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JOSEPH SCH I.LL INGER
CORRESPOND E ·N.CE COURSE
Subject: Mu�j.c
With: Dr. Jerome Gross
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Lesson LXV.
SPECIAL THEORY OF HARMONY
Introduction
Special Theory of Harmony is confined to
E, of the First Group of Scales, which contain·, all
musical names (c, d, e, f, g, a, b) and without
repetition.
There are 36 such scales in all.
The total
number of seven-unit scales equals 462.
The uses of E, refer to both structures and
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progressions in the Diatonic System of Harmony.
The
latter can be defined as a system which borrows all its
pitch units for both structures and progressions from any
one of the 36 scales.
Whi'le the structures are limited
to the above scales, the progressions develop through all
the semi-tonal relations of the Equal Temperament.
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The
latter comprises all the Symmetric Systems of Pitch,
i. e. , the Third and ti1e Fourth Group.
Chord-stDuctures, contrary to common notion,
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do not derive from harmo_ nics .
·I f the evolution of
chord-structures in musical harmony
would parallel the
evolution of harmonics, we would never acquire the
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developed forms of harmony we now possess .
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2.
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To
· begin with, a group of harmonics,
simultaneously produced at equal amplitudes, sounds
like a saturated unison and not like a chord.
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other words, a perfect harmony of frequencies and
int ensities does not result i n musical harmony but in
a unison.
we
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In
�his means that through the use of harmonics,
v«:>uld never have arrived at musical harmony.
But
we do get harmony, an d exactly for the opposite reason.
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The relations of sounds we use in Equal Temperament
are not s.µnple ratios (harmonic ratios).
When acousticians and music theorists
advocate nJust intonationn , that is, the intonation of
harmonic ratios, they are not aware of the actual
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situation .
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On the other hand, the ratios they give
.
for certain trivial chords, like the major triad
(4+5+6) , the minor triad (5+6+15) , the dominant seventh
chord (4+5+6+7), do not correspond to the actual intona
tions of the Equal Temperament.
Some of these ratios,
like¼, deviate so much from the nearest i ntonation,
lik� the �inor seventh, which we have adopted through
habit, that it sounds to us out of tune.
Habits in
music, as well as in all manifestations of life, are
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more important tha.n the natural• phenomena.
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If the
problem of chord-structures i n harmony would be confined
to the ratios nearest to Equal Tem perament, we could
have offered 16+19+24 for the minor triad for example,
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as it approaches the tempered.minor triad much better
than 5+6+15.
But this, if accepted, would "discredit
the approach commonly used in all textbooks on
harmony, and for this reason.
If such high harmonics
as the 19th are necessary for the construction o f· a
minor triad, what would chords of superior complexity,
,
which are in use today, look like when expressed
through ratios.
When a violinist plays b as a leading
tone to c and raises the pitch of b above the tempered
b, his claims for higher acoustical perfection are
nonsense, as the nearest harmonic in that region is
the 135th •
Facing faets, we have to admit that all
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the acoustical explanations of chord-structures as
being developed from the simple ratios, are pseudo-
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scientific attempts to rehabilitate musical harmony,
t o give.the·latter a greater prestige.
l.
Though the
original reasoning in this field was caused by the
ho11est ·spirit of investigation of Jean Philippe Rameau
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("Generatio� Harmonique", Paris, 1737), his successors
overlooked the -development
. of acoustical science.
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Their inspiration was Rameau plus their own mental
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laziness and cowardice.
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The whole misunderstanding in the field
of musical harmony is due to two main factors:
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4.
(1) the underrating of habit;
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(2) the confusion of the term "hermonic" in its
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mathematical connotation,
pertaining
to simple ratios with "harmony" in its
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mu.sical connotation, i.e., simultaneous
pitch-assemblages varied in time sequence.
Thus, musical harmony is not a natural
phenomenon, but a highly conditioned and specialized
field.
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It is a material of musical expression, for
which we, in our civilization, have an inborn
inclination and need.
This need is cultivated and
furthered by the existing trends in our music and
musical education.
I. Diatopic System of Harmony.
Chord-structures and chord progressions in
the Diatonic System of Harmony have a definite interdependence: chord-structures develop in the direction
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opposite to their progression�.
Thi-s s-t;;atement brings about the practical
classification of.the Diatonic System into two forms:
the posi�ive and the negative.
As the term Diatonic implies, all pitch•
units of a given scale constitute both structures and
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progressions, without the use of any other pitch-units
(not existing in a given scale) whatsoever.
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5.
In the form which we shall call positive,
all chord structures (S) are the component parts of
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the entire structure (E,) emphasizing all pitch-units
of a given scale in their first tonal expansion (E,)
)
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and in position Ci).
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In the same form chord progressions
derive from the same tonal expansion but in position
In the negative
form of the Diatonic
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System, it works in the opposite manner.
Chord
structures derive from the scale in E, and in position
(§) , wb�le the prog
, ressions develop from E, @ .
According to the qualities we inherited
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and developed, the positive form produces upon us an
effect of greater tonal stability.
It is chrono
logically true that the negative form is an earlier
one.
It predominates in the works where the effect o f
tonality,_ as we know and feel it today, is, rather
vague.
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Such is the XIV and XV Century Ecclesiastic
music, developed on contrapuntal and not on harmonic
foundations.
Many theorists confuse the negative form
of the Diatonic System with "modal" harmony.
As by
Diatonic Tonality they mean, in most cases, Natural
Major or Harmonic Minor scales moving in the positive
form, they miss the tor1al stability when harmony moves
backwards.
Losing tonal orientation they mistake such
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6.
progressions for modes, which are merely derivative
scales, and may also have the positive, as v1ell as
the negative form.
But as we have seen in the
Theory of Pitch Scales, modes can be acquired from
any original scale through the introduction of
accidentals (sharps and flats).
In the following table, MS represents
"melody scale" (pitch-scale), and MR represents
"harmony scale" (i.e. , the fundamental sequence of
chord progressions) .
Diatonic System
Positive Form
Negat:l,.ve For!D
L = MSEr@
= .MSE,@
L
HS= MSE, @
HS= MSE,@
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Figure I.
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Example (Natural Major)
Positive Form
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In the positive form, chords are
constructed upward, in the negative, on the cont rary,
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downward.
The matter is greatly simplified by the
fact that any prog ression, originally written as
positive, becomes negative, when read backward.
All the principles of structures and motion involved
•
are therefore reversible.
No properly constructed
harmonic continuity can be wrong in backward motion.
Some composers without training in
harmony (for example, Modest Moussorgsky) as v,ell as
beginners, due to inadequate study, confuse the
positive and the negative forms in writing their
harmonic progressions.
The resulting effect of such
music is a vague tonaltty.
The admire rs of Moussorgsky
consider such styl� a virtue (in Moussorgsky's case it
is about half-and-half positive and negative), and do
not realize that
all the incompetent students of a
-harmony course incompetently taught possess full
command over such style.
t.
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8"
Lesson LXVI.
A, Diatonic Progressions (Positive Form)
Expansions of the original Harmony Scale
produce the Derivative Harmony Scales.
The original
HS and its expansions form the Diatonic Cycles.
Diatonic (or Tonal) Cycles repre sent all the funda
mental chord progressions.
There are three Tonal Cycles in the
Positive Form for the seven-unit scales.
The First
Cycle, or Cycle of the Third (C 3 ) , corresponds to
HSE0; the Second Cycle, or Cycle of the Fifth (Cs-),
corresponds to HSE, ; the Third Cycle, or the Cycle
of the Seventh (C7), corresponds to H-SE ., Beyond
2
these expansions of HS lies the Negative Form o f
Diatonic Pro gressions.
,,,
In addition to both forms of progressions,
there may be changes in a chord pertaining to the
same root (axis).
modified S of the
Connections of an S with its
s:ime root will be considered a
Zero Cycle (co) • •
In the follov1ing table notes are used
merely for convenience: they indicate the sequepce of
roots; their octave position was dictated by purely
raelodic
•
considerations and by the neces.sity to
moderate
the range•
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The respective interva·ls. .represening
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Cycles must be constructed downward for the Positive
Form, regardless of their actual position on the
musical staff.
Figure II.
Diatonic Cycles (Positive Form)
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Cadences:
Starting
Ending
Cycle of the Third·· (C3 )
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cadences of the �espective cycles
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Cadences consist
of the axis-chord moving into its adjac��t chord and
back.
It is interesting to note, that what is usually
kno\m as Plagal Cadences are the Star.ting Cadences
and.that Cadences known as Authentic are the Ending
Cadences.
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Ending
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The immediate seque11ce of Starting and
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10.
Ending Cadences produces Combined Cadences (the axis
chord is omitted in the middle ) .
Progressions of constant tonal Cycles
(C3, or Cs, or C ., cons t·. ) produce a sequence of seven
chords each appearing once and none repeating itself.
The repetition of the axis-chord either completes the
Cycle or star· t s a new one.
The addition of Cadences
to the Cycles is optional, as Cycles are self-sufficient.
Considering constant Cycles as a form of
Monomial Progressions, we can devise Binomia.l and
Trinomial Progressions by assigning a sequence pf two
or three Cycles at a time.
In Binomial Progressior1s each chord appears
twice and in a different combination with the preceding
and the following chord�
Thus, a complete Binomial
Cycle in a seven-unit scale consists of 2 x 7 = 14 chords.
Figure III.
Binomial cycles
Gs- +
C3
Cs- + c1
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In Trinomial Progressions each chord
appears three times and in a different combination
with the preceding and . the following chord. Thus,
a complete Trinomial Cycle in a seven-unit scale
consists of 3 x 7 � 21 chord.
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12.
Figure IV.
TrinomiaJ Cycles
C3 + C� + C1
c, + c, + c.,
C3 + C1 + Cf
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Lesson LXVII.
Both Binomial and Trinomial Cycles produce
the ultimate
variety combined with the absolute
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consistency of the character (style) of harmonic
progressions.
Being perfect in this respect they are
of little use when a personal selection of character
becomes a paramount factor.
In order to produce an individual style
of harmonic progressions, it is necessary to use a
selective continuity of Cycles.
This can be accom
plished by means o f the Coefficients of Recurrence
applied to a selected combination of Cycles.
A
combination of Cycles can be either a Binomial or a
Groups producing coefficients of
Tr,inomial.
recurrence can be Binomial, Trinomial or Polynomial.
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The materials for these can be fo und in the Theory of
Rhythm.
Rhythmic resultants of different types and
their variations provide various groups which can be
used as coefficients of rec�rrence.
Distributive
Power-Groups as well as the different Series of Growth .
and Acceleration can be used for the same purpose •
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14.
Figur�- Y�
Binomial Cycles, Binomial Coefficients
Cycles: C 3 + Cr; Coeffic ients: 2+1
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Binomial Cycles, Coefficient-Groups producing interference with the
Cycles (not divisible by 2)
Cycles: C.r +
Coefficients: 3 + 1 + 2
C3
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Synchronized Cycles: 3C; + c3 + 2Cr + 3C 3 + C; + 2c,
Synchronized coefficients: 6t x 2 = 12t; 12 x 7
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Coefficients: 4 + 1 + 3
Synchronized Cycles: 4C3 + Cr + 3C7;
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(Figure V, Cont.)
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Cycles: C 7 + c�., + Ce;
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17.
The style of harmonic progressions depends
entirely on the form of cycles employed.
No composer
confines himself to one definite cycle, yet it is the
predominance of a certain cycle over others that makes
his music immediately recognizable to the listener.
In one case it may be that the beginning of a progression
is expressed through the cadences of a certain cycle, in
another case it may be a prominent coefficient group
that makes such music sound distinctly different from
the other.
The style of harmonic progressions can be
defined as a definite form of Selectiv� Cycles.
Both
the combination of cycles (their sequence) and the
coefficient group determining their recurrence are the
factors of a style of harmonic progressions.
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•
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18.
u
Lesson LXVIII.
There is much to be said about the
historical development of the cycles, as there are
already some wrong notions established in this field.
Though the common belief is that the
progressions from the tonic to the dominant and back
to the tonic (ending cadence in c ), is the foundation
5
of diatonic harmony, historical, evidence, as well as
mathematical analysis prove to the oontrary.
During
the course of centuries of European musical history,
parallel to the developwent of counterpoint, there
was an awakening of harmonic consciousness.
-
can be traced, in its apparent
XV . Century A.D.
The latter
forms, back to
At that time harmony meant concord,
an agreea.ble, consonant, stabilized sonority of
•
several voices- simultaneously sustained.
Concordant
prog_ressio11s could be accomplished therefore throug h
consonant chords moving in consonant relations.
Obviously such progressions require common tones, and
the latter can be expressed as c3• As the -tonality,
i.e., an organized progression of tonal cycles was at
that time in the state of fermentation, it is natural
to expect the cycle of the third to appear in both
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positive (c3) arid negative (C-3) form.
The following are a fevw illustrations
tak en from the music of XV and XVI Centuries.
C.
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19.
Figure VI.
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Cycle of the Seventh, on the other hand,
has a purely contrapuntal derivation.
When the two
leading tones (the upper and the lower) move in
cadence into their
a
respective tonics (like b.--+c and
d > c) by means of contra.ry motion
in two voices, we
•
,
obtain the ending cadence o f c 7• Further development
of· the third part was undoubtedly 11ecessitated by the
d€sire for fuller sonorities.
This introduced an extra
tone (f in a chord of b) with which
tones form S(6) i.e., a
the remaining
third-sixth-chord or a sixth
chord, the first inversion of the root-chord: S(5).
Figure VII.
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domi11ance of
It is only natural to ex�ect the pre
the c7 in contrapuntal music.
Cadences
as in F!tgure VII are most standardized in the XIII
and XIV Century European music.
Machault (1300-1377)
11
See Guillaume de
Mass for the Coronation o f
Cha rles V11 (phonograph recording published by the
Gramophone Shop)..
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•
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21 ..
The appearance of the cycle of the fifth
date, when c 3 and c 7
I of fer the following hypothesis
must be referred to a later
were already in use.
of the origin of C6.
The positive form might have
•
occurred as a pedal point development, where by
sustaining the tonic and changing the remaining two
tones to their leading tones, the sequence would
represent c,.
Another interpretation of the origin
of Cs is the one which this system of Harmony is based
•
upon, i.e., omission of intermediate links in a series •
This principle ties up musical harmony with harmonic
structure of crystals, as used in crystallographic
analysis.
Figure VIII.
Cs
..
•
The origin of the negative form of the
cycle of th e fifth (C-5) is due to the desire of
acquiring a concord supporting a leading tone.
be a leading tone in the scale of c.
Let b
The most con
cordant combination of tones in the pre-Bach - time,
i.e., in the •mean temperament (the tuning system
v.
officially recognized in Europe before the advent of
•
22.
equal temperament) , harmonizing the tone b was the
G-chord (g, b, d) .
But when movi ng from G-chord to
C-chord the form of the cycle is positive.
In
are the beginning and the ending cadences.
Compare
reality both forms, the positive and the ne gative,
Figure IX with Figure VIII.
Figure IX .
•
,
•
•
•
,
•
•
•
,,
J O SE P H
S C H IL L I N G ER
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 LXIX.
•
The development of harmonic progressions ·
in the European music of the last three centuries c an
be easily traced b ack to their sources.
.
The style of
every composer is hybrid , yet the quantitative predominance of certain ingredients (like the cycles
appearing with the different coefficients of recurrence)
produces individual characteristics.
In the following exposition I will confine
the concept of nstylen to harmonic progressions in the
diatonic system.
I.
Richard Wagner was the greatest representa
tive of C 3 in th e XIX Century.
This statement is
backed by the statistical analysis of tonal cycles in
•
•
his works as compared to his contemporaries and
predecessors: c 5 v,as the universal vogue of a whole
century preceding Wagner� In fact, it is not necessary
to analyze all works of Wagner.
The most characteristic
progressions may be found at the beginnings of his
preludes to musical dramas and also
cadences.
:in the various
The beginnings of major works of any
L
•
•
,..
•
2.
•
composers
are important, fo r the reason that they
composer.
The importance of cadences as determinants
cannot b e casual: it is the "calling card" of a
of harmonic styles was stressed upon by our contempo
rary, Alfredo Casella, in his paper, "Evolution of
Harmony from the Authentic Cadence".
Wagner, being German and intentionally '
•
Germanic composer, undoubtedly has done some research
of tt1e earlier German music, as he intended to deal
with the subjects of German mythology, in which he was
well versed.
The XV Century German music discloses
such an abundance of C 3 , that it is only natural to
expect the influence of such an authentic source of
Germanic mus ic upon Wagner's creations.
In his time,
Wagner 's harmonic progressions sounded revolutionary
'
because many things were forgotten in four hundred
years, and archaic acquired a flavor of modernistic.
So far as the development of diatonic progressions in
Wagner ' s music appears to the unbiased analyst, the
whole mission of Wagner 's life was to develop a
consistent combined cadence in C 3 •
Starting with an early work like
"Tannhauser", we find that already the very beginning
of the Overture is typical in this respect.
..
•
3.
Figure x.
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Later on v,re find more extended pro
gressions of C 3 , as in the Aria of Wolfram von
•
Eschenbach (the scene of Minnesingers contest):
Figure XJ.
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C
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C3 than
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C�
C. �
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"Logengrin" is even more abundant with
11 Tannhauser 11 •
In "Farewell to S wan", as in
many other places of the same opera, vre find the
•
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4.
characteristic back-and-forth fluctuation: C 3+C-3.
figure XJI.
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Forming his cadences, Wagner paid some
time his tributes to the dominating "dominant'' of
This produced combined b,ybrid
cadences, which are characteristic of "Lohengrin".
The first part of such a cadence is the beginning
cadence in Ca, while the second part is the ending
cadence in c5 : I - VI - V - I .
Figure XIII .
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5.
Dealing with other type s of progressions
than diatonic in the cour se of his career, Wagner
came back to diatonic purity in its complete and
consistent form in his last work "Parsifal".
The
beginning of the "Prelude to Act I" reveals that the
composer came to the realization of the combined
cadence of C 3 : I - VI - III;
Figure XIV.
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The more extensive sequences of C3 are:
I - VI - IV - II;
Figur� xv_.
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C3
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.A nd the complete combined cadence ( nProcession of
the Noblemen of Graal"): I - VI - III - I •.
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Figure XVI.
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v,
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The second half of the XVIII Century and
the first half of t:t1e XIX Century cover the period of
the hegemony of the dominant and c5 in all its aspects
in general. The latter are: continuous progressions o�
c5; starting, ending and combined cadences ( I - IV - I;
I - V - I; I - IV - V - I). The main sources of music
possessing these characteristics are: the Italian Opera
and tr1e Viennese School..
To the first belong :
Monteverdi, · scarlatti, Pergolesi, Rossini, Verdi.
The
second is represented by Dittersdorf, Hayd n, Mozart,
Beethoven, Schubert.
Today th is style disintegrated
into the least imaginative creations in the field of
popular music.
Nevertheless it is the stronghold of
harmony in the educational music institutions.
•
•
7.
Here are a few illustrations of c 5 style
in the early Sonatas for the Piano by Ludwig van
Beethoven: Sonata Op. 7, Largo; Sonata Op. 13,
Adagio Cantabile.
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Figure-XVII.
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Any number of illustrations can be found
in Mozart 's and Beethoven 's symphonies, particularly
in the conclusive p�rts of the last movements.
Assuming that the historical origin of the
cycle of the seventh can be traced back to contrapuntal
cadences, it would be only logical to expect the
evidence of c7 in the works of the great contrapuntalists.
•
8.
I choose for tl1e illustration of c 7, as characteristic
starting progressions, some of the well knovm Prelud es
to Fugues taken from the First Volume of
11
Well Tempered
Clavichord" by Johann Sebastian Bach: Prelude I;
Prelude III; Prelude V.
Fig_ure XVIII.
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Bach ' s famous "Chiacopna in D-minor n for
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Violin, discloses the same characteristics, as the
first chord is d and the s econd chord is e, which makes
•
--
,
•
9.
A consistent and ripe style of diatoni c
progressi ons corresponds to a consis tent use of one
form, either positive or negati ve and not to an
indiscriminate mixture of both.
Many theorists con£use
•
the hybrid of positive and negati ve forms wi th modal
progressi ons, which the theorists have never defined
clearly.
In reality, modal progressi ons are in no
respect different from tonal progressions, except for
the scale structure.
Both types (tonal and modal) can
be eit her positive, or negative, or hybrid.
Modes can
be obtained by the direct change of key si gnatures, as
descri bed in the "Theory of Pitch Scales 11 (transposi t i on
to one axi s) .
Here is an example, typical of Moussorgsky,
from "Bori s Godounov 11 (opera):
Figure J{IX.
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In the a bove example the mode (scale) i s Cd5 ,
the fi fth derivati ve scale of the Natural Major i n the
key of c, known as Aeolian mode, while
the progression
of tonal cycles i s a hybrid of positive and negative forms.
/
•
10 •
Lesson LXX.
•
B, Transformations of S(52
• .
· In the traditional courses of harmony the
problems of progressio�s and voice-leading are
inseparable.
Each pair of chords is described as
sequence and a f orm of voice-leading .
Thus each case
becomes an individual case where the movement of voices
is described in terms of melodic intervals (like: a
fifth down, a second up, a leap in soprano, a sustained
tone in alto, etc.).
No person of normal mentality can
ever memorize all the rules and exceptions offered in
such courses.
In addition to this unsatisfactory form
of presentation of the subject of harmony, one finds
out very soon that the abundance of rules covers a very
limited material (mostly the harmony of the second-rateL
XVIII Century European composers).
The main defect of the existing theories
of harmony is in the use of the descriptive method.
Each case is analyz.ed apart from other cases and
without an� general underlying principles.
The mathematical treatment of this subject
discloses the general properties of the positions and
movements of the voices in terms of transformations of
the chor dal fun ctions.
Any chord, n-0 matter of what structure,
from a mathematical standpoint, is an assemblage
• of
•
•
•
11.
pitch units, or a
(elements).
gro up of conjugated functions
These functions are the different pitch
units distributed in each group, assemblage or chord
according to the different number o f voices (parts)
and the intervals between the latter.
In groups with three · functio ns known as
three-part structures (S = 3p) the fu nction s are a, b
a nd c.
These functi ons behave through general forms
o f transformations and not throug h any musical
specifications.
As in thi s branch we are deal ing with so
called fo ur-part harmony, we have to define the meaning
•
o f this expression more precisely •
When an S(5) constitutes a chord-structure,
the functions o f the chord are: the root, the third 8.Ild
the fifth or 1, 3 and 5.
In their general form they
correspond to a, b and c, i. e. , a
=
l, b
L
= 3, and c = 5.
The bass of such harmony is a constant root-tone,
i.e,'
co nst. 1 or co nst. a.
Thvs the transformation of fu nctions
affects all parts except the bass.
Here, therefore,.we
are dealin g with the gro ups consisting of three functions.
formation.s:
Such gro ups have two fundamental trans
•
•
•
12.
(1) clockwise (Z, ) a.nd (2) counterclockwise (� )
The clockwise transformation is :
The counterclockwise transformation
l.S :
Each of these transform ations has two
meanings: the first to be read -a is followed by b
•
b
n
C
tt
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n
"
C
" a
for the ,.-�
and
f:__..
a is followed by c
C
n
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b
" a
b "
for the �
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d·iscloses the mechanism of the !)OSi tions of a chord;
the second to be read -a transforms into b
b
C
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C
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n
a
for the ;::! and
a transf orrns into c
C
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b
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n
for the 11::
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b
a
•
13.
constitutes the forms of vo ice-leadi ng,
Positions.
The different positions of S (5)
= 1, 3, 5
can be obtained by constructing the chordal functions
dovmward from each phase of the transformations.
a
b
c
b
c
a
c
a
b
and
a
c
b
C
b
a
b
a
c
......___
____,➔
Substituting 1, 3 , 5 for a, b, c, we
obtain
1
3
3
5
5
1
5
1
and
3
1
5
5
3
3
1
Ir;;
......
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3
1
-
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The cloclcwise positions are commonly _known
as open, and the counterclockwise as close.
Here are
the positions for S(5)
=4 + 3 =
Bass is added for the doubling o f the
= c - e - g.
root.
•
•
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Figure
14.
XX.
P o s i +i o ,.,., 6
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Voice-Leading
•
The movement o f the individual voices
follows the groups of transformation in this form:
u
a of the first chord transforms into b of the
following chord; b of the first chord transforms
into c of the following chord; c of the first chord
•
transforms into a of the following chord.
The
above three forms con stitute the clockwise voice
leading.
For the counterclockwise vo ice-leading
the reading must follov1 this order: a of the first
chord transforms into c of the following chord; c
of the first chord transforms into b of the
following chord; b of the first chord transforms
into a of the following chord.
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...., ,
a
,, 1
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b
)C
C
)a
}C
and
\
C
,, b
b
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Applying the above transformations to
•
1, 3, 5 of the 8 (5) , we obtain:
�
1 )3
3
and
)5
5--) 1
,,
1
)5
5
)3
3
)1
•
CJ.ock,,ise form:
The root of the firs t chord becomes the
•
third of the next chord; the third o f the first
,
chor d becomes the fif�� of the next chord; the fifth
of the first chord becomes the root of the next chor d .
Counte�clockwise form:
The root of the first chord becomes the
fifth of the next chord; the fifth of the first
chord becomes the - third o f the next chord; the third
of the first chord becomes the ro ot of the next
chord.
Both forms apply to all tonal cycles.
Let us take C3 in the natural major, for
'
example.
The
first chord is C= c - e - g and the
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u
16 .
next chord is A= a - c - e.
Clockwise for m gives the following
reading :
C
>C
g
)a
Counterclockwise form gives the
•
following reading:
C
➔e
g
)C
e
?a
Let us take c5 in the sam e scale.
The chords are: C = c - e - g and F = f - a - c.
r ➔
C
e
F .,,,,
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--
g -=, a
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Let us take c7 in the same scale.
chords are : C = c - e - g and D = d - f - a.
The
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Both forms of F,. are acceptable in
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nearly equidistant.
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18.
Lesson LXXI.
Each tonal cycle permits a continuous
progression through one form of transformation.
In
the following table const. 1 in the bass is added .
Apostrophies indicate an octave variation when the
extension of' range
becomes impractical.
In c7 both directions are combined,
offering the most practical form for the range.
(please see
Figure XXI.
following page)
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19 .
Tonal Cycles
Clockwise and Counterclockwise Transformations.
....
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D o'" 15""
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20.
The clockwise and the counterclockwise
transformations are applicable to all positions
for the starting chord.
When the first cl1ord is in
the ic:� (open) position, the entire progression
remains automatically in such a position.
•
first chord is in
G
When the
(close) position, the entire
progression remains in such a position.
The co11s tancy of
position ( open or
close) is not affected by the co11stancy of the
tonal cycles, neither is it affected by the lack of
their constancy.
The transition from close to open position
and vice-versa can be accomplished through the use of
the following formula:
Constant b transformation
Const� 3
a· > C
l
>5
b·
)b
3
�3
C
)a
5= ) 1
It is ·best to have 3 in the upper voice
for such purposes, as in some positions voices
cross otherwise.
Function 3 from close to open
followi ng chord.
Reverse the procedure from open to
position moves upward to the function 3 of the
close.
L
•
•
21.
Figure XXII.,
Const. 3 Transformation
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v i ,,:,,-
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s-
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Continuous application of const. 3
transformation produces a consistent variation of
the
2
and the � positions, regardless of the
sequence of tonal cycles.
The following table offers continuous
progressions through canst. cycles and const� 3
transformation.
Fig ure XXIII,
(please see next page)
•
•
C3 Const .. 3
"'
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22.
Figure �I I I .
Constant 3 Transforma tions
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J OSE P H S C H I L L IN G ER
C ORRES P O N D ENCE CO UR S E
Subject: 1viusic
With: Dr. Jerome Gross
Lesson LXXII,
There are four forms of relationship
between the cycles and the transformations with regard
to the variability of both.
(1) const. -cycle, const.-transformation;
(2) const.-cycle, variable transformation;
•
(3) variable cycle , const�-trans formation;
(4) variable cycle, variable transformation.
The forms of transformation produce their
own periodic groups, which mpy be superimposed on the
groups of cycles.
Monomial forms of transformations (const.
transformations):·-
( 1)
(2)
(3) const.- 3
Binomial forws of transformations:
(1) �
Here Const. 3 is excluded on account of
tl1e crossing of inner voices.
•
Coefficients of recurrence being applied
to the forms of transformations produce selective
transformati9n-groups.
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•
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2.
For example: 2 ; ! + ! ; ; 3 � ; + 2 ;� ;
=
! + 2 �; ; 4 i� + � +
2 ; °! + �; + -;..
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4 -':: ..
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+
+ 2 ;.: + 3 ....� + �, +
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.... ,.... ➔ + I:: "" .
+ 8 .. ,, ; f"+
.. ..., .c. p .... ,
Though the groups of tonal cycles, as well
...1
as the forms of transformations, may be chosen freely
with the writing of each sub sequent chord, rhythmic
planning of both guarantees a greater regularity and,
therefore, greater unity of s tyle. .
Examples of variable transformations
applied to constant tonal cycles.
Figure XXIV.
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Examples of variable transformations applied
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Figur� XXV .
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4.
All forms of harmonic continuity, due to
their property of redistribution, modal variability
and convertibility, are subject to the following
modifications:
(1) Placement of the voice representing
constant function, and originally appearing
in the bass, into any other voice, •l. . e . ,
•
tenor, alto or soprano .
There are four
forms of such distribution:
s
A
T
B
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B
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A
T
B
Red letters re present the voice functioning
-
as const .. 1.
(2) General redistribution (vertical permuta
tions) of all voices according to 24
L
variations of 4 elements.
(3) Geometrical inversions : @ , @, © and
@ for any or all forms of distribution
of tlie four voices .
(4) Modal variation by means of modal trans
positj.on, i.e., direct change of key
signature, without replacing the notes on
the staff.
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7.
tesson LXXIII.
c. The Negative Fprm.
As it was previ ously defined, the
negative form of harmony can be obtained by direct
reading of the positive form in position @ •
Here, for the sake of clarity in the
entire matter, I am offering some technical details
which explain the theoretical side of the negative
form.
According to the definition given to
the harmony scale in the negative form, we obtain
the latter by means of further expansions of HS.
In the positive form we have used : H SE0 (= C 3 ),
HSE, =
( Cs ) and HSE2 (= C 7).
Novv by further expanding HS, we acquire
the cycles of the negative form: H SE 3 (= C - 7) ,
HSE =
( C - 5), HSE (= C - 3) .
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Figure x;xv_r�.
(please see ne�t page)
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As chord-structures are built downward
fro m a given pitch unit, such a pitch unit becomes
L
the root-tone of the negative structure : the
negative root ( - 1) .
All chord-structures of the negative
form, according to the previ ous definition, derive
from HS @ •
Thus in order to construct a negative
S (5),
-
it is necessary to take the next pitch-unit dovmwa rd,
which becomes the negative third ( - 3) and the next
tt
T
unit do wnv.rard fro m the latter, which bec omes the
negative fifth ( - 5) .
•
•
•
•
For example, starting from c as a - 1,
we obtain a negative S (5), where a is - 3 and
f is - 5.
Figure )Q.CVIII .
Natural C- Major.
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Positions of chords, as they were
expressed through transformations, remain identical
in the negative form, providing they are constructed
upv,ard.
•
In such a case, the addition of a cons t. 1
in the bass must be, strictly speaking, transferred
to the soprano.
Here is how a negative CS (5) would
appear in its fo ur-part settings.
Figure XXIX .
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10.
If, u11der such cor1ditions, the chord
were constructed downward, the reversal of ;;.! and
reading would
take place.
Transformations as applied to voice
leading possess the same reversibility : if everything
is read downv.rard, the
'ic-..?
and the ': ...., tra.nsformations
correspond to the positive form, while in the upward
,+
Jr:
reading the ,:::::� becomes the ...., � and vice-versa.
Let us connect two chords in the negative
cycle of the third: CS (5) + C 3 + ES (5).
= C - a - f.
ES (5) = -1, -3, -5 = e - c - a.
CS (5) = -1, -3, -5
Figure XXX.
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It is easy to see that in the upward
reading chord C corresponds to F, and chord E
corres�onds to A.
Transposing this upward reading
to C, we notice that th is progression is c ---) E.
This proves the reversibility of tonal cycles and
the correctness of reading the positive form of
progressions in positi on @ , when the negati�e form
L
•
•
is desired.
11.
The mixture of positive and negative
forms i.n continuity does not change the situation,
but merely reverses the characteristics of voice
leading with regard to positive and negative forms.
For example, C 3 in ;:-.! in the positive system
produces two sustained comwon tones.
In order to
obtain an analogous pattern of vo ice-leading in C- 3 ,
it is necessary to reverse the transfor mation, i.e. ,
to use the : � form in this case.
•
•
•
•
12.
Lesson LXXIV.
II. Symmetric S, y stem.
Diatonic harmony can be best defined as
where chord-structures as well as chorda system
••
progressions derive· from a given s�a�� -
Structural
consti tution of pi tch assemblages, known as chords,
as well as the actual intonation of the sequences
of root-tones, knovm as tonal .cycles, are enti rely
conditioned by the structural constitution of the
scale, which is the
s:Jurce of intonation.
Symmetric harmony is a system of pre
•
selected chord-structures and pre-selected chord
progressions , one indepe ndent from an other.
In the
symmetr ic system of harmony scale is the result, the
conseguence of chords in motion.
The selection of
L
intonati on for structures is independent from the
selection of intonati on for the progressions.
A. Structures of 8(5).
In this course of harmony only such
three-part structures vvill be used, whic� satisfy
•
the definition of "special theory of harmonyn.
The
:ingredients of chord-structures here are limited to
3 and 4 semitones r
Under such limitations only
four forms .of_ 8 (5) are possib le.
It should be
remembered, though, that the number of all possible
•
•
•
three-part structures would amount to 55, which is
the general number of three-unit scales from one axis.
Table of S(5)
s , (5) = 4 + 3, knovm as major triad;
S2 (5) = 3 + 4, kno,m as minor triad;
·s 3 (5) = 4 + 4, known as augme nted triad;
S'f (5) = 3 + 3, known as diminished triad.
"
s,(s)
-J -
Figure XXXI .
-
-i-
-
i:;s
M
.
I
.,,
-
.- �
So long as S(5) will be the only structure
I..
for th e present use, we shall simplify the abov e
•
expressions to the
•
following form:
Whatever th e ch ord-progression may be,
structural constitution of chords appe aring in such
progression may be either constant or variable �
Constant structures will be considere d as monomial
progressions of structures, while the variable
structur es will be considered as binomial, trinomial
and polynomial structural groups.
•
•
•
14.
Monomial forms of S(5)
• •••
••••
••••
••••
Total: 4 forms
Binomial forms of 8_(5)
Sa + S "I
s , + Sa
s , + s'f
,
6 combir1ations, 2 permutations each.
•
Total : 12 forms
•
Trinomial forms of 8(5)
•
S ' + S ' + S3
s,
+
s,
+
s ,..
•
S., + Sa + S a
s , + s,, + s�
12 combinations, 3 permutations each.
Total: 36 forms
•
•
15.
s, +
S2
+ Sa
s, +
S2
+ Sy
S2
+ Sa + S &f
•
•
4 combinations, 6 permutations each.
Total : 24 form s .
The total of all trinomials: 36 + 24 = 60.
S ' + S t + S t + S2
Quadrinomial forros of 8(5).
•
L
12 combinations, 4 permutations each.
Total: 48 forms
•
6 combinations , 6 permutations each.
Total: 36 forms
•
•
16.
s , + S, + S2 + Sa
s , + s , + S 2 + S�
s , + s , + Sa + Sy
s , + S 2 + S 2 + Sa
s , + S2 + S2 + �
S , + Sa + S 3 + S�
s , + S 2 + Sa + Sa
•
s , + s2 + s� + s�
s , + S 3 + S� + 8¥
•
12 combinations, 12 permutations each.
•
•
Total : 144 forms •
•
1 combination, 24 permutations.
Total: 24 forms.
•
The total of all quadrinomials: 48 + 36 + 144 + 24 = 252.
In addition to all these fundamental forms of
the groups of S (5), which represent a 11eutral harmonic
continuity of str�ctures, there are groups with coefficients
of recurrence, which represent a selective harmonic
•
•
17.
continuity of structures.
individual selection.
The latter are subject to
Any rhythmic groups may be used
as coefficients of recurrence.
Examples
(1)
2S, + Sa
(2)
3Sa + S2
(3) 3S ,
+ 2Sa + S2·
(4)
2S 2 + s , + S2 + 2S 1
(6)
3S f + S2 + 2S '
(8 )
2S 1 + S2 + S, + S2 + S, + S2 + 2S 1 + 2Sz + s , · + S2 +I.· S 1 +
(9)
+ S2 + S , + 282
\
4S, + 2s 2 + 2s� + 2s , + s 2 + s� + 2s , + s 2 + s�
(10)
•
B. Symmetric Progressions .
•
Symmetric · zero CYcle (C0 )
•
A group of chords with a common root-tone but
positions and variable structures produces
with variable
•
a symmetric zero cycle (C0 ).
•
•
18.
Such a group may be an independent form
of harmonic continuity, as wel l as a portion of other
symmetric forms of harmonic continuity.
Coefficients of recurrence in the groups
of structures, when used in a continuity of C0 ,
acquire the following meaning: a structure with a
coefficient greater than one changes its positions,
The change of
until the next structure appears.
structure requires the preservation of the position
of the chord.
This can be expressed as a form of
interdependence of structures and their positions
in the C0 :
position var.
S const•
•
S var. ------- position const.
=
s,
+
s,
•
t.
For instance, in a case of 3S, + Sa + 2$ 2 =
+
s,
+ Sa + S 2 + S 2 , the constant and
variable positions appear as
var.
var.
s , + s' + s,
follows:
con st. con st.
+
+
Sa
•
•
Ex.amples of harm onic cont inuity in C 0 •
Figure XXXII •
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20.
Lesson LXXV.
Diatonic-Symmetric System of Harmony
(Type II).
•
Diatonic-Symmetric system of harmony
must satisfy the following two requirements:
(1) all root-tones of the diatonic-symmetric
system belong to one scale of the First
Group;
(2) all chord structures must be pre-selected ;
they are not affected by the intona tion of
scale formed by the root-tones.
•
In this system of h armony structur al
groups must be superimposed upon the progressions of
the root-tones belonging to one scale.
This form of
h armony has some advan tages over the Diatonic System
( to which I will refer as Type I).
Like the diatonic
system, the diatonic-symmetric system produces a
united tonality, which is due to the structural unity
of the scale.
Unlike the diatonic system, the
diatonic-symmetric system is not bound to use the
structures wh ich are considered defective in the Equal
Temper ament [ like S � (5) , for example ] , as the
individual struc tures and the structur al groups . are
a matter of free choice.
Unlike the di atonic system, the di atonic-
•
•
L
•
21 .
symmetric system has a greater variety of intonations,
as the pre-selected structures unavoidably introduce
new accidentals (alterations), 1,mich implies a
modulatory character without destroying the unity of
the tonality.
Examples of Harmony TYJ?e ;r.
Figure XXXIII.
(please see following pages)
C
•
•
•
v
.,
,
r
....
I:}
Pitch-scale :
+
-
0
,;::
? 0 �3
!'?
-
,-
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r,'
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-....
"£
.
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-..
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::::I
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-.,,
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.,
_..,
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....
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�c
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- -.
...'.
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"'L
ba
... t,J
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+ C3 +
-
--
,.,
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_,,
- � q�
-
.....
.
"
-
-
ul
-
-
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-'
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0
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,'
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.__
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--
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-fV
Structural group: S , . + S 3 + 2S 2
...
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t.
:
..
l'
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'...
- -
..,
L.
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�
-.-
- ---
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·-
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r-
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V
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.-
-s,, ...: .- j ·-
�
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- et.:
0
i�'' '
Tonal cycles: 2C 3 + Cr
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c,
I
- --
- [,...
"
.
- - . .,,
.,
,
--...
��
,..
.....
1._ _i L..,. �
-
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2c1
Structural group : S , (5) co11st.
-. - -- ....
-e- �
�
,.
....
''
=- -
....
�
,.._
I
0
o�� ,...
Pitch-scale :
'
,.,
Q
�
�
Tonal cycles;
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22 •
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S C H I L L I N G E R
J O S E P H
C o ·u R S E
C O R R E S P O N D E N C E
Subject: Music
With: Dr. Jerome Gross
Lesson LXXVI.
•
Symmetric �ystem of Harmony
0
(Type III)
Symmetric System of harmony must satisfy
the following requirements:
(1) the root-to nes and their progress ions are
the roots of two (i.e. ,ff,, 3,12, "./2, "../2, 1':/2) ,
that is the points of symmetry of an octave.
(2) chord structures are pre-selected .
As a consequence o f motion through symmetric
roots, each voice of harmony produces one of t.he pitchscales of the Third Group.
1.
Symmetric C0 represents one tonic;
•
./2 represents two to nics;
"
three "
'./2
"J2
�
'':/2
"
"
"
four
•
Sl.X
"
"
twelve n
The correspondences of the tonal cycles
and ti1e symmetric roots are as follows:
•
•
•
2.
One ton ic:
C
Two tonics : 0
Co
Cr
C
Four tonics : C
C
Six tonics:
f'
F
,
•
Three tonics: C
•
C
C
C
c-,
E}
C
Alz C
C.3
3
c,
E-
C-3
A
Ci
C-3
C7
C- 7
Twelve tonics: C
C
ER
D
B)?
C7
C-3
c,
A!? ·c
·-3
F it
C
EiP
c,
F,:
C-1 A
C- 1
C '1
C-7
-D�
E
Ai?
C7
C- 7
c,
C-J
C
-c
Ft
V
A
C7
C7
BJz
Ftt
E
D
C- 7
C-7
. D�
E�
C7
C7 E� • • •
C7
B
B�
C
A'
A
C-7
C- 7
C-7
C-7
..
Transformations with regard to positi ons
and voice-leading remain the same as in the diatonic
system.
In case of do ubt cancel all the accidentals.
Two Tonics.
Two tonics break up an octave into two
uniform intervals.
The second to nic (T�) being the
.[§, produces the center of an octave.
makes the t wo-to nic system reversible.
0
This property
All points of
intonation in the � � as well as in the � transformations are identical, i.e�, both the clockwise and the
c,
C-7
C
•
•
3.
I
counterclockwise voice-leading produce the same
pattern of motion.
This is true only in the case of
two tonics.
Two tonics form a continuous system,
i.e., the recurring tonic does not appear in its
original position.
Two tonics produce a triple
recurrence-cycle before the original position: falls
on the
first tonic (T ,) for the �� and the �
Const. 3 produces a closed system.
1.
Figure XXXIV_.
•
�
I,,,
5 1 ' �--. A•
.,
'
«
�
' •
�
I.-
•
•
0I
,1
i
,
,,
,
�
-
ff
•
.,,
T1
,,
'
.�
'
�
�
�
••
,�
•
.
'-
�
-e
--
<
I
-·- ----
------
The upper voice of harmony produces the
following scale: c
.__.- d'- e -
r"
L=
g - a - (c) =
�
•
4.
= (1+3 ) + 2 + (1+3 ) + 2.
All other vo ices of the
above progression pr oduce the same scale starting
from its different phases.
I t is easy to see that this scale belongs
, to the Thir d Gro up and is constructed on two tonics.
By selecting other structures and
structural groups of 8(5) one can get some other
scales of the Third Group.
For example, the use of S 2 cor1st. produces
the follov1ing scale: c - d'7 - e � - fM'- g - a - (c) =
= (1+ 2) + 3 + (1+ 2) + 3 .
Structural gro ups may be used in two ways:
(1) S changes with each t o nic;
( 2) the groups of S produce C0 on eact1 tonic.
Illustrations of the first method
Figure
-XX.XV •
-----------� -----------------I
•
.,
ff!i:.'
·�
.
..p
At .:e-
-
:i!
,_
•
=e
�
' "I...
-s
I
I-
•
'(
.
fi
.
•
.I
..
••
-�
•
•
5•
Illustrations of the second met hod
•
Figure XXXVI.
•
Combinations of the preceding two
methods with regard to the structural selection for
· each tonic of one symmetric system are applicable t o
all symmetric systems •
•
•
6.
Example
f�gtn' e ;xxxvrr .
•
•
Longer progressions can be obtained
through the use of longer structural groups, such
as rhythmic resultants, power-groups, series of
growth, etc.
In some cases the number of terms in the
structural group produces interference against the
number of tonics in the symmetric system.
Example
'
•
7.
Three Tonics.
Three tonics produce a closed system
for ; � and 1:
"'
., , and a continuous sy stem (two
recurrence-cycles) for const. 3 .
Figure XXXVIII.
•
••
•
Four Tonics.
..
Four tonics produce a continuous system
(three recurrence-cycles) for
closed system for const. 3 .
..-=;
,:.,,
and
(please see next page)
•
le ,
.�
, and a
•
....
s , const •
,
.,.,
'
.1
Const.
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3
,
.f
/
-- -�
,
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. r�I
-
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1-
i
Six Tonic s .
s•
Figure XXXIX.
•
•
•
I rI
r,1 <�
-
�
•
r..
r-
Six tonics produce a closed system for
as well as for the const. 3.
f;lg11tp
S , const.
-�
,c:,...
I:; -
and _ � ,
x;r...
I
�
I�•
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tit
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•
Const. 3
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•
•
9•
Twelve Tonics.
l:. Twelve tonics produce a closed system for ,�
__ and ...__:::,
as well as for the const. 3 .
Figure, X}:,I.
s , const.
�
�-
t-,....., <
••
Const. 3 ��---...9-"-----------------'------�_,.
l!t:���f
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,
•
•
10 .
I
Lesson LXXVII.
Variable Doubl ings
•
Harmony, 1n many cases conceived as an
accompaniment, may be given a self-suf ficient
character by means of variab.le doublings.
This
device attributes to �hord progressions a greater
versatility of sonority and voice-leading than the
one usually observed.
Variable doublings comprise the three
functions of S (5) .
Thus · the root, the third or the
The corresponding notation
fi ft h can be doubled.
to be used is: 8(5)© , 8(5)@ and 8(5) ©
.
As the root-tone remains in the bass,
S ( 5 ) u) is the only case of doubling where all three
functions (1, 3, 5) appear in the upper three parts.
The followi ng represents a comparative
table of functi ons in the three upper parts under
. various forms of doubling.
8 (5)
©
=
1, 3, 5
8 (5)© � 3, 3, 5
S (5)©
=
3 , 5, 5 •
Figure XLII.
.
V
In cases S (5 ) @ and 8 (5) @ only three
positions are possible for each case.
Black notes
•
•
•
11.
represent variants where unison is substituted for
an octave.
,
Positions
Jf
I)
&I
�-
..
-
r1
-I
••
-
*'
..
,
I
I
•
•
--
••
•
j
---
---
·-
,.
•
,I
•
TI
•
•
••
I,
Fi&!!_re •XLIII .
Tr.ansf ormati ans
8(5)©
5 ( .- 5
3�
)3
c� �
8 (5 )
@
5( ...): 3
5( � 3
3( � 5
3 �e t3
lf > 3
1 ( :)3
ti -
::,;.
A-
••
-
•
.,
,i
�
..
,l, �
l < >5
.
c s- �
t7 �
_
_
____
__ __,_ - - -
. ,,..
. , ._
'
•
•
12.
S (5)
(!)
<
➔
8(5)@
5( ) 3
3(
3 f�� 3
,�
•
•
>5
8 (5 )
(i)
<
>3
5' )5
5f
3 ( .> 5
3( ) 5
1( >i 3
1(
i�
e1 �
-
cf��-
e 3 �-
)5
>
S (5) {j)
5(
�5
3( ➔ 3
1( ) 5
e,,
e�
•
•
•
s (5 )
•
+ ---�> s { 5 ) ©
ev...-
3( > 5
•
•
•
•
•
•
( -------t) 8 ( 5 ) (l)
8 (5 ) @�
3(
➔6
3 !-} 3
..
•
When r eading these tables, consider
identical directions of the arrows for the sequen�e
of structures and for the c orrespondi ng transforma
tions.
Notice that there always are three
transformations when S(5) © participates and only one
when it does not .
•
•
•
14 .
Musical tables in the above Figure
are devised fro m the initial chord being in the
same position.
frow all
Similar tables can be• constructed
positions as well as in rev�:rse sequence
and also in the cycles of the negative form.
Variable doublings are subject to
distributive arrangement and can be superimposed on
any desirable cycle-gro up.
Figure XLIV.
Example : 2C.3 + C , + C 7 ;
8 (5)© + 28(5)
(1)
Ir = 8 (5)0 + C 3 + 8 (5)
+
C5
®
+ 8 (5) �
+ C 3 + 8 (5) (V +
+ S(5)(i) -. C ? + 8(5) (D •
(S)
•
•
Example: 2C� + C 3 + C, + 2C 7 ;
+ 8 (5) @ + 8 (5) © .
Ir = 8 (5)
@
Ci)
8 (5)
+
0
8 (5)
+
+ c, + �(5)(D + c, + S (5)
@
+
+ c, + 8 (5)@ + c, + 8(5)© + c ,, + 8 (5) 0 +
+ c 7 + s (5)@ + c, + 8(5)© + er + 8 (5)@ +
+ C3 + 8 (5)0 + c, + 8 (5)© + C '7 + S (5)@ +
+ c, + 8 (5)�
•
•
15.
I
I
I
I
I
Variable doublings are applicable to all
types of harmonic progressions , thus including types
II
and III .
Figure XLV •
•
Type II (diatonic-symmetric) .
Ir
as in the preceding example.
:: 2S2 + Sa + s ,
I
I
I
I
•
Figure XLVI,.
Type III (symmetric ) .
©+
(i) + T S,Q + T� S
- - T, S1(i)+ T�S�
� (6T) 3
3
+
TsS at + T. S1G) + T, S -a,(j).
©
•
•
•
•
16 •
•
•
•
•
•
.
•
•
..
•
17.
Lesson LXXVIII,
Inversions of 8(5)
The usual tech nique of i nversions,
•
The
strictly speaking, is unnecessary to a co mposer.
reason for this is, that by vertical per mutations of
the pos itions of parts i n a ny harmonic continuity of
8(5) , the inversions appear automati cally, as inner
or upper parts beco me the bass parts u nder such
conditions.
This teclmique was fully described in my
"Geometrical Pro jections of Mllsic", in the branch
dealing with the co ntinuity of geometrical i nversions.
For a n analyst or a teacher, however, a
thorough systematization 0£ the classical te�hnique
of inversi ons is a necessity.
There is no other
branch of harmony I know of, where- confusion is great�r
and t he information less reliab le •
The first inversion of 8(5) is know n as a
"sixth-chord" or a "third-sixth-chord" a nd is
expressed i n th is notati on by the symbol 8 (6).
The
only condition under which 8 (5) becomes an 8 (6) is
when the th ird (3 ) appears in the bass.
The'
positions
•
of the upper voices are not affected by suc h a cha nge,
. th e foFms of do ublings -- are..
Which do ublings are
appropriate in each case, will be discussed later.
Assuming that �ny S(6) may be eit her 8 (6) ©, or 8 (6) @ ,
or 8(6) @ , we obtai n the fo llowi ng Tab le of Positions:
•
•
18.
u
-
1.
�
,
--
.
-
-
•
,_
V.
1
I
:.;
,Ir
..
-
-
.,.. .,.
,,i
r�
,
•
Figure XLVII.
.
,.
-
�
�
·�
••
•
�
s
.
'I
i
�
-
i!'
$
r•
�
. It is easy to memorize the above table,
as $(6 ) © and 8 (6 )© positi ons are systematized
through the followi ng cliaracteristics: (1) the doubled
function appears above the remaining function; (2)
the doub led fu nc tion surrounds the remaining function;
(3 ) the doubled fu nction appears below the remaining
function.
8 (6 )@ is identical with 8 (5) positions,
except for the bass having constant 3 •
. Harmonic progressions (Ir ) consisting of
,
8 (5) and 8 (6 ) are based on the followi ng combinations
by two:
,,
-
•
•
19.
(1) 8(5)
(4 ) s (6)
) 8 (5);
, s (s ) .
( 2) 8 (5)
> S (6) ;
(3) S (6)
J S (5) ;
As the first case is covered by the
previous technique, we are concerned, for the present,
with the last three cases.
All the following transformations, being
applied to vo ice-leading, are reversible, as in the
case of Variable Doublings of 8 (5) .
always measured thro ugh root-tones•
•
Figure XLVIII .
S(5) �-----4
•
5( � 5
3 • ll
l• ) 1
• es-
8(5)
3• -. 5
1.- � 1
S (6) ©
5 � ►l
5�1
l< > 5
•
S (6) (l)
5E�l
5< �5
5• )5
lt·) 5
1� ) 5
lf •l
3• )5
•
5< ll
Tonal cycles are
3� ) 1
3( ) 5
•
20 •
•
5 (5 )
8(6) ®
5'(" � 3
5� )l
3.-• , l
3 fl ) 3
lf � 5
1.- ) 5
Const .,. 3
Const- 3 � =
C,5'
•
•
•
21 •
•
,,,
'
-
�
J
-
+
,�
-
... ..,.
.
,.
•
·�
rI
,
,I
,,I
t.
8(6-fD
5(
•
l"
1�
e�
.., 1
S(6) @
)5
�s--
e, S'
e,
•
,
S (6)(q _____,.;,...___
> 8 (6)©
5 ( )1
5( ) 5
1( ) 5
•
22 •
•
©
S(6)
5 f-) 5
.( -----�
� S (6) (D
5f-�l
5(""3
l'( ) 3
1( >t 5
1� > 5
l" ) 3
lf' > 1
�s
�.----.----.----,--o---:_r====:Jc:::;;====
c�
S (6) (D
S(6) (D
5� 1
5( )Z
3 < )1
3� ) 5
l< ) 5
l'f ·) 3
➔ ,. .;
F ....
4
..._,
I
,
•
•
-
,j
j -
�
,
-
r.
-
=-
_;:;,,
-
'I
__,
_r
�
I
a
�
;:
-
�
�
�
•
V
�
8
�I
$
�
•
@
8(6) (
)5
5( =. 1
5( ) 5
5( > 5
5 t'
l< > l
l< ) 3
)
23.
©
S (6)
5< ) 3
5( ) 1
1( � 5
0
•
.Any variants conformed to identical
transformations (like the black notes in some of
the preceding tables) are as acceptable as the
ones in the tables.
•
•
•
24 •
•
, v
Lesson LXXIX.
Doublings of 8(6)
rapidly.
Musical
habits are formed comparatively
Once they assume a form of natural
reactions, they infl uence us more tha n the purely
acoustical factors .
This is particularly true in
the case of doublings of 5(6 ) .
The mere fact that
identical doublings in the different musical contexts
affect us in a different way, sh ows that our auditory
reactions i n music are not natural but conditioned.
The principles offered here are based on
a comparative study of the respective forms of music.
There are two technical factors affecti ng
•
the doubling i n an 8 (6 ) :
(1) the structure of the ch ord;
( 2) the degree of the scale (on which the
chord is co nstructed) .
These two influences are ever-present
regardless of the type to which the respective
harmonic continuity belong s.,
Yet, while in harmonic progressions of
type II and III the structure of the chord is the
most infl uential factor, i n the diatonic progressio ns
(type I) it is exactly the reverse.
The influence of
a constant pitch-scale is so overwhelming, that each
•
•
25.
chord becomes associated with its definite position
in the scale.
Thus, one chord begins to sound to
us as a do minant and another
or a leading tone.
as a tonic, a mediant
This hierarchy of importance of
the various chords calls for the different forms of
doubling, particularly when the respective cl1ords
appear in the different inversions.
The fo1-lowing is most practical for use
in diatonic progressions •
•
Figure XLIX.
Stro ng Factor
The degree
of the scale
I,
IV, V, VI
Regular
Doubling
CD , ©
Irreg.
Doubling
Weak Factor
The structure
of the chord
s, (6)
Regular
Doubling
G) , @
II, III, VII
S 14 (6)
Regular doublings are statistically pre
dominant.
Irregular doublings, in most cases are the
result of melodic tendencies.
In reading the above table, give preference
to the strong factor, except in the case of S 3 (6) and
6 � (6) .
It is customary to believe that an s, (6) must
have doubled root or fifth.
But in reality it seldom
happens when such a cl1ord belongs to II, III or VII.
Irreg .
Doubling
@
•
26.
Naturally, all our habits with regard to doublings
are formed o n more customary maj or and minor scales.
The above table will work perfectly when applied
to such scales.
There will be no discrepancy when
8 3 (6) and S� ( 6) will be compared with the data on
the left side of the table, as such structures do
•
not occur o n the main degrees of the usual scales •
When using less familiar scales, one or another type
of doubling will not make as much difference.
•
in such oases
the structure may become a more
Yet
influential factor, though the sequence - is diatonic.
In the types II and III the most practi cal
u
forms of doublings are:
Structure
Figure L •
Regular
Doubling
s , (6)
CD,®
S 2 (6)
(D, @
8 3 (6)
@
(y , @ , @
s.. ( 6)
Irregular
Doubli ng
@
@
Continuitz of 8(5) and 8(6)
�he comparative characteristic of S (5)
is its stability, due to the presen�e of the
root-tone i n the bass.
The absence of the root-tone
in the bass of S (6) deprives this structure of such
stability.
•
•
•
27.
Composition of continuity consisting of
S (5) and S ( 6 ) results in an interplay of stable and
unstable units or groups.
The following fundamental
forms of co nti nuity with ut ilization of the above
mentioned structures are possible :
(1)
( 2)
(3)
( 4)
8 (5) const.- --- stable
8 (6 ) const. ---- unstable
( 8 (5) + S (6 ) ] + . . .
alternate
2S(5) + 8 (6 ) + S (5) + 2S (6 )
3S(5) + S (6 ) + 2S(5) + 28(6) + S ( 5) + 38(6)
48 ( 5) + S (6 ) + 38(5) + 28(6) + 2S(5) + 3S (6) +
+ S (5) + 48 ( 6 )
•
•
increasing stability
increasing instability
(5) 4S(5) + 28(6) + 28(5) + S { 6 )
---� proportionately decreasing ratios
proportionately increasi ng ratios
1-------
(6) 8 ( 5) + 28(6) + 38(5) + 5S (6) + 85 (5) + 138 (6)
progressive over-balanci ng .pf unstable
e,lements
S ( 6) • + 28 (5) + 3S ( 6) + 5S ( 5) + 8S ( 6) + 13S ( 5)
progressive over-balancing of stable
elements
Many other forms of distribution of 8 (5)
V
and S(6) may be devised o n the basis of t he "Theory
of Rhythm " .
•
•
Examples of Progressions
28.
Figure LI.
Diatenie
S ( 6) Const . ; 2C '1 + 2C !' + Ca + Cs�
t
'!,
•
l.
•
Figure LII.
Diatonic-�nnmetri�
a •
•
-
2C S' + C"1 + C '° + 2C 7
2S 2.. ( 6) + S 1 ( 6) + S 3 ( 6) + S 1 ( 6) + S" ( 6) ;
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Figure LIII.
D!2tonic
38(6) + 8 (5) + 28(6) + 2S(5) + S(6) + 3S(5) ;
2C 1 + C 1 •
••
lo
Diatonic-Symmetric
+ s , (6) + 2S 2 (5) ; 2·c· -r + c,; Scale- of roots: Aeolian
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+ s2(6)] T, + [sq(6) + S,(5)] T2} + . . .
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Tonic�.
•
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
With: Dr. Jero me Gro ss
Less9n LXXX.
C O U R S E
Subject.: Mu�ic
Groups with, fa ssip.g Ch9rds
A. Passing Sixth-chprd;;
A group with a passing S (6 ) is a pre-.set
c ombinati m of three chords, namely: 8 (5) + 8 ( 6 ) +
•
+ S (5) .
Every papsi ng c�prd occupies. the_�enter of
its group, appears, on a �eak beat and has � doubled
bass.
•
The complete expression for a group (G) with
passing sixth-chord is:
G = 8(5) + 8 ( 6 )@ + 8 ( 5) .
6
This formula is not reversible in actual intonation.
between the extreme chord,s of G6 is
This relationship reruain� constant in all cases
The relati onship
C-5.
of classical music ..
all cycles.
We - shall extend this principle to
Under such conditions G6 r�tains the
following characteristics:
(1) The transf'orma tion between the extreme 'chords
of the group is always c�ockw�� e for both the
positive and. the negative cycles
(2) The . bass progression is: 1-�> 3 �> l, which
necessitates
the first condition.
•
•
2.
In the classical form of Gs, bass moves
by the thirds. Thus, 3 in the bass under S (S) is a
third above its preceding position under the rirst
S (5), and a third below its following position under
the last S(5) .
•
In order t.o obtain Gs , it is necessary to
connect S (5) with the next S (5) through C-5
and
add the intermediate third of the first chord in the
bass, without moving the remaini ng voices.
G
Figure LIV.
q
I
= 8 (5) + S (6) 3 + S(5)
1::
G
•
There are three melodic for ms for the
bass movement.
Figure LV.
0
•
•
•
•
,
•
3.
Combinations of these three forms in
sequence produce a very flexible bass part and,
being repeated with one G6 , make expressive cadences
of Mozartian flavor .
fig:ure LVI •
•
•
•
•
Continuity of G6 •
Continuity of such groups can be obtained
by connecting them through the tonal cycles.
•
while C8
Connecting by c5 closes the sequence,
and c produce a progression of 70 •
7
6
Figure J..,Y.I.I_.
(please see next page)
•
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.. . . - - -
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Further versatility of G6 progressions can
be achieved by varying the cycles betv,een the groups.
Any time a decisive cadence is desirable C5 must be
introduced, as this cycle closes the progression.
u
•
5.
Figure LVIII.
P
+
•
= G6 + c7 + G6 + c 7 + Gs + c3 + Gs + C 7 + Gs +
C3 + Gs + C3 + Gs + C5
c,
c,
•
Generalization of G6
•
In addition to . the classical form of
G6 ,
other forms can be developed through the use of other
Of course, each cycle
than C-5 cycles within the gro up.
produces its own eharacteristic bass pattern.
Figure LIX.
Various forms of Gs:
,�� (c.� )
�I ·
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r�
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6.
u
The respective variations of the bass
pattern will be as
follows:
•
Figure LX.
•
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-
•
7
r
,
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Q
�1.. ( � - 5)
--
•
�
4lr> ( � -1)
����������1��!������������3�;���
G
•
Continuity of the generalized G6
Such a continuity can be developed through
the selective progressions of the various forms of G6
combined with the various cycle connections between
the groups.
8·
•
7•
•
Example:
figure LXI.
� = G6 (C-5 ) + C7 + G6 (C3) + c5 + G(C-7) + c3 +
+ G(C-5) + C5 + G(C7 ) + C5
I.
•
Generalization_ of �pe Passing Th�rd
It follows fro m · the technique of gro ups
•
•
•
with a passing sixth-chord, that the first two chords,
i.e. , S (5) and 8 (6)@ , belong to C0 , and that as the
position of the three upp er parts does not change until
the last chord of the gro up appears.
This last chord ,
8 (5) , can be 1n any r elation but C0 with the preceding
chord.
If we think about the appearance of the thi rd
in the bass dur ing 8 (6)'.V merely as a passing third, it
is easy to see that this entire technique can be
general ized.
providing the
The passing 3 can be used after any 8 (5) ,
transformation betwe en the latter and
•
8.
the following S {5) is clockwise for all the cycles.
Such a device can be applied to any progressions of
root-tones in the bass.
S (5) w1 th the
Figure LXI):,
Ex.ample :
•
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The effect of such harmonic continuity
is one of overlapping gro ups of G6 , as marked in the
above Figure.
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- ---- ------
-- . ··-
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•
Lesso n LXXXI,
Applications of Gs to Diatonic-Symmetric
•
(Type II) and Symmetric 1Typ e JII)
Progressions,
The use of structures of S(5) and 8 (6)@
in the groups with a passing sixth-chord must satisfy
the following requireme nt : �he a�ja�en� 8(5) and S(6)(j)
of one group, must hav� identical structures •
This re,quirement does not affect the form
•
.
of the last S(5) of a group; neither does it influence
the selecti on of the forms of S (5) in the� adjacent
groups.
V
•
As each G6 consists of three places, two
of which are identical, the number of structural
combinations for the individual groups equals 42 = 16.
S1 + Sl
S2 + S1
81 + S 3
S 2 + S3
S1 + 8 2
S1 + S4
S2 + S2
S 2 + 84 .
S3 + S 1
S4 + Sl
S3 + S3
S4 + S 3
S3 + S 2
83 + S4
S4 + S 2
S4 + S4
Thus we obtain 16 forms of G6 with the
following distribution of structural co mbinations •
•
•
•
10.
G = S1 (5) + sl (6)® + 81 (5)
6
G6
S1 (5) + s1 (s)® + s 2 (5)
=
Ga = s1 (5) + S 1 (6)® + s3 (5)
G6 = 81 (5) + s1 (6)® + s4 (5)
G6
=
s2 (5) + s 2 (6)® + s 1 (5)
G6 = 82 (5) + s2 (6)@ + s 2 (5)
•
G
6
=
s 2 (5) + s 2 (6)@ + s3 (5)
G = s (5) + s (6) 2
2
6
u
r
Ga = s3 (5) + s3 ( a )@ + s 1 (5)
Ga = s3 (5) + s 3 (6) + ·S 2 (5)
Ga = s3 (5) + s 3 ( a)@ + s3 (5)
G = s3 (5) + s3 ( a)� + 84 (5)
6
G6 = s4 (e,) + s4 (6)® + 81 (5)
Gs
=
S4 (5) + S4 (6)® + S 2 (5)
G6 = 84 (5) + 84 (6)@ + 83 (5)
G6
•
u
=
84 (5) + 84 (6)® + s4 (5)
As the melodic interval in the bass, while
moving from the root (1) in 8 (5) to the third ( 3 ) i n
•
11.
S(6 )@ is identi cal for the forms s 1 and s 3, as
well as s2 and s4, the total qUanti ty of intonations
in . the bass part for one type of 0 is ½ = 2 .
6
81 + 8 1
S1 + 82
S2 + S 1
82
+
62
As each intonation has 3 melodic forms
•
and there are two different intonations, the total
number of intonations combined with melodic forms
in the bass part is 2 x 3 = 6 .
•
,_
-.
,�
-s
-G
Progressions 9f. �e ty-pe
t.
-
IJ.
Figure LXIII.
Example:
Forms of S: s 2 (5) + s 2 (6)® + 81 (5)
r = G6 (C-5)
+
c3
+ G6 (C-5) + C7 + G6 (C-5) + C5 •
•
fl .,,.
-
•
12.
Example :
Forms of S : [s1 (5) + s1 (6)® + s2 (5) ] + [S3(5) +
+ S3(6:® t S2 (5) ]
� = as in the preceding example •
•
•
•
u
•
Example :
Forms of S :
s2 (5)
+
s2 (6)@ + s2 (5)
r = as in Figure LXI.
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13 •
•
Generalization of the passing third 1s
applicable to this type of harmonic progressions as
The following is an application of the
well.
structural group
Figure LXII.
2s 1
+
s2
+
2s1
+
s2
2s1
+
to the
Figure J;XJ.V.
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14.
• Lesson LXXXII.
Progressi�ns of_ the type IJI.
Applications of G6 to symmetrical systems
of tonics disclose many u nexplored possibilities,
among which the two-tonic S)'stem deserves a particular
As intervals formi ng the two tonics
are
•
attention.
equidistant, the passing tones of S ( 6 'fiJ, which i n
turn may also be equidistant from T 1 and T2 , thus
produce, in the bass movement, diminished seventh
•
chords in symmetric harmonization, a device heretofore
unknown.
•
The justification for the use of G6 in the
symmetrical systems of tonics is based on the following
deduction from the original classical form, i.e.,
•
--- --
----
(Symmetric)
(Diatonic)
•
The abovementioned equidistancy of the two
tonics permits t o obtain
r
= 3G6 until the cycle
•
15.
•
Selecting
closes.
•
,.
-y
�
II
�
�
-
•
•
�
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.
-t:
�
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s1
for the entire G6 , we obtain:
Figure LXV.
1�
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7'7
,7
t;
.,
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r
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l
.,
I
.·•
h
r•
t.
u
The overlapping of groups, indicated by
•
the brackets in the above Figure, is an invariant of
the symmetrical systems.
Thus, the passing third can
be considered a general device for progressions of
the type
•
III.
The number of bass patterns for the
cycle of the two tor1ics equals: 22 = 4.
The number of intonations in each cycle
of the two tonics equals : 22
:;::
4.
The latter is due
to the use of the different forms of S (5) .
The
interval between 1 and 3 equals 4 and is identical for
The interval between l and 3 equals
It
•
•
3 and is identical for s2 (5) and s4 (5) .
Thus, by
•
•
16.
distributing the different structures through two
tonics, we obtain the following co mbinations :
81 (T 1) + S 1 ( T2)
Sl (Tl ) + S 3 (T 2 )
�3 (T 1) + 81 (T 2)
•
S2 (Tl ) + S4 (T2 )
S4 (T l) + S 2 ( T 2)
•
•
identical intonations
in the bass part
S3 (Tl ) + S 3 (T 2 )
S 2 (Tl ) + S2 (T 2)
•
.-
- identical intonations
i n the bass B_art
S4 (T l) + S4 (T 2)
61 (T l) + S 2 (T 2)
Sl (Tl) + S4 (T2 )
S 3 (T l) + S 2 (T 2)
ident ica l intonations
in the bass part
S 3 (T l) + S4 (T 2)
S2 (T 1) + S1 (T2 )
S2 (Tl ) + S3 ( T2 )
S4 (T l) + S l (T l)
S4 (Tl) + S 3 (T'2 )
•
identical intonations
in the bass part
•
\
•
17 .
The following is a table of intonations
and melodic forms in. the bass part on two tonics.
Total : 4 2 = 16.
Figure LXVI•.
•
s,.
•
.r
54-
I.,
L
The above combinati ons can be incorpor
. ated
into a versatile continuity of 06 on two tonics.
'
•
•
18 •
•
LXVII.
Fig11£e
-.
Example :
•
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produces
V
Application of G to three t9pic§
6
8
melodic
forms in
the
bass par t : 2 3 = a .
Fi&1re LXVII I .
a
a
•
,
•
19.
Figure LXVIII (cont . )
.r
•
I ••
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�1 --
:
-
-
•�
-
�
---------
-
•
different S
A
,�
•
----.
•
,
.-
'
-
-----------------,.,-----
The number of distributions of the
through thr ee tonics is 43 = 64, while
the number of non-identical intonati ons is 23 = a.
.
Non-identical intonations:
S1(T1 ) + s1 (T2) + s1 (T a )
Sl ( T1) + Sl (T...� ) + S 2 ( T3 )
S l( Tl) + S 2 ( Tz ) + S l (T a )
Sa ( T ) + S ( T2 ) + S (Ta �
1
1
1
(j
.
•
-
-
•
u
· 1 (T a)
Sa ( Tl ) + S2 (T2 ) + S
•
Sa (T l) + Sl (T 2 ) + S2 {T 3 )
Sl (Tl) + S\( T2 ) + S2 (T 3 )
S2 (Tl) + S2 (T 3 ) + S2 (T 3 )
The total number of different intonations
and melodic forms in the bass part is 8 2 = 64 •
Examples of continuity of G6 on three tonics
•
Figure LXIX,
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Application of 06 to four tonics
If
produces 2 = 16 melodic forms in the bass part .
21.
The number of distributions of the four
forms of S through four tonics produces 4� = 256
intonations�
The number of intonation$ in the bass
part is limited to 2'4 = 16.
Thus the total number of intonations and
melodic forms in the bass part is 16 2
•
Examples of continuity of G6
on four tonics.
•
Figure LXX.
(ple.ase see next page)
•
•
,
•
0
= 256 •
•
22.
Figure LXX,
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Application of G6 to six tonics produces
= 64 melodic forms in the bass part.
The number of distributions of the four
°
4
forms of S through four tonics produces
intonations.
2
4
= 64.
= 409"6
The number of intonati ons in the bass part is
The total number of intonations and melodic
forms in the bass part is 64 2 = 4096.
Examples of continuity of Ga on s·ix tonics.
¥1gur� LXXI •
•
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24.
1
produces 2 �
Applicati on of G6 to twelve tonics
=
4096 melodic forms i n the bass part.
lhe number of distributions of the four
forms of S through four tonics produces 4
It.
•
=
16,777, 216 •
The number of intonations in the bass part
is 2 1 � = 4096.
The total nu mber of intonations and
melodic forms in the bass part is 40962 = 16,777, 216.
Examples of conti nuity of G6 on twelve tonics.
'Figµre
LXXII.
(please see next page)
•
•
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25.
s , const .,
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26.
Lesson LXXXIII.
B. Passing Fourth-Sixth Chords: S (4) .
of S (5) is a fourthThe second inversion
•
This name derives from the old
sixth chord: s(!).
basso continuo or generalbass,
where intervals
were measured from the bass.
7
•
-
-,�
�
I-I
8 ( !) has a fifth (5) in the bass, while� t he three
u
upper parts have the six usual arrangements�
•
The use of s(:) in classical music is
a very peculiar one.
This chord appears only in
definite pre-set combinatiODS r
group with a p?ssing
One of theru is the
fourth��ixth chord : G�.
As in the case of G 6, the passing chord
itself' appears on a weak beat, being surrounded by
the two other chords, and has a doubled fifth: s6
4 ®•
The two other chords of G: are : 8 (5) and 8 (6).
The
latter can have two forms o f cbubling (regardless of
the chord-structure): S (6) 0 an� 8 (6)® .
The group v,ith a passing fourth-sixth
0
chord, contrary to G , is reversible.
6
•
27.
u
,
This property being COD)bined with the
•
choice of two possible doublings produces four
••
variants.
•
+ 8 (6)
CD
•
•
l
•
The arrows in the above formulae specify
•
the directions of the bass pattern which is always
scalewise, and therefore can be either ascending or
descending.
The bass pattern is developed on three
adjacent pitch-units, which correspond to the three
•
a
chords of
9:
•
C)
�
C
l
8 (5)
-
*;,
5
3
......
s(!)
\\
8 (6)
Arabic numberals :represent the respective chordal
functions.
,
•
-
28.
Transformations between 8 (5) and s (:)
in the G4 : as the bass moves from 1 to 5, when read
in upward motion, the three upper voices must move
clockwise, in order to get the transformation of 1
into 3.
n
'
.,
,
•
....,
-
•
$
rJ
•
l.
G·
•
The transition from s (:) into 8 (6)
©
or S (6){f) follows the forms of transformations,
where two identical functions participate, as in the
cases of S ( 5) <
) S (6)
©
and S (5) (
> S ( 6) ©
However, classical technique adopted
.
definite routines concerning this transition:
•
(1) one part must carry out a melodic form
reci;proc,!11_ to the bass (i.e., position �
of the bass melody);
(2) it is ·th is reciprocal part that deviates
from its path in order to supply the
(j
d9Jl,bling
of the fifth in an 8 (6).
'
•
•
29.
f
'-'
Under such conditions G! acquires the
followi ng appearances:
•
"
I
'
•
.,.�
-
.
'-- .J
•
.
...._
,...,
...,
,
,�
r
-- -
--c:::,.
0
•.
,
-
--
-
-
"""!
_,.,,,,.
-s.
-t.
The following sequence of operations
•
is recommended :
(1) bass
(2) part reciprocating the bass
•
(3) coxumon tone
(4) part supplying the thir d for s ( !)
The relations between the chords of G4
6
are as follows:
Co
'
l,
8(5) . + C-5 + s (!) + c 5 + S (6 )
•
Co
+
8( 6) + C-5 + $ ( 4
6)
•
30.
Each group can be carried out in 6
positions which· depend on t he starting position.
a!
forms of
•
l
...
-
-
'�
�
0
'..
,I
•
0
--
I
-
The following is the table of all four
in one position.
Figure
LXXIII.
'
-
..
,
::,.
0
-
-
-
�
�
�
I•
.
,
:..,
-
,
,.;
�
.'
,
-
-
•
0
-
.
,
6 can be
The dif ferent forms o f G4
connected by means of tonal cycles and their
coefficients of recurrence can be specified.
It is desirable to make the following
tables:
c7
c3
(2)
const. ;
"
"
lf
"
"
n
(3)
const . ;
"
"
11
"
"
n
const. ;
ff
n
"
,,
"
"
(4)
0
c5
const.;
(1)
6
G4
!
©
-
const.,
const.,
const.
•
31.
•
(5)
•
(6 )
(7)
(8)
(9)
•
G4
rr ©
a!!©
6
T
G
4
ati
©
(0
6T 0
G4
COilSt . ;
c+ = C3 + C5 + C7
const.; c-t = C3 + C5 + C7
const .; cot = C
3
COils t . ;
:!©
+ G
=
c""
+ G�
T
C
3
(f)
+ c
+
C
"
"
"
"
"
"
TT
"
"
n
"
(11)
(12)
5
6
G
+
"
(10)
+ c
7
5
+
0
C
7
,
,•
C3
c
5
COilSt •
const ..
,• C7 const .
•' � =
C3 + C5 + C7
� is the symbol of a group of cycles (cycle
COiltinuity ) .
Continuity of G64, when connected through
a constant tonal cycle, consists of seven cycles:
� =
7C.
e
Figure LXXIV,
Example : G4 r
•
.,,
..,
-e
-s,-..
'
I
•
-
-
0
-
•
•
V
..
...,,
,-..
-8--
�
©
const.
•
i.....
.._,
.,
•
....
,�
'
-
.'
,.i
.., •
'.J
.,
•
�
-
�
.0..
••
-
- -e- - - .
�
-
�
�
I
,,i
q
••
�
• -i
;a;- .a=:.
•
,c ,;
••
�
•
•
32.
·v
a:
Continuity of
of different forms and
co11nection through different cycle-groups can be
applied in its present form to Diatonic progressions.
a:
in symmetric progressions of the
types I I and III require identical structures for the
two
" extreme chords of one group.
-
This requirement
does not affect the middle chord of the group,
s (:), nor do es it influence the selection of
•
structua,es for the following groups.
Examples of continuity with
i • e •,
a:
in progr�ssions of the types• I a.nd I I .
Ir' =
•
,
�
" 'I•
•
-
-
�
.,,
.a..
.....
-
�
••
-s•
-
·�
,
-9-..,,
C'
_,
6'
6
2G4t + G4
.....
•
---
.,,
C
•
r
�
�
!
�
r•
!; � :::
61T'
6
+ G4 J + 2G4
....
- 9-- .w.
-- .'
�
FiBur e LXXV •.
-....
..0-
'
-
..-:
.,
,.
• •
--
-
..,
,
i
,.
•
',
._,
-!r -e- .,
...
�
-
'
A 0
- -
�
,.
r,
Figure LXXVI.
1-r and � as in the preceding example•
•
--------
-
....
,.., I •
-
-s- .... -5--
.
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,
-e.
--
�
,.,
,....
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__• r,
-
•
•
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
With : Dr. Jerome Gross
C O U R S E
Subject: Music
Lesson µcxxiv,
Application of G� to symmetric systems
requires the following seq uence of tonics:
GW � (T, + T 2 + T , ) + (T 2 + T 3 + T2 ) +
• • •
For example, the three-tonic system must
be distributed a s follows :
0
G� = (T + T 2 + T ) + (T 2 + T 3 + T 2 ) +
,
,
•
+ (T 3 + T + T 3 ) .
,
The quantity of to nics in tl1e respective
system specifies the cycle.
either 8(5) or S (6 ) .
Eac11 gro up may begin with
Each group acquires the following distri
•
bution of inversions:
Under such conditi ons, each tonic appears
in all the three inversions.
•
6
Table of G4
•
applied to all symmetric systems
"
Figure tXXv!l.
I
1
.,
,
..I
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,,
�
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�-
•
,
,I
�-
I
,�
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. CI
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a,
L1
r
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•
5
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rouR To N l C. � .
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-r�
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•
4.
Other negative forms are not as practical: inversions
weaken tonality.
Example of variation of structures
a
l
$
and directions.
Figure LXXVIII.
Four Tonics.
G� - [s, (5)
+
s 2 (:)
+
s , (s) J
+(s 2 (6) +
s , ( : ) + s 2 (6) )
+
+ [S 3 (6) + S� ( :) + 8 3 (5) ] + [ S 2 (5) + S 3 (:) + S 2 (6) )
•
•
11[.___
___
b i;
_q
.,__
'l:
_P
_
�
_
_
>'
#
a,.
>< +
�t:
_---tl
•
•
•
c. Cyc�es. �d ��oups Mixed.
Tonal cycles can be introduced into the
continuity of groups, as well as groups can be intro
duced into the continuity of cycles.
It is convenient to plan the mixed form of
cycle-group continuity by the bars (T).
Bars of cycles and bars of groups can be
•
-
•
•
assigned to have different coefficients of recurrence.
,
When planning such a continuity i n advance,
it is i mportant to co nsider that there is always a
cycle-connection between the bars •
•
Examples:
•
Figure LXX}:;x.
Ir = 2TC + TG + TC + 2TG = (C� + C3) + C 7 + (Ca + C 7) +
+ c, + G6 + C 7 + (C3 + C3 ) + Cr + G:1© + C 1 + G! _l,(f)+ C3 •
L
••
•
•
T'(PE It
•
,
...__.,,,
,
•
•
•
,
•
JOSEPH SCH I.LL INGER
CORRESPOND E ·N.CE COURSE
Subject: Mu�j.c
With: Dr. Jerome Gross
•
Lesson LXV.
SPECIAL THEORY OF HARMONY
Introduction
Special Theory of Harmony is confined to
E, of the First Group of Scales, which contain·, all
musical names (c, d, e, f, g, a, b) and without
repetition.
There are 36 such scales in all.
The total
number of seven-unit scales equals 462.
The uses of E, refer to both structures and
..
progressions in the Diatonic System of Harmony.
The
latter can be defined as a system which borrows all its
pitch units for both structures and progressions from any
one of the 36 scales.
Whi'le the structures are limited
to the above scales, the progressions develop through all
the semi-tonal relations of the Equal Temperament.
•
The
latter comprises all the Symmetric Systems of Pitch,
i. e. , the Third and ti1e Fourth Group.
Chord-stDuctures, contrary to common notion,
\
•
do not derive from harmo_ nics .
·I f the evolution of
chord-structures in musical harmony
would parallel the
evolution of harmonics, we would never acquire the
•
developed forms of harmony we now possess .
,
•
2.
•
To
· begin with, a group of harmonics,
simultaneously produced at equal amplitudes, sounds
like a saturated unison and not like a chord.
•
,
•
other words, a perfect harmony of frequencies and
int ensities does not result i n musical harmony but in
a unison.
we
•
In
�his means that through the use of harmonics,
v«:>uld never have arrived at musical harmony.
But
we do get harmony, an d exactly for the opposite reason.
'
The relations of sounds we use in Equal Temperament
are not s.µnple ratios (harmonic ratios).
When acousticians and music theorists
advocate nJust intonationn , that is, the intonation of
harmonic ratios, they are not aware of the actual
(
situation .
•
On the other hand, the ratios they give
.
for certain trivial chords, like the major triad
(4+5+6) , the minor triad (5+6+15) , the dominant seventh
chord (4+5+6+7), do not correspond to the actual intona
tions of the Equal Temperament.
Some of these ratios,
like¼, deviate so much from the nearest i ntonation,
lik� the �inor seventh, which we have adopted through
habit, that it sounds to us out of tune.
Habits in
music, as well as in all manifestations of life, are
I
more important tha.n the natural• phenomena.
•
If the
problem of chord-structures i n harmony would be confined
to the ratios nearest to Equal Tem perament, we could
have offered 16+19+24 for the minor triad for example,
,
•
•
3 ..
L
as it approaches the tempered.minor triad much better
than 5+6+15.
But this, if accepted, would "discredit
the approach commonly used in all textbooks on
harmony, and for this reason.
If such high harmonics
as the 19th are necessary for the construction o f· a
minor triad, what would chords of superior complexity,
,
which are in use today, look like when expressed
through ratios.
When a violinist plays b as a leading
tone to c and raises the pitch of b above the tempered
b, his claims for higher acoustical perfection are
nonsense, as the nearest harmonic in that region is
the 135th •
Facing faets, we have to admit that all
•
the acoustical explanations of chord-structures as
being developed from the simple ratios, are pseudo-
•
scientific attempts to rehabilitate musical harmony,
t o give.the·latter a greater prestige.
l.
Though the
original reasoning in this field was caused by the
ho11est ·spirit of investigation of Jean Philippe Rameau
I
•
("Generatio� Harmonique", Paris, 1737), his successors
overlooked the -development
. of acoustical science.
,
Their inspiration was Rameau plus their own mental
I
laziness and cowardice.
•
The whole misunderstanding in the field
of musical harmony is due to two main factors:
•
•
4.
(1) the underrating of habit;
i
(2) the confusion of the term "hermonic" in its
I
mathematical connotation,
pertaining
to simple ratios with "harmony" in its
•
mu.sical connotation, i.e., simultaneous
pitch-assemblages varied in time sequence.
Thus, musical harmony is not a natural
phenomenon, but a highly conditioned and specialized
field.
•
It is a material of musical expression, for
which we, in our civilization, have an inborn
inclination and need.
This need is cultivated and
furthered by the existing trends in our music and
musical education.
I. Diatopic System of Harmony.
Chord-structures and chord progressions in
the Diatonic System of Harmony have a definite interdependence: chord-structures develop in the direction
•
.
•
•
opposite to their progression�.
Thi-s s-t;;atement brings about the practical
classification of.the Diatonic System into two forms:
the posi�ive and the negative.
As the term Diatonic implies, all pitch•
units of a given scale constitute both structures and
'
progressions, without the use of any other pitch-units
(not existing in a given scale) whatsoever.
l
•
•
5.
In the form which we shall call positive,
all chord structures (S) are the component parts of
I
the entire structure (E,) emphasizing all pitch-units
of a given scale in their first tonal expansion (E,)
)
•
and in position Ci).
,
In the same form chord progressions
derive from the same tonal expansion but in position
In the negative
form of the Diatonic
0
System, it works in the opposite manner.
Chord
structures derive from the scale in E, and in position
(§) , wb�le the prog
, ressions develop from E, @ .
According to the qualities we inherited
•
and developed, the positive form produces upon us an
effect of greater tonal stability.
It is chrono
logically true that the negative form is an earlier
one.
It predominates in the works where the effect o f
tonality,_ as we know and feel it today, is, rather
vague.
'
. '
Such is the XIV and XV Century Ecclesiastic
music, developed on contrapuntal and not on harmonic
foundations.
Many theorists confuse the negative form
of the Diatonic System with "modal" harmony.
As by
Diatonic Tonality they mean, in most cases, Natural
Major or Harmonic Minor scales moving in the positive
form, they miss the tor1al stability when harmony moves
backwards.
Losing tonal orientation they mistake such
..
•
I
6.
progressions for modes, which are merely derivative
scales, and may also have the positive, as v1ell as
the negative form.
But as we have seen in the
Theory of Pitch Scales, modes can be acquired from
any original scale through the introduction of
accidentals (sharps and flats).
In the following table, MS represents
"melody scale" (pitch-scale), and MR represents
"harmony scale" (i.e. , the fundamental sequence of
chord progressions) .
Diatonic System
Positive Form
Negat:l,.ve For!D
L = MSEr@
= .MSE,@
L
HS= MSE, @
HS= MSE,@
•
Figure I.
•
Example (Natural Major)
Positive Form
.g.
(\
I
,
I
..
-,. -
•
�
-�·
,.
•
�
...-
Negative Form
..- -e--
A
.A.
•
-!I
-9-
t.
.c.
-.
-...
.-
·==========================================================----------- - - - ---=
..
'
••
• HS
-
L.
• ,
..
•
.,
....
-9-
♦
-
-.,
-
-
�
•
-
•
7.
L
In the positive form, chords are
constructed upward, in the negative, on the cont rary,
'
downward.
The matter is greatly simplified by the
fact that any prog ression, originally written as
positive, becomes negative, when read backward.
All the principles of structures and motion involved
•
are therefore reversible.
No properly constructed
harmonic continuity can be wrong in backward motion.
Some composers without training in
harmony (for example, Modest Moussorgsky) as v,ell as
beginners, due to inadequate study, confuse the
positive and the negative forms in writing their
harmonic progressions.
The resulting effect of such
music is a vague tonaltty.
The admire rs of Moussorgsky
consider such styl� a virtue (in Moussorgsky's case it
is about half-and-half positive and negative), and do
not realize that
all the incompetent students of a
-harmony course incompetently taught possess full
command over such style.
t.
•
8"
Lesson LXVI.
A, Diatonic Progressions (Positive Form)
Expansions of the original Harmony Scale
produce the Derivative Harmony Scales.
The original
HS and its expansions form the Diatonic Cycles.
Diatonic (or Tonal) Cycles repre sent all the funda
mental chord progressions.
There are three Tonal Cycles in the
Positive Form for the seven-unit scales.
The First
Cycle, or Cycle of the Third (C 3 ) , corresponds to
HSE0; the Second Cycle, or Cycle of the Fifth (Cs-),
corresponds to HSE, ; the Third Cycle, or the Cycle
of the Seventh (C7), corresponds to H-SE ., Beyond
2
these expansions of HS lies the Negative Form o f
Diatonic Pro gressions.
,,,
In addition to both forms of progressions,
there may be changes in a chord pertaining to the
same root (axis).
modified S of the
Connections of an S with its
s:ime root will be considered a
Zero Cycle (co) • •
In the follov1ing table notes are used
merely for convenience: they indicate the sequepce of
roots; their octave position was dictated by purely
raelodic
•
considerations and by the neces.sity to
moderate
the range•
•
•
•
The respective interva·ls. .represening
'
Cycles must be constructed downward for the Positive
Form, regardless of their actual position on the
musical staff.
Figure II.
Diatonic Cycles (Positive Form)
••
•
•
•
Cadences:
Starting
Ending
Cycle of the Third·· (C3 )
.
. �3
.....
I
-
•
•.
I
..
c
Starting
. ..
'
....
..
,----;
�
-
-. ,.
•
0
0
••
'
· Starting
;
Ci
c
�I =
Ending
•
Combined
c
'.
Combined
--- ·---
In the above table·arrows indicate
.
cadences of the �espective cycles
•
Cadences consist
of the axis-chord moving into its adjac��t chord and
back.
It is interesting to note, that what is usually
kno\m as Plagal Cadences are the Star.ting Cadences
and.that Cadences known as Authentic are the Ending
Cadences.
0
Combined
•
Ending
Z!
The immediate seque11ce of Starting and
•
I
-
•
10.
Ending Cadences produces Combined Cadences (the axis
chord is omitted in the middle ) .
Progressions of constant tonal Cycles
(C3, or Cs, or C ., cons t·. ) produce a sequence of seven
chords each appearing once and none repeating itself.
The repetition of the axis-chord either completes the
Cycle or star· t s a new one.
The addition of Cadences
to the Cycles is optional, as Cycles are self-sufficient.
Considering constant Cycles as a form of
Monomial Progressions, we can devise Binomia.l and
Trinomial Progressions by assigning a sequence pf two
or three Cycles at a time.
In Binomial Progressior1s each chord appears
twice and in a different combination with the preceding
and the following chord�
Thus, a complete Binomial
Cycle in a seven-unit scale consists of 2 x 7 = 14 chords.
Figure III.
Binomial cycles
Gs- +
C3
Cs- + c1
(please see next page)
,
•
•
11
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•
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... .
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-
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,...
-
0
.
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-
,...
....,
-
;::;:
�
-
e
+ Cr
-
,...
.,
,;;
0
..,
I If
•
-
;;,
�
�
-
..
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with the preceding and . the following chord. Thus,
a complete Trinomial Cycle in a seven-unit scale
consists of 3 x 7 � 21 chord.
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Figure IV.
TrinomiaJ Cycles
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c, + c, + c.,
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Lesson LXVII.
Both Binomial and Trinomial Cycles produce
the ultimate
variety combined with the absolute
•
consistency of the character (style) of harmonic
progressions.
Being perfect in this respect they are
of little use when a personal selection of character
becomes a paramount factor.
In order to produce an individual style
of harmonic progressions, it is necessary to use a
selective continuity of Cycles.
This can be accom
plished by means o f the Coefficients of Recurrence
applied to a selected combination of Cycles.
A
combination of Cycles can be either a Binomial or a
Groups producing coefficients of
Tr,inomial.
recurrence can be Binomial, Trinomial or Polynomial.
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The materials for these can be fo und in the Theory of
Rhythm.
Rhythmic resultants of different types and
their variations provide various groups which can be
used as coefficients of rec�rrence.
Distributive
Power-Groups as well as the different Series of Growth .
and Acceleration can be used for the same purpose •
•
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14.
Figur�- Y�
Binomial Cycles, Binomial Coefficients
Cycles: C 3 + Cr; Coeffic ients: 2+1
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15.
Binomial Cycles, Coefficient-Groups producing interference with the
Cycles (not divisible by 2)
Cycles: C.r +
Coefficients: 3 + 1 + 2
C3
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Synchronized Cycles: 3C; + c3 + 2Cr + 3C 3 + C; + 2c,
Synchronized coefficients: 6t x 2 = 12t; 12 x 7
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Cycles: C3 +Cr + C7
Coefficients: 4 + 1 + 3
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(Figure V, Cont.)
Trinomial Cycles, Co efficient-G-roups with tl1e number of terms
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17.
The style of harmonic progressions depends
entirely on the form of cycles employed.
No composer
confines himself to one definite cycle, yet it is the
predominance of a certain cycle over others that makes
his music immediately recognizable to the listener.
In one case it may be that the beginning of a progression
is expressed through the cadences of a certain cycle, in
another case it may be a prominent coefficient group
that makes such music sound distinctly different from
the other.
The style of harmonic progressions can be
defined as a definite form of Selectiv� Cycles.
Both
the combination of cycles (their sequence) and the
coefficient group determining their recurrence are the
factors of a style of harmonic progressions.
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•
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18.
u
Lesson LXVIII.
There is much to be said about the
historical development of the cycles, as there are
already some wrong notions established in this field.
Though the common belief is that the
progressions from the tonic to the dominant and back
to the tonic (ending cadence in c ), is the foundation
5
of diatonic harmony, historical, evidence, as well as
mathematical analysis prove to the oontrary.
During
the course of centuries of European musical history,
parallel to the developwent of counterpoint, there
was an awakening of harmonic consciousness.
-
can be traced, in its apparent
XV . Century A.D.
The latter
forms, back to
At that time harmony meant concord,
an agreea.ble, consonant, stabilized sonority of
•
several voices- simultaneously sustained.
Concordant
prog_ressio11s could be accomplished therefore throug h
consonant chords moving in consonant relations.
Obviously such progressions require common tones, and
the latter can be expressed as c3• As the -tonality,
i.e., an organized progression of tonal cycles was at
that time in the state of fermentation, it is natural
to expect the cycle of the third to appear in both
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positive (c3) arid negative (C-3) form.
The following are a fevw illustrations
tak en from the music of XV and XVI Centuries.
C.
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Figure VI.
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Cycle of the Seventh, on the other hand,
has a purely contrapuntal derivation.
When the two
leading tones (the upper and the lower) move in
cadence into their
a
respective tonics (like b.--+c and
d > c) by means of contra.ry motion
in two voices, we
•
,
obtain the ending cadence o f c 7• Further development
of· the third part was undoubtedly 11ecessitated by the
d€sire for fuller sonorities.
This introduced an extra
tone (f in a chord of b) with which
tones form S(6) i.e., a
the remaining
third-sixth-chord or a sixth
chord, the first inversion of the root-chord: S(5).
Figure VII.
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domi11ance of
It is only natural to ex�ect the pre
the c7 in contrapuntal music.
Cadences
as in F!tgure VII are most standardized in the XIII
and XIV Century European music.
Machault (1300-1377)
11
See Guillaume de
Mass for the Coronation o f
Cha rles V11 (phonograph recording published by the
Gramophone Shop)..
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21 ..
The appearance of the cycle of the fifth
date, when c 3 and c 7
I of fer the following hypothesis
must be referred to a later
were already in use.
of the origin of C6.
The positive form might have
•
occurred as a pedal point development, where by
sustaining the tonic and changing the remaining two
tones to their leading tones, the sequence would
represent c,.
Another interpretation of the origin
of Cs is the one which this system of Harmony is based
•
upon, i.e., omission of intermediate links in a series •
This principle ties up musical harmony with harmonic
structure of crystals, as used in crystallographic
analysis.
Figure VIII.
Cs
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The origin of the negative form of the
cycle of th e fifth (C-5) is due to the desire of
acquiring a concord supporting a leading tone.
be a leading tone in the scale of c.
Let b
The most con
cordant combination of tones in the pre-Bach - time,
i.e., in the •mean temperament (the tuning system
v.
officially recognized in Europe before the advent of
•
22.
equal temperament) , harmonizing the tone b was the
G-chord (g, b, d) .
But when movi ng from G-chord to
C-chord the form of the cycle is positive.
In
are the beginning and the ending cadences.
Compare
reality both forms, the positive and the ne gative,
Figure IX with Figure VIII.
Figure IX .
•
,
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J O SE P H
S C H IL L I N G ER
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 LXIX.
•
The development of harmonic progressions ·
in the European music of the last three centuries c an
be easily traced b ack to their sources.
.
The style of
every composer is hybrid , yet the quantitative predominance of certain ingredients (like the cycles
appearing with the different coefficients of recurrence)
produces individual characteristics.
In the following exposition I will confine
the concept of nstylen to harmonic progressions in the
diatonic system.
I.
Richard Wagner was the greatest representa
tive of C 3 in th e XIX Century.
This statement is
backed by the statistical analysis of tonal cycles in
•
•
his works as compared to his contemporaries and
predecessors: c 5 v,as the universal vogue of a whole
century preceding Wagner� In fact, it is not necessary
to analyze all works of Wagner.
The most characteristic
progressions may be found at the beginnings of his
preludes to musical dramas and also
cadences.
:in the various
The beginnings of major works of any
L
•
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•
2.
•
composers
are important, fo r the reason that they
composer.
The importance of cadences as determinants
cannot b e casual: it is the "calling card" of a
of harmonic styles was stressed upon by our contempo
rary, Alfredo Casella, in his paper, "Evolution of
Harmony from the Authentic Cadence".
Wagner, being German and intentionally '
•
Germanic composer, undoubtedly has done some research
of tt1e earlier German music, as he intended to deal
with the subjects of German mythology, in which he was
well versed.
The XV Century German music discloses
such an abundance of C 3 , that it is only natural to
expect the influence of such an authentic source of
Germanic mus ic upon Wagner's creations.
In his time,
Wagner 's harmonic progressions sounded revolutionary
'
because many things were forgotten in four hundred
years, and archaic acquired a flavor of modernistic.
So far as the development of diatonic progressions in
Wagner ' s music appears to the unbiased analyst, the
whole mission of Wagner 's life was to develop a
consistent combined cadence in C 3 •
Starting with an early work like
"Tannhauser", we find that already the very beginning
of the Overture is typical in this respect.
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3.
Figure x.
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Later on v,re find more extended pro
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Eschenbach (the scene of Minnesingers contest):
Figure XJ.
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"Logengrin" is even more abundant with
11 Tannhauser 11 •
In "Farewell to S wan", as in
many other places of the same opera, vre find the
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4.
characteristic back-and-forth fluctuation: C 3+C-3.
figure XJI.
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Forming his cadences, Wagner paid some
time his tributes to the dominating "dominant'' of
This produced combined b,ybrid
cadences, which are characteristic of "Lohengrin".
The first part of such a cadence is the beginning
cadence in Ca, while the second part is the ending
cadence in c5 : I - VI - V - I .
Figure XIII .
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5.
Dealing with other type s of progressions
than diatonic in the cour se of his career, Wagner
came back to diatonic purity in its complete and
consistent form in his last work "Parsifal".
The
beginning of the "Prelude to Act I" reveals that the
composer came to the realization of the combined
cadence of C 3 : I - VI - III;
Figure XIV.
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The more extensive sequences of C3 are:
I - VI - IV - II;
Figur� xv_.
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.A nd the complete combined cadence ( nProcession of
the Noblemen of Graal"): I - VI - III - I •.
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Figure XVI.
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The second half of the XVIII Century and
the first half of t:t1e XIX Century cover the period of
the hegemony of the dominant and c5 in all its aspects
in general. The latter are: continuous progressions o�
c5; starting, ending and combined cadences ( I - IV - I;
I - V - I; I - IV - V - I). The main sources of music
possessing these characteristics are: the Italian Opera
and tr1e Viennese School..
To the first belong :
Monteverdi, · scarlatti, Pergolesi, Rossini, Verdi.
The
second is represented by Dittersdorf, Hayd n, Mozart,
Beethoven, Schubert.
Today th is style disintegrated
into the least imaginative creations in the field of
popular music.
Nevertheless it is the stronghold of
harmony in the educational music institutions.
•
•
7.
Here are a few illustrations of c 5 style
in the early Sonatas for the Piano by Ludwig van
Beethoven: Sonata Op. 7, Largo; Sonata Op. 13,
Adagio Cantabile.
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Any number of illustrations can be found
in Mozart 's and Beethoven 's symphonies, particularly
in the conclusive p�rts of the last movements.
Assuming that the historical origin of the
cycle of the seventh can be traced back to contrapuntal
cadences, it would be only logical to expect the
evidence of c7 in the works of the great contrapuntalists.
•
8.
I choose for tl1e illustration of c 7, as characteristic
starting progressions, some of the well knovm Prelud es
to Fugues taken from the First Volume of
11
Well Tempered
Clavichord" by Johann Sebastian Bach: Prelude I;
Prelude III; Prelude V.
Fig_ure XVIII.
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Bach ' s famous "Chiacopna in D-minor n for
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Violin, discloses the same characteristics, as the
first chord is d and the s econd chord is e, which makes
•
--
,
•
9.
A consistent and ripe style of diatoni c
progressi ons corresponds to a consis tent use of one
form, either positive or negati ve and not to an
indiscriminate mixture of both.
Many theorists con£use
•
the hybrid of positive and negati ve forms wi th modal
progressi ons, which the theorists have never defined
clearly.
In reality, modal progressi ons are in no
respect different from tonal progressions, except for
the scale structure.
Both types (tonal and modal) can
be eit her positive, or negative, or hybrid.
Modes can
be obtained by the direct change of key si gnatures, as
descri bed in the "Theory of Pitch Scales 11 (transposi t i on
to one axi s) .
Here is an example, typical of Moussorgsky,
from "Bori s Godounov 11 (opera):
Figure J{IX.
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In the a bove example the mode (scale) i s Cd5 ,
the fi fth derivati ve scale of the Natural Major i n the
key of c, known as Aeolian mode, while
the progression
of tonal cycles i s a hybrid of positive and negative forms.
/
•
10 •
Lesson LXX.
•
B, Transformations of S(52
• .
· In the traditional courses of harmony the
problems of progressio�s and voice-leading are
inseparable.
Each pair of chords is described as
sequence and a f orm of voice-leading .
Thus each case
becomes an individual case where the movement of voices
is described in terms of melodic intervals (like: a
fifth down, a second up, a leap in soprano, a sustained
tone in alto, etc.).
No person of normal mentality can
ever memorize all the rules and exceptions offered in
such courses.
In addition to this unsatisfactory form
of presentation of the subject of harmony, one finds
out very soon that the abundance of rules covers a very
limited material (mostly the harmony of the second-rateL
XVIII Century European composers).
The main defect of the existing theories
of harmony is in the use of the descriptive method.
Each case is analyz.ed apart from other cases and
without an� general underlying principles.
The mathematical treatment of this subject
discloses the general properties of the positions and
movements of the voices in terms of transformations of
the chor dal fun ctions.
Any chord, n-0 matter of what structure,
from a mathematical standpoint, is an assemblage
• of
•
•
•
11.
pitch units, or a
(elements).
gro up of conjugated functions
These functions are the different pitch
units distributed in each group, assemblage or chord
according to the different number o f voices (parts)
and the intervals between the latter.
In groups with three · functio ns known as
three-part structures (S = 3p) the fu nction s are a, b
a nd c.
These functi ons behave through general forms
o f transformations and not throug h any musical
specifications.
As in thi s branch we are deal ing with so
called fo ur-part harmony, we have to define the meaning
•
o f this expression more precisely •
When an S(5) constitutes a chord-structure,
the functions o f the chord are: the root, the third 8.Ild
the fifth or 1, 3 and 5.
In their general form they
correspond to a, b and c, i. e. , a
=
l, b
L
= 3, and c = 5.
The bass of such harmony is a constant root-tone,
i.e,'
co nst. 1 or co nst. a.
Thvs the transformation of fu nctions
affects all parts except the bass.
Here, therefore,.we
are dealin g with the gro ups consisting of three functions.
formation.s:
Such gro ups have two fundamental trans
•
•
•
12.
(1) clockwise (Z, ) a.nd (2) counterclockwise (� )
The clockwise transformation is :
The counterclockwise transformation
l.S :
Each of these transform ations has two
meanings: the first to be read -a is followed by b
•
b
n
C
tt
"
n
"
C
" a
for the ,.-�
and
f:__..
a is followed by c
C
n
"
"
b
" a
b "
for the �
•
- ,,,
d·iscloses the mechanism of the !)OSi tions of a chord;
the second to be read -a transforms into b
b
C
"
"
C
"
n
a
for the ;::! and
a transf orrns into c
C
"
"
b
"
n
for the 11::
"""
., �
b
a
•
13.
constitutes the forms of vo ice-leadi ng,
Positions.
The different positions of S (5)
= 1, 3, 5
can be obtained by constructing the chordal functions
dovmward from each phase of the transformations.
a
b
c
b
c
a
c
a
b
and
a
c
b
C
b
a
b
a
c
......___
____,➔
Substituting 1, 3 , 5 for a, b, c, we
obtain
1
3
3
5
5
1
5
1
and
3
1
5
5
3
3
1
Ir;;
......
�
•
3
1
-
..,
The cloclcwise positions are commonly _known
as open, and the counterclockwise as close.
Here are
the positions for S(5)
=4 + 3 =
Bass is added for the doubling o f the
= c - e - g.
root.
•
•
•
•
Figure
14.
XX.
P o s i +i o ,.,., 6
•
\
.....
,. I
.,, ..•
,
•
--
4
-
•, V
.�
•
',
e
rJ
.' I
V
-
••
-
.!
.
.;
.....
�
'
I
�
15"
'S'
<•
l
t
.
•
Voice-Leading
•
The movement o f the individual voices
follows the groups of transformation in this form:
u
a of the first chord transforms into b of the
following chord; b of the first chord transforms
into c of the following chord; c of the first chord
•
transforms into a of the following chord.
The
above three forms con stitute the clockwise voice
leading.
For the counterclockwise vo ice-leading
the reading must follov1 this order: a of the first
chord transforms into c of the following chord; c
of the first chord transforms into b of the
following chord; b of the first chord transforms
into a of the following chord.
'
•
...., ,
a
,, 1
�
I::
)b
b
)C
C
)a
}C
and
\
C
,, b
b
,a
Applying the above transformations to
•
1, 3, 5 of the 8 (5) , we obtain:
�
1 )3
3
and
)5
5--) 1
,,
1
)5
5
)3
3
)1
•
CJ.ock,,ise form:
The root of the firs t chord becomes the
•
third of the next chord; the third o f the first
,
chor d becomes the fif�� of the next chord; the fifth
of the first chord becomes the root of the next chor d .
Counte�clockwise form:
The root of the first chord becomes the
fifth of the next chord; the fifth of the first
chord becomes the - third o f the next chord; the third
of the first chord becomes the ro ot of the next
chord.
Both forms apply to all tonal cycles.
Let us take C3 in the natural major, for
'
example.
The
first chord is C= c - e - g and the
�
•
u
16 .
next chord is A= a - c - e.
Clockwise for m gives the following
reading :
C
>C
g
)a
Counterclockwise form gives the
•
following reading:
C
➔e
g
)C
e
?a
Let us take c5 in the sam e scale.
The chords are: C = c - e - g and F = f - a - c.
r ➔
C
e
F .,,,,
', a
)c
•
C- ) C
--
g -=, a
-"
' IJ
e -- ) f
('I
- ,_'
;:,
... -
-
➔
Let us take c7 in the same scale.
chords are : C = c - e - g and D = d - f - a.
The
•
•
17.
r
C
)f
e
)a
�d
g
.,
"'
·•
-
.. -- r,. -
.' I
•
Both forms of F,. are acceptable in
r' �
•
this case, as the intervals in bo th directions are
nearly equidistant.
Jt:.
•
�
C
,a
e
)d
g- ) f
-�
0 I
u
t.
•
•
•
u
18.
Lesson LXXI.
Each tonal cycle permits a continuous
progression through one form of transformation.
In
the following table const. 1 in the bass is added .
Apostrophies indicate an octave variation when the
extension of' range
becomes impractical.
In c7 both directions are combined,
offering the most practical form for the range.
(please see
Figure XXI.
following page)
I.
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Figure XXI,
19 .
Tonal Cycles
Clockwise and Counterclockwise Transformations.
....
0 .a.
,..,
.Q.
,,.. ,... I"'\ -911'
D o'" 15""
•
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•
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20.
The clockwise and the counterclockwise
transformations are applicable to all positions
for the starting chord.
When the first cl1ord is in
the ic:� (open) position, the entire progression
remains automatically in such a position.
•
first chord is in
G
When the
(close) position, the entire
progression remains in such a position.
The co11s tancy of
position ( open or
close) is not affected by the co11stancy of the
tonal cycles, neither is it affected by the lack of
their constancy.
The transition from close to open position
and vice-versa can be accomplished through the use of
the following formula:
Constant b transformation
Const� 3
a· > C
l
>5
b·
)b
3
�3
C
)a
5= ) 1
It is ·best to have 3 in the upper voice
for such purposes, as in some positions voices
cross otherwise.
Function 3 from close to open
followi ng chord.
Reverse the procedure from open to
position moves upward to the function 3 of the
close.
L
•
•
21.
Figure XXII.,
Const. 3 Transformation
•
..
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�
v i ,,:,,-
..
,,.
,
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� ,'f V i1
-, .
-r .'
' :: I ➔-9-f
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-•
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r ;,. 0.
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:;.. I
s-
-
Continuous application of const. 3
transformation produces a consistent variation of
the
2
and the � positions, regardless of the
sequence of tonal cycles.
The following table offers continuous
progressions through canst. cycles and const� 3
transformation.
Fig ure XXIII,
(please see next page)
•
•
C3 Const .. 3
"'
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Const. 3
.,,
-5-
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.£.
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22.
Figure �I I I .
Constant 3 Transforma tions
•
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:a.
.....
·-
�
..0..
?
�
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-
.
c7 Const. 3
�
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•
•
J OSE P H S C H I L L IN G ER
C ORRES P O N D ENCE CO UR S E
Subject: 1viusic
With: Dr. Jerome Gross
Lesson LXXII,
There are four forms of relationship
between the cycles and the transformations with regard
to the variability of both.
(1) const. -cycle, const.-transformation;
(2) const.-cycle, variable transformation;
•
(3) variable cycle , const�-trans formation;
(4) variable cycle, variable transformation.
The forms of transformation produce their
own periodic groups, which mpy be superimposed on the
groups of cycles.
Monomial forms of transformations (const.
transformations):·-
( 1)
(2)
(3) const.- 3
Binomial forws of transformations:
(1) �
Here Const. 3 is excluded on account of
tl1e crossing of inner voices.
•
Coefficients of recurrence being applied
to the forms of transformations produce selective
transformati9n-groups.
L
•
•
2.
For example: 2 ; ! + ! ; ; 3 � ; + 2 ;� ;
=
! + 2 �; ; 4 i� + � +
2 ; °! + �; + -;..
'= "'
rJ
4 -':: ..
<'
3 � ! + 2� � +
➔
Jc. ·
5 -J
2
.. ,, ; ., ,_, + i,.. + 3 .. , + t=:--
+
+ 2 ;.: + 3 ....� + �, +
«"'
4r 'j + 2 l:;; ,.
.... ,.... ➔ + I:: "" .
+ 8 .. ,, ; f"+
.. ..., .c. p .... ,
Though the groups of tonal cycles, as well
...1
as the forms of transformations, may be chosen freely
with the writing of each sub sequent chord, rhythmic
planning of both guarantees a greater regularity and,
therefore, greater unity of s tyle. .
Examples of variable transformations
applied to constant tonal cycles.
Figure XXIV.
'
-
•
'
,J
.l:1.
•
c,
,... �
const ., 4
,
-
•
,
�
,.
..
- -'
+ 2 .t
T
�
,_
<>
•
.....•
-�
-
..,..,
,
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u
..
•
-·.
�.,,
-
-
.
'
.
,
'
.
..,
-
'
-,SI
'
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,.
..,
.
,:
-
,.
-
'
-€
...,
-
.,;. ,J
'
+
.. ? + 2 � t
C r const. 3. ..j;.,' + ,,.
,- .,,
nI
•
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..,
-
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ls
t.
,
·-
+ 2 .=: ,. + .. >
II:
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,.,.
�
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ic::' added for the ending.
-e,....
-�
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3
.,
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-,:,
---
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Examples of variable transformations applied
to variable tonal cycles.
Figur� XXV .
--
CO + C1 + C 3 ; 2 � + ��
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4.
All forms of harmonic continuity, due to
their property of redistribution, modal variability
and convertibility, are subject to the following
modifications:
(1) Placement of the voice representing
constant function, and originally appearing
in the bass, into any other voice, •l. . e . ,
•
tenor, alto or soprano .
There are four
forms of such distribution:
s
A
T
B
s
A
T
B
S·
A
T
B
s
A
T
B
Red letters re present the voice functioning
-
as const .. 1.
(2) General redistribution (vertical permuta
tions) of all voices according to 24
L
variations of 4 elements.
(3) Geometrical inversions : @ , @, © and
@ for any or all forms of distribution
of tlie four voices .
(4) Modal variation by means of modal trans
positj.on, i.e., direct change of key
signature, without replacing the notes on
the staff.
,
•
Original
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5.
Example of variations .
Figu;re XXVI.
•
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7.
tesson LXXIII.
c. The Negative Fprm.
As it was previ ously defined, the
negative form of harmony can be obtained by direct
reading of the positive form in position @ •
Here, for the sake of clarity in the
entire matter, I am offering some technical details
which explain the theoretical side of the negative
form.
According to the definition given to
the harmony scale in the negative form, we obtain
the latter by means of further expansions of HS.
In the positive form we have used : H SE0 (= C 3 ),
HSE, =
( Cs ) and HSE2 (= C 7).
Novv by further expanding HS, we acquire
the cycles of the negative form: H SE 3 (= C - 7) ,
HSE =
( C - 5), HSE (= C - 3) .
�
�
Figure x;xv_r�.
(please see ne�t page)
•
•
•
...
•
a.
C -
•
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C -J
-e-
Cadences
Cycles
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e
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--
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.'
As chord-structures are built downward
fro m a given pitch unit, such a pitch unit becomes
L
the root-tone of the negative structure : the
negative root ( - 1) .
All chord-structures of the negative
form, according to the previ ous definition, derive
from HS @ •
Thus in order to construct a negative
S (5),
-
it is necessary to take the next pitch-unit dovmwa rd,
which becomes the negative third ( - 3) and the next
tt
T
unit do wnv.rard fro m the latter, which bec omes the
negative fifth ( - 5) .
•
•
•
•
For example, starting from c as a - 1,
we obtain a negative S (5), where a is - 3 and
f is - 5.
Figure )Q.CVIII .
Natural C- Major.
,..
�
"""
/
•
/
'
lo..
J
�, -J
Positions of chords, as they were
expressed through transformations, remain identical
in the negative form, providing they are constructed
upv,ard.
•
In such a case, the addition of a cons t. 1
in the bass must be, strictly speaking, transferred
to the soprano.
Here is how a negative CS (5) would
appear in its fo ur-part settings.
Figure XXIX .
•
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,,
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-�
-T
,-,
.0...
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=;
10.
If, u11der such cor1ditions, the chord
were constructed downward, the reversal of ;;.! and
reading would
take place.
Transformations as applied to voice
leading possess the same reversibility : if everything
is read downv.rard, the
'ic-..?
and the ': ...., tra.nsformations
correspond to the positive form, while in the upward
,+
Jr:
reading the ,:::::� becomes the ...., � and vice-versa.
Let us connect two chords in the negative
cycle of the third: CS (5) + C 3 + ES (5).
= C - a - f.
ES (5) = -1, -3, -5 = e - c - a.
CS (5) = -1, -3, -5
Figure XXX.
I
,
.,
'
...
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�,;
..
......
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_-$
-
-
--
.,
,.__
-
-- -
•
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-
x -•
,
�
•
It is easy to see that in the upward
reading chord C corresponds to F, and chord E
corres�onds to A.
Transposing this upward reading
to C, we notice that th is progression is c ---) E.
This proves the reversibility of tonal cycles and
the correctness of reading the positive form of
progressions in positi on @ , when the negati�e form
L
•
•
is desired.
11.
The mixture of positive and negative
forms i.n continuity does not change the situation,
but merely reverses the characteristics of voice
leading with regard to positive and negative forms.
For example, C 3 in ;:-.! in the positive system
produces two sustained comwon tones.
In order to
obtain an analogous pattern of vo ice-leading in C- 3 ,
it is necessary to reverse the transfor mation, i.e. ,
to use the : � form in this case.
•
•
•
•
12.
Lesson LXXIV.
II. Symmetric S, y stem.
Diatonic harmony can be best defined as
where chord-structures as well as chorda system
••
progressions derive· from a given s�a�� -
Structural
consti tution of pi tch assemblages, known as chords,
as well as the actual intonation of the sequences
of root-tones, knovm as tonal .cycles, are enti rely
conditioned by the structural constitution of the
scale, which is the
s:Jurce of intonation.
Symmetric harmony is a system of pre
•
selected chord-structures and pre-selected chord
progressions , one indepe ndent from an other.
In the
symmetr ic system of harmony scale is the result, the
conseguence of chords in motion.
The selection of
L
intonati on for structures is independent from the
selection of intonati on for the progressions.
A. Structures of 8(5).
In this course of harmony only such
three-part structures vvill be used, whic� satisfy
•
the definition of "special theory of harmonyn.
The
:ingredients of chord-structures here are limited to
3 and 4 semitones r
Under such limitations only
four forms .of_ 8 (5) are possib le.
It should be
remembered, though, that the number of all possible
•
•
•
three-part structures would amount to 55, which is
the general number of three-unit scales from one axis.
Table of S(5)
s , (5) = 4 + 3, knovm as major triad;
S2 (5) = 3 + 4, kno,m as minor triad;
·s 3 (5) = 4 + 4, known as augme nted triad;
S'f (5) = 3 + 3, known as diminished triad.
"
s,(s)
-J -
Figure XXXI .
-
-i-
-
i:;s
M
.
I
.,,
-
.- �
So long as S(5) will be the only structure
I..
for th e present use, we shall simplify the abov e
•
expressions to the
•
following form:
Whatever th e ch ord-progression may be,
structural constitution of chords appe aring in such
progression may be either constant or variable �
Constant structures will be considere d as monomial
progressions of structures, while the variable
structur es will be considered as binomial, trinomial
and polynomial structural groups.
•
•
•
14.
Monomial forms of S(5)
• •••
••••
••••
••••
Total: 4 forms
Binomial forms of 8_(5)
Sa + S "I
s , + Sa
s , + s'f
,
6 combir1ations, 2 permutations each.
•
Total : 12 forms
•
Trinomial forms of 8(5)
•
S ' + S ' + S3
s,
+
s,
+
s ,..
•
S., + Sa + S a
s , + s,, + s�
12 combinations, 3 permutations each.
Total: 36 forms
•
•
15.
s, +
S2
+ Sa
s, +
S2
+ Sy
S2
+ Sa + S &f
•
•
4 combinations, 6 permutations each.
Total : 24 form s .
The total of all trinomials: 36 + 24 = 60.
S ' + S t + S t + S2
Quadrinomial forros of 8(5).
•
L
12 combinations, 4 permutations each.
Total: 48 forms
•
6 combinations , 6 permutations each.
Total: 36 forms
•
•
16.
s , + S, + S2 + Sa
s , + s , + S 2 + S�
s , + s , + Sa + Sy
s , + S 2 + S 2 + Sa
s , + S2 + S2 + �
S , + Sa + S 3 + S�
s , + S 2 + Sa + Sa
•
s , + s2 + s� + s�
s , + S 3 + S� + 8¥
•
12 combinations, 12 permutations each.
•
•
Total : 144 forms •
•
1 combination, 24 permutations.
Total: 24 forms.
•
The total of all quadrinomials: 48 + 36 + 144 + 24 = 252.
In addition to all these fundamental forms of
the groups of S (5), which represent a 11eutral harmonic
continuity of str�ctures, there are groups with coefficients
of recurrence, which represent a selective harmonic
•
•
17.
continuity of structures.
individual selection.
The latter are subject to
Any rhythmic groups may be used
as coefficients of recurrence.
Examples
(1)
2S, + Sa
(2)
3Sa + S2
(3) 3S ,
+ 2Sa + S2·
(4)
2S 2 + s , + S2 + 2S 1
(6)
3S f + S2 + 2S '
(8 )
2S 1 + S2 + S, + S2 + S, + S2 + 2S 1 + 2Sz + s , · + S2 +I.· S 1 +
(9)
+ S2 + S , + 282
\
4S, + 2s 2 + 2s� + 2s , + s 2 + s� + 2s , + s 2 + s�
(10)
•
B. Symmetric Progressions .
•
Symmetric · zero CYcle (C0 )
•
A group of chords with a common root-tone but
positions and variable structures produces
with variable
•
a symmetric zero cycle (C0 ).
•
•
18.
Such a group may be an independent form
of harmonic continuity, as wel l as a portion of other
symmetric forms of harmonic continuity.
Coefficients of recurrence in the groups
of structures, when used in a continuity of C0 ,
acquire the following meaning: a structure with a
coefficient greater than one changes its positions,
The change of
until the next structure appears.
structure requires the preservation of the position
of the chord.
This can be expressed as a form of
interdependence of structures and their positions
in the C0 :
position var.
S const•
•
S var. ------- position const.
=
s,
+
s,
•
t.
For instance, in a case of 3S, + Sa + 2$ 2 =
+
s,
+ Sa + S 2 + S 2 , the constant and
variable positions appear as
var.
var.
s , + s' + s,
follows:
con st. con st.
+
+
Sa
•
•
Ex.amples of harm onic cont inuity in C 0 •
Figure XXXII •
•
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20.
Lesson LXXV.
Diatonic-Symmetric System of Harmony
(Type II).
•
Diatonic-Symmetric system of harmony
must satisfy the following two requirements:
(1) all root-tones of the diatonic-symmetric
system belong to one scale of the First
Group;
(2) all chord structures must be pre-selected ;
they are not affected by the intona tion of
scale formed by the root-tones.
•
In this system of h armony structur al
groups must be superimposed upon the progressions of
the root-tones belonging to one scale.
This form of
h armony has some advan tages over the Diatonic System
( to which I will refer as Type I).
Like the diatonic
system, the diatonic-symmetric system produces a
united tonality, which is due to the structural unity
of the scale.
Unlike the diatonic system, the
diatonic-symmetric system is not bound to use the
structures wh ich are considered defective in the Equal
Temper ament [ like S � (5) , for example ] , as the
individual struc tures and the structur al groups . are
a matter of free choice.
Unlike the di atonic system, the di atonic-
•
•
L
•
21 .
symmetric system has a greater variety of intonations,
as the pre-selected structures unavoidably introduce
new accidentals (alterations), 1,mich implies a
modulatory character without destroying the unity of
the tonality.
Examples of Harmony TYJ?e ;r.
Figure XXXIII.
(please see following pages)
C
•
•
•
v
.,
,
r
....
I:}
Pitch-scale :
+
-
0
,;::
? 0 �3
!'?
-
,-
.
r,'
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-....
"£
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-..
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I.J
,..
-e- ·- ..,
•
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::::I
:..I
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-.,,
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.,
_..,
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9
....
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�c
r;.;.
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- -.
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ba
... t,J
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+ C3 +
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- � q�
-
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ul
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Structural group: S , . + S 3 + 2S 2
...
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-s,, ...: .- j ·-
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Tonal cycles: 2C 3 + Cr
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- [,...
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- - . .,,
.,
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--...
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.....
1._ _i L..,. �
-
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2c1
Structural group : S , (5) co11st.
-. - -- ....
-e- �
�
,.
....
''
=- -
....
�
,.._
I
0
o�� ,...
Pitch-scale :
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,.,
Q
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Tonal cycles;
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S C H I L L I N G E R
J O S E P H
C o ·u R S E
C O R R E S P O N D E N C E
Subject: Music
With: Dr. Jerome Gross
Lesson LXXVI.
•
Symmetric �ystem of Harmony
0
(Type III)
Symmetric System of harmony must satisfy
the following requirements:
(1) the root-to nes and their progress ions are
the roots of two (i.e. ,ff,, 3,12, "./2, "../2, 1':/2) ,
that is the points of symmetry of an octave.
(2) chord structures are pre-selected .
As a consequence o f motion through symmetric
roots, each voice of harmony produces one of t.he pitchscales of the Third Group.
1.
Symmetric C0 represents one tonic;
•
./2 represents two to nics;
"
three "
'./2
"J2
�
'':/2
"
"
"
four
•
Sl.X
"
"
twelve n
The correspondences of the tonal cycles
and ti1e symmetric roots are as follows:
•
•
•
2.
One ton ic:
C
Two tonics : 0
Co
Cr
C
Four tonics : C
C
Six tonics:
f'
F
,
•
Three tonics: C
•
C
C
C
c-,
E}
C
Alz C
C.3
3
c,
E-
C-3
A
Ci
C-3
C7
C- 7
Twelve tonics: C
C
ER
D
B)?
C7
C-3
c,
A!? ·c
·-3
F it
C
EiP
c,
F,:
C-1 A
C- 1
C '1
C-7
-D�
E
Ai?
C7
C- 7
c,
C-J
C
-c
Ft
V
A
C7
C7
BJz
Ftt
E
D
C- 7
C-7
. D�
E�
C7
C7 E� • • •
C7
B
B�
C
A'
A
C-7
C- 7
C-7
C-7
..
Transformations with regard to positi ons
and voice-leading remain the same as in the diatonic
system.
In case of do ubt cancel all the accidentals.
Two Tonics.
Two tonics break up an octave into two
uniform intervals.
The second to nic (T�) being the
.[§, produces the center of an octave.
makes the t wo-to nic system reversible.
0
This property
All points of
intonation in the � � as well as in the � transformations are identical, i.e�, both the clockwise and the
c,
C-7
C
•
•
3.
I
counterclockwise voice-leading produce the same
pattern of motion.
This is true only in the case of
two tonics.
Two tonics form a continuous system,
i.e., the recurring tonic does not appear in its
original position.
Two tonics produce a triple
recurrence-cycle before the original position: falls
on the
first tonic (T ,) for the �� and the �
Const. 3 produces a closed system.
1.
Figure XXXIV_.
•
�
I,,,
5 1 ' �--. A•
.,
'
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�
' •
�
I.-
•
•
0I
,1
i
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,,
,
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ff
•
.,,
T1
,,
'
.�
'
�
�
�
••
,�
•
.
'-
�
-e
--
<
I
-·- ----
------
The upper voice of harmony produces the
following scale: c
.__.- d'- e -
r"
L=
g - a - (c) =
�
•
4.
= (1+3 ) + 2 + (1+3 ) + 2.
All other vo ices of the
above progression pr oduce the same scale starting
from its different phases.
I t is easy to see that this scale belongs
, to the Thir d Gro up and is constructed on two tonics.
By selecting other structures and
structural groups of 8(5) one can get some other
scales of the Third Group.
For example, the use of S 2 cor1st. produces
the follov1ing scale: c - d'7 - e � - fM'- g - a - (c) =
= (1+ 2) + 3 + (1+ 2) + 3 .
Structural gro ups may be used in two ways:
(1) S changes with each t o nic;
( 2) the groups of S produce C0 on eact1 tonic.
Illustrations of the first method
Figure
-XX.XV •
-----------� -----------------I
•
.,
ff!i:.'
·�
.
..p
At .:e-
-
:i!
,_
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=e
�
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-s
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I-
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.
fi
.
•
.I
..
••
-�
•
•
5•
Illustrations of the second met hod
•
Figure XXXVI.
•
Combinations of the preceding two
methods with regard to the structural selection for
· each tonic of one symmetric system are applicable t o
all symmetric systems •
•
•
6.
Example
f�gtn' e ;xxxvrr .
•
•
Longer progressions can be obtained
through the use of longer structural groups, such
as rhythmic resultants, power-groups, series of
growth, etc.
In some cases the number of terms in the
structural group produces interference against the
number of tonics in the symmetric system.
Example
'
•
7.
Three Tonics.
Three tonics produce a closed system
for ; � and 1:
"'
., , and a continuous sy stem (two
recurrence-cycles) for const. 3 .
Figure XXXVIII.
•
••
•
Four Tonics.
..
Four tonics produce a continuous system
(three recurrence-cycles) for
closed system for const. 3 .
..-=;
,:.,,
and
(please see next page)
•
le ,
.�
, and a
•
....
s , const •
,
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Const.
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3
,
.f
/
-- -�
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-
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1-
i
Six Tonic s .
s•
Figure XXXIX.
•
•
•
I rI
r,1 <�
-
�
•
r..
r-
Six tonics produce a closed system for
as well as for the const. 3.
f;lg11tp
S , const.
-�
,c:,...
I:; -
and _ � ,
x;r...
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Twelve Tonics.
l:. Twelve tonics produce a closed system for ,�
__ and ...__:::,
as well as for the const. 3 .
Figure, X}:,I.
s , const.
�
�-
t-,....., <
••
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No. 230 Loose Leaf 12 Stave Style - Standard Punch
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•
•
10 .
I
Lesson LXXVII.
Variable Doubl ings
•
Harmony, 1n many cases conceived as an
accompaniment, may be given a self-suf ficient
character by means of variab.le doublings.
This
device attributes to �hord progressions a greater
versatility of sonority and voice-leading than the
one usually observed.
Variable doublings comprise the three
functions of S (5) .
Thus · the root, the third or the
The corresponding notation
fi ft h can be doubled.
to be used is: 8(5)© , 8(5)@ and 8(5) ©
.
As the root-tone remains in the bass,
S ( 5 ) u) is the only case of doubling where all three
functions (1, 3, 5) appear in the upper three parts.
The followi ng represents a comparative
table of functi ons in the three upper parts under
. various forms of doubling.
8 (5)
©
=
1, 3, 5
8 (5)© � 3, 3, 5
S (5)©
=
3 , 5, 5 •
Figure XLII.
.
V
In cases S (5 ) @ and 8 (5) @ only three
positions are possible for each case.
Black notes
•
•
•
11.
represent variants where unison is substituted for
an octave.
,
Positions
Jf
I)
&I
�-
..
-
r1
-I
••
-
*'
..
,
I
I
•
•
--
••
•
j
---
---
·-
,.
•
,I
•
TI
•
•
••
I,
Fi&!!_re •XLIII .
Tr.ansf ormati ans
8(5)©
5 ( .- 5
3�
)3
c� �
8 (5 )
@
5( ...): 3
5( � 3
3( � 5
3 �e t3
lf > 3
1 ( :)3
ti -
::,;.
A-
••
-
•
.,
,i
�
..
,l, �
l < >5
.
c s- �
t7 �
_
_
____
__ __,_ - - -
. ,,..
. , ._
'
•
•
12.
S (5)
(!)
<
➔
8(5)@
5( ) 3
3(
3 f�� 3
,�
•
•
>5
8 (5 )
(i)
<
>3
5' )5
5f
3 ( .> 5
3( ) 5
1( >i 3
1(
i�
e1 �
-
cf��-
e 3 �-
)5
>
S (5) {j)
5(
�5
3( ➔ 3
1( ) 5
e,,
e�
•
•
•
s (5 )
•
+ ---�> s { 5 ) ©
ev...-
3( > 5
•
•
•
•
•
•
( -------t) 8 ( 5 ) (l)
8 (5 ) @�
3(
➔6
3 !-} 3
..
•
When r eading these tables, consider
identical directions of the arrows for the sequen�e
of structures and for the c orrespondi ng transforma
tions.
Notice that there always are three
transformations when S(5) © participates and only one
when it does not .
•
•
•
14 .
Musical tables in the above Figure
are devised fro m the initial chord being in the
same position.
frow all
Similar tables can be• constructed
positions as well as in rev�:rse sequence
and also in the cycles of the negative form.
Variable doublings are subject to
distributive arrangement and can be superimposed on
any desirable cycle-gro up.
Figure XLIV.
Example : 2C.3 + C , + C 7 ;
8 (5)© + 28(5)
(1)
Ir = 8 (5)0 + C 3 + 8 (5)
+
C5
®
+ 8 (5) �
+ C 3 + 8 (5) (V +
+ S(5)(i) -. C ? + 8(5) (D •
(S)
•
•
Example: 2C� + C 3 + C, + 2C 7 ;
+ 8 (5) @ + 8 (5) © .
Ir = 8 (5)
@
Ci)
8 (5)
+
0
8 (5)
+
+ c, + �(5)(D + c, + S (5)
@
+
+ c, + 8 (5)@ + c, + 8(5)© + c ,, + 8 (5) 0 +
+ c 7 + s (5)@ + c, + 8(5)© + er + 8 (5)@ +
+ C3 + 8 (5)0 + c, + 8 (5)© + C '7 + S (5)@ +
+ c, + 8 (5)�
•
•
15.
I
I
I
I
I
Variable doublings are applicable to all
types of harmonic progressions , thus including types
II
and III .
Figure XLV •
•
Type II (diatonic-symmetric) .
Ir
as in the preceding example.
:: 2S2 + Sa + s ,
I
I
I
I
•
Figure XLVI,.
Type III (symmetric ) .
©+
(i) + T S,Q + T� S
- - T, S1(i)+ T�S�
� (6T) 3
3
+
TsS at + T. S1G) + T, S -a,(j).
©
•
•
•
•
16 •
•
•
•
•
•
.
•
•
..
•
17.
Lesson LXXVIII,
Inversions of 8(5)
The usual tech nique of i nversions,
•
The
strictly speaking, is unnecessary to a co mposer.
reason for this is, that by vertical per mutations of
the pos itions of parts i n a ny harmonic continuity of
8(5) , the inversions appear automati cally, as inner
or upper parts beco me the bass parts u nder such
conditions.
This teclmique was fully described in my
"Geometrical Pro jections of Mllsic", in the branch
dealing with the co ntinuity of geometrical i nversions.
For a n analyst or a teacher, however, a
thorough systematization 0£ the classical te�hnique
of inversi ons is a necessity.
There is no other
branch of harmony I know of, where- confusion is great�r
and t he information less reliab le •
The first inversion of 8(5) is know n as a
"sixth-chord" or a "third-sixth-chord" a nd is
expressed i n th is notati on by the symbol 8 (6).
The
only condition under which 8 (5) becomes an 8 (6) is
when the th ird (3 ) appears in the bass.
The'
positions
•
of the upper voices are not affected by suc h a cha nge,
. th e foFms of do ublings -- are..
Which do ublings are
appropriate in each case, will be discussed later.
Assuming that �ny S(6) may be eit her 8 (6) ©, or 8 (6) @ ,
or 8(6) @ , we obtai n the fo llowi ng Tab le of Positions:
•
•
18.
u
-
1.
�
,
--
.
-
-
•
,_
V.
1
I
:.;
,Ir
..
-
-
.,.. .,.
,,i
r�
,
•
Figure XLVII.
.
,.
-
�
�
·�
••
•
�
s
.
'I
i
�
-
i!'
$
r•
�
. It is easy to memorize the above table,
as $(6 ) © and 8 (6 )© positi ons are systematized
through the followi ng cliaracteristics: (1) the doubled
function appears above the remaining function; (2)
the doub led fu nc tion surrounds the remaining function;
(3 ) the doubled fu nction appears below the remaining
function.
8 (6 )@ is identical with 8 (5) positions,
except for the bass having constant 3 •
. Harmonic progressions (Ir ) consisting of
,
8 (5) and 8 (6 ) are based on the followi ng combinations
by two:
,,
-
•
•
19.
(1) 8(5)
(4 ) s (6)
) 8 (5);
, s (s ) .
( 2) 8 (5)
> S (6) ;
(3) S (6)
J S (5) ;
As the first case is covered by the
previous technique, we are concerned, for the present,
with the last three cases.
All the following transformations, being
applied to vo ice-leading, are reversible, as in the
case of Variable Doublings of 8 (5) .
always measured thro ugh root-tones•
•
Figure XLVIII .
S(5) �-----4
•
5( � 5
3 • ll
l• ) 1
• es-
8(5)
3• -. 5
1.- � 1
S (6) ©
5 � ►l
5�1
l< > 5
•
S (6) (l)
5E�l
5< �5
5• )5
lt·) 5
1� ) 5
lf •l
3• )5
•
5< ll
Tonal cycles are
3� ) 1
3( ) 5
•
20 •
•
5 (5 )
8(6) ®
5'(" � 3
5� )l
3.-• , l
3 fl ) 3
lf � 5
1.- ) 5
Const .,. 3
Const- 3 � =
C,5'
•
•
•
21 •
•
,,,
'
-
�
J
-
+
,�
-
... ..,.
.
,.
•
·�
rI
,
,I
,,I
t.
8(6-fD
5(
•
l"
1�
e�
.., 1
S(6) @
)5
�s--
e, S'
e,
•
,
S (6)(q _____,.;,...___
> 8 (6)©
5 ( )1
5( ) 5
1( ) 5
•
22 •
•
©
S(6)
5 f-) 5
.( -----�
� S (6) (D
5f-�l
5(""3
l'( ) 3
1( >t 5
1� > 5
l" ) 3
lf' > 1
�s
�.----.----.----,--o---:_r====:Jc:::;;====
c�
S (6) (D
S(6) (D
5� 1
5( )Z
3 < )1
3� ) 5
l< ) 5
l'f ·) 3
➔ ,. .;
F ....
4
..._,
I
,
•
•
-
,j
j -
�
,
-
r.
-
=-
_;:;,,
-
'I
__,
_r
�
I
a
�
;:
-
�
�
�
•
V
�
8
�I
$
�
•
@
8(6) (
)5
5( =. 1
5( ) 5
5( > 5
5 t'
l< > l
l< ) 3
)
23.
©
S (6)
5< ) 3
5( ) 1
1( � 5
0
•
.Any variants conformed to identical
transformations (like the black notes in some of
the preceding tables) are as acceptable as the
ones in the tables.
•
•
•
24 •
•
, v
Lesson LXXIX.
Doublings of 8(6)
rapidly.
Musical
habits are formed comparatively
Once they assume a form of natural
reactions, they infl uence us more tha n the purely
acoustical factors .
This is particularly true in
the case of doublings of 5(6 ) .
The mere fact that
identical doublings in the different musical contexts
affect us in a different way, sh ows that our auditory
reactions i n music are not natural but conditioned.
The principles offered here are based on
a comparative study of the respective forms of music.
There are two technical factors affecti ng
•
the doubling i n an 8 (6 ) :
(1) the structure of the ch ord;
( 2) the degree of the scale (on which the
chord is co nstructed) .
These two influences are ever-present
regardless of the type to which the respective
harmonic continuity belong s.,
Yet, while in harmonic progressions of
type II and III the structure of the chord is the
most infl uential factor, i n the diatonic progressio ns
(type I) it is exactly the reverse.
The influence of
a constant pitch-scale is so overwhelming, that each
•
•
25.
chord becomes associated with its definite position
in the scale.
Thus, one chord begins to sound to
us as a do minant and another
or a leading tone.
as a tonic, a mediant
This hierarchy of importance of
the various chords calls for the different forms of
doubling, particularly when the respective cl1ords
appear in the different inversions.
The fo1-lowing is most practical for use
in diatonic progressions •
•
Figure XLIX.
Stro ng Factor
The degree
of the scale
I,
IV, V, VI
Regular
Doubling
CD , ©
Irreg.
Doubling
Weak Factor
The structure
of the chord
s, (6)
Regular
Doubling
G) , @
II, III, VII
S 14 (6)
Regular doublings are statistically pre
dominant.
Irregular doublings, in most cases are the
result of melodic tendencies.
In reading the above table, give preference
to the strong factor, except in the case of S 3 (6) and
6 � (6) .
It is customary to believe that an s, (6) must
have doubled root or fifth.
But in reality it seldom
happens when such a cl1ord belongs to II, III or VII.
Irreg .
Doubling
@
•
26.
Naturally, all our habits with regard to doublings
are formed o n more customary maj or and minor scales.
The above table will work perfectly when applied
to such scales.
There will be no discrepancy when
8 3 (6) and S� ( 6) will be compared with the data on
the left side of the table, as such structures do
•
not occur o n the main degrees of the usual scales •
When using less familiar scales, one or another type
of doubling will not make as much difference.
•
in such oases
the structure may become a more
Yet
influential factor, though the sequence - is diatonic.
In the types II and III the most practi cal
u
forms of doublings are:
Structure
Figure L •
Regular
Doubling
s , (6)
CD,®
S 2 (6)
(D, @
8 3 (6)
@
(y , @ , @
s.. ( 6)
Irregular
Doubli ng
@
@
Continuitz of 8(5) and 8(6)
�he comparative characteristic of S (5)
is its stability, due to the presen�e of the
root-tone i n the bass.
The absence of the root-tone
in the bass of S (6) deprives this structure of such
stability.
•
•
•
27.
Composition of continuity consisting of
S (5) and S ( 6 ) results in an interplay of stable and
unstable units or groups.
The following fundamental
forms of co nti nuity with ut ilization of the above
mentioned structures are possible :
(1)
( 2)
(3)
( 4)
8 (5) const.- --- stable
8 (6 ) const. ---- unstable
( 8 (5) + S (6 ) ] + . . .
alternate
2S(5) + 8 (6 ) + S (5) + 2S (6 )
3S(5) + S (6 ) + 2S(5) + 28(6) + S ( 5) + 38(6)
48 ( 5) + S (6 ) + 38(5) + 28(6) + 2S(5) + 3S (6) +
+ S (5) + 48 ( 6 )
•
•
increasing stability
increasing instability
(5) 4S(5) + 28(6) + 28(5) + S { 6 )
---� proportionately decreasing ratios
proportionately increasi ng ratios
1-------
(6) 8 ( 5) + 28(6) + 38(5) + 5S (6) + 85 (5) + 138 (6)
progressive over-balanci ng .pf unstable
e,lements
S ( 6) • + 28 (5) + 3S ( 6) + 5S ( 5) + 8S ( 6) + 13S ( 5)
progressive over-balancing of stable
elements
Many other forms of distribution of 8 (5)
V
and S(6) may be devised o n the basis of t he "Theory
of Rhythm " .
•
•
Examples of Progressions
28.
Figure LI.
Diatenie
S ( 6) Const . ; 2C '1 + 2C !' + Ca + Cs�
t
'!,
•
l.
•
Figure LII.
Diatonic-�nnmetri�
a •
•
-
2C S' + C"1 + C '° + 2C 7
2S 2.. ( 6) + S 1 ( 6) + S 3 ( 6) + S 1 ( 6) + S" ( 6) ;
,:
I� I
,,
r
l=f -
"
.--
•
-
,
,.,.
I:,J
,�
,�
Ii)
/.i'I
I
-....
-�
•
/.i'i
I�
I
-
/?)
(.i)
-
' �
•
✓
,�
-:,.,,
'-�
•
§YHH:!!!#tric
+ S: ( 6) + S Ji ( 6) + 2S 1 ( 6) ; Six tonics
1�==i�t=
::i
3
•
)
•.,..,
-aCl
:��
�0 No. 230 Loose Leaf 12 Stave Style - Standard Punch
...,..
,
•
29.
Figure LIII.
D!2tonic
38(6) + 8 (5) + 28(6) + 2S(5) + S(6) + 3S(5) ;
2C 1 + C 1 •
••
lo
Diatonic-Symmetric
+ s , (6) + 2S 2 (5) ; 2·c· -r + c,; Scale- of roots: Aeolian
..
I
Symmetric
+ s2(6)] T, + [sq(6) + S,(5)] T2} + . . .
== ...
·o�...a
u:si!-=
BRA,.._.D
No. 230 Loose Leaf 12 Stave Style - Standard Punch
Fo111·
Tonic�.
•
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
With: Dr. Jero me Gro ss
Less9n LXXX.
C O U R S E
Subject.: Mu�ic
Groups with, fa ssip.g Ch9rds
A. Passing Sixth-chprd;;
A group with a passing S (6 ) is a pre-.set
c ombinati m of three chords, namely: 8 (5) + 8 ( 6 ) +
•
+ S (5) .
Every papsi ng c�prd occupies. the_�enter of
its group, appears, on a �eak beat and has � doubled
bass.
•
The complete expression for a group (G) with
passing sixth-chord is:
G = 8(5) + 8 ( 6 )@ + 8 ( 5) .
6
This formula is not reversible in actual intonation.
between the extreme chord,s of G6 is
This relationship reruain� constant in all cases
The relati onship
C-5.
of classical music ..
all cycles.
We - shall extend this principle to
Under such conditions G6 r�tains the
following characteristics:
(1) The transf'orma tion between the extreme 'chords
of the group is always c�ockw�� e for both the
positive and. the negative cycles
(2) The . bass progression is: 1-�> 3 �> l, which
necessitates
the first condition.
•
•
2.
In the classical form of Gs, bass moves
by the thirds. Thus, 3 in the bass under S (S) is a
third above its preceding position under the rirst
S (5), and a third below its following position under
the last S(5) .
•
In order t.o obtain Gs , it is necessary to
connect S (5) with the next S (5) through C-5
and
add the intermediate third of the first chord in the
bass, without moving the remaini ng voices.
G
Figure LIV.
q
I
= 8 (5) + S (6) 3 + S(5)
1::
G
•
There are three melodic for ms for the
bass movement.
Figure LV.
0
•
•
•
•
,
•
3.
Combinations of these three forms in
sequence produce a very flexible bass part and,
being repeated with one G6 , make expressive cadences
of Mozartian flavor .
fig:ure LVI •
•
•
•
•
Continuity of G6 •
Continuity of such groups can be obtained
by connecting them through the tonal cycles.
•
while C8
Connecting by c5 closes the sequence,
and c produce a progression of 70 •
7
6
Figure J..,Y.I.I_.
(please see next page)
•
•
•
• '
•
•
•
�
l
C
,I
ILI
•
It=
•
•'
'.
••
,,
4•
,
.
�
.
,
•
'
C1
.. . . - - -
-·-
Further versatility of G6 progressions can
be achieved by varying the cycles betv,een the groups.
Any time a decisive cadence is desirable C5 must be
introduced, as this cycle closes the progression.
u
•
5.
Figure LVIII.
P
+
•
= G6 + c7 + G6 + c 7 + Gs + c3 + Gs + C 7 + Gs +
C3 + Gs + C3 + Gs + C5
c,
c,
•
Generalization of G6
•
In addition to . the classical form of
G6 ,
other forms can be developed through the use of other
Of course, each cycle
than C-5 cycles within the gro up.
produces its own eharacteristic bass pattern.
Figure LIX.
Various forms of Gs:
,�� (c.� )
�I ·
u
�
r,
r�
...
•
�b (t,$ )
.,.
"7
�
,
-
�L, (e,
r1
-
-...
) -
2
�lo (�-3)
•
..
r
'
7
4£.( t-
�' (e-s)
i
...
F
-
', ,.
$
fil
-
$8
(If)
•
6.
u
The respective variations of the bass
pattern will be as
follows:
•
Figure LX.
•
,
-
•
7
r
,
'
Q
�1.. ( � - 5)
--
•
�
4lr> ( � -1)
����������1��!������������3�;���
G
•
Continuity of the generalized G6
Such a continuity can be developed through
the selective progressions of the various forms of G6
combined with the various cycle connections between
the groups.
8·
•
7•
•
Example:
figure LXI.
� = G6 (C-5 ) + C7 + G6 (C3) + c5 + G(C-7) + c3 +
+ G(C-5) + C5 + G(C7 ) + C5
I.
•
Generalization_ of �pe Passing Th�rd
It follows fro m · the technique of gro ups
•
•
•
with a passing sixth-chord, that the first two chords,
i.e. , S (5) and 8 (6)@ , belong to C0 , and that as the
position of the three upp er parts does not change until
the last chord of the gro up appears.
This last chord ,
8 (5) , can be 1n any r elation but C0 with the preceding
chord.
If we think about the appearance of the thi rd
in the bass dur ing 8 (6)'.V merely as a passing third, it
is easy to see that this entire technique can be
general ized.
providing the
The passing 3 can be used after any 8 (5) ,
transformation betwe en the latter and
•
8.
the following S {5) is clockwise for all the cycles.
Such a device can be applied to any progressions of
root-tones in the bass.
S (5) w1 th the
Figure LXI):,
Ex.ample :
•
•
I
,
'd
,
r
tI
-· '
-
.
r�
I,
•
l!Si •
,
�
I
-
I
r:;I
C
c
I
•
'
�
I
•
•
•
,
i
I
L
[
]
r
•
.
J
�
I
r
.I
•
-•
•
The effect of such harmonic continuity
is one of overlapping gro ups of G6 , as marked in the
above Figure.
u
'
-
'
-.
•
- ---- ------
-- . ··-
�
•
-i-
•
:j
-i-
-'t
(ii
--
,.
U·
•
,
•
Lesso n LXXXI,
Applications of Gs to Diatonic-Symmetric
•
(Type II) and Symmetric 1Typ e JII)
Progressions,
The use of structures of S(5) and 8 (6)@
in the groups with a passing sixth-chord must satisfy
the following requireme nt : �he a�ja�en� 8(5) and S(6)(j)
of one group, must hav� identical structures •
This re,quirement does not affect the form
•
.
of the last S(5) of a group; neither does it influence
the selecti on of the forms of S (5) in the� adjacent
groups.
V
•
As each G6 consists of three places, two
of which are identical, the number of structural
combinations for the individual groups equals 42 = 16.
S1 + Sl
S2 + S1
81 + S 3
S 2 + S3
S1 + 8 2
S1 + S4
S2 + S2
S 2 + 84 .
S3 + S 1
S4 + Sl
S3 + S3
S4 + S 3
S3 + S 2
83 + S4
S4 + S 2
S4 + S4
Thus we obtain 16 forms of G6 with the
following distribution of structural co mbinations •
•
•
•
10.
G = S1 (5) + sl (6)® + 81 (5)
6
G6
S1 (5) + s1 (s)® + s 2 (5)
=
Ga = s1 (5) + S 1 (6)® + s3 (5)
G6 = 81 (5) + s1 (6)® + s4 (5)
G6
=
s2 (5) + s 2 (6)® + s 1 (5)
G6 = 82 (5) + s2 (6)@ + s 2 (5)
•
G
6
=
s 2 (5) + s 2 (6)@ + s3 (5)
G = s (5) + s (6) 2
2
6
u
r
Ga = s3 (5) + s3 ( a )@ + s 1 (5)
Ga = s3 (5) + s 3 (6) + ·S 2 (5)
Ga = s3 (5) + s 3 ( a)@ + s3 (5)
G = s3 (5) + s3 ( a)� + 84 (5)
6
G6 = s4 (e,) + s4 (6)® + 81 (5)
Gs
=
S4 (5) + S4 (6)® + S 2 (5)
G6 = 84 (5) + 84 (6)@ + 83 (5)
G6
•
u
=
84 (5) + 84 (6)® + s4 (5)
As the melodic interval in the bass, while
moving from the root (1) in 8 (5) to the third ( 3 ) i n
•
11.
S(6 )@ is identi cal for the forms s 1 and s 3, as
well as s2 and s4, the total qUanti ty of intonations
in . the bass part for one type of 0 is ½ = 2 .
6
81 + 8 1
S1 + 82
S2 + S 1
82
+
62
As each intonation has 3 melodic forms
•
and there are two different intonations, the total
number of intonations combined with melodic forms
in the bass part is 2 x 3 = 6 .
•
,_
-.
,�
-s
-G
Progressions 9f. �e ty-pe
t.
-
IJ.
Figure LXIII.
Example:
Forms of S: s 2 (5) + s 2 (6)® + 81 (5)
r = G6 (C-5)
+
c3
+ G6 (C-5) + C7 + G6 (C-5) + C5 •
•
fl .,,.
-
•
12.
Example :
Forms of S : [s1 (5) + s1 (6)® + s2 (5) ] + [S3(5) +
+ S3(6:® t S2 (5) ]
� = as in the preceding example •
•
•
•
u
•
Example :
Forms of S :
s2 (5)
+
s2 (6)@ + s2 (5)
r = as in Figure LXI.
-�
II
-
" -.
�
!'
,_
�
'
-
�·
I, �
- ·-- ·
_..
•
.
•
••
....
I
I
r
�
'
�
�
,
+-
j,,i t"t
'
JI
I
•
I
I
-
,
l3l
Ir
-(l
+
',
•
••
.
•
-
..�
""
�I
•
-
•
..
•
13 •
•
Generalization of the passing third 1s
applicable to this type of harmonic progressions as
The following is an application of the
well.
structural group
Figure LXII.
2s 1
+
s2
+
2s1
+
s2
2s1
+
to the
Figure J;XJ.V.
•
,.
•
'-
::.--�
,,
r
-,.j!
�
fil
�f
r,
I
pi
l�
I
-
r-•
i
L
,.
••
'
•
•
•
•
u
•
14.
• Lesson LXXXII.
Progressi�ns of_ the type IJI.
Applications of G6 to symmetrical systems
of tonics disclose many u nexplored possibilities,
among which the two-tonic S)'stem deserves a particular
As intervals formi ng the two tonics
are
•
attention.
equidistant, the passing tones of S ( 6 'fiJ, which i n
turn may also be equidistant from T 1 and T2 , thus
produce, in the bass movement, diminished seventh
•
chords in symmetric harmonization, a device heretofore
unknown.
•
The justification for the use of G6 in the
symmetrical systems of tonics is based on the following
deduction from the original classical form, i.e.,
•
--- --
----
(Symmetric)
(Diatonic)
•
The abovementioned equidistancy of the two
tonics permits t o obtain
r
= 3G6 until the cycle
•
15.
•
Selecting
closes.
•
,.
-y
�
II
�
�
-
•
•
�
,I
I
""
.
-t:
�
Ii,
I
•
J
s1
for the entire G6 , we obtain:
Figure LXV.
1�
I
•
$,_
Ia:
7'7
,7
t;
.,
,�
I
r
I
•
l
.,
I
.·•
h
r•
t.
u
The overlapping of groups, indicated by
•
the brackets in the above Figure, is an invariant of
the symmetrical systems.
Thus, the passing third can
be considered a general device for progressions of
the type
•
III.
The number of bass patterns for the
cycle of the two tor1ics equals: 22 = 4.
The number of intonations in each cycle
of the two tonics equals : 22
:;::
4.
The latter is due
to the use of the different forms of S (5) .
The
interval between 1 and 3 equals 4 and is identical for
The interval between l and 3 equals
It
•
•
3 and is identical for s2 (5) and s4 (5) .
Thus, by
•
•
16.
distributing the different structures through two
tonics, we obtain the following co mbinations :
81 (T 1) + S 1 ( T2)
Sl (Tl ) + S 3 (T 2 )
�3 (T 1) + 81 (T 2)
•
S2 (Tl ) + S4 (T2 )
S4 (T l) + S 2 ( T 2)
•
•
identical intonations
in the bass part
S3 (Tl ) + S 3 (T 2 )
S 2 (Tl ) + S2 (T 2)
•
.-
- identical intonations
i n the bass B_art
S4 (T l) + S4 (T 2)
61 (T l) + S 2 (T 2)
Sl (Tl) + S4 (T2 )
S 3 (T l) + S 2 (T 2)
ident ica l intonations
in the bass part
S 3 (T l) + S4 (T 2)
S2 (T 1) + S1 (T2 )
S2 (Tl ) + S3 ( T2 )
S4 (T l) + S l (T l)
S4 (Tl) + S 3 (T'2 )
•
identical intonations
in the bass part
•
\
•
17 .
The following is a table of intonations
and melodic forms in. the bass part on two tonics.
Total : 4 2 = 16.
Figure LXVI•.
•
s,.
•
.r
54-
I.,
L
The above combinati ons can be incorpor
. ated
into a versatile continuity of 06 on two tonics.
'
•
•
18 •
•
LXVII.
Fig11£e
-.
Example :
•
.)
�
,
--
�
,.
,
•
d,
'.
�
�
,=
�
-
:p
I
•
�
]
-,
-
tfi
•
I.
produces
V
Application of G to three t9pic§
6
8
melodic
forms in
the
bass par t : 2 3 = a .
Fi&1re LXVII I .
a
a
•
,
•
19.
Figure LXVIII (cont . )
.r
•
I ••
'I
�1 --
:
-
-
•�
-
�
---------
-
•
different S
A
,�
•
----.
•
,
.-
'
-
-----------------,.,-----
The number of distributions of the
through thr ee tonics is 43 = 64, while
the number of non-identical intonati ons is 23 = a.
.
Non-identical intonations:
S1(T1 ) + s1 (T2) + s1 (T a )
Sl ( T1) + Sl (T...� ) + S 2 ( T3 )
S l( Tl) + S 2 ( Tz ) + S l (T a )
Sa ( T ) + S ( T2 ) + S (Ta �
1
1
1
(j
.
•
-
-
•
u
· 1 (T a)
Sa ( Tl ) + S2 (T2 ) + S
•
Sa (T l) + Sl (T 2 ) + S2 {T 3 )
Sl (Tl) + S\( T2 ) + S2 (T 3 )
S2 (Tl) + S2 (T 3 ) + S2 (T 3 )
The total number of different intonations
and melodic forms in the bass part is 8 2 = 64 •
Examples of continuity of G6 on three tonics
•
Figure LXIX,
u
?
I
•
•
-' .,,
•
,.
.,, )
I
�·
1
-
-�
•
,,
., ::,.
J
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,,
•
'I•
'I I
" •I
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·--
1I
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--
s,
--
-Ir
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V
I
�-
V
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• .-�
I
•I
.
�"
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ifi:
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•
•
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n
--
S:i..
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�
•
"' -
•I
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/}
•
•
•
•
•
-+-
,I
I
�7
•
I�
••
,
•
Application of 06 to four tonics
If
produces 2 = 16 melodic forms in the bass part .
21.
The number of distributions of the four
forms of S through four tonics produces 4� = 256
intonations�
The number of intonation$ in the bass
part is limited to 2'4 = 16.
Thus the total number of intonations and
melodic forms in the bass part is 16 2
•
Examples of continuity of G6
on four tonics.
•
Figure LXX.
(ple.ase see next page)
•
•
,
•
0
= 256 •
•
22.
Figure LXX,
� (J
L
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l ,.
I
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1
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�·
.
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1f
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l
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Application of G6 to six tonics produces
= 64 melodic forms in the bass part.
The number of distributions of the four
°
4
forms of S through four tonics produces
intonations.
2
4
= 64.
= 409"6
The number of intonati ons in the bass part is
The total number of intonations and melodic
forms in the bass part is 64 2 = 4096.
Examples of continuity of Ga on s·ix tonics.
¥1gur� LXXI •
•
L
S1 �-
I
S4
I
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0
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24.
1
produces 2 �
Applicati on of G6 to twelve tonics
=
4096 melodic forms i n the bass part.
lhe number of distributions of the four
forms of S through four tonics produces 4
It.
•
=
16,777, 216 •
The number of intonations in the bass part
is 2 1 � = 4096.
The total nu mber of intonations and
melodic forms in the bass part is 40962 = 16,777, 216.
Examples of conti nuity of G6 on twelve tonics.
'Figµre
LXXII.
(please see next page)
•
•
•
u
•
25.
s , const .,
,�
I
•
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�
-
1
i..
"'
�
� "'
�·�
:n-
�
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•
-�
•
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u I'
•
'-
,
n-
S
0/I
�
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•
1•
-,,
118
'
cons t .
I ,,
1�'7
..�
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1;;•
:a
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--
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==
==
(>========== =========== ===================
•
I
26.
Lesson LXXXIII.
B. Passing Fourth-Sixth Chords: S (4) .
of S (5) is a fourthThe second inversion
•
This name derives from the old
sixth chord: s(!).
basso continuo or generalbass,
where intervals
were measured from the bass.
7
•
-
-,�
�
I-I
8 ( !) has a fifth (5) in the bass, while� t he three
u
upper parts have the six usual arrangements�
•
The use of s(:) in classical music is
a very peculiar one.
This chord appears only in
definite pre-set combinatiODS r
group with a p?ssing
One of theru is the
fourth��ixth chord : G�.
As in the case of G 6, the passing chord
itself' appears on a weak beat, being surrounded by
the two other chords, and has a doubled fifth: s6
4 ®•
The two other chords of G: are : 8 (5) and 8 (6).
The
latter can have two forms o f cbubling (regardless of
the chord-structure): S (6) 0 an� 8 (6)® .
The group v,ith a passing fourth-sixth
0
chord, contrary to G , is reversible.
6
•
27.
u
,
This property being COD)bined with the
•
choice of two possible doublings produces four
••
variants.
•
+ 8 (6)
CD
•
•
l
•
The arrows in the above formulae specify
•
the directions of the bass pattern which is always
scalewise, and therefore can be either ascending or
descending.
The bass pattern is developed on three
adjacent pitch-units, which correspond to the three
•
a
chords of
9:
•
C)
�
C
l
8 (5)
-
*;,
5
3
......
s(!)
\\
8 (6)
Arabic numberals :represent the respective chordal
functions.
,
•
-
28.
Transformations between 8 (5) and s (:)
in the G4 : as the bass moves from 1 to 5, when read
in upward motion, the three upper voices must move
clockwise, in order to get the transformation of 1
into 3.
n
'
.,
,
•
....,
-
•
$
rJ
•
l.
G·
•
The transition from s (:) into 8 (6)
©
or S (6){f) follows the forms of transformations,
where two identical functions participate, as in the
cases of S ( 5) <
) S (6)
©
and S (5) (
> S ( 6) ©
However, classical technique adopted
.
definite routines concerning this transition:
•
(1) one part must carry out a melodic form
reci;proc,!11_ to the bass (i.e., position �
of the bass melody);
(2) it is ·th is reciprocal part that deviates
from its path in order to supply the
(j
d9Jl,bling
of the fifth in an 8 (6).
'
•
•
29.
f
'-'
Under such conditions G! acquires the
followi ng appearances:
•
"
I
'
•
.,.�
-
.
'-- .J
•
.
...._
,...,
...,
,
,�
r
-- -
--c:::,.
0
•.
,
-
--
-
-
"""!
_,.,,,,.
-s.
-t.
The following sequence of operations
•
is recommended :
(1) bass
(2) part reciprocating the bass
•
(3) coxumon tone
(4) part supplying the thir d for s ( !)
The relations between the chords of G4
6
are as follows:
Co
'
l,
8(5) . + C-5 + s (!) + c 5 + S (6 )
•
Co
+
8( 6) + C-5 + $ ( 4
6)
•
30.
Each group can be carried out in 6
positions which· depend on t he starting position.
a!
forms of
•
l
...
-
-
'�
�
0
'..
,I
•
0
--
I
-
The following is the table of all four
in one position.
Figure
LXXIII.
'
-
..
,
::,.
0
-
-
-
�
�
�
I•
.
,
:..,
-
,
,.;
�
.'
,
-
-
•
0
-
.
,
6 can be
The dif ferent forms o f G4
connected by means of tonal cycles and their
coefficients of recurrence can be specified.
It is desirable to make the following
tables:
c7
c3
(2)
const. ;
"
"
lf
"
"
n
(3)
const . ;
"
"
11
"
"
n
const. ;
ff
n
"
,,
"
"
(4)
0
c5
const.;
(1)
6
G4
!
©
-
const.,
const.,
const.
•
31.
•
(5)
•
(6 )
(7)
(8)
(9)
•
G4
rr ©
a!!©
6
T
G
4
ati
©
(0
6T 0
G4
COilSt . ;
c+ = C3 + C5 + C7
const.; c-t = C3 + C5 + C7
const .; cot = C
3
COils t . ;
:!©
+ G
=
c""
+ G�
T
C
3
(f)
+ c
+
C
"
"
"
"
"
"
TT
"
"
n
"
(11)
(12)
5
6
G
+
"
(10)
+ c
7
5
+
0
C
7
,
,•
C3
c
5
COilSt •
const ..
,• C7 const .
•' � =
C3 + C5 + C7
� is the symbol of a group of cycles (cycle
COiltinuity ) .
Continuity of G64, when connected through
a constant tonal cycle, consists of seven cycles:
� =
7C.
e
Figure LXXIV,
Example : G4 r
•
.,,
..,
-e
-s,-..
'
I
•
-
-
0
-
•
•
V
..
...,,
,-..
-8--
�
©
const.
•
i.....
.._,
.,
•
....
,�
'
-
.'
,.i
.., •
'.J
.,
•
�
-
�
.0..
••
-
- -e- - - .
�
-
�
�
I
,,i
q
••
�
• -i
;a;- .a=:.
•
,c ,;
••
�
•
•
32.
·v
a:
Continuity of
of different forms and
co11nection through different cycle-groups can be
applied in its present form to Diatonic progressions.
a:
in symmetric progressions of the
types I I and III require identical structures for the
two
" extreme chords of one group.
-
This requirement
does not affect the middle chord of the group,
s (:), nor do es it influence the selection of
•
structua,es for the following groups.
Examples of continuity with
i • e •,
a:
in progr�ssions of the types• I a.nd I I .
Ir' =
•
,
�
" 'I•
•
-
-
�
.,,
.a..
.....
-
�
••
-s•
-
·�
,
-9-..,,
C'
_,
6'
6
2G4t + G4
.....
•
---
.,,
C
•
r
�
�
!
�
r•
!; � :::
61T'
6
+ G4 J + 2G4
....
- 9-- .w.
-- .'
�
FiBur e LXXV •.
-....
..0-
'
-
..-:
.,
,.
• •
--
-
..,
,
i
,.
•
',
._,
-!r -e- .,
...
�
-
'
A 0
- -
�
,.
r,
Figure LXXVI.
1-r and � as in the preceding example•
•
--------
-
....
,.., I •
-
-s- .... -5--
.
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,
-e.
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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
With : Dr. Jerome Gross
C O U R S E
Subject: Music
Lesson µcxxiv,
Application of G� to symmetric systems
requires the following seq uence of tonics:
GW � (T, + T 2 + T , ) + (T 2 + T 3 + T2 ) +
• • •
For example, the three-tonic system must
be distributed a s follows :
0
G� = (T + T 2 + T ) + (T 2 + T 3 + T 2 ) +
,
,
•
+ (T 3 + T + T 3 ) .
,
The quantity of to nics in tl1e respective
system specifies the cycle.
either 8(5) or S (6 ) .
Eac11 gro up may begin with
Each group acquires the following distri
•
bution of inversions:
Under such conditi ons, each tonic appears
in all the three inversions.
•
6
Table of G4
•
applied to all symmetric systems
"
Figure tXXv!l.
I
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No. 230 Loose Leaf 12 Stave Style - Standard Punc:h
•
4.
Other negative forms are not as practical: inversions
weaken tonality.
Example of variation of structures
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l
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and directions.
Figure LXXVIII.
Four Tonics.
G� - [s, (5)
+
s 2 (:)
+
s , (s) J
+(s 2 (6) +
s , ( : ) + s 2 (6) )
+
+ [S 3 (6) + S� ( :) + 8 3 (5) ] + [ S 2 (5) + S 3 (:) + S 2 (6) )
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11[.___
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b i;
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a,.
>< +
�t:
_---tl
•
•
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c. Cyc�es. �d ��oups Mixed.
Tonal cycles can be introduced into the
continuity of groups, as well as groups can be intro
duced into the continuity of cycles.
It is convenient to plan the mixed form of
cycle-group continuity by the bars (T).
Bars of cycles and bars of groups can be
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-
•
•
assigned to have different coefficients of recurrence.
,
When planning such a continuity i n advance,
it is i mportant to co nsider that there is always a
cycle-connection between the bars •
•
Examples:
•
Figure LXX}:;x.
Ir = 2TC + TG + TC + 2TG = (C� + C3) + C 7 + (Ca + C 7) +
+ c, + G6 + C 7 + (C3 + C3 ) + Cr + G:1© + C 1 + G! _l,(f)+ C3 •
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Media of
