonority imply
the diatonic intervals�
And all cases when
J
0
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I
25.
consonances and dissonances do not correspond t o
their original names produce chromatic intervals.
The group of diatonic cor1sonances includes perfect
unisons, perfect octaves, perfect fifths, perfect
•
fourths, major thirds, minor thirds, major sixths,
minor sixths.
The group of diatonic dissonances
includes major and minor seconds, major and minor
sevenths, major and minor ninths.
All the chromatic
intervals are classified into augmented and diminished.
•
The Augmented Intervals:
Unison, 2nd
3rd, 4th, 5th, 6th.
The Diminished Intervals:
Octave, 7th, 6th, 5th, 4th, 3rd.
;Figure V.
'
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•
•
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26•
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2nd
n
The augmented unison
"
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"
n
n
3rd
4th
equivalent to minor 2nd by sonority.
"
n
n
n
n
n
n
n
n
"
n
major 3rd
"
perfect 4th n
no diatonic interval.
n
"
minor 6th by sonority.
minor 7th n
"
The diminished octave "
n
n
major 7th "
n
n
n
n
n
n
n
n
"
n
"
"
"
"
n
n
5th
6th
7th
6th
5th
4th
3rd
n
n
n
n
"
"
n
"
n
"
major 6th "
ferfect 5th"
n
no diat-0nic interval.
maJor 3rd by sonority.
•
n
major 2nd n
Thus, the following intervals are
•
n
n
n
"
•
"
n
n
•
n
•
•
consonanc-es by sonori ty. The augmented 2nd, 3rd, 5th;
the diminished 7th, 6th, 4th.
All other chromatic
intervals will be treated as dissonances with the
•
resoluti ons corresponding either to diatonic or to
chromatic dissonances.
r
•
0
0
27.
Lesson CXLIX.
B. Resoluti on of Harmonic Intervals
The necessity of varying tension implies
the procedl.ll'e known as resoluti on of intervals.
It
is important to realize that the variation of tension
may be g radual as well as sudden, i . e. the transition
•
from a more dissonant harmonic interval to a less
dissonant one and finally i nto a fully consonant one
i s as desirable as a di rect transit ion from e.xtreme
tension to full consonance .
In the followi ng tables i ntervals such
as perfect 4th and 5th are included as well, not
for the purpose of relievi ng them from tensi on, but
for the pur�ose of devising different useful manipula
tions forming contrapuntal sequences.
The quantity of
resolutions known to a composer has a definite effect
upon the harmonic versatility of his counterpoint.
For e:xaniple, if one knows only four re.solutia>ns • of a
major 2nd (which is the �sual case) as compared to
the twelve possible resoluti ons, the amount of
musical possibi lities is considerably less.
Thinking
in terms of vari ati ons one can see that the number
of permutati ons available from four or from twelve
. .
'
elements is so diffebent i n quantity that they cannot
twentyeven be compared (the first giving/four variations
•
0
0
28.
and the s eco nd givi ng 479, 001, 600 vari ations) .
i s easy to see that having such losses on the.
It
quantity of resoluti ons of each harmoni c interval,
the loss on the total of v.ersatili ty of counterpo int
i s incalculable..
There is no need i n memorizing all
the details of the reso luti on o f intervals , as there
are general underlyi ng pri nciples evolved thr oug h
the
tradi ti on of centuries•
1.
All diatonic i ntervals resolve
thr oug h either outward or i nward or o�lique motion
of each voice on a semi tone or a wh ole tone. *
2. When a resoluti on is ob tained
through oblique moti on the sustained voice may
produc e a leap on a melodic i nt erval of a perfect
4th, either up or dow n.
3. All i ntervals known as 2nds have a
tendency to expand.
All i ntervals kn1.,,wn as 7ths
have a tendency to contract.
All 7ths are the
exact equivale nts of 2nds i n the oc tave i nversi on
(i.e. all pi t ch-uni ts are identi cal with those of
the 2nds) . All the 9ths have a tendency to contract..
All the 4ths and 5ths are neutral, i .e. they either
expand or contract. ·
Thus,. the enti re range of permutations of
semi tones and whole tones, wi th their respective
* An 1 = 3 is also correct when such an i nterval
- represents two adjacent musical nan1es (c - dff,
for example) .
•
0
0
29 ..
directions, constitutes the entire manifold of
resolutions ..
Refer to Resolution of Diatonic Intervals
chart be.low.
Resolution of Diatonic Intervals
•
.
•
•
Seconds
Ninths
Sevenths
---
Fourths
- --
and Fifths
..,
-
--
•
(enharmonic)
The following is a complete table of
resolutions of diatonic intervals ..
The i11tervals in
parentheses are the secondary resolutions.
They are
used in all cases when the first resolution produces
a dissonance.
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All chromati c i ntervals which are
augmented have a tendency of expansion•
.And all
chromatic int ervals which are dimi nished have a
tendency of contraction.
The method of reasoni ng
in resolving augmented or diminished intervals is
d-f
is a 2nd derived throug h augmentation
as follows:
C
of a major second, either through alteri ng of d t o'd�
or of c� to c�.
d
d�
a 2nd
or ,J.,�
c¥"
C
Thus, ori gi nally it mi g ht have been
Consi dering the dual ori gin of such
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interval we fi nd the respective resolu�ions: if d:4P
•
i s the alteration of d, it has the inertia of m oving
further in the same di recti•..,n, i . e .. to e; or if c t,
is the alterati on of c� it has the inertia of
moving to b�.
Such two steps taken individually or
simultaneo usly constitute the fundamental resolutions.
An
analogoos procedure must be·applied to the
diminished intervals where the diminutions are
produced throug h inward alteration.
The following is a complete table of
resol uti ons of chromatic i ntervals.
When a chromatic
interval resolves into a consonance by sonority, the
si g n "enh. " is placed above it (enharmonic).
When
the interval of resolution is surrounded by paren
•
thesis, the interval of resolution is a dissonance.
Figure
VII,
(please see next page)
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In the old counterpoint we often find a
different type of resolutions from the ones described
above.
They were known as kambiata resolutions,
which are conceived as a melo<:lic step of 'a 3rd instead
of a 2nd.
No good reason bas ever been given why such
resolutions would be used.
••
I offer an hypothesis for
the explanation . of these resolutions which I believe is
the only one to be correct •
As the tradition of old counterpoint
•
was developed, while the pentatonic (5 units) scales
'
.
were in use, some of the pitch-units o'r full diatonic
(heptatonic = 7 units) scales were absent. Thus, if
d
we find that in an interval
, d moves to e, while
C
•
c moves to a (instead of b) , a kambiata takes place
merely because such scale may be a pentatonic scale
and the unit b does not exist.
This approach offers us a definite
principle of resolution of int ervals in the scales
which have not been in use in the classical
traditional music confining all the resolutions
merely to the next step with the f'o.llow�g musical
name ,,,
For example, in harmonic a minor, the interval
a
1f may be resolved through moveme· nt of th� lower
g
u
voice only to f,, as no other pitch-unit with the name
f exists in such scale,,,
· This concludes the Theory of Harmonic
Intervals ,,,
0
0
•
•
S C H I L L I N G E R
J O S E P H
C O R R E S P O N D E N C E
C O U R S E
With: Dr . Jerome Gross
Subject : Music
Lesson CL.
Theory of Correlated Melodies .
(Counterpoint)
As counterpoint represents a system of
correlation of melodies in simultaneity and continuity,
it is absolutely essential to be thoroughly familiar
with the constitution of melody.
.
Only by being
familiar with the ma terial of the Theory of Mel ody is
the successful accomplishment of such task possible.
Correlation of melodi es is usually considered to be
one of the most difficult procedures.
As the structural
constitution of one melody is unknown theoretically, the
combination of two unknown Quantities is an entirely fan
•
tastic task to undertake.
It is not only a problem of putting two
voices together, but a problem of either combining two
melodies already made, or a composition of two melodies
•
with distinct individual characteristics .
As each
melody consists of several components, such as the
rhythm of duratior1s, attacks, melodic forms, the forms
of tra jectorial mo tion, etc., the correlation of two
·C
melodies in addition to the above described components
•
0
0
•
•
•
2.
adds one more : harmonic correlation.
Thus, counterpoint
can briefly be defined as a system of correlation of
rhythmic, melodic and harmonic forms in two or more
conjugated melodies.
As the' forms concerning one individual melody
,
are known thro ugh the previous material , we will first
cover the field of harmonic correlation which is based .
.•
on tJ:1e Theory of Harmonic Intervals .
After covering this
particular branch we shall return to the other forms of
correlation for the purpose of achieving t�e final
results offered by the contrapuntal technique.
A. Two-Part Counterpoint
L
The fundamental technique in writing two-part
counterpoint is based on writing one new melody to a
given melody.
A given melody is usually abstracted
from its rhythm of durations, thus producing a purely
melodic form which may be taken from a choral as well as
from a popular song.
The usual way of presenting such
an abstracted melodic form is in whole notes.
Such a
melodic form is usually known as Cantus Firmus (firm
cl1ar¢: = canonic or established chant) .
Our abbrevia
tions for Cantus Firmus will be C .F. and for the . melody
written to it, counterpoint or C. P.
The first forms of
counterpoint will be classified through the quantity of
attacks in C.P. as against one attack in C.F.
Thus,
0
0
•
•
0
all the fundamental forms of counterpoint will be as
follows:
CP
-
CP
CF
- a
. . . . . . n
2,
3
1,
er- •
This form of counterpoint, through inter
national agreement for a number of centuries, implies ·
the usage of co nsonances only.
As we shall have four
fundamental forms of harmonic correlation and some
of these forms will be polytonal (i.e., there will be
two different keys used simultaneously), we will have
to use consonances by name and by sonority.
The
positive requirements for harmonic correlation in
2-part C.P. are:
a. The variety of types of intervals (i. e.,
intervals expressed by different numbers).
b. The variety of density.
c. Well defined cadences expressed through the
leading tones moving into the ir axes.
d. Crossing of C.P. and C.F. is permissible
when necessary.
The negative requirements are:
a. The elimination of co nsecutive intervals whioh
are perfect unisons, octaves, 4ths and 5ths.
C
0
0
•
No consecutive dissonances.
Thus, the only
intervals to be used in parallel motion are
thirds and sixths.
b. Motion toward such intervals only through
contrary (outward or i nward) directions.
c. No repetiti on of the same pitch-unit in CP
unless it i s i n a di fferent octave.
The forms of harm onic relations previously
••
used in time continuity (see Theory of Pitch Scales)
will be used in counterpoint as the forms- of
'
simultaneous harm onic correlation.
•
1.
u. - u.
2.
u. -
3. P. -
Forms of Harmonic Correlation
Unitonal - Unimodal (identical scale
structure and key si gnat ure).
P. Unitonal - PolyJjlodal (a family scale
u.
wit h identical key signature).
Polyt onal - Unimodal { i dentical scale
structure, dif ferent key signature) .
4. P. - P. Polyt onal - Polymodal (di fferent scale
structure, different key signature).
::( n the XIV Century, . in the ca.se of
Guillaume de .Machault* we find a fully developed type
2, and in some cases an undeveloped type 3.
Only the
* The phonograph records of a Mass written by this
composer for the coronation of Charles V are
available. (Les Paraphonistes de St. Jean des Matines
and Brass E nsemb le conducted by Van) . The reconstruction
of Machault ' s 2 and 3-Part Madrigals in our musical
not ation is publi shed by the Hi·storic Musi cological
Society of Leipzig in 1926. Not aYailable in U.S.A •
'--"'
•
•
•
0
0
•
L.
ignorance and vanity of the contemporary composers
make them believe that they are the discoverers of
polytonal counterpoint.
The greatest joke is on
the modern French composers who make the claim of
priority, not being aware that their direct musical
ancestors were the originato rs o f this style centuries
ago.
••
It is also unfortunate that the idea of poly
tonality goes hand in hand with the so-called
"dissonant counterpoint", i.e., the counterpoint of
•
continuous tension without release.
Music based on
polytonality with resolutions is a v ery fruitful,
highly promising and almost undiscovered field .
The usual length of C. F . is about 5, 7, 9
L
or more bars, p referably in odd numbers (this require
ment is traditional) .
The selecti on of different key
signatures for the types 3 and 4 is entirely optional.
Any two scales, the root tones o f which produce a
conso nance, may be used for this type of counterpoint.
The best way of cor1struc ting exercises is the placement
of C.F. on a central staff surrounded by two staves
below, and two staves above, assigning each staff for
•
a different type of counterpoint.
In the followi ng group o� exercises each
part mus t, be played individually with C.F.
Thus, each
example produ.ces four types o f counterpoint with a
I
•
•
0
0
6•
•
historical emphasis of eight centuries, as the first
and second types were consid erably developed during
the middle ages, and the third and the fourth types
are mostly used in the music of today •
•
It is important to realize that all forms
of traditional contrapuntal writing were based on the
••
conception of each melody being in a different mode,
and w e can even trace the polytonal forms (though in
their embryonic form) as far back as the XIII Century.
•
•
•
•
•
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7.
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Lesson CLI.
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accompaniment , doub le pedal point may be used in
.
additi on t o 2-part cou nterpoint . The root tones of
both cont rapuntal part s become the axes which must
be assig ned as chordal functions of a double pedal
For exam ple, cou nterp oint type l (giving t he
same pitch-units for poth voi ces) may be consi dered
as a root t one or a 3rd or a 5t h, etc., of a simple
c hord structure.
Then, havi ng c as a axi s for bot h
cont rapu ntal parts, the pedal point will become
...._,,
.
,_
'
As a temporary devic e for harmonic
point.
•,
.
-
•
.
0
0
a.
C
g or
C
, c,
a f
e
e tc.
This device is applicable to all four
types of counterpoint.
For example, in type 2, if one
contrapuntal part is ionian c and the other ae olian a,
the y may represent a root and a 3rd, or a 3rd and a
•
The pedal point in such case
5th, etc., respectively.
will be
e
C
or
f'
a
two axes as c and
•
pedal points.
In the types 3 and 4 with such
e tc.
a P,
w e may use
e ),
al,
or
C
f'
e tc.
as
Each double pedal point must last
through the entir e co1:1trapuntal continu:i:ty.
More £lexible forms o f harmo�1zation of
the 2-part counterpoint will be offered later.
CP
CF
2 a
In devising two attacks of a counterpoint
against one attack of the C.F . , the following combina
tions of barillonic intervals ar e possible :
(c - consonanc e ;
C
- C
d - dissonance )
c - d
d -
C
d - d*
In the old counterpoint all the se cas e s were used in
both strict and free style, with the
*In scal ewise contrary motion only.
•
exc eption
of a
0
0
dissonance being on the first beat.
Thus, each bar may start with either a
consonance or a dissonance.
And, in the case of
�� == 2, all dissonances require immed iate resolutions .
•
Here are a few examples of such contrapuntal exercises.
Figure II.
2
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12..
Lesson CLII.
CP
CF
= 3 a
Three attacks of CP against one attack
of CF offer the following combinations of harmonic
intervals:
C - C - C
•
C
-d
-· C
d
- C
C
C - C
-
- d � resolution
d - C - d
d - d*-
�
resolution
C
c - d - d*
The d - c - c combination offers a new
device which becomes possible with three and more
attacks .;
\Ale shall call it a del.ayed (or indirect)
resolution.
Instead of resolving a tense interval we
move it to another cot1sonance, after which we resolve
the dissonance.
This device accomplishes two things:
(1) it produces a psychologica l suspense, thus
making music more intriguing;
* I n scalewise contrary motion only
L
•
0
0
13 .
(2) it produces ipso facto a more expressive
melodic form.
Examples of Delayed Resolut�ons
Figu,re III,.
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Examples of
CP
CF
= 3a
Figure IV .
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,
CP
CF
=
14 ..
4 a
•
Four attacks of C.P. against one attac-k of
C.F . offer the following combinations of harmonic
intervals:
C - C - C - C
c - c - c - d � resolution
C - C C -
d -
d -
C
C - C
·d - c - c - c
C - C C -
d - d*
d - d*-
d - d**
V
•
C
- C
-
d
C--
d
d
- C - C
C
- d -
d - C - d - C
+
-t
resolution
resolution
There are wider possibilities in the field
CP
of delayed resolution for CF = 4 •
Parallel axes, centrifugal and centripetal
forms become more prominent.
*In scalewise co11trary motion only.
�Either as * or two independent dmssonances, both of
which are resolved by the follo wing c - c in any
order.
0
0
15 ..
Exawples of Delayed Resolutions
Figure V.
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It is also useful to know all the
advantageous starting points for the scalewise
passages ending with a consonance.
Examples of Passages Ending with a
Consonance.
Figure VI.
'-.....,,-
c5
,
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0
0
16.
CP = 4
a
CF
Figure VII.
Examples of
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17 .
•
Lesson CLIII.
CP
CF
5 a
It is no longer necessary to tabulate all
the possible combinations of c and d.
Best melodic qual ity of CP results from
••
an extensive use of delayed resolutions .
The latter ,
being combined with the variety of intervals and v-,ith
the scalewise passages produce most versatile forms
of melody.
The devices for delayed resolution,
impossible for less attacks than five, are as follows :
•
d, d
c
..!,_;,i
d , c , i.e. : the first dissonance is
"�
followed by the second dissonance with its resolution,
then by the repetition of the first dissonance with
its resolution;
d...,
, d2 c
71
d 2 c , i .e.: the first dissonance is
�
followed by the second dissonance without resolution,
followed by the resolution of the first dissonance,
then by the repetition of the second dissonance
followed by its resolution.
Examples of Delayed Resolutions.
Figure VIII.
(please see next page)
L
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0
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18.
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19 .
•
CF
CP
6 a
The new devices for delayed resolutions
possible with six attacks:
d , d2 d , c
d 2 c , i. e . : the first dissonance, the
...__::,,
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second dissonance, the repetition of the first
.
.
dissonanc-e with its resolution, the repetition of the.
•
•
second dissonance with its resolution;
d1 d
o
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d
c
c , i.e. : the first dissonance, the
�
second dissonance, the resolution of the 'first
dissonance, the repetition of the second dissonance,
the delay, the resolution of the second dissonance;
•
d , d2 c
'--=="
d, c
.....
c , i.e. : the first dissonance, the
:,,
second dis sonance with its resolution, the repetition
of the first dissonance, a delay, resolution of the
•
first dis sonance;
d1 c
....,_
c
_,,
d
�
c
c , i . e . a combination of two groups
�
by three, each consisting of a dissonance, a delay and
a resolution.
Other combinations can be devised in a
similar way.
For example: d 1 c
..,
a combination of 2 + 4 .
"'
'
d2 c
d 2 c , which is
;,,
While using six attacks agai nst CF, it i s
easy to devise a great variety of melodic forms and
interference pattern (see: Melodi:za.tion of Harmony) .
.,
0
0
20.
•
Exam ples of De layed Resolutions.
Fip;ure XI .
•
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Examples of Scalewise Passages Ending wit h
a C onsonance.
Figure XII.
•
· Examples of
CP
CF
= 6 a
Figur e XIII.
•
(please see next page)
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0
0
21.
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0
0
22.
•
CP - 7 a
C, Seven attacks of CP against one of CF offer
new forms of delayed resolutions.
•
The number of new
comb inations grows, and it becomes quite easy to
develop vari ous melodic forms, built on parallel,
converging and diverging axes.
Examples of Delayed Resolutions.
Figure XIV,
•
"
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.....
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j
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•
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=L::. ·-
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Examples of Scalewise Passage,s Ending
aI
with a Consonance.
Figure XV.
•
•
j
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0
0
23.
u
- CP
Examples of CF
= 7 a
Figure XVI .
(4)
I•
•
CP =
CF
8
a
Eight attac k s of CP against one of CF offer
a great variety of melodic form s.
The latter can be
obtained through the technique of delayed resolutions.
It is equally fruitful t o devise melodic forms by
mea n s of attac k-groups.
For example, thinking of 8 as
0
0
24.
i
series represented through its binomials a.nd
Interference groups can be carried out
trino�ials.
in counterpoint in the same way �s in the Melodization
of Harmony, where such groups were used against the
attacks of H.
Examples of Delayed Resolutions.
Figure XVII .
•
•
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,
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All scalewis e passages ending w ith a
1�
•
•
•
•
consonance must start and end with the same pitch
unit, as such is the property of our seven-name
musical system.
Example s of Scalewise Passages
Ending with a Consonance,
Figyre XVIII.
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Examples of
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25.
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Fig ure
= 8 a
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CP = 8 a gives sufficient technical
CF
equipment for any greater quantity of att acl<:s. It is
CP - 12 a an d
desirable to devise such cases as CF
CP = 16 a, as they provide very usable material for
CF
the animated forms of passage-like obligato. Under
usual · (traditional) treatment, such groups with many
attacks of CP agains t CF remain uniform or nearly
•
-
0
0
26.
uniform in durations.
The most important conditions f or
obtai ning an expressive counterpoint:
(1) abundance of dissonances;
(2) delayed resolutions;
(3) interference attack- g roups•
••
•
•
•
•
•
0
0
. .
J O S E P H
S C H I L L I N G E R
C O R R E S P O N D EN C E
With: Dr,• Jerome Gross
C O U R S E
Subject: Music
•
Lesson CLIV.
Composi tion of the Attack-Groups in
Two-Part Counterpoint.
In all the previous forms of counterpoint the
attack-group of CP against each attack of CY was
P =
A const.
constant: G
CF
The monomial attack group cons�sted of any
•
desirable number of attacks: A = a, 2a, 3a, . . . ma .
•
CP.
Now we arrive at binomial attack-groups for
This can be expressed as
counterpoint written to two successive attacks of the
ca11tus firmus consists of two . differ.ent attack-groups.
For instance :
'
(1 )
(3)
CP ,
CF, +
CP2
CF 2
_
-
a •
2a + a
a '
(2 )
(4)
CP 2
+ CF
2
_
-
3 a + 2a .
a
a '
-- a
a ' •••
8a
The selection of number values for the attacks
of CP against the attacks of CF depends on the amount of
contrast desired in the two successive attack-groups of
CP.
All further details pertaining tothis matter
0
0
2.
•
are in the respective chapter of the Theory of
Melodization .
•
Binomial attack-groups are subject to permut.ations.
For example :
CP, + C.P 2- = 4a + 2a
CF ,
a
a "
CF 2
This
•
binomial attack group can be varied further through the
Suppose CF has 8a.
permutations of the higher order.
Then the whole contrapuntal continuity will acquire the
following distribution of the att ack-groups:
CP, + CP,
CFa
CF,
CP 1-t
CF , -r
-
+
_CP, +
CF,-
4a + 2a + 2a + 4a +
a
a
a
a
2a+
a
4a +
a
4a +
a
2a
a
•
Polynomial attack-groups of CP against CF can
be devi.sed in a similar fashion.
The resultants of interference , their variations,
involution groups and series of variable velocities can be
used as material for this purpose.,
Examples of polynomial att ack-groups of
(1)
OP4-b
CF , - t,
_
-
3a + � + 2a + 2a + �
a
a
a,
a
a
(2 )
CP ,-8
=
2a + � + ,! + � + 2a + � + ,! + ,!
a
a
a
a
a
a
a
a
(3)
CP &->
CF ,-5'
=
(4)
CF ,_,.
+ 3a •
a '
+ 2a •
a
.! + 2a + 3a + 5a + 8a .
a
a
a '
a
a
= 9a
a
+ 6a + 6a + 4a
a
a
a
CP .
CF .
•
Simplest duration-equivalents of attacks
will be used in the following examples.·
'
•
0
0
,
- ·..'\·
Figµre XX •
•
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4.
.
At this stage it shou ld not be difficu lt to
develop the t echn ique of writing one att ack of CP t o a
g rou p of attacks of CF.
In an exercise CF must be so
constructed as to permit the matching of one attack
agai nst a g iven attack- group.
I n a g iven melody, when
compos:l.ng a counterpa.rt, it is necessary to compose the
.
attack�g roups first • This shou ld be accom plished with
.
a view upon the possibilities of the t reat ment of
harm onic int ervals.
,
Whenever the assumed g rou p does
not permit t o carry out the resolut ion• requirements
'
(such as expanding of the second, contracting of the
seventh or the ninth, et c. ), the attack-group itself
must be reconst ructed •
•
As it was mentioned before, it is qu ite
pract ical to re-wri.te the given melody into u niform
durations first, and then t o assig� the advant ageous
attack- g rou ps.
After the cou nterpoint is written,. the
original scheme of durations ean be reeonstruc�ed.
Vf ith the pr.esent equipment , only such
melodies can be used as cantus f irmu s which are built
on one scale at a time, and the scale itself must belong
t o the Fir st Grou p (see Theory of P itch Scales) .
The procedure it self of dist ributing the
att ack- g roups of a given melody is a nalogous t o that
used in the branch of Harmo.nization of Melody, where
•
0
0
5.
attacks of a given melody were distributed in
relation to the quantity of chords accompanying
..
them.
The follov,ing is a melody subjected to
different attack treatments for the purpose of
writing a counterpart to it.
••
•
Figure XXI,
(please see next page)
•
•
0
0
Fig . XXI.
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7.
Lesson CLV.
In writing a counterpart to a given melody
(but without any considerations of the given harmonic
accompaniment) it is important to consider :
(1) the composition of attacks, and
(2 ) the composition of duratioas.
Composition of attacks depends upon the
'
degree of animation of the given melody.
If a lively
melody is to be compensated, the countermelody should
•
'
be devised on the basis of reciprocation of' attacks and,
finally, durations.
•
•
All the techniques pertaining to
variations of two elements serve as material for the
two part compensation (counterbalancing) •
If a lively melody is to be contra.sted, the
countermelody should be devised by summing up groups
of attacks together with their durations.
The sums of
durations of the given melody, with the specified number
of attacks ag.ainst eaeh attack of the countermelody,
define the durations of the counterpart.
If a slow melody is to be compensated
(counterbalanced) by a slow co unterpart, the technique
. of reciprocation of attacks and durations should take
place.
Variations of two elements provide such a
technique.
If a slow melody is to be contrasted, the
countermelody should be devised fir st by defining the
•
0
0
8.
number of attacks in the countermelody against each
indiv idual attack of the given melody, after which
the sum of the attacks of the counterpart will
represent the duration, equivalent to the duration of
one attack of the given melody.
Melodies where animated portions alternate
••
with the slow ones, or with cadences, are particularly
suited for the compensation method.
In such a case
when one melody stops, the other moves and vice versa.
We shaJ 1 analyze now th.e prob·lem of writing
the counterpart to a given melody.
Let us take Ben Jonson rs "Drink to Me Only
With Thine Eyes".
The melody reads as follows:
,
•
,''
"
'
..
�
�
•
-,
I
•
•.
....
...
-c
-
�
Reconstruction of this melody into a CF
gives it the following appearance :
)I
•
�
•
'
j
....
'
•
.....
This is a fairly animated type of melody.
0
0
•
Let us devise the scheme of durations for CP.
One
be to make each attack of CP correspond to T.
Thus
•
of the •simplest solutions for a contrasting CP wo� ld
we would obtain CP
=
4a and a
=
6t.
For a less
moderate contrast we could assign CP = Sa and a = 3t.
To obtain CP of the counterbalancing ty pe wou ld require
the assignment of two contrasting elenents, if such
can be found in CF.
As T, = 2a and T2
=
6a, and as T3 =
= 5a and T� = a, this CF provides suf f ic ient material
•
for assig ning two elements an d f or compensating them in
CP. There is of course no way to counterbalance the
original v·ersion of this melody.
•
Thus, we have obtained the following three
solutions, each different but equally acceptable.
Figure XXII,
(please see next page)
•
0
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•
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12.
•
Lesson CLVI.
Direct Composition of Durations in
0
Two-Part Counterpoint,
In composing an original two-part counter
point it is often desirable to compose the two counter
parts rhythmically first.
;
The entire technique
concerning binomials and their variations (see Theory '
of Rhythm) is applicable in this case.
Counterbalancing (compensation) is achieved
tbro-ugh the permutation of binomia1s. ·, and ',this may
follow through the higher orders .,
For example:
•
3
J.
,,,
J ,I J
q.
iJJ
q.
d.
i11
jJJ J.
4,
J.
JJJ
11l i1l �-
•
Which part is written first (thus becoming
CF) is not essential in such a case.
It is essential,
however, to write one part completely, and not section
by section.
GP must be written after CF is completed.
For more diversified rhythmic continuity,
C
•
0
0
•
resultant s with an even number of terms can be us ed.
T he binomials cons tantly reci procate i n such a case.
8
J.
For exam ple: T = r
8t ) .
8+7(+
I J I' J . J J
,,r] J
J . PJ
-
.
·
q
,�
u
,-..
, • V'�
q
�
J J.
- .
'I ,., ,
l) •
,
-
,,rJ J .
'I ·
In all such cases (c ont inuous· reciprocat ion
of the variable binomials ) , the number of at tacks of
CP against - CF remains constant , while the durations
,
vary .
Still more homogeneous effects of rhythm
in both counterparts may be achieved through t he use
of variations of rests or s plit-unit groups. The
groups thems elv es do not have to be binomials; the
two best of any poly nomial groups take place .
F or exam ple: (a) rests
4 .L. 1 J J
l�J J
J J J .i.
(b) tied rests
or:
1 1 , , '1 , ,
L
J J�J
t '-1
•
1
u
0
0
0
0
14.
•
(c ) sp lit- unit g roups
nJ J 1
4
4
, 1 1 u
JJ nJ
1 u, i
l !1 J l
, , u,
JLl n
u, 1 1
Any rhythmic group set against its c o nv erse
••
provides satisfact ory c ounterparts •
•
4
4
-
For _examp le:
2
(r5+4) .
2
JJ.
J J
114
4
T = 4t.
'
dJ
'9 -
.1
,
,J
0
Any o f the series of variable velocities
can be used for such a purp ose.
4
4
iJJ
::;
For examp le: summat ion Series I :
cl ol
J . •I
0
�'
, 4 ..
0
•I
0
-
,4
'j
,-..
C
9 11 l
Adjacent c ontrasts for two mut uall.y .
c ompensating parts can be achieved by any sy nchronized
involut ion-groups p lac ed in sequence.
The two p ower s
supv lY the a and b elements, and thus are treat ed
thl.· ough the permutations o.f two eleme nt s (any order) •
...._,,
I
0
0
15•
•
For example: (2+1) 2 + 3 (2+1) .
a :::: (2+1) 2 •,
-89 J·'-. J) J J
q-
,
b
- 3 (2+1)
p d.
J---.
J
i · v 1 1 -V 1 • v ,
I""'.
J
, 'v
J . s, ., J r
·-
q
1
•
•
For example : 4 (2+1+1) + (2+1+1) 2
a :::: 4 (2+1+1) ,•
-4
•
J J
�
, , ,u,u
b :::: (2+1+1) 2
J JJ
0
•
J
•
n J .rJ
9 4
All the above described d.evices permit
to start w ith the composition of either part as CF,
and they all refer to counterbalancing (compensation).
The technique of simultaneous harmonic
contrasts between CF and CP is based on the
distributive involution for the two synchronized parts
used simultaneously .
as a group.
Any number of terms can be used
The limitation of two parts corresponds
to the two power-groups (adjacent or non-ad jacent
0
0
16.
povrers).
In all such cases the number of attacks
of CP against CF is constant, and such a number
equals the quantity of terms in the polynomial.
CP = 2a,
Thus a binomial squared gives CF
a trinomial squared gives
gt
= 3a, etc.
Still greater cont�asts can be acl1ieved
either by using larger polynomials, or by synchronizing
•'
non-adjacent powers...
•
In tl1e latter case a binomial
cubed and used against its synchronized first pov1er
_ 4a, i.e . , 22 ; a trinomial� cubed and
gives CP
CF
=
used against its synchronized first power gives·
g:
= 9a, i.e. , 3 3 , etc.
Nothing prevents the composer from using
adjacent higher powers, like cubes against squares,
•
fourth power groups against e ubes, etc.
In all these cases the lower power
employed represents CF, as it is easier to match
several attacks agai nst a given one attack, than vice
versa.
Examples :
(a) CF = 3 (2+1) ;
9
8
CP = (2+1) 2
d.
J.
•
J-
J.
0
0
17 .
\_)
(b) CF = 9(2+1) ; CP = ( 2+1) 3
J .'--.�
-
,,,---.
,l .
J .'--
.I - J _..__...... ,I .
r'
, ,.
, V'
8
•
,,..-...
v 1 , · v 1 1 11
(c) CF = 8 ( 2+1+2+1+2) ; CP
8 �
•
= (2+1+2+1+2) 2
C
•
C
0
q 1i 1 1 4
( d) CF = 16(2�1+1) ;
4 J
4
0
CP = (2+1+1) 3
•
J ,rJ J fl J n rn m 1 n
J J
C
C
0
rn m
•
In addit i on to involuti on- g roups,
coeff icient s of dur ation can be used, like
_ 2( r4+3)
r
8+6
I
g; =
= (3+ 1+2+2+1+3) + (3+1+2+2+1+3) , as well
6 + 2 + 4 + 4+ 2 + 6
as the resµl tants of inst rumental interference composed
• for two parts.
Figure XXIII,
Exampa.es of Two-Part C ounterpoint wit h Pre
Compos ed Duration-Groups.
0
0
Fig . XXI I I .
In all the following examples the int0nation of CF was
composed fir st. •
•
I,
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20.
Lesson CLVII,
Chromatization of the Diatonic Counterpoint
It seems to be easy to write a chroma tic
counterpart to any diatonic melody, as any suitable
pitch-units can be chosen from the entire chromatic
scale.
Such countermelodies, however, contain one
general defect: the neutral character which comes with
a uniform soale.
To an average listener it sounds as
if any pitch-unit would be equally as acceptable in
place of the ones already set.
This peculiarity of
musical perce1,Jtion is due to the inl1eri ted and
0
cultivated diatonic orientation.
•
An average listener hears chromatic units
as an ornamental supplement to a diatonic scale.
Such
chromat.ic units are commonly used as auxiliary tones
moving into the diatonic units of a given scale, thus
forming directional unit$.
Diatonic uni ts are
perceived as independent pitches (though in a certain
grouping in sequence) .
Chromatic units are perceived
as dependent pitches leading il1to diatonic pitches.
Music constructed entirely chromatically, i.e., without
diatonic dependence usually belongs to a different
cat�gory than the dia· tonic music with directional
units ..
It is known under the name of
• "twelve-tone" music.
•
11
atonal'', or the
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21.
For this reason, we shall use chromatic
counterpoint with diatonic dependence only.
Such a
counterpoint can be devised at its best by means of
inserting the passing or the auxiliary chromatic units
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post factum.
This technique is applicable to all four
••
types of harmonic relations .
It is important that the
conversion of a diatonic counterpoint into chromatic
does not affect the est-ablished forms of resolutions.
The remodeling of durations can be
accomplished by means of split-unit groups.
This
device allo-vi,s to preserve the character of rhythm
•
which was originally set •
Figure XXIV,
Examples of Chromatic Variati o ns of
the Diatonic Coun�erpoint.
(please see next page)
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J O S E P H
S C H I L L I N G E R
C O R R E S P O N D E N C E
C O U R S E
With; pr. Jerome Gross
Lesson CLVIII.
Subject: Music
•
Composition of C ontrapuntal C on tinuity.
Extension of any given contrapuntal continuity
is based on geometrical .mutations.
The fundamental technique of geometrical
•
muta.tions for the two-part counterpoint is the inter
change of music assigned to CF and CP .
Assuming that
CF represents the actual part and CP -- the actual
\.._,/
•
counterpart , we obtain the two variants for each voice:
CP
CF
+
CP , where both CF and both CP are identical, but
CF
appear in a different octave .
In the old systems of counterpoint it was
known as "vertical convertibility in octave" .
We shall
look upon it merely as two variants of exposition for any
co�nterpoint and consider such a convertibility to be an
inherent property of counterpoint as such.
By applying the princi ple of variation of
two elements ad infinitum, i. e . , through permutations of
the higher orders, we can compose an entire piece of
music from one contrapuntal exposition.
Figure XXV .
Example of Contrapuntal Cont inuity of the Thir§
Order Produced Through the Permutation of Parts
2_f t!le Original EtcPosi tion •
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As any musical exposition, when conceived
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geometrically, becomes sub j ect to gua�.ra11t •rotation (see:
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Geometrical Pro j ections of Music ) , we obtain the four
variations of t:t1e georlletrical positi ons :
(i) , @ , © ,
@ .
Through t:t1e verti cal permutation of parts
•
As each variant
two-part exposi tion yields two variants.
has four rotatior1al posi ti o11s, the total number of variants
fo r 011e two-par t contr_apuntal exposition is ei ght :
CF @
a '
GP
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CF
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b
CP
CF In"'\
CP � '
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Wl:1en making a transit ion from one form into
another in the same part, place the respective pi tch-unit
in its nearest pitch position.
This is true of both : the
octave and the geometrical inversion.
for
©
The axis of inversion
and @) is the axis of CF (or the part assumed to
bear its meaning) .
Figure µvr.
Examples of the Variants of One Exposition .
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The eight variants of contrapuntal exposition
p;n y
combination of the selected variants produces a complete
form of coatinuity, i.e. , a whole composition.
The sele ction of various geometrical inversions
•
must be guided by a definite tendency with reg.ard to the
amount and distribution of corlt rasts..
All the considera
tions pertaining to this matter were discussed in the
•
Geometrical Pro jections of Music.
The most importa11t principle to remember is;
(1 )
@ a nd @ are identical in intonation and converse
in temporal structure;
(2)
•
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(4)
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(5)
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(6)
in temporal structure;
•
.
and
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are converse in intonation and converse
in temporal structure;
and ide11.tical
in temporal structure;
•
a11d
@
are converse J.Il iritonation and converse
•
in temporal structure.
There is a way to obtain identical temporal
structures for all geometrical inversions: any
symmetrical group is identical with its converse .
For ·
instance:
(1)
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= 4 + 1 + 3 + 2 + 2 + 3 + 1 + 4
u , , , 1 u , , ,, u, ,, , u
•
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10.
There is also a way to obtain an identical
\....J
pitch-scale for all geometrical inversions, when
desirable.
The original seale must be symmetrically
constructed (whi.ch does not necessarily place it into
the Third or the Fourth Group).
pitch units in
©
In such a case the
and @ are not idei.-itical but the
scale structur e (that is, the set of intervals) is •
•
•
For instance:
•
C - e �- f - g -
®
@
b v _ g - f - e t;, -
@
C -
,
b V (3 + 2 + 2 + 3)
C
d - f -· g - a - C
a - g - f - d
(3 + 2 + 2 + 3)
(3 + 2 + 2 +
1'
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3) 1'
( 3 + 2 + 2 + 3)
i
Examples of complete forms of contrapu·ntal
•
continuity based on geometrical inversions:
CP /4:" + CF @ + CF @ + CP @ + CF /4.' + CF 'c" + CP @ •
CP
CP
CF
CP�
CP �
CF · '
(1) CF \!:/
CF -'a' + CP 'b' + CF 0 + CP ' (2) cp
'
lS-'
CF \V
CP �
CF�
CP
(3) CP© + CF 'b' + . C P 'a', + CF /:;"\ + CF
CP @ '
cp '-!V
CF �
CP �
CF
•
We shall apply t.t1e first of the above schemes
of continuity to the theme based on the exposition in
type II of Fig. XX.VI.
The theme will be used in its
original ST version (i .e. , without the added balance).
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12.
As we have seen before, the interchangeability
of CF and CP produces two forms for each geometri cal
p osition.
This property can be utilized for the purpose
of producing conti nui ty based on imitati on.
The two
reciprocal expositi ons follow i ng one another are planned
i11 su ch a man ner, that the fir st one cor1 sists of an
unaccompanied CF only, whi le the second has both parts.
••
When CF exchanges its positions, the resulting effect is
im itation •
•
In• the fol lowing example, Fig,. XXVI type III,
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The com plete continuity will follow this
v
scheme: CF @ + gf@ +
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gi@ + g;(g) + g�@ •
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(please see next page)
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•
j
..
L.. .
'
.-
l:l
I
•
.
,_
·-
-- -
..
•.
ii! ".
]
.-
•
•
.
■ •-
.
•
•
•
•
'•
•
J_:
-
0
0
14.
Lesson CLIX.
Correlation of Melodic For�s in Two-Part Counterpoint
We have achieved s0 far the harmonic and the
temporal correlation of .t�o me:,lodic parts.
•
Melodic fo1-- ms
have been plam1ed in some gene ral way, and many details
were merely the outcome of the harmonic treatment of intervals.
•
•
Now we arrive at the point where systematic
treatment in correlating melodic f9rms becomes necessary .
As melody is expre. ssed fundamentally by means of an axial
co mbination, the correlation of two melodies becomes
essentially tile problem of coordination between the twQ
axial groups.
We shall start this analytical survey with
monomial axes for both CF and CP.
Under such conditi ons the following 25 forms
become possible.
·' -0a ·' -Oa ·' -bo ·' -Ob '· -c0 '· -Oc ·' -d0 '· -0d '·
-
- - - - - - - -
a . b . a . C . a . d . a . b . C • b .
a ' .a , b ' a ' c ' a , d ' b ' b ' c '
-,- -,- - -.
d . b . C • d . C • d
o , d , d
b
d , c
It is important to note that the various
f orms of balancing and unbalancing are inherent with the
above combinations.
The analysis of two parts being
parallel or .contrary is not suf' ficient, as, under either
conditions, one voice may be balancing and the other may
0
0
15.
be unbalanci ng, or both voices may be balancing as
well as unbalancing.
For example:
CF _ b . d . b . a
CP - b ' b '
' ct •
c
In the first case both voiaes are parallel
and balancing ; in the seco11d case both voices are
parallel, out CF is unbalancing and CP is balancing;
in the third case
both voices are contrary, but both
•
•
are balancing; in the fourth case both voices are
contrary, but both are unbalancing.
It · follows from the above consi._derations,
that in order to achieve continuous motion in two-part
counterpoint, it is necessary to introduce ah unbalancing
axis in one of the parts when the othe r part is moving
toward balance, unless a c.adence is desired.
J.s. Bach
Aad more of parallel motion than it is usually believed
to be, but he always managed to avo id cadencing, except
where it is obviousl y intended.
•
On the other hand,
many academic theoreticians advocate an abundance of
contrary motion as being· essentially contrapuntal.
This
in itself is of little importance, and beeorues a source
of monotony, unless coupled with the composition of
balance relations between CF and CP.
Thus, the selection of axial combinations
for the two counterparts (or for one · counterpart to a
u
given part). depends upon t.he form of expression.
0
0
16.
Axial relations with regard to their
directions are: (1) parallel ; (2) contrary; ( 3 ) oblique.
Axial relations with regard to their balancing
tendencies are:
(1) � ; (2)
i;
( 3)
( )
;
4
�
i .
In addition to this, the zero-axis expresses
a coritinuous state of balance ..
•
All further development of oorrelating axial
•
combinati ons of two melodies follows the ratio developIDent
•
of the quantities of axes in one part
111
r�la tion to
another.
•
Under such cond iti ons, all the above described
· cp
cases refer to one category only: CF = ax, i.e. , one
secondary axis of counterpoint corresponds to one
secondary axis of ca11tus firmus; ax is an abbreviation of
the word axis •
Nov; we arrive at the binomial relations of
axial groups of the counterpoint in rela tion to the
C &"l.tus firmus :
CP
CF
2ax and
ax
ax
2ax
Under such conditio11s, a monomial axis of one
part corresponds to · a binomial axial combination of
another.
For instance:
0
0
•
17.
CP
-
CP
CF
-
O+a • a+b • c+d • b+O • d+a
0 , b , a , C ' 0 ' • • •
0
O+a
•
,
b • a • C
a+b , c+d , b+O
,
•
0
d+a
'
• • •
It is easy to see that the re are 200 such
simultaneous combinati ons, as the re are 10 origi nal
binomial axial combinations, each havi ng 2 permutations.
•
20 combinations are now combi ned verti cally with 5
monomials (O, a, b, c, d).
This produces 20 • 5 = 100.
Finally, · 100 must be multiplied by 2, as each simultaneous combination can be inverted.
·.
The period of duration of or1e axis equals to
the sum of durations of the two axes constituting the
binomial.
Thus, in a combination:
axnt - T C� = 2
ax
mt
+
1, the time period
Y
�
C
axpt
for both parts is the same.
Time rati os for the binomial axes must be
selected in accordance with the seri e s which the
•
monomial axis represent.
For instance, the duration of ax of CF is 8T;
then, CP can be matched as any binoroial of 8 series .
Let us select the 5+3 binomial of this �ries..
Now we
can define the simultaneous temporal relations as follows :
CP _ ax5T + ax3T
CF
ax8T
In a simultaneo us combination of a binomial
0
0
18 ..
versus monomial axial combina tion it acquires the
following significanc e : during the period of duration
of a monomial axis (balanced, balancing or unbalancing)
its counterpart has two phases which may be: :U+u; U+B;
B+U; B+B.
If we sing le out a continuous balance
(O-axis) as an independent form, we obtain 12 forms of
••
balance relations between CP and CF, when one of them
is a binomial and the other a monomial.
CF _ ax _ O . . 0 .
CP - 2ax - U+U ' U+B '
•
u .
u . ...._;.,.,..
U+U ' U+B '
B
.
B
.
U+U ' U+B '
•
0
.
B+U '
u
0
.
B+B '
B+U '
B+B '
B '
B+u
B
B+B •
The same quantity is available for
ax
GP - -28x •
CF
..;;__.
If 0-axis partic ipates in a binomial , there are 15 more
combina.�ions, as O+U, O+B, B+O, O+O would have to be
multiplied by 3.
Let us select one of the possible combinations,
U+U
CP
_
_
d+a
_
2ax
and let it be:
•
C
CF
ax
Suppose CF = BT and we matc h the previously
selected time-ratio1 for CP.
CP
CF
appears as follows :
Then the correlation of
CP _ d5T + a3T
c8T
CF -
In this case CP unbalances for 5T in the
direction below its P.A., and unbalan� es still furt her
•
•
0
0
19 .,
in the di rection above its P . A. for 3T .,
While this
happens, CF moves. steadily toward its own P.A. in the
upward direction, during the course of 8T.
FifillE e XXIX .
•
CP
-·---- - · - - - -
- - - - - · ··
CF -- - - - - - - - - - - - - - - - - •
•
In the same fashion, trinomial axial combina
•
tions of one part can be correlated with a monomial axis
of another.
The quantities of simultaneous combinations
equal the number of trinomials times 5 .,
There are 60 trinomials with two identical
terms (see Theory of Melody) and 60 trinomials with all
This yields: 120 • 5 = 600 for CP and
CF
CF
the same quantity for CP •
terms dif" ferent.
As t.h e number of axes in one part is three
and in the other part -- one, we can write:
CP -- 3ax
CF
ax or
CP
CF
ax
- .......
3ax
T
•
In each case� the tri nomial requires three
temporal coefficients, the sum of which equals to that
of monomial.
0
0
20 •
•
g� = 3� = axmt + a�� + axpt ,
+ pt =
�"lhere mt + nt +
T.
Let T equal 5.
Then, by selecting 2+2+1,
which is one of the trinomials of
ax2T + ax2T + axT •
ax5T
CP
CF
i
series, we obtain :
The trinomial distribution of the o, U arid .B
•.
gives the following number of the forms of balance.
O+O+U;
O+O+B;
U+U+O;
U+U+B ; B+B+O ;
B+B+U.
Each of the above 6 combinatio�s has 3
permutations, giving the total o f 6 • 3 = 18.
When each
of these variations is placed against O, U or B in the
counterpart, the number of forms becomes tri pled : 18•3 =
•
= 54.
g� and g� have 54 forms each.
Thus both
But the above forms contain trinomials with
The addition of trinomials without
two identical terms.
identical terms produces one combination: O+U+B, which
has 6 permutations�
•
These 6 forms, being placed ag ainst
the three poss-ible forms of the counterpart, produce
CP and CF
CF
CP
•
balance of
for
CF
CP
•
have 18 forms each.
The total of trinomial combinations of
CP is 54 + 18 = 72, and the same number
CF
0
0
21 .,
When secondaI"'"f axes are su bstituted for
the forms of
each case gives more than one
For example: CP = U+O+B •
CF
U
solution.
balance,
CF
CP
- a '· u =
(1 ) u (2) 0
(3) B
CP
CF
-
-
0
u
d;
= a ,·
u-
•
d.
B = c;
b;
Then
the following solutions are available:
a+O+b • a+O+b • a+O+c • a+O+c •
' a ' d ,
'
a
d
d+O+b
a
d+O+ b
, d
•
,
•
d+O+c • d+O+c
•
,
a
d
5
5
We yield the following
Let us assign the pr eviously discussed
series trinomial time ratio.
solutions:
T_.;.........;
+ 0;::.;2=..:T::;..,...;+�b:.=..
T • a2T + 02T +
a�2-=CP = =
'
d5T
CF
a5T
a2T + 02T + cT . d2T + 02T +
'
a5T
d5T
bt
a2T + 02T + cT
'
'
a5T
bT
. d2T + 02T +
'
d5T
•
bT
d2T + 02T + cT • d2T + 02T + oT •
'
a5T
d5T
Figure XXX.
(please see next page)
•
•
.
•'
0
0
22.
•
1
t
I
T
1
CP
I
T
OF
.
•
GP
•
CF
•
•
I
I
-
.
� � "I:
t
J,f
..
•
,r
l
•
l
I
;,-
•j
t
· · · · · · --··
..
�
•
t"I
•
:r
I
i
1
i
�
-.;;,
- • -, • - • ·_! _ __
'
l-
•
- - • - • j••
1
t
•
1
-
. . . - - .. - . •- •··
�
1 .•
•
• I.
.t
,
lr -
•
I
l
1
[
I
•
\._)
•
•
0
0
23.
Lesson CLX.
Ultim ately a polynomial axial combination
Cfu"'l
serve as the counterpart to a monomial axis.
The
effect of such � correlation is instability (poly
nomial) versus stability (monomial).
•
The selection of
forms of O, U and B depends upon the effects of balance
necessary in each particular case.
The abundance of
unbalancing axes results in restless, disquieting,
unstable melodies.
Such melodies are termed as
dramatic, passionate, ecstatic, etc .
The abundance of
balancing and the 0-axes produces the restful, quiet,
stable melodies.
They are usually termed as contem
plative, epical, serene.
Examples of composition of
Let m
U+B+U+B+U.
= 5; then:
g� = 5axax •
CP
CF
_ max
ax
•
balance-group:
Let us consider the following
..
Let us assume that the two extreme terms
are identical, but different from the middle one.
Then
. the possibilities for the u , s are:
(1) a+d+a
and (2) d+a+d
Let us select the fir st combination .
Let us
assume that both B ' s are identical but on the opposite
side of P.A. from the two identical urs.
c+c for the B+B.
Then we get:
The entire axial combination for the
CP appears as follows:
0
0
•
24 •
CP = a+c+-d+c+a
Let CF be represent ed by B, and let it be
b, in order to achieve greater variety of balancing
forms of CP in relation to CF .
CP _ a+c+d+c+a
CF
b
..
•
Let the duration of the entire group be 16T.
!
Let the temporal coe fficients correspond to
series
on the basis of t = 2T. Then, by selecting a quinti
nomial ( for the five axes of CP) , we obtain the
following temporal scheme:
CP
a4T + c2T + d4T + c2T + a4T
CF = _____b_,,,,1_
6=
T------•
Figur e XXXI .
r + L • r-- �i-,;"; ;
CP
1
•
CF
I
1- - 1
.i
..
!
. • .i • • • • __.• • • • -· _. • ....
�•._., -1•-•
-
1
• • ,. • .. • e;
I
1
l
The temporal ratios, discussed so far,
CP = 1, 2, 3, ... m.
referred to the form CF
.
Such axial relations can be further developed
into polynomial groups in both CF and CP: .
•
/
0
0
25.
(1) Through the technique prev1ously applied to the
. composition of attack-groups (see Melodization
of Harmony) ;
(2) By the direct application of ratios producing
interference.
The first technique makes it possible to
.
•
match any desirable number of axes of the CP against each
axis of the
cf.
Let us take CF with 4 axes.
Vve can match
2, 3 or more axes of CP against each axis of CF and in
any desirable sequence�
For example:
•
CP = 2ax + 2ax + 2ax + 2ax
ax
ax
ax
CF
ax
By assigning temporal coefficients in such
a way that the sum of durations in each 2ax of CP
corresponds to the duration of ax of CF, we acquire a
With the temporal coefficients based
synchroni zed CP
•
CF
on r5+4 , for instance, we obtain the following correlation :
CP = ax4T + axT + ax3T + ax2T + ax2T + ax3T + axT + ax4T
ax5T
ax5T
ax5T
CF
ax5T
Let O+b+c+a be th e axial combination of CF,
and (O+a) + (O+b) + (b+O) + (a+O) -- the axial combination
acquires the f olloYv"ing appearance.
of CP. Then CP
CF
CP = 04T + aT + 03T + b2T + b2T + 03T + aT + 04T
CF
05T
b5T
c5T
a5T
•
•
0
0
26.
Figure XXXII.
•
I
I
. .. ... _... .J . ... .. . -�
•
.
•
- ......___.I,__..t�_
'I, .]__..,._..'· ___
L
l
I
When proportionate relations of the temporal
coefficients of
g�
are desirable and a 'constant number
of the axes of CP is assigned against each axis of CF,
the technique of distributive involution solves the
problem •
•
For example :
CP
3ax
3ax
3ax
.,.. = 9ax ..,.
CF
3ax - ax + ax + ax •
To carry out this form of correlation in
proportions, we shall select the square o f 2+1+1 of the
series.
CP _ ax4T + ax2T + ax2T + ax2T + axT + axT + ax2T + axT + axT
axl:T
CF
ax8T
ax4T
Let the axial combination for both CP and CF
be the trinomial a+b+c.
Then:
+___
T_
+_
T + __
c_
b_
T_
a__
2__
+__
T_
T_
+_
CP _ a4T + b2T + c2T + __
c_
T •
a__
b_
2_
a8T
b4T
CF
c4T
•
•
0
0
27 •
•
r1ggre XXXIII.
i· • .. f-{ 'f.l
...
I
.t -
CP
•
i
I
4
t
i . ...... . -'• ._
I,
•r
-
----......
I
'
t •�
.
"·
•. 1I
"
j
·I
J.
•
•
� ·J
'f.
•
i
1
-•
I 1
I I
t
l
,..
..
i
l
Most complex temporal relations result from
the quantities of axes in CP and CF, whi ch' produce
'interference ratios .,
We shall discuss here only the
simplest f orms of such interference, which require
•
uniform temporal coefficients for both CP and CF, only
different in value.
This corresponds to Binary SY!}
chronization as described in th e Theory of Rhythm .
th is sense an
%
In
ratio represents the number of
secondary axes in the two counterparts.
2 ratio.
2
Let us take
CP _ 3ax
CF - 2ax
CP = 2ax
or �
3ax •
Under such conditions
After synch ronization, the
first expression appears as follovvs:
•
CP _ -----=---=-ax2T + ax2T + ax2T
ax3T + ax3T
CF Let CF consist of O+d and CP -- of a+d+O.
•
•
Then :
CP _ a2T + d2T + 02T
CF
03T + d3T
•
•
0
0
V
28.
Figure XXXIV.
r
� :t
r
•
CP
l
--
r
1
•
• ..
;
.
Series of acceler·ations used in• their
•
•
reciprocal directions serve as another material for
the temporal coefficients of CP
CF • This technique
produces two counterparts in the form of growth versus
decline.
An example:
CP - axT + ax2T + ax3T + ax5T
CF - ax5T + ax3T + ax2T + axT
Axial combinations:
CP _ a+b+c+d
CF - a+b+c+d •
Hence:
- ---------------
CP - aT + b2T + c3T + d5T
CF
a5T + b3T + c2T + dT
Figure x:t.rv .
CP '
- . . ..
....... ,.
r
l
. ··- . . ·· ···
CF
I
r
l
L
0
0
This case illustrates the f.act that even
identical axial combinations in both counterparts can
be made contrasting by the reciprocation of temporal
coefficients.
An obvious contrast of some axial combinations
against their ovm magnified versions can be achieved by
means of the coefficients of duration applied to the
•
original group of temporal coefficients�
•
An example :
•
CP = 2(ax3T + axT + ax2T + ax2T)
- ax6T + ax2T + ax4T + ax4T
CF
CP _ a+b+c+d
CF
a+b+c+d •
Axial combination:
•
Hence:
CP = a3T + bT + c2T + d2T + a3T + bT + c2T + d2T
abT + b2T + c4T + d4T
CF
�
r+
Figure XXXVI.
t
I
'
'
1
-1
II
l
I
i
•
I
....
·
.·
,.
.
GP ,
',
Cf
i
..
j
�
..•.
+-
L
�
'I-
--
I
I
�
i �4 f 1
.
. . . . ' .. . ..' . I
- . .. . .
1
.
l -.
..,, ---- . . .
t
t
I
�
.I
t
1
•
I
0
0
30.
Lesson CLXI,
After the correlation of temporal coefficients
has been established, the cqrrelation of pitch ranges of
both counterparts must follow.
Identi cal secondary axes may have a different
rate of speed.
In terms of pitch ranges it means that a
greater range may be covered in the same period of time
as the smaller range.
Identi cal axes having dif ferent pi tch-ranges
produce noticeable amount of contrast.
CP _ axT2P
axTP •
CF
Then:
Let a be the axis in both parts.
CP _ - aT2P
aTP •
CF
Figµre XXXVI
.- I .
CP
CF
IiI 'I '
• • ---- - ...
1 · • ., - - - -
i - • - ---�·
•
•
·-· ·+ . ......
•
·r· ··
I
.l . . . . � . .
•
When the two counterparts are repr.esented by
•
the axes identical with respect to balance, but non
identical in structure, the contrast becomes still more
obvious.
,
J
0
0
V
31 .,
(1 )
CP _ B
CF - B •
CP _ b2P . c2P . b3P . c3P . b3P . c3P . • • •
CF - cP ' bP ' cP ' pP ' c2P ' b2P '
•
Figur e XXXVIII.
'
I
r r
t
I
1
•
I.
,
l
· .r :_
(2)
l
f
'
I
CP _ U
CF - U •
CP = a2P . d�P . a3P . d3P . a3P . d3P . • • •
CF
dP ' aP ' dP ' aP ' d2P ' a2P '
Figure XXXIX .
•
•
. . . .. . . . ...., . . . .
'
\.J
-
1
0
0
32.
Still greater contrasts result from juxta
position of pitch ranges of the two counterparts,, when
the axial structures differ with respect to balance.
CP - U
CF B .
CP -_ -a2P . a2P . -d2P . d2P
CF
bP ' cP ' bP ' cP '
•
• •
•
Figure XJ, .
•
I
•
•
•
0-axis is not to be concerned with, when
correlating pitch-ranges of the two counterparts.
As pitch-ratios may be in direct, oblique or
inverse relatior1s with the time-ratios in each part,
correlation of the tv10 counterparts offers the following
fundamental possibilities:
CP = T+P direct .
CF
T+P direct '
T+P oblique .
T+P oblique '
T+P opl�que
T-;-P direct '
T+P inverse
T+P inverse .
T+P inverse
T+P inverse ,,
T+P oblique '
T+P direct
'
The second, the third and the fifth forms
u
have another varia nt each (by inversion).
Thus, the
•
0
0
33.
total number of the above relati ons is 6+3 = 9.
E xamples:
CP _ T+P direct
T+P direct
CF
(1)
CP _ bTP + o2T2P + a4T4P '.
d4T4P + b 3T3P
CF -
(2)
CP _ aTP + b2T2P + a 3T3P + d4T4P
•
CF
04T + a3T3P + c2T2P + bTP
Fi ure XLI.
•
(•)
T
I
•
(�)
C p . . . . ..�.. . . . . . . .. . . . . . . .. . . . . .
...
•
.•
CF
•
t
I
L
t
CP
T+P direct
CF - T+P oblique
�
-
-----=;.._..,;._;;;.
CP _ a4T4P + c2T2P
( l) CF
- dT3P + c2T2P + d2TlP
;
CP = b 3T3P + dTP + c2T2P + a2T2P
(2) CF
dT4P + b3T3P + c4TlP
•
l
I
•
0
0
Figure XLI I .
I
CP .. • • • • • • • • • • • •
l
I -
(p . . . . . .
•
..
•
•
!
Cf
•
Cf" ..... ... . . . . . . . . . . . .. . . .
•
CP _ T+P inverse
CF - T+P direct
(1)
CP
CF
(2 )
dT
P_+
T�
T=
P
2�
+ a=T=P-=CP = =a
2T
�d=2�T�2�
�2�P;,---+_.;.d�2=
�
2P�___
+ =
��a=2=
=P
;.__,,+
•
=
T
c
+
CF
c4 l P + c3T2P + 2T3P
cT4P
=
a6T2P + b3T4P
b4T4P + d2T2P + c2T 2P + dTP
Figur e ,XLI I I .
-,.------:------- __,..-----
(1)
.,
CP
CF
CP
.... • • • .. • • • .. • • •
•
• • •• • • • • •
c;:, F . - . . . . . . . . . . . .. ... . . . . •·
+
+
0
0
35.
CP _ T+P obligue
CF - T+P oblique
(1 )
CP _ a3TlP + a2T2P + bT3P + b3TlP + b2T2P + aT3P
c3T5P + d4T4P + c5T3P
CF
(2)
_
CP �
CF
bT5P + a2T4P + d3T3P + b4T2P + a5TlP
a7T3P + b5T5P + c3T7P
Figure
• XLIV .
••
•
(t)
(l)
. ..
r•
'
.
;,
C p -----.
. .:. . . . . . T .. ..
•
'
Cp .. . . -.
- • • . .....
j
..
•
....
•
•
•
CF · · · · ·- · · · · · · · · · · · • • ..J. ... . . . . .
·
CF
t
. . . .... . -. . . __. . . . . . . . . . . . .... . . . . . . ......
•
•
f
•
CP == T-:-P oplig!}e
T+P i nverse
CF
•
(1)
CP _ b3T2P + · c3T3P + b2T3P
CF - aT2P + b2TlP + c2TlP + d3TlP '
(2 )
CP _ a4T3P + d3T3P + a3T4P
CF - cT4P + b2T3P + b3T2P + c4TlP
0
0
r
36.
Figure XLV .
(l.)
..
CP
. . . . . . .. ... . . . . ..... ... .. . . . . .
p
C
CF
C F ···· · · · · · · · · · · : · . .. . . . ...
•
•
•
CP _ T+P inv erse
CF - T+P inverse
(1 )
CP _ a3T lP + cT3P + c3T lP + aT3P
a5T3P + b3T5P
CF -
(2 )
CP -_ cT2P + c2TlP + b2TlP + b4T2P
CF
d6T3P + d3T6P
•
'
•
0
0
37 .
Fig ure XLVI_.
•
-
-
·CP
f"
,.
•
•
• • • •
+C F
I
•
•
•
•
(2.)
;
•
t
t- T -+
i
•
..+-'"·
,
CF
._
�t�,t
__,. T
---;---1--.--;--,-
..,.-.....-1i_
..-f ...,_I_____
I --1-[-\
.
.
.
--
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..
I
J
I '
•
-.,1'
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r
•
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•
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_,....,..
• •
-. ,, ..
. . .. ... . . .'.. •.. . . - • . .. .
....
_.
_
..
••
.....
•
Example
of App lication
'
CP _
T+P direct
�
CF T+P inv erse
CP - a4T4P + b3T3P + a3T3P + b2T2P
b8TlP + d4T2P
CF
T(CF) - (4+3+3+2) 2
-
( 16+12+12+8) + (12+9+9+6) +
+ ( 12+9+9+6) + (8+6+6+4 ).
T(CP )
-
•
( [I] +l+l+ l+ ltl+l+ l+ l+l+ l+l)
•
r ➔
1
0
0
•
38 ..
A�ial combination of
'
CP
CF
i n its general form:
f
•
C p . . .... . .� · . ... . . . . .
t
..
' . ..,
•
•
'
Let CF be constructed from C-ma j . na t. d0
scale
a11d
CP -- from A�- maj . nat . d� s cale.
P = 5p wit h approxim ation.
•
Let
Under such conditions, the
range of CF will be abo ut an octave an d a half, and
the range of CP -- about two octaves . .
Figure XLVII.
'
( please see next page)
0
0
39 ..
(Fig ., XLVII)
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.
Composition of the counterp�rt to a given melody
by means of axial correlation.
In order to accomplish the process o f
correlation of counterparts by means of axial
correlati on, it is necessary to reconstruct the axial
group of the given melody first.
After the analysis
of TP ratios of CF has been accomplished, it is
I
'ti •
•
,
•
0
0
•
40.
I
important to detect whether the T+P is of direct,
�
After this, the general
oblique or inverse form.
planning of the CP axial combination must follow.
Fir st -- with respect to T+P correlation, and second -
with respect to the axial combination itself and its
T+P ratios.
•
•
The following graph is a transcription of
Ben Jon son ' s "Dr ink to Me Only With Thine Eyes" .
Figyr e XLVIII.
•
.
'-.__,)
C
,
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.
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--- .
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•
==
This melody contains a modal modulation.
P.A. 1 is .. Phrygian (d 2 ) ar1d P .A . 2 is Ionian (d0 ) .
The entire axial gr.oup gradually gravitates toward
.
P .P. • 2 , where it forms its absolute balance.
If vve take
into account all the minute crossings, analysis of the
axial group appears as follows.
0
0
P . A . , = a6t + b2t + dt + ct + a2t + b3t + d3t
P.A. 2 = b3t + 05t + llJ .
The modulation is performe d by establishing
the correspondence between d3t (P.A. , ) and b3t (P.A. 2 ) .
We can say that: d3t (P.A. , ): b3t (P.A. 2 ).
As pitch
ranges are approximately equal, the P- ratio may be
regarded as constant .
Let us devise a counterpart in 1+4 time•
ratio • . This would mean that CP v,ould have only. one
secondary axis.
As the general tendency, of CF is
gradual gravitation toward balance in the course of
two oscillations (which correspond to four directions
•
and eight ;individual axes), we shall introduce b-axis
for the counterpart..
Then CP will consist of one
gravitating to,vard balance.
direction, consistently
'
such conditions
development .
gi
Under
represents a complete cycle of
This counterpart corresponds to the case
(2) in group (a) of Fig. XXII , where CP has an Aeolian
P.A . (ds ) .
figure XLIX.
(please see next page)
0
0
•
42.
(Fig . XLIX)
•
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o
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0
0
S C H I L L I N G E R
J O S E P H
C O R R E S P O N D E N C E
C O U R S E
Subject : Music
With: Dr. Jerome Gross
Lesson CLXII.
The Use of Symm�tric Scales
in Two-Part Coupterpoint
•
The unity of style requires th.at both counter
•
•
parts are based on symmetric scales, if one of them is •
•
Scales of the Third Group and scales of the
'
. Fourth Group, mostly in contracted form, s�rve as
material for counterpoint.
It is acceptable to have one
counterpart in the Third Group and another either in the
u
•
Third or in the Fourth Group.
When the two counterparts
belong to the different groups, two cases can be observed :
of pitche�;
(1) both scales have identical set
"
set' .of pitches.
(2) both scales have different
'
•
Example :
�
(1)
�
�
(2)
't'
\.__/
�
-
Tr
-
T,
-
T,
-
T,
C
C
C
T,
Ta
T
a;_ d�- e - a4 - C
-
f
-
T2
dl,- e - f -
-
C
L.....
-
..._
�
d
-
·e � - f
d - f - g
!•- a L..-
T,
C
-
T2
- g4F- a - b
f'f'
L
T,
-
!�- b
Ta
\
V-
C
d' - ef- e� '--
r*-
a - b
-
Ti
C
0
0
2.
Relations of the harmonic axes of the two
counterparts can be carried out in all four forms
previously used.
Their meaning with regard to symmetric
scales appears as follows:
Type I (U.u.)
: both scales have the same T , , the same
number of tonics and an identical set
of pitch-units;
•
•
Type II (U. P.)
•
: both scales have the same number of
tonics, their sets of pitch-units are
identical, but their harmonic axes are
on different tonics;
Type III (P .U.) : both scales have an identical form of
symmetry (the quantity of tonics) and
•
an identical set of pitch-units; none
of the tonics of one scale have common
pitches with the tonics of the other,
. .
i . e . , the two sets of tonics belor1g to
the mutually exclusive sets of pitches;
Type IV (P.P.)
: the two scales belong to either identical
or non-identical forms of symmetry;
their sectional scales are of nonidentical structure, yet belonging to
•
one family (according to the classifica
•
u
tion offered in the First Group of
Scales) ; the two sets of · tonics belong to
the mutually exclus,ive sets of pitches.
•
0
0
3.
E xampJ�s of two-par t counterpoint executed
in the scales of the Thir d and the Fourth Gr oup
F igure L .
,. .
IIU,.
+
..
'-"
-
,_
L
TYPE. I.
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0
0
10.
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(Fig • L cont • )
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4 a.
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(Fig . L, cont . )
Type IV.
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•
5•
•
Lesson CLXIII .
Continuous (Canonic) Imitation .
•
The source of Continuous Imi t_ation, usually
known as Canoni c , is a well known phenomenon of
acoustical resonance, bearing the name of Hellenic
Before any composer existed on this planet,
nymph Echo.
•
•
•
•
nature created by chance a quintuple echo "Lorelei"
(which can be justly called five-part canon) dis covered
on the Rhein.
Admiral Wrangel (Russian) describes a
- a canyon
place in Siberia, where the river Lena enters
.,
about 600 feet high and where a pistol shot rapidly
repeats itself more than a hundred times.
u
•
How would you like that for a canon?
Music theorists, which is typical of their
species, think canon to be a purely esthetic development.
VVhatever they think, it is a natural phenomenon and tl1e
most ancient form of musical continui ty.
Th e
. re is a common belief that it requires a
great skill to write a canon.
In reality, the real cause
of any dif.ficulty in writing in this form is methodo
logical incompetence.
Both music theorists and composers
are guilty, because they have not been able to forroulate
the principles of continuous imitation.
· I will not discuss the case of Sergei Ivano
vic h Taneiev, as his interpretation of the canon
'
0
0
6.
•
requires the knowledge of his "Con�ertible Counter
point of Strict Style " , which is a highly complicated
system and deals with the Strict Style only.
Besides,
it does not bring the solutiun to melodic and rhyth mic
forms, being mostly preoccupied with the vertical and
horizontal convertibility of intervals in the harmonic
•
serise.
••
Canon is a complete composition written in
the form of continuous imitation.
•
The usual academic approach to this form is
such that the student is taught first how to write an
11
ordinaryn imitation (scientifically: discontinuous
imitation ) .
•
After not getting anywhere with this form
of imitation, he begins to struggle with the canon .
As from the start the principles of any imitation are
not disclosed . to him, it doe s not make any difference
whether the imitation is discont inuous or continuous.
Once such principles are defined and the technique is
specified, it becomes obvious that the discontinuous
imitation is merely a special case of continuous
0
imitation.
With this in view, we shall establish the
principles of continuous imitation .
Continuous imitation consists of one melody,
coexisting in two different parts in its different
and at a constant velocity.
phases
.
-
•
,
)
0
0
'
• 7.
•
This melody, being of identical structure
in both parts, may vary in intonation.
The latter
condition takes place only when the scale-structure
itself varies.
The temporal organization of cont inuous
imitation has no direct influe11ce on the duration of a
canon.
Longer rhythmic groups are preferable, however,
as continuous recurrence of the same rhythmic structure
.
becowes, eventually, monotonous.
The main sour ce of con tinuous - self-stimula-tion
'\.
in a canon is its melodic form, i.e. , the axial group.
With the devices offered in the Theory of Melody (see
•
Chapter II) it is possible to evolve an axial group of
great extension and, if necessary, wit.bout any repe t.itions.
Thus, the continuance of melodic flow becomes completely
protected.
The correlation of harmonic types and the
treatment of harmonic intervals remains the same as for
all· other forms of contrapuntal technique.
This permits
to compose canons in unitonal as well as in polytonal types.
..
Temporal Strucj:;ure of Cop;tinuous Imitatiop.
A complete composition based on continuous
imitation is known as canon.
The duration of continuous imitation or of a
canon is the multiple of its temporal structure.
The
temporal structure of a two-part canon is related to the
0
0
8•
theme of the canon as 371.
The first third is the
•
aru,.ouncement, the second third is the imitation of
announcement in the first voice and the counterpoint in
the seco:t1d voice, and the last third is the imitation of
the first portion ·of counterpoint in the second voice and
the second portion of counterpoint in the first voice .
..
After the temporal scheme is exh austed, it begins to
repeat itself with new iritonations.
If we designate the first entering voice as
Pi
(whether upper or lower) , the second , entering voice
as "P! r , the first ann ouncement as CPj , the first portion
of counterpoint as CP a, the second portion of counter
point as CP, etc. , the temporal structure of a canon
•
ap�ears as follows :
The continuation of the
temporal structure does not alter the process, merely
increasing the subnumerals of CP in the original
relation :
+ CP,
+
+ CP-.
CP�
• • •
The temporal structure of any two-part canon
is based on two elements, which appear as reciprocal
permutations.
All forms of variation of two elements
are applicable therefore to two part canons (see Theory
,
I
0
0
Let a and b be two elements representing
of Rhythm).
any kind of duration-groups.
a + b + a
Then,
, and the continuation of tbe
a + b
temporal structure assumes the following appearance :
a + b + � + b + � + ·b + � +
.
•
•
a +
a +
a +
=
•
The duration of a temporal structure is t�e
real factor controlling the flow of the canon .
The
longer the structure (not by speed, b y the quantity of
attacks) , the greater the fluidity of the canon.
Duration-groups of all kinds are acceptable as temporal
struct�res for continuous, . imitation and for the canon.
,
A. Temporal structures
compose� from the parts
•
of resultants.
•
(1)
(2 )
J.
I
O·
J J nn J J J J n
, , 1J u , ,
Ot,
I
0
0
10.
B. Temporal s�ructures c9mppsed from
\.._)
co�Elete resultants.
4
4
•
J . J J J J J·J J J J . J . J J J J
9 1 , 1 1 '\ , � -
r· ,
.
•
•
•
(2)
r5+4
, r· r
•
r 'f •
•
(J
•
,0
0
0
I
11.
CL
'
'
t
£ J.:. n J . J J..,rJ. J 01 J. J'J J d . nJ . J�n J. J JJJ .
J
r , u r ,,� t1· r 9 , · i- � , , 4 ul 9
-� -
,,
�
.
•
•
•
1 1,; 'i ·
- ·u
9.
,-..
111 9
,,--,.
q-
, v 'l · ,,.,
---- "
, , • , \.--1 . 1 · 1,1 1 ·
1 - 'f .
q.
,....,
,.
, v4
L-€
C . Temporal structures evolved by
mean? of permutations.
4
u11,
•
ot.
1 u,,
, , u, , , , u u, , ,
1.1 1 , t 1 u , 1
,u,1
, r u t rt t u
.......... ......
1. • , v'l •
,.,. , q .
I
0
0
12.
D. Temporal structures composed from
synchronized involution-groups .
•
(1)
3(2+1) + (2+1) 2
a..
-
3
4
J .'-" d .
...g
J J J j J ....__ J . J .
f · _, f • f · f .-../ , f r ,
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15.
Lesson CLXIV.
Composition of canons in. all four typ��
of harmonic correlation.
As canon is a duplication of melody at a
certain
tj.me interval, the differences o f intonation
in the two counterparts are due to scale-structures.
•
•
•
Thus, type I produces identical intonations, type II -
non-identical ir1tonations, type III -- identical
intonations and type IV -- non-identical intonations.
The choice of axes in all four forms of correlation
remains based on the . original principle: cor1sonance
between the axes of two counterparts.
In types II and
IV the starting P.A. can be in a dissonant relation with
•
the P.A. of the first voice, but it must end on a
•
consonance.
As continuous imitation can go on indefinitely,
•
we have to know the exact technique of bringing it to a
close .
Cadences are produced by ·the leading tones moving
into their primary axis.
As the first moving voice
defines what happens to the second voice, all that
l.S
necessary is to produ ce a leading tone in the first .
.
.
moving voice.
•
When this portion of melody
•
l.S
transferred
second voice, the first voice produces it•s own
to the
•
leading tone, after which both voices close on their
primary axes.
•
0
0
16.
The use of symmetric pitch-scales is
applicable to cano11s as well.
Exa,oples of two-part canons in all four tYJ?es
•
of harmonic correlation.
Figure LI.
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Compositi on of Canonip Cont�nuity by means
of Geometrical Inversions .
•
•
The original canon can be considerably extended
by means of geowetrical inversion.
Tl1e voice entering first produces the axis of
inversio11 for the positio11s
©
and @ .
The final
balance of the last cadence must not participate in the
sequence of inversions, as this would disrupt the
It must be used only at
continuous flov, of the canon .
the very end of the coroposition, if the canon er1ds in
position @ or @ .
Otherwise a new balance must be
added.
Under sue� conditions, the canon consists of
several cor1trasti11g and indevendent sections of
continuous imitation.
Example of a canon developed thrOU£h the
use of geometrica� inversions.
-
•
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0
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21 .
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readjustment of the parts becomes a necessity under such
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•
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two vertical
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�-
0
0
•
23 .
Lesson CLXV.
Fugue
A complete -compositi o n based on discontinuous
imitation constitutes a Fugue.
,
A fragmentary (incomplete) composition based
on discontinuous imitati on constitutes a Fugato.
All other names established - in the past, like
Sinfonia, Invention , Praeludium, Fughetta refer to the
same fundamental form , i . e . , Fugue.
The difference is
mostly in the magnitude of the composition\. (Fugue,
Fughetta) or in the type of harmonic correlation of the
counterparts.
Thus a Fugue which is unitonal-unimodal is
called Invention, Praeludium or Sinfonia.
Praeludium
'0eing the loo sest term o f all, as in many cases it has
not.hing in common with the Fugue.
A Fugue which is
.
unitonal-polymodal (and of a specified polymodal ity) is
called Fugue.
As in my opinion the presence or absence of
polymodality · as well as or polytonality is a matter of
harmonic specificati ons, which vary with time and place,
any complete compositi on based on disc9ntinuous imitation
can rightly be cal�ed fugue.
Fugato usually appears as a polyphonic
episode in a homophonic composition.
•
•
•
0
0
The Form of a Fugue
The temporal struc ture of a fugue depends
on the quantity of themes (sub je cts).
It is customary
to call the fugue with one theme a nsingle fugue n and
the fugue with two themes a "double fugue".
Triple
fugues are very rare, and a real triple fugue requires
many parts (voices), otherwise th e idea that each part
is a theme becomes nonsensical.
For this reason it is expedient to confine
the two-part counterpoint to. fugues with one theme only.
Tr1e general characteristic of all fugues is
the appearance of the theme in all parts in sequence.
The complete thematic cycle is known as an exposition.
•
In two-part counterpoint the first entering voice
announces the theme (we shall call it CF, for the sake
of unity in terminology) , after which the second voice
enters w ith the imitation.
called
necho".
11
This imitatio11 is usually
reply" and might as well have been called
In fact, it is tr1e same theme, with the possible
difference caused by the form of harmonic correlation.
Thus, reply in the types I and III is identical v1ith the
theme, whereas in the types I I and IV it is non-identical,
insofar as the scale-structure is modified.
At the time the second entering voice makes
its announcement (CF ) , the first entering voice evolves
a counterpart (CP) to it .
Thus the form of the first
0
0
25.
exposition (E , ) is as follows:
Pr - ---"'+---CFCP
-CF
Prr -
and the form of any other
--
exposition (En ) is : En =
-
CF + CP
•
CP + CF
In both cases the definition of the first
entering voice (P1 ) and second er1tering voice (P 11 ) ca�
be inverted.
•
In a fugue where CF and CP represent the only
thematic material and no int erludes are used, the ent ire
\.
composition acquires the following form :
F = E , + E 2 + E 3 + • • • + En •
•
In homophonic music this corresponds to a
theme with variations.
In the fugue the variation
technique consists of geometrical inversions of the
•
original exposition.
The counterpoint to the theme can be either
thru
constant (i.e . , the CP is carried out/the a1tire fugue),
or variable (i.e. , a new CP is composed for each
exposition).
Statistically, the use o f constant or
variable CP is about fifty-fift y.
In the XVII and XVIII
Centuries constant CP was somewhat o f a luxury , as the
counterpoint which we consider to be general technique ,
at that time was known as vertically convertible
counterpoint, which was believed to be more difficult to
'-J
0
0
26.
execute.
On the other hand, old composers did not lmow
the technique of geome trical inversi ons, but used tonal
inversions instead and merely as a trick, on some
special occasi ons .. , With the systematic use of geome trical
inversions, fugue becomes greatly diversified.
Under
such a conditi on, constant CP becoroes merely a practical
convenience.
••
Once the theme and the count erpoint are
composed (preparati on of one expositi on) , you get the
entire fugue by means of quadrant rotation arranged in
any desirable sequence ..
If rotatior1s refer to the entire
E, the fugue assumes the follo,ving appearance:
F = E
+ . • • , where m, n and
' \!!!.I
t::'\
p ar e any of the geometric al inversi ons.
F or example:
•
•
Such scheme s are subject t o composers r
i.J.1ventiveness.
Quadrant rotation may affect each appearance.
of the theme, then theme and reply appear in the different
geome trical positions.
. .
0
0
27•
•
For example :
(1 ) E =
(2) E =
=
CF @ + CP
CP
+ C�
(3) E
••
I t is important to note that position is
_always identical for two Si1DUltaneous parts.
''-
Thus,
CF @ means that CP set against it �s also in position @ .
Quadrant rotation applied to theme and
reply pr9duc es al together 16 geometrical forms of
exposition.
Forms o f Imitation Evolved
Through Four_ Quadrants
Figure LIII.
b
•
d
b
C
•
d
0
0
28 •
All cases referring to one geometrical
position for the ·entire . E form the diagonal arrange
ment (heavily outlined) on the above table and appear
to be special cases of the general rotary system.
It is easy to see that with this technique a
fugue of any length can be composed without any effort.
An example of fugal scheme employing
•
•
•
guadrant rotation •
,
CF @ + CP @)
CP @ + CF@
+
•
+
+
c� + CP @
+
E
3
CP@) + CF @
CF @ + GP @
CP @ + CF G)
E 41
+
CF G) + cp·
CF G) + �P G)
CP @ + CF G)
Ee
+
(Q)
CP @ + CF ©
E.r +
+
CP G) + CF @
CF � + CP @
+
CF @) + CP �
E,
CP @ + CF @
CP {li) + CF @
CF @ + · CP @
As this example shows, CF may appear in
the same voice successively, when its geometrical
position alters.
The form of fugue where counterpoint is
varied vdth some or with each of the expositions can
•
0
0
29.
r
'--'
also be subjected to quadrant rotation.
The general scheme of such a fugue appears
as follows :
+
••
CF
CP a
+
+
+ CP 1)
CF
CF + CP ,
F =
....- - �F
V
CPa
CF s
+ CF
E2
+
CF
+
+
CP
CF
E3 +
E 'I + • • •
µi example with application of the quadrant rotation
F =
•
CF
CF (E) + CP 3 ®
E3
CF
CP
2@+
I
CF + CP 3
+
CP 2 + CF
@
{g) E.(
+
+
CF
+ CP 2
CP 1 + CF
E &.4 +
CP� + CF �
CF (ii) + CP , (g)
E6
In· the old fugue the· elimination of monot ony
was usually achieved by means of Interludes .
An inter
lude (we shall term it : I ) is a contrapuntal sequence
of the imitation or of the general type.
Statistics
show that about 50 out of 100 interludes are thema tic
•
(i.e., based on elements of CF or CP) and the rest
neu tral (i.e. , using themat ic element.s of its own) .
As in th e case of coun terpoint itself, I
may be COHJ,Posed once and rotated af'tervfards..
other cases a new I may be composed each time.
In
In the
0
0
30 .
old classical fugues i nterludes served mostly as a
bridge between ��e E • s , and leading into new key.
In our fugues of �fpes I and II they can serve the
Sallle purp ose, whereas in types I I I and IV the interludes
are hardly necessary, as the key variety i s already
inherent with the group of different symmetric tonics.
.
•
•
As Vfe shall see late1"' , the fact that we h·ave two parts
does not limit the quantity of tonics .
The general scheme of a fugue with interludes
appears as follows:
This form is equivalent to the First Rondo
of the homophonic music.
I f , I 2 , I 3 , . . . may be either identical
(though in different geometrical posi tions) or totally
different.
I n , i . e . , the last interlude is quite a
common featur e iri the old fugues and has the rnea11ing
of a conclusion (coda) .
By rotating the same interlude
we acquire new modulaton directi ons.
The method of composing a fugue by this
system consists of the foll o,vi ng stages :
(1) Com_posi tior1 of the theme;
(2) Composition of the counterpoint (one or more)
to the theme; this is equi,ralent to the
u
,,,,
preparation of an exposition;
0
0
31.
(3) Prepa rati on of the exposition (or of all
expositions if there is more than one
count erpoint) in fo·ur geomet rical posi t1ons:
CF @
'
CP
CF @ •
'
CP
CF @ •
'
CP
(4) Composition of the interlude ( s) ;
CF @ •
,
CP
(5) Preparation of the four geometrical positions
••
'
of the interlude( s) ;
(6) C om position of the scheme of F ;
(7) Assembling the fugue.
•
•
•
•
(_J
•
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J O S E P H
S C H I L L I N G E R
C O RR E S P O N DEN C E
•
C O U R S E
-
With: Dr. Jerome Gro� s
pubject: :U.u si c
Lessoµ CLXVI,
C omposi tion of the, Theme
Theme in a fugue i s of utmost importance, as
it constitutes at least one half of the entire composition.
Nobody yet has defined clearly
•
fug al theme.
the requirements for a
A good fugal theme is usually ascribed to
the composer' s genius, and this is neither help nor
consola tion to the student of tl1is subject.
We want to
know precipelY_, what makes the_ melody a sµi tabl� fugal
•
theme, as experience shows that: (1 ) not every g ood or
•
expressive melody makes a sui table fugal theme, and (2)
•
not every sui table fugal theme is a good melody for any
other purpose. C ompos.ers, who were. outstandi n g melodists,
fai led to show important achieveme nts as contrapuntalists
(Chopi n, · sohuman n� Liszt, Chaikovsky and others) .
-The
first regu1r�ment for a fugal theme i s.
that it must be an ipcomplet� m elodic form .
In the best
and most typical fugues by J.S. Bach we find that su ch
incomplete melodic forms follov1 their com pletion
as
=
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e
a
a
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=
•
counterpoint evolvin g during the ann ouncement of the
theme i n the second voice.
An i ncomplete melodic form i n this case means
that at the moment the second voice starts the theme� the
1
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0
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•
2.
first voi ce does not arriv� at its primary axis,
For an illustrati on, let us take Fugue II,
Vol. I, Well Tempered Clavi chord (later to be referred to
as w.T.C.) by J.s. Bach.
Figure LIV.
Pl:
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The theme ends on the first sixteenth of the
third bar, while the melodic form com pletes itself on the
third quarter of the same bar.
It is interesting to note
that the theme (and the melodi c f orm) is constructed on
0
In order to present his announcethe binary axis:
a.
ment clearly, Bach uses
½(
= � ) at the point where the
theme might have stopped otherwise, reserving the eig hth
until the reply i s far on its way of developing.
Thus
Bach eliminates the danger of stopping, which, indeed, i f
reali zed, would have sppiled the en tire fugue.
Another
important detail i s the j uxtaposition of db- axis i n CP
f
versus 0-axis in CF.
0
0
•
All other requirements for a fugal theme
really derive from the first one: all such resources
of tempo ral rhythm and axial. forms ca n be used which
•
•
demonstrate an unfinished melodic structure in the
process of its fo rmation,
The presence of any one of the f ollowing
..
structural characteristics, as well as of !AY combinatio ns
of the latter, produces a suitable fugal theme.
$
(
(l) The p resence of rests; particularly a decreasing
series of rests, combined with an inc�easing number
of attacks; stop-and- go effects; gaining momentum
effects.
•
(2) The sequence of decreasing durati on-values: rhythmic
•
acceleration in the broadest sense.
(3) Dialo gue effects achieved by means of binary axes,
and by means o f atta ck-groups contrasting in f orm,
like legato-staccato.
(4) Effects of g rowth, ach ieved by mea ns of binary and
ternary diverging axes.
•
(5) The p resence of resistance forms (including
repetitio n, phasic and periodic rotation), particularly
leadi ng to climaxes.
Coµibinati ons of the above techniques applied to
p
. ne theme make the latter more saturated and tense, which
increases the fugal characteristic •
I
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•
4.
Figure LV,
Fugal theme:5 b)'.' . J, s_. Bach and by just J. s.
(Numbers in musical examples refer to the
preceding classifications ) .
w. ,: e .
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8.
As it follows from the abov� examp les, the
total duration of a theme (i n terms of quantities of
attacks, or in terms of bars) largely depends up on
the eomposer t s deci sion.
However, the following
generalization is true for most classical fugues: the
duration of the fugal theme i s i n 1nvers� p�oportion
to the number of parts.
Indeed, the first theme of Fugue IV, Vol. I,
W.T.C. has only five attacks; the theme in Fugue XXII,
•
Vol. I, W. T.c. has six attacks.
•
written i n five parts.
Both of � thes . e fugues are
On the other hand, Fugue X of the
sam e volume, wri tten in two parts, has a theme of
twenty-six attacks.
It is not important that the reply enters
on the same time-unit of the measure as the theme.
Quite
• to the contrary, difference in the starting moments (in
relati on to the bar) adds interest . to the whole composi
tion, as it produces an element of surprise.
unsuitable for fugues can be subjected
Themes
,
to some alterati ons , which will make them sui table.
•
It can be demonstrated, by reversing the
procedure, that the mere addi ti on of 0-axis to any
melodic form cap render it sui table as a fugal theme.
J.S. Bach t s theme from the "Toccata and Fugue" i n
D- minor for Organ, being deprived of its 0- axis, loses
..
all its fugal quali ty. When 0-axis is taken out, the
0
0
9.
axial combinati on becomes: b+a+c+a.
Thi s theme seems
t o have nothing but rotati on in relatively narrow range.
The inclusion of 0-axis produces a n effect . ·or- growing
resi stance, and the axial combination becomes:
- 0
d+c+c
•
Figure
•
• LVI.
•
•
The number of bars i n a fugal t heme is an
optional qua ntity.
It may be pair or odd.
integral or f ractional.
It may be
Both odd and fractional are
preferable to pai r and integral, becau se t he latter two
suggest a cadence at the end of the t heme.
I believe
one of the factors that influ enced Buxtehude and all
the Bachs is the awareness of cantus firmus (in a
strict sense) as a theme.
odd number of attacks.
Canti firmi usually had an
0
0
.-
•
10.
Lesson CLXVII,
•
Prepa ration of, the Expo�ition
After se·lecti ng the theme, th.e com.f)os'er must
dev9 te himself to the preparati on of fugal exposition.
As it is easy, with this method, to write
· f our types of fugues on one trieme, i t becomes desirable
to prepare f our expositi ons for the future f ugues.
•
In
a two-part fugue, the enti re preparati on of E consists
merely of wri ti ng CP to CF.
It is advisable that the
expositi on prepared f or each ty pe would be written out
i n all geometrical positi9ns.
This saves time during
the period of assembling the fugue.
Fugues of type IV
often require preparation of two exposi tions, as when
CP
the axes exchange in cf , CP may not fit, and a new
G
counterpoint must be written (CP11 ) .
To make the demonstrati on of all techniques
pertaining to fugue concise, we shall use
theme •
•
&
very brief
Figyre LVII,
•
( please see followi ng pages)
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Composition of th� E2rn osi tions
Compo sition of the expositions in ty pe I
does not require any special co nsiderations, as bo th
parts have an identical P.A.
In ty pe II, the modal modulations o f CF,
and its respectively related
of modal sequence.
CP,
must be in one system
For example, if P.A. of CF , is £
and P .A. of CP , is !_, the axis of CF2 (reply ) must be �
and CP2 (counterpoint to rep ly ) must have P.A. o n .Q.,
in order to retain the axial unity in the first part
for the course of one exposition, an d in order to preserve
•
0
0
•
g� as it was origi nally
the vertical relation of
conceived.
The entire structure of the fugue ( from the
above relations) appears as follows:
F =
•
(CF3 + CPa)�
CP 2£ + CFlf !
(CF, + CP , )£
CF:a �
(CF,r + CP.r) !
CI!, � + CF6 �
where .Q., .!., f, .!!, • • • are the primary axes of the
respective parts.
-
Likewise i f
becomes:
C
+
A
A
-c+d-
-
, the sequence of P.A. • s
C
+ • . •
•
I n type III, the tonal ( key) modulations of
CF, and its respectively related CP, must be in one
•
This sequence preserves
system of symmetric sequence.
CP
its constant CF relation only when CP2 (the reply )
forms its P.A. in symmetric inversion to the original
setting.
Let us take the symmetry of '!m,.
For example:
CP
_
§. y
--c •
CF - --
-
In order to
preserve the axial relation where CP is 3 semitones
above CF, the reply must appear from the opposite
•
•
equidista nt point,
from a.
Th is allows the
relativ e stability of both parts, as CP, being three
semitones abov e CF requires th e �-axis.
The structure of such a fugue, evolved on
(_)
four points of symmetry (tonics) , appears as follows:
•
)
0
0
•
15.
A simi lar case evolved from t h ree points of
symm.etry (3J2 ), where
§� =
sequence of- P.A. ' s:
C
-a
••
•
a► +
+
-c+e-
� , g ives t he following
-
-aV+c-
In type IV, in order to carry out t he sequence
of P.A. ' s in symmetric in versi on of t he original setting,
it often becomes necessary t o prepare two 'independent
expositions:
•
E = �:I
and E' = �;��- , as CP may be 1n a
different intervallic relation to CF2 tha n it is t o CF, •
The differen ce usually appears i n vari ations on semit one
or whole tone, which results in most disturbing relat ions,
such as a seco nd instead of a thi rd.
For t his · reason,
example in Fig. LVII offers t wo expositions.
It is easy t o see t he unfitness of CP I as a
counterpoint to reply, by exchangi ng it wit h P.A. of CF.
The sequence of symmet ric P .A. • s• in type IV
of Fig. LVII would develop on t he basis of its pre-set
expositi ons:
E =
CP1
CF
-
and E• =
CPII
CF'
•
0
0
16.
Consi deri ng the enharmonic equality of e4F
and f, a:#f and bp e-tc . � and the fac t that CF i s ev olv ed
in natural major d0 and CF• in natural major <4 , we
obtain the following structure for the fugue:
•
F ::
•
t
I
'
I n the old classical fugues reply appears on
•
•
(CF + CP11 )t
(CF + CP1 1 )£. E + (CF + CP 11) f. E
+
,
1
(CPr + CF )!,if. a
CF � '
(CP1
CF•)g_#
the dominant (i.e., seven semi tones abov e or fiv e semi
tones below the theme) . If there was a sequence of
expositions before the interlude took place, the theme
would usually have returned to the tonic . Acc ording to
our type II, if CF , =
•
�
and CF a = t, CF3 should have been
CF'f should have been a etc . However, this was not the
C
c ase in the fugues of the classi c al period, and there was a
g ood reason for it. As the tQning of mean temperament
! and ! )
(the two-c oordinate system:
developed abberation,
while deviati ng from the tuning center ( =· 1 ) , i t was not
possible to get satisfactory intona ti on in the co urse of
travelin g throug h
Cs
or
C-s-
P.A.• s. And thoug h equal
temperament has overcome this defec t, the habit remained
with the c omposers till the end of XIX Century .
•
•
•
0
0
17 •
.
Le s s on CLXVIII .
,Prepareation ¢' �he Inter lud e s
•
Inter ludes (I, , I 2 , • • • Im) s erve as bridges
between the expos itions .
The las t interlude, if the
fugue end s with one, i s a pos tlu�� (coda) .
s erve two pur pos es :
Interludes
(1) to diver t the li s tener • s attenticn from the
per s i s tenc e of theme ;
•
(2) to produc e a modulatory transi tion fr om one
•
key-axis to another.
•
T he fir s t form i s confi ned to one key, but may
have any number of s ucces s ive P.A. • s , thus produc ing
modal modulati ons (U .-P.) between the two adjacent
expositions having the s ame key-axis (U. -U. and U.-P. ) .
The second form contains different key-�es (P .-U. and
P.-P . ) and c onnec ts the two adjacent expos itions having
different key-axes (P. -U. and P.-P. ).
•
Both form s of
interludes may be either neutral or thematic. Neutral
'
interludes are bas ed on the material of r hythm, or
intonation, or both, not appeari ng i n any of the
expos i tion.
T hematic inter ludes borr ow their material
of rhytbm, or i ntonati on, or both from either CF or CP of
the expos ition. Fur thermore, any of the above des cribed
types of interludes can be executed either i n general or in
imitati ve c ounter point.
•
•
•
0
0
18.
The dur ation of an interlude depends on the
duration of the exposition and the quantity of interlude�.
The form of an in terlude itself has an influence upon its
duration.
In order to construc t a pe,rfect fugue, the
duration of int erludes must be pu t into some def inite
correspondence with the duration of exp ositions..
••
Assuming
one exposition as a temporal unit, we arrive at the
following fu ndam ental schemes for the temporal organization
of in terlu des:
•
(1) T (E) = T (I) , i.e., the duration of � interlude
equ als to that of an exposition.
This presupposes
an equal duration for each of the- interludes;
•
(2 ) T (E) ) T (I), i.e., the dur ation of an exposition
is longer than that of an interlude.
An exact
ratio must be established in ea.ch case;
(3) T (E) < T (I), i.e. , the duration of an interlude
is lo nger than that of an exposition.
ratio mu st be established in each case. •
An exact
(4) r-' = I,T + I2 2T + 1 3 3T + . .• , i. e. , each successive
in terlude becomes longer.
T he du rations of
consecutive interludes may evolve in any desirable
type of progression (natural, arithmetic, geometric,
involu tion, summation etc. ).
The resulting effect
of su ch fugue- structures is tha t the interludes in
cour se of time, begin to dominate the theme.
the persistence of the theme diminishes.
Thu s
0
0
19.
(5)
\..._J
P
= I,nT + I 2 (n-l) T + I8 (n-2) T + • • • , i . e. ,
each successive in terlude becomes shorter.
The
res ulting effect is opposite to that of . (4) : the
domination of th eme oVcer in terludes grows i n the
cours e of time.
(6)
•
•
r ,
i.e., the s equence of interlud& s o.evelops
along s ome form of rhythmic grouping.
•
As convertibi lity an d quadrant rotation are
general properties , the same interlude ma y be used
several times , during
the cou rse of a fugtle.
•
This, being
combined wi th key-transpositi ons, offers a n enormou s
variety of resou rces, at the same time cons ervi ng the
•
compos erts energy •
Non-Modu lating Inte� ludes
(Types I and II)
N on-modulating interludes can be either neu tral
or thematic and they can be evolved i n general or imitative
counterpoint.
Figure LVIII,
(1) An example of Interlude type II executed in general
counterpoint.
Non- th�matic (Neu traJ. ) .
(2) An example of interlude type II executed in
imitative counterpoint.
This one is thematic wi th
reference to CF of Fig. LVII.,
(pleas e see next page)
0
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20.
(Fi g. LVIII)
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Modulating Interludes
I. •Modulating Counterpoint Evolved throug h Harmonic
Technique.
Contrary to the general notion, J.S. Bac h ' s
c ounterpoi nt is less "contrapuntal" than it i s believed
to be.
And especially so when it comes to tonal (key
to-key) modulati ons.
It i s obv.ious that Bac h as well as
many other important contrapunta1 is ts thoug ht of key-to
key transitions i n terms of modulating c h ords.
See, for
example, J.S. Bac h • s W.T.c. , Vol. I, Fugue No. X (a two
part fugue) in E- minor.
The harmonic background of
this fugue is very distinct, and this fugue is rather
typical and not excepti onal.
(
I
•
j=
0
0
21 •
..
I t is easy to convert any modu lating chord
progre.ssion wri tten in four-part harmony into two-part
harmony.
Chord structures of two-part harmony have the
followi ng functions:
(1) 8 (3)
=
1, 3 ; used instead of S (5) of the three-part
structure;
(2) 8 (5) = 1, 5 ; used instead of S(5) of the three-part
structure;
.
(3) 8(7) = 1, 7; used ins tead of 8(7) of the four-part
structure.
G
•
s (,)
Figur e LI
• X1
s
...
In order to obtain an in terlude from a four
pau- t chord-progression i t is necessary to select the
corresponding chor dal fu ncti ons which wou ld translate
the fou r-part structures into. two-part structures.
The
voice-leading pertai ning to two-part harmony will not
be discussed here, as any posi tion of two functions is
equally as acceptable for the present purpose.
.,,.
Both
•
•
parts are more or less i n the vicinity of the fou r-part
0
0
22.
harmony range.
The fin al step consists of developing
melodio figurati on in both parts, but with somewhat
contrasting rhythms of durations and attacks.
Modulating interludes can be either neutral
(general counterpoint) or thematic (imitative counter
In the latter case, thematic material is either
point).
borrowed from CF or CP of the expositions, or is enti�ely
indepen dent.
•
Examples of Modulating Interlu�es
r1gure LX.
(1) Neutral and (2 ) Them atic.
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An
interlude can be � ed i n the same fugue
more than once, appearing i n the different geomet rical
positi ons.
It also can be t ra nsposed t o any desi rable
key-axis, in any of the four qua drants.
II. Modulating Counterpoint Ev olved throug h Me lodic
Technique.
•
This new technique is bei ng offered i n order
t o enable t h e c omposer to ca rry out the pure c ontrapuntal
st y le, even when a key-t9- key tra nsition is desi rable.
•
Modulating counterpoint consists of two
independently modulati ng melodies (see modulation in the
Theory of Pit ch Scales), whose prima ry axes are i n a
•
c onstant simu_ltaneous relationship ax an y given key-poi nt
of the sequence.
After t he vertica l dependence has been
established (the harm onic interval between CP and CF),
it becomes necessary to assign to the primary axis of CP
the meaning of t he tonic which is near�st t o CF th roug h
th� scale of key-signatures.
Let t h e exposition end in t he key of c, an d
-
-
let CF end on c an d CP end on a . Then a becomes a - minor
-
(as the key nearest t o t he key of C t hru t he scale of
key signatures; A- major would be far more rem ote).
Thus
we have established a constant dependence where CP is the
minor key t h ree semi tones. below CF.
The next step c onsists of planning the
modulation of P I ( originally: CF). Let t he modulation be
0
0
u
to the key of f- minor •
Then :
IT
= C + d + G + f
Now we assume that i n ord&r to retain the
original vertical dependence between Pr and P11 , each
axis of a major key must be reciprocated by a minor • key,
and vice versa. Then:
•
pf
II
•
_
C + d + G
a + F + e
, i .e., while P r
modulates
from Q to .5!, P11 modulates from .!. to E, ax:1d when P
I
modul� tes from A to Q, P11 modulate·s f rom E, to ,!:!;
finally both parts arrive at CF havi ng an A►-axis and
CP having an f-axis.
The period of modulation f rom key to key in
both parts i s approximateli the same.
Examples of Modulati ng Interludes
Figure LXI .
(1) Neutral and (2) Thematic
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25.
The easiest way to a ompo.s e modulating
interludes by the contrapuntal technique is through a
sequence of procedures:
(1 ) PI modulates to the first in termediate key;
( 2 ) Prr
(3) PI
(4) PII
n
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•
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Lesson C;LXIX,
26.
Composition of the Fugue
The p rocess of assembling a fugu e c onsists of
planning the general sequence of expos itio ns, interludes,
the ir geometrical p ositions and their p rimary axes (key-axes) .
In the following group of fugues only such
materia ls were used, wh ich were p repared in advance (see
Fig. LVII, LVIII, LX and LXI).
The f irst three fugues have inte rlude s ( of both
harmonic and melodic type), while the fourth has none, as
key-variety is suf f ic iently great without it. The la st
'
fugue has indepe ndent c ounterpoints for the theme a nd the
\.
G
•
rep ly.
The la tter. �re interchanged in E s.
E , (i)
It + E 2 @
+
The form of Fugue I (F. ig. LXII):
+
Ea @
+
I a€)• + E i# @@
+
I a G) + E.r@
The f orm of Fugue II (Fig.· LXII) :
6
•F
C
E , @+ E 2 @ + I , + E3 @ + E t1 @ + E.r@) + 1 2 + E6 �
©
The form of Fugue III (Fig. LXII):
0
AP
The form of Fugue IV (Fig. LXII):
0
(E, + E2 + E 3 + E'I + E_, )@ + E& @ + E 7 @ + E8 G)
Figure LXII.
(please see following pages)
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