JOSEPH S CH I L L I N G E R

COURSE

S P O N D E N CE
C O R RE
With: Dr. Jerome Gross

Subject: Music

Lesson CLXX.
TWO-PART MELODIZATION
This technique consists of writing two correlated
melodies (two-part oou.nterpoint) to a given chord­
progression.

The counterpoint itself must satisfy all

the requirements pertaining to harmonic intervals.

Each

of the melodic parts (to be designated as Mr and M 11, or
as CPr and CPrr ) must satisfy the requirements pertaining
to melodization.
The sequence in which two-part melodization
should be performed is as follows:
(1) the writing of�;

•

(2) the writi2�g of M with the least number of

attacks per H;
(3) the writing of M with the most number of

attacks perH.
It is not essential which melody is designated
as Mr and which as M rr .
Considering the natural physical scale of
frequencies as increasing in the upward direction of
musical pitch, we shall evolve t he melody with the least
number of attacks ii.Dmediately above harmony, and the

•

-

0

2•
•

melody with the most number of attacks above the first
melody.

Such schemes will be considered fundamental

and could be later rearranged.
Thus we arrive at the two possible settings:

(1)

and

(2)

Octave-convertibility (exchange of the
positions of MI and M11) is possible only when the
harmonic intervals of both melodic parts are chosen with
consideration of su�h a convertibility.

This mainly

concerns the necessity of supporting certain higher
functions (such as 11) by the immediately preceding
function (such as 9).
All forms of quadrant rotation ( G), @, @ and

@)) are acceptable on one condition: Mr and Mrr always
remain above the chord progression (Ir).
As melodization of harmony by means of one part
produced different types of melody in relation to the
different types of harmonic progressions, the same
possibilit ies still exist for the two-part melodization.
•

It is to be remembered that some types o f melody
in one-part melodization were the outcome of new techniques.
For instance, the technique of modulating symmetric melody
above all forms of symmetric harmony, or the technique of
diatonic melody evolved fro m a quantitative scale above

0

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•

all forms of chromatic harmony.

All such new techniques

shall be applied now to the two-part melodization.

This,

naturally, will result in the new types of counterpoint.

Ir

The distribution of attacks of�' M
11 and
is a matter of co nsiderable complexity and will be

discussed later.

For the present, we shall distribute

the attacks for all three parts (M1, MII and r)
uniformly and by means of multiples.
Some elementm forms of the gistribution o f attacks.
a 2a

a a

a

a 2a

H

H

a 3a

H

H

MI.

9a 3a 12a 3a

Mrr

3a 9a
H

H

a 4a 2a 6a 2a 8a 2a 6a 3a 8a 4a

a 4a
H

a 4a 2a 4a 2a

a 2a 8a 3a 6a 4a Sa

H

H

H

2a 4a

H

H

5a 3a

H

6a 4a

H

H

H

H

H

H

• • •

3a 12a 4a 12a 3a 15a 4a 16a
H

H

H

H

H

H

H

H

Here the quantities of attacks in

are

designated per chord.
Each original setting of two simultaneous
melodies accompanied by a chord-progression offers
seven forms of exposit_?.on,.
(5)

L

(6)

•
•

0

4.
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L

Melodization of DiatonicHarmony by means of
Two-Part Diatonic Counterpoint.
(Type I and II)
The melody with least number of attacks and
appearing immediately above harmony m u st conform with
the principles of diatonic melodization.

It is desirable

not to include higher functions (9, 11) into this melody
(we shall call it Mrr), for the reason that the latter
cou ld be spared for the use in melody with the most
number of attacks (we shall call it Mr ).

Thus the high

functions of M1 will be supported by MII• pcales of both
melodies must have common source of derivation. This
common source is the diatonic scale of harmony.

Any

derivative scales of the original d c an be employed.
Harmony can be devised in four or five parts.

Four-part

harmony is preferable as the textu re of a duet accompanied
by five parts is somewhat heavy.
None of the melodies must produce consecutive
octaves with any of'. the harmoo:m .parts.
should be written as counterpoint to M11
and as melodization of the chord-progression.
M1

Identical as well as non-identical scales
(which derive thro ugh permutation of the pitch-units of
d0) can be used in Mr, Mrr and

r.

Under such

conditions any d0 produces 35 possibilities of modal

relations between the abovementioned three components.

0

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As we are employing se ven-unit scales,
'7

c3 =

°

3!

7t

= 5040
6 •24

('1-3) I

=

5040
144

=

35

The number of two-part melodizations which
is possible to evolve to one chord-progression (written
in one definite d)

l.S:

-

5040
2•120

-

5040
240

21

Examples of Diatonic Two-Part Melodization
Figure I .
(please see pages 6 and 7)
Chromatization of the Diatonic
Two-Part Melodization.
In order to produce a greater contrast between
M1 and MII either one can be subjected to chromatic
variation. If desirable, both melodies can be used in
their chromatic version.
Chromatic variation is achieved by means of
passing or auxiliary chromatic tones.
Example of Chromatic V ariation.
Figure II, Var. I �d II.
By mean s of combining the two variations of
Fig. II, we can obtain a new version, where chromatic
sections alternate with the diatonic ones.
r1gure II, Var, II�.
(please see page 8)

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9.

Lesson CLXXI.
Melodization of Smmetric Harmony

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(Type II, III and Generalized) by m�ans of
Symmetric Counterpoint
-Two-Part
Symmetric melodization is based on the pitch­
scale which is the contracted
each individual H.

a new scale.

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13 corresponding to

Theoretically, each cl1ord requires

The quality of the melody, bov,ever, depends

on the quantity of common tones between the successive
'I:. 13 upon which the� are based.

This concerns both�
and M11 of the two-part melodization.
The ultimate requirements for two-part

symmetric melodization may be stated as follows:
(1) Adherence of one M to a particular set of
pitch-units thus prod ucing a scale.
(2) The graduality of modulation, which is executed
by means of common tones, chromatic alterations
and identical motifs.
(3) Strict adherence to contrapuntal treatment of
harmonic intervals between Mr a nd MII·
Exapiples of Symmetric Two-Part Melod�zation
figure III.
(please see next pages)
10 and 11

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Figure III.

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Chromatization of theSymmetric
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Two-Part Melodization.
This technique is identical with chromatization
of the diatonic counterpoint.

P assing and auxiliary

chromatic tones are not the part of

L 13.

two contrapuntal parts can be chromatized.

E
ither of the
Alternation

of chromatic and symmetric sections in both melodies is
fully satisfactory.
Example of Chromatic Variation
;F,igure IV,
(please see ne�t page)

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Fifillr e. J.V,

TliEMC:: F l � - m: (1.)

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14.
Lesson CLXXII.
Melodization of Chromatic �armony by
means of Twp-Part Counterp�int

•

As one-part melodization of chromatic harmony
is possible from two distinctly different sources :
(1) directional units and
(2) quantitative scale,
chromatic melodization in two parts is possible in the
following combinatior1s of the above techniques :
di

ch

di

ch

where di (diatonic) represents

di

di

ch

ch

the quantitative scale; ch of

ch

ch

ch

ch

M represents the directional
units method and ch of !Pstands
for chromatic harmonic continuity .

If there is a contrast to be achieved between
Mr and M11 , one of them becomes di and the other ch.
If a similarity is preferable (the co11 trast
still can be achieved by juxtaposition of the quantities
of attacks of

:r )
II

both melodies are either di or ch.

The first J:1as a diatonic character ( due to adherence to
one particular pitch-scale) and the second has a
modulating character abundant with semitonal directional
units.

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17.
Lesson CLXXIII.
Composition of Attack-Groups for
I

4

the Two-Part Melodization
Mr
The quantity of attacks of Mr r
H
either constant or variable.

can be

A constant form of the attack-group talces
place when every individualH has a definite corresponding
number of attacks in Mr and Mrr , v1hich remains the same
for every consecutive H.
Mr
Mrr =

H

•

A canst.

L

A constant A does not necessitate an even
a(Mr )
distribution in a(Mrr) · An even distribution may be
considered merely as a special case •
E
xamples of an even distribution of A:
4a

6a

6a

8a

8a 9a

2a

2a ·3a

2a

4a

a

a

a

a

a

12a

12

3a

3a

4a

a

a

a

E
xamples of uneven distribution of A :
Mr

2a+3a

4a+2a

4a+2a

4a+6a

MII

a+a

a+a

2a+a

2a+2a

a

a

a

a

H
Mr

4a+2a+3a+6a

6a+3a+6a+4a+2a+9a

M rr

2a+a+a+2a

3a+a+2a+2a+a+3a

a

a

H

•

0

18.
A variable form of the attack-group takes
place when A emphasizes a group of· chords, and when
each consecutive H has a specified number of attacks for
a definite quantity of chords.
For example :
MI - 2a+a
Let A , = II _ a+a
a
•

and let A 3

Mr

Mrr
= ----H

� = A

'

+

2

+ A3
MI
= 4a+3a
M rI - 2a+a
H
a

and let A 2

_ 4a+6a+3a
_ -------2a+2a+a
a

then :

4a+3a
2a+a

2a+a
a+a
a

A

4a+6a+3a
2a+2a+a
a

a

Ha

All other considerations concerning the
distribution and quantities of attacks are identical with
one-part melodization (see : "Composition of the Attack­
Groups of Melody" in the branch of Melodization o f
Harmony) .
Example of Correlated fotta�k-Grou£s
in Two-Part Melodization
Figµre VI.

Mr Mrr -�

-

2a+3a
•
a+a
a

+
H, +

3a+4a
�+a
a

H�

+

+

4a+3a+2a
a+ a+ a
a

H�

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=
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20.
Composition of, Durat�ons for the Attack­
Groups of Two-Part Melodizatio
n
•
Selection of durations and duration-groups
satisfying the attack-groups com posed for two-part
melodization can be based either on theSeries of the

-

E
volution of Rhyth m Families (in which case there is no
i nterference between the attacks of the attack-group and
the attacks -of the duration-group) or on a direct
compos
• ition of duratio�-g;:ouFs (which may or may not produce
•
an interference between the attacks of the attack-group a nd
the attack of the duration-group) which would be super­
imposed upo·n the attack-gro ups.
When the respective attack-groups are represented
S eries, and the number
by the durations selected fromStyleof individual attacks in the attack-sub-groups does not
correspond to the number of attac ks in the duration-groups,
it is necessary to split the respective duration-units.
This consideration concerns the first technique only (i.e. ,
the matching of attack- groups by the series of durations).
Musical example of Figure VI is a translation of
its corresponding attack-group into

j

series, where three

1, 1 a nd 1 . One
3
4
2
exception to the series was made at the cadence , where a
4 series binomial, i.e.,
musical quarter was split in to 4
3+1.
The numerical representation of this example of
types of split-unit groups were used :

melodization appears as follows:

0

21 ..

1/2t+l/2t+l/2ttl/2�tt
t

+

+

+

+ 2 t

3t

+

H, +

l/3t+l/3t+l/3t+l/2t+l/2t+l/2t+l/2t

+ 2

t

3t

t

1L4t+l/4t+l/4t+�/4t+l/3�+1/3t+lf3t+l/2t+l/2t
+ t
+ t
t
3 t
The ab undance of split units and split-unit

groups in this instance is due to the abundance of attacks
over each H and to relatively low value of tha series.

With

a series of higher value, the splitting of units would be
•

greatly reduced •
We shall translate now the same example into
i
9

series :

MI = t+3t+t+3t+t
+ t+2t+t+t+2t+t+t
M II - 4t +5t
+5t
H ' + 4t
- _..::::.;;_.......,;�-9t
�
9t

-----

+ t+t+t+t+t+t+t+t+t
+3t
+2t
+ 4t
9t

-Figure

H3

VII.

(please see next page)

0

22.

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Now we shall take a case where the attack
and the duration-groups are composed independently.
Let r5+4 represent the quantities of attacks
of M 1 to each attack of Mrr, and let ever.y 2 attacks of
M 11 correspond to one attack of � .
Then the distribution of attacks for all

three parts takes the following appearance:
a (M 1)

- 4a+a

aCllrr) a(H '')

3a+2a
2a+3a
+
+
a+ a H 2
a+a H,
a+ a
+
+
a
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•

a+4a
+
Ha a+ a
a

H'f

Let us superimpose the following duration-group :
T

L

Then :

= r4+3 =

$=

f§ =

Hence, T • = 16t •2
Let T"

=

f;

1 ( 20 )
2 (10)

32t

= at, then : NT"
-

l6t; lOa

=

32 = 4
8

Each a(M1) corresponds to an individual term
of T; each a (Mrr) corresponds to the sum of the respective
durat ions of Mr; each a (�) corresponds to the sum of 2
durat ions of M II .
The final temporal scheme of this two-part
melodization takes the following form :
Mr M II +
+

3t+t+2t+t+t
+t H,
7t
8t
3t+t+2t+t+t
4t +4t
Ha
8t

+ t+t+2t+t+3t
+4t
+ 4t
8t
+ t+t+2t+t+3t
+ t+7t
8t

H2

+
+

d

0
�

C

24 .

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25.
Direct Composition of Durations for the

C

C

4

Two-Part Melodization
Direct composition of durations becomes
particularly valuable, when a prop9Ftiona�e distribution
of durations for a constant number of attacks between the
component parts (Mr, Mrr and �) is desired.

Distributive

involution of three synchronized powers solves this
problem.

As it follows from the Theory of Rhythm, the

cube of a binomial produces an eight-term polynomial, the
square of a b inomial produces a quadrinomial and the
first -power group remains a binomial.
of attacks of the two adjacent parts
two.

Thus, the quantity
Mr
Mrr
is
and
MII

�

Cubing of a trinomial gives a twenty-seven-term

polynomial, the synchronized square producing nine and the
first-power group -- three terms.

The quantity of attacks

between the two adjacent parts remains three.

Thus, the

number of terms of the original polynomial equals the
•

quantity of attacks between the adjacent parts .
We shall devise now a correlated proportionate
system of duration-groups.

The distributive cube will

serve as T for Mr , the synchronized distributive square
as T for Mrr and the synchronized first-power group as T
for Ir.
We shall operate from the trinomial of the
series.

This secures the following attack- group correlation:

0

26.
a (M1)

a ( ia:11

9a
-) =

;·(a 5)

3a

The entire temporal scheme assumes

a

the following form:

T (Mr) = [(8t+4t+�t) + (4t+2t+2t) + (4t+2t+2t) ] +
T (M 11) = (16t
) +
+ at
. + 8t
32tH ,
T (Ir )
+ [(4t+2t+2t) + (2t+t+t)_+ (2t+t+t) ] +
+ 4t_
) +
+ 4t
+ (8t
16t
H2
2t+2t) + (2t+t+t) + (2t+t+t) ]
+
t
4
[(
+
+ 4t
)
+ 4t
+ •(at
16t
H3
L

Fig_yr� ,ItC.
(please see page 27)

In addition to this technique, coefficients
of duration can be used for correlation of durations in
the two-part melodization.
Example :
Mr

Mrr
•

=
==

(3t+t+2t+2t)+(3t+t+2t+2t)+(3�+t+2t+2t)+(3t+t+2t+2t)
(6t+2t+4t+4t) + (6t+2t+4t+4t)
•

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27.

Figure IX.

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28.
Lesson CLXXIV.
Compopition of_pontinuity in
Two-Part Melodization

•

The seven forms of expositions previously
classified can be now incorporated into continuity of
two-part melodization.

The applied meaning of these

seven forms can be expressed as follows :

(1) MI
(2) M11
(3) �

-- Solo

---

melody : theme A;

Solo melody: theme B ,·
Solo harmony: theme C;

(4) � -- Solo melody with harmonic accompaniment
(theme A accompanie d) ;
(5) ;; -- Solo melody with harmonic accompaniment
(theme B accompanie d) ;
(6)

Theme A
Duet
-of two melodies ( The me B )
•

(7)

Duet of two melodies with harmonic
accompaniment

Theme A
Theme B
Theme C

The above seven forms serve as thematic elements
of a composition, in which they appear in an organized
sequence producing a comple te musical whole.
Themes A, B and C must be conside red as
component parts of the whole in which the y e xpress the ir

0

29.

L·

particular characteristics.

These characteristics

which distinguish A from B and C are :
(1) High mobility of A (maximum quantity of attacks) ;
(2) Medium mobility of B (medium quanti ty of attacks) ;
(3) Low mobility of C (minimum quantity of attacks)
combined with maximum density (four or five parts) .
The planning of continuity must be based on a
definite pattern of the variati on of density combined
with the vari ation of the quanti ty of attacks.
scale of densi ty can be arranged from

The

low to high as follows :
A
Iv.
•
B
A,
c,
(1)
,
B'
C' C
A
B, B •
(2 ) B, B' c, C
C

L

-

-

More or less extreme points of any such scale
produce contrasts.
(1 )

-B +
A

C

(2) A +

For instance:

A
B+ A + C + B + C + B
B + A + B + ,·
C
A

A

C

C

C + B + C + A + B + B + B + A + B
c
c
A

Durations corresponding to one individual
attack of the component of lowest mobility (mostly H4 )
become time-units of the continuity .

Such units (we

shall call them T) can be arranged in any form of rhythmic
distribution.
Correlati on of the thematic duration-groups

0

30.
( T • s with their coefficie.nts) with the different forms
of density constitutes a composition.
Assuming that there are three forms o f density
and three forms of mobility, we obtain the following
combined thematic forms (Low, Medium, High):
Low
Low

Density
Mobllity

I

Low

Medium

Medium
High

Medium
Low

High
Medium

Low
High

High
Low

�- g�
g

Thus, for instance:

Density = High = Ir' .
Mobility
Low '

Medium
Medium

3 2 = 9.

_M
Densi:tY _ Low =
II '
Mobility - Low

Density _ High
= MII
Mobility - Medium -

r

etc.

We shall now devise a composition which will
combine �he gradual and the sudden variations of mobility
and of density.
It i s desirable to have such a scheme of
two-part melodization which is cyclic
-. and recapitulating,
i.e. , one permitting a correct transition from the end
to the beginning for all three components ..
For the present, we shall not resort to any
additional techniques (such as inversions, expansions
etc.) , as the complete synthesis will be accomplished in
the branch of •Composition •
Let Figure VIII serve as the fundamental
scheme of two-part me lodization, as this material is

0

31.

cyclic and recapitulating.
Let us adopt the following scheme of density
and mobility:
Density = Low + Low + Medium + �i gh
+ �igh + Medium +High
Mobility
Low High High
Medium
Low
High
High
The sequence of thematic elements and their
combinations, corresponding to the seven forms of
�xpositions and satisfying the above scheme of thematic
forms may be selected as follows:

r E + Mr 1 E
M
E
+
r
M 2 � 3
�
't

+ �E
•

J,..

+

Mr
Mrr

We shall make T correspond to H and establish
the following sequence for the T • s: T

= r 5+3 •

� :;: 7T 15H.
The 7T of � produce no interference in relation
to the 7E of W.

There is an interference between � V and

8 .
:rr-' , however, as � == H
8 ( 7) .
7 (8 ) '

r'

= 7 • 8 = 56 TE.

As 7 TE corresponds to 15 H, there will be
7 TE• 8 = 56 TE and 15 H•8 = 120H.
Thus the complete composition after synchronization
evolves into the following form:

� ' �• =

56 TE 120H;

0

32.
As in Figure VIII T11 = TH, the entire
composition consumes 120 measures, which is 15 times
the duration of the original scheme of melodization.
•

Here is the final layout of the composition :
Figure• X.

+

Mrr (
if""" H7 +

H,9 + H , ) T., E44 + ff'(H2 ) T£ ES" +

M�

M I

(H3 + H�)T6 E6 +

Mr

+

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!rt

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Ir+

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' (H5) T,0E 1 0 + � (H& + H7 + Hg)

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t/

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0

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J O S E P H

S C H I L L I N G E R

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With: Dr. Jerome Gross

C O U R S E
pubject : Music

Lesson CLXXVI.

'

TWO-PART HARMONIZATION
The principle of writing a harmonic accompaniment
to the duet of two contrapuntal parts consists of assigning
harmonic consonances as chordal functions .
Every combination of two pitch-units producing a
simultaneous consonance becomes a pair of cho.rdal functions.
This premise concerns all types of counterpoint and all
types of harmonization.
Pitch-units produc ing dissonances are p erceived
throug h the auditory association as auxiliary and p assing
tones .

Justification of the consonance as a pair of

chordal functions gives meaning to the harm onic acc ompaniment.
Diatonic Harmonization of the
Diatoni,c Two-Part Counterpoint,
Under the conditions imposed by Special Harmony,

•

two-part counterpoint , which can be harmonized by the latter,
must be constructed from seven-unit scales of the first
grotip, not containing identical intonatioris.
As all three components must belong to one key,

u

according to the definition o f diatonic, the only types of
counterpoint which can be diato nically harmonized are types
I and II .

0

2.
It is important for the composer to realize
the modal versatility of relations which exist between
the three component s.

As M1 may be written in any of
the seven modes ( do , d , , d 3 , d�, dq , %-, d0 ) of one scale,
and so may M11 and the i:r-t, th e total number of modal
variations for one scale is : 73 = 343. This, of course,
includes all the identical as well as non-ident ical
combinations.

Practically, however, this qua ntity must be

somewhat limited, if we want t o preserve th e consonant

•

relation between the P.A. • s o f M1 and M1 1 •
It is important to remember that the number of
seven-unit scales not containing identical units is 36.
Therefore the total manifold of relations of Mr :
in the diatonic counterpoint of types I and II is:
343•36

M1 1 : �

= 12,348.

Any given combination can be modified into a
new system of intonations, i.e. , into a new scale, by mere
readjustment of the accidentals.
All th e above quapt ities, nat ura lly, do not
include the attack-relations which have to be est ablished
for the harmonization.

r
M
are fixed groups, the
As the attacks of
MII
only relation th at is necessary to establish concerns � .
The most refined form of harmonization results from

I

assigning each harmonic consonance to one H.

If counter-

0

3 .,

\_

point contains many delayed resolutions of one dissonance,
then the number of attacks of MI is quite great and the
changes of H are not as frequent. On the other hand,
direct resolutions produce frequent chord changes.

The

assignment of two suc cessive harmonic consonances to one H,
amplifies the number of chords satisfying such a set, but
at the same time neutralizes somewhat the character of P .
This technique, however, permits a greater variety of
'

attack-relations between the three components.
We shall now proceed with the two-part diatonic
harmonization.
Let us harmonize counterpoint type I I , where

= a . In s uch a case all the harmonic intervals are
consonances .
of attacks:

Therefore we can have the following mat ching
Mr = a
Mr = 3a
Mr = 2a
= 2a
Mrr = a
Mir = 3a etc.
Mir
�= a
= -;
� = 'a

r

E
x�mples of Diatonic Harw9nization of �he Two-Part
Coun�erpoint Mr = a .

Mrr

Figure �.

(please see pages 4 and 5)

E
xamples of Diatonic Harmonization of the Two-Part
3a
_
- --Counterpoint
4C

I

-Figure

I

I

II. (please see pages 5 and 6)

E
xamples of Dia�onicHarmonizati9n of the Two-Part
Counterpoint

MI - 4a
M11

and

6a
a

Fig,ure III. (please see pages 6 and 7)

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8.
Lesson CLXXVII .
Chromatization of Harmony AccompanYi,ng
�wo�Part Diatonic Counter£oint (Types I an� II).
A chromatic variation of the diatonic harmony
accomp_anying two-part counterpoint can be obtained by
means of auxiliary and passing chromatic tones.

Of course

such altered tones shall not confl ict in any way with the
two melodies.
For our example we shall take the two-part
counterpoint diatonically harmonized from Figure III (2) .
�mple of the Chropiatization

t.

pf Harmonic_ Accompaniment
Figure IT.
(please see page 9)
Diaton ic Harmonization of the Chromatic
Counterpoint_ Whose Origin is Diato�ic (Types I and II)
The principle of this form of harmonization
consists of assigning the diatonic co11sonanc,es as chordal
functions.

Chromatic consonances as well as all other forms

of harmonic intervalsshall be neglected.
The quantity of suc cessive consonances
corresponding to one H is optional.

It is practical to

make T or 2T, or 3T correspond to one H.
When harmonizing a chromatic counterpoint,
whose diatonic original is known, one can assign chordal

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10.
functions directly from the diatonic original .

This

measure obviously eliminates any possible confusion of the
diatonic and the chr.omatic consonances.
We shall now harmonize a duet where both parts
are chromatic.

The theme is taken fro m Figure XXIV of

the Two-Part Counterpoint.

For clari ty's sake, we shall

write out both the or iginal and the chromatized version.
We shall choose the following relationship between W and
••

which is a modified version of the r3+ 2 , and �which permits
to demonstrate the diversified forms of attacks groups of

'

Mr and Mrr i n relation to � .
Example of Diatopic �rmonization of
the Chrom�ti�. Count�rp9int
Figure V,
(please see page 11)
When the diatonic origin of chromatic counter­
point is unknown, the analysi s of diatonic consonances
must precede the planning of harmonization.

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12.
Lesson CLXXVIII .
Symmetric Harmonization of
Diatonic
. the ..
Two-Part Counter�oint (TYPes I1 II, III and IV) .
The principle of gmmetric harmonization of
the two-part counterpoint consists of assigp.ing all
harmonic intervals as chordal functions .
The fewer attacks of M r and Mir correspond to
one H, the easier it is to perform such harmonization by
means of one i:' 13.

When a considerable number of attacks

(even i n one of the two melodies) corresponds to one H,
it becomes necessary to introduce two, and sometimes
three L 13 .

The forms of the latter should vary only

slightly, servir1g the only purpose of rectifying the
non-corresponding pitch-unit.

For instance, when usirlg

!: 13 XIII as � , , correction of tbe eleventh to f tJ
gives satisfactory solution for most cases.

Thus,

-r- 2

in this instance differs from !:_ only with respect to 1 1 .

'

The selection of the original 2 13 is a
matter of harmonic character .

For example, the use of

X- 13 XIII attributes to music a definitely Ravelian
quality.

However, harmonic quality still remains virgin

territory awaiting the composer 's exploration .

Most of

the 36 forms of the L 13 have not been utilized.
Whether counterpoint belongs to types I and

0

II, or to types III and IV, it does not give any clue
to any particular � 13.

And whereas symmetric

•

0

13 .,
harmonization of the counterpoint of types I and II
is a luxury, it is a bare necessity for types III and
IV, as the latter correlate two different key-axes.
The fact that two different keys with identical or
with non-identical scales can be united by one chord
is of particular importance.

This is so because the

quality of a selected Z: 13 is capable of influencing
the two melodies .,

The ear in our musical civilization

is so much conditioned by harmony, that most of our
listeners have lost the ability of enjoying melodic
line per se.

And if the ear of an average music-lover

can relate one diatonic melody to some chord progression,
the harmonic association of two melodies belonging to
two different keys becomes impos sible.

Therefore the

role of a harmonic master-structure ( -Y: 13 in this case)
is one of a synthesizer.
0

The simplest way to assign harmonic functions
is by relating the latter to consonan ces first .
The master-structure used in the following
harmonizations is

r 13 XII I .

Symmetric Harmonization of the Diatonic
Two-Part Counterpoint of Types I and II.
F,igµre VI .
(please see next page)

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15 .
Chromatic variation of

r

in the above

example is obtained through the usual technique:
the insertion of passing and auxiliary units.
Symmetric Harmonization of the Diatonic
T�o-P�rt Counterpoint of Types III and IV.
Figure VII.
(please see page 16)
Symmetric Harmonization of the Chromat�c
Two-Par� Counterpo�nt Whose Origin is Diatonic
(Types I , II, III and IV) .

The principle of symmetric harmonization o f
the chromatic two-part counterpoint consists of
assigning all the diatonic pitch-units of both melodies
as chordal functions of the master-structure ( r 13)
and neglecting all the chromatic pitch-units, as not
belonging to the scale.

It does not matter whether the

chromatic units belong to the master-structure or not .
When the diatonic original of the two-part counterpoint
is unknown, the diatonic units of both melodies should
be detected first.
Figure VIII.
(please see page 17)
Counterpoint executed h1 symmetric scales
of the Third and the Fourth Group can be harmonized by

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16.

Figure VII .

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means of a symmetric master-structure.

This master­

structure is independent of the system of symmetry of
the pitch-scales involved.

As in the previous cases,

all units corresponding to oneH must belong to one

r 13.

After the harmonizati on is performed, it may

be sub jected, if desirable, to chromatic variation.
Symmetric Harmonization of the

•

Symmetric Two-Part Counterpoint.
figure IX .
(please see page 19)

t.

All forms of contrapuntal continuity as well
as complete composit ions in the form of canons and
fugues can be harmonized accordingly to this technique.
Any of the above described correspondences between
counterpoint and harmony can be established by the
composer.

One sh ould remember that overloading harmonic

accompaniments is more a sin than a virtue.

For this

reason the technique of variable density should receive
utmost consideration.

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19.

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Lesson CLXXIX.
Ostinato
Forms of ostinato or ground motion have been
known since time immemorial.

They appear in different

folk and tradit ional music as a fundamental form of
improvisation around a given theme.

The characteristic

of ostinato (literally obstinate) is a continuous
repetition of a certain thematic group, which may be either
rhythm, or melod y, or harmony.

For example, the dance

beat of 4/4 in a fox-trot is one of such fundamental forms
of ostinato .
•

As a matter of fact, a rhythmic ostinato is

ever present in all the developments in classical
symphonies.

Take, for example, Beethoven's Fifth Symphony,

the first motiI of it consisting of 4 notes, and follow it
up through the development (middle section of the first
movement).

The motif, rhythmically the same, changes its

forms of intonation either melodically or in the form of
accompanying harmony.
Repetitions of groups of chords, as well as
repetitions of melodic fragments accompanied by continuously
changing chords, are forms of ostinato.

Ostinato is one

of the traditional forms of thematic growth and, as such,
is very well knovm in the form of ciaconna and passacaglia.
In many Irish jigs, ostinato appears in forms of pedal
point as well as in repetitious melodic fragments.

When

0

21.
portions of the same melody appear in succession, being
harmonized every time anew, (which may be found even in
such works as Chopin 's Mazurkas,) we have a case of
ostinato.
I . Melodic Ostin ato
. (BasJ30 Ostdnato)
Melodic ostinato, better knovm under the name
of "Ground Bass " , is a harmo nization of an ever-repeating
melody with continuously changing chords.

•

Ostinato groups

produce one uninterrupted continuity where the recurrence
of the bass form produces unity, and the acco�panying
harmony - variety.
•

All forms of harmonization can be

applied to the conti nuously repeating melody, and regardless
as to whether it appears in the bass or in any of the
middle voices, or in the upper voice (above harm ony).
As every harmonic setting of chords is subject
to vertical permutations, a basso ostinato can be trans­
formed into tenor, or alto, or soprano ostinato, i . e.,
it may appear in any desirable voice and in any desirable
sequence after the harmonization has been completed.
In the f9llowing example the ostinato of the
theme is a melody in whole notes in the bass (the first
itself
four bars) , after which it repeats;two more times. The
form of harmonization is symmetric in this case, though
it could be diatonic or any of the chromatic forms.
This device can be used as a form of thematic developme nt,

0

22 ..

•

and in arranging for the purpose of constructing intro­
ductions or transitions, as any characteristic melodic
pattern can be converted into basso ostinato either with
the preservation of its original rhythm or in an entirely
new setting. *
Figure I .

Melodic Ostinato
Basso Ostinato (Ground Bass)
Symmetric Harmonization of the Bass .

•

u

*See: Arensky's "Basso Ostinato" for Piano.

•

0

23.
Il. Harm onic Ostinato
Harmonic Ostinato may be also called, by
analogy, "ground harmony".

It consists of the repetition

of a group of chords in relation to which a continuously.
changing melody is evolved.

This form of ostinato is the

one J. S . Bach employed in his D-minor "Ciaconna 11 for
Violin, besides numerous other compositions by Bach and
other co�posers.
•

Among my students, a successful use of

this device occurred in an exercise made by George Gershwin,
and which later , at my suggestion, was put into the musical
comedy, "Let 'Em Eat Cake", of which it became the h�t
song ( 11Minen).
This form of ostinato can be applied to any
type of harmonic progressions.

The technical procedure ·

is exactly the opposite of the first one.
we deal with melodization of harmony.

In this case

As in the previous

case, the melody evolved against chords may be transferred
to a different position in relation to chord by means o f
vertical permutation.

Naturally, not every melody will be

equally as good under such conditions if it appears in the
bass and in the soprano, as th.e chordal functions
represented by melody may be more advantageous for an upper
part than for the lower , or vice versa.
In the following example, the harmonic theme
of ostinato emphasizes four different chords (the first
two bars ), and is based on a '£ 13 XIII .

The melody

0

24 .

evolves through the principle of symmetric melodization
forming its axi s points in relati on to the chord
structure itself.

The main resource of variety is the

manifold of melodic forms.
Figure II.

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Harmon¾£ Ost!neto (Ground Harmony)
Symmetric Melodization of Harmony

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•
The form of contrapuntal osti nato is well
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If a C .F.

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25.

produc ing different relations with every appearance of
the C. F. , we have a contrapuntal ostinato .
In the following example, the theme of
ostinato is taken from Figure I , and the accompanying
counterpoint is evolved thro ugh Type II, adhering t o a
rhythmic ostinato as well (except for a few intentional
permutations).

Naturally both voices can be exch.anged as

well as subjected to any of the variations through
geometrical positions � ,

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Fig!)re III.
Contrapuntal Ostinato
Basso Ostinato (Gro11nd Bass)
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Likewise, a counterpoint can be evolved to
the soprano voice through the use of the same principle .
In Figure IV, the same theme is employed except that it
is altered rhythmically, and the counterpoint, in its
rhythmic settiqg, produces a constant interference against
the C.F ., as it consists of a 3-bar group.

The harmonic

setting of this example is in Type III : the C .F. is in
natural C major, and the counterpoint is in natural A �
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mot if and yet introduce variety into an obligato.

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characteristics make the above described device extremel y
useful for introductions, transitions and codas, when
applied to arranging.

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J OSEP H S C H I L L I N G ER
C O R RE
S P O N DE N C E
Wit h : Dr . Jerome Gross

C O U RS E
Subject:
Music
•

Lesson CLXXX.
IN
STRUMENTAL FORM
S OF MELODY AND HARMONY
The meaning of Instrumental Form implies a
modification of tl1e original which renders the latter fit
for execution on a.n instrument .

Instrumental can be

defined as an applied form of the pure.

Depending on the

degree of virtuosity which is to be expected from the
performers, instrumental forms may be applied to vocal
music as well.
The main technical characteristic of the
instrumental (i. e . , of applied versus pure) form is the
development of the quantities (multiplication) and forms
of attacks from the origi nal attack.

This branch will be

concerned only with the first, i . e . , with quantities and
their uses in composition, leaving the second, i . e . , the
forms of attacks (such as durable, abrupt, bouncing,
oscillating, etc.), to the branch of Orchestration.
Multiplication of attacks can be applied
directly to single pitch-units as well as to pitch­
assemblages.

u

The quantity of the instrumental forms is

dependent upon the quantity of pitch-units in an assemblage.

0

2.
When the quantity of pitch-units (parts) in an assemblage
is scarce, the number of instrumental forms is low.

When

the number of pitch-units (parts) in an assemblage is
abundant, the number of instrumental forms is high.

The

latter permits to accomplish greater varie ty in a
composition, insofar as its instrumental aspect is
;

concerned.

The scarcity of instrumental forms derived from
one pitch-unit (part) often makes it compelling to resort
to couplings.

By addition of one co upling to one part we

achieve a two-part setting, with all its instrumental
<
implications.

Likewise, the addition of two couplings to

one part transforms the latter into a three-part assemblage,
etc.

This branch consists of an exhaustive study of

all forms of arpeggio and their applications in the field
of melody, harmony and correlated melodies.
Nomenclatur e :

---Score (Group of instrumental strata)

s
p

a

---Stratum (instrumen tal stratum)
--- part (function, coupling)
--- attack

•

Pre.liminary Data:

=a • p
•
s
=
(2)
p , s
(3) r = s t• �
(1) p

-

2a ,• • • • p = na

=

2S '• • •

2p ,• • • •

s = np

• z:- = S
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3 .,

Sources of Instrumental Forms
(a) Multiplication of S is achieved by 1 : 2 : 4 : 8 : • • •
ratio (i.e., by the octaves)
(b) Multiplication of p in S is achieved by coupling or
by harmonization.

It is applicable to melody (p),

correlated melodies (2p, • • • np) and harmony (2p... 4p).
The material for p is in the Theory of Pitch Scales
and the Tr1eory of Melody.

The material for 2p, ... np

acting as melodies is in the Theory of Correlated
•

Melodies (Counterpoint) .

The material for 2p, ... np

acting as parts of harmony is in the Special Theory of
Harmony and in the General Theory of Harmony.
(c) Multiplication of a is achieved by repetition and
sequence of P ' s (arpeggio).
(d) Different S • s and different p • s, as correlated melodies
of 2: may have indepe nd e nt instrumental forms .,
Definition of the Instrumental Forms:
I. (a) •Instrumental Forms of Melody: I (M = p) :
repetition of pitch-units represented by the duration­
group and expressed through its common denominator.
The number of a equals the number of t.
1
If n'"t = nt, the11 nt = na
Rhythmic composition of dur ations assig11ed to each at tack.
(b) Instrument�! Forms of M elody: I (M = np):
repetition of pitch-units (Pr) and their couplings
(Prr , Prrr , ... PN ) and transition (sequence) from one

0

p to another, represented by the duration group and
expre ssed through it s common denominator.

Instrumental

groups of p ' s consisting of repetitions and sequences are
subject to permutations.
Groups 9f Melody :
•
(�) Instrumental Forms of the Simultaneous
M = PII ,•
PI

Pr ,•
PII

P 1rr
Pr
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Prr

P 11
Pr
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PIII

Pr
Pr
•
Prrr;
,
Prr
PrrI Prr

P rrr

Prr
Pr

Prr
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Pr

Instrumental Forms of the Seguent Groups 9f Melody :

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=
M
(a) I ( rr= P ) : correlation of instrumental forms
Mr

p
melodies (Mr and M11) by means
of the two uncoupled
of correlating their a ' s .

MI (nt = na) ;

MI I (nt

Mrr ( t = a)
MI (t = 2a)

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M r (t

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Mr

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= 2na; 3na; ••• mna)
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= a)

M rr (t
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= 3a)
= a)

Mir ( t = 4a)
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{t = 4a) ' Mr
(t = 2a) ' Mr (t = 4a) ' Mr (t = 3a)'
• • • Mr I ( t = na)
i{1 '{ t"� ma)

(b) r

Mrr = np
Mr • - mp

: this form corresponds to combinations

of ( ex, ) , ( � ) and ( � ) of I ( b ) .
Mrr C °"' )

MI

( o<. )' ;
( ol )

III. Instrumental Forms of Harmony:
I

b

pt

I S
( = p, 2p, 3p, 4p) : this corresponds to one part
harmony, which is the equivalent of M ; two-part harmony ,

0

6.

which is the equivalent of two correlated uncoupled
melodies ; three-part harmony, which is the equivalent
of three correlated uncoupled melodies; four-part
harmony, which is the equivalent of four correlated
uncoupled melodies.
The source of Harmony can be the Theory of
Pitch Scales, Special Theory of Harmony and General
Theory of Harmony.

Parts (p rs) i n their simultaneous

and sequent groupings correspond to a, b, c, d.
= d.
Instrumental Forms of S = p .
Material :
(a) melody;
(b ) any one of the correlated melodies;
(c) one-part harmony;

(d) harmonic form of one unit scale;
(e) one part of any harmony.

I = a; 2a; 3a; ma; A var.
nt = na

Figure I .
(please see pages 7 and 8)

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9.
Lesson CLXXXI .
peneral Classificatiop of_ I (S = 2p) .
(Table of the combinations of attacks for a and b)
A = a; 2a ; 3a ; 4a ; 5a; 6a ; 7a ; 8a; 12 a.
The following is a complete table of all forms
of I (s = 2p).

It includes all the combinations and

permutations for 2, 3, 4, 5, 6, 7, 8 and 12 attacks.
0

A = 2a; a + b.

Total in general permutations: 2
•

Total in circular permutations: 2

A

- 3a; 2a + b; a + 2b.

-

p3

31
6
21 ;:; 2

-

3

Eac h of the a bove 2 permutations of the
coefficients has 3 general permutations.
Total: 3 • 2 = 6
The total number of cases

A =

3a

•

General permutations : 6
Circular permutations: 6
A = 4a

Forms of the distribution of coefficients :
4 = 1+3;
A = a + 3b ;

p'-t

2+2

3a + b .
1 _ 24 _
= 4
3! - 6 -

4

•

0

10.

u

Each of the above 2 permutations of the first
form of distribution of the coefficients of recurrence
has 4 general permutations.
Total : 4 • 2 == 8
A

=

2a + 2b

24
p = 4!
=
�
2 •2
21 2!

6

The above invariant form of distribution has
6 general permutations.
The total number of cases: A = 4a
General permutations: 8 + 6 = 14
Circular permutations: 4•3

= 12

A = 5a
Forms of the distribution of coefficients:
5

= 1+4; 2+3 .

A = a + 4b; 4a + b
p
�

= 5 1 = 120 =
41
24

5

Each of the above 2 permutations of the first
form of distribution has 5 general permutations.
Total: 5·2 = 10
A = 2a + 3b ; 3a + 2b .
_ 5!
= 120 = 10
pS" -21
3!
2•6
Each of the above 2 permutations of the second

0

11.
form of distribution has 10 general permutations.
Total : 10•2 = 20

The total number of cases: A = 5a

General permutations: 10 + 20 = 30
Circular permutations: 5•4 = 20

A = 6a
Forms of the distribution of coefficients :
•

6

=

1+5; 2+4; 3+3 .

A = a + 5b; 5a + b .
p

6

= fil = 720 = 6
120
51

Each of the abov e 2 permutations of the first
form of distribution has 6 general permutations.
Total: 6 • 2 = 12

A = 2a + 4b; 4a + 2b .
p

6

=

61
21 4 1

720 = 15
2•24

E
ach of the above 2 permutations of the second
form of distribution has 15 general permutations,
Total: 15 • 2 = 30

A = 3a + 3b
_ 720
61
6•6
3! 3 !

= 20

The above invariant (third) form of
distribution has 20 general permuta tions.
The total number of ca ses : A = 6a

General permutations: 12 + 30 + 20 = 62
= 30
Circular permutations: 6 • 5

0

12.
A = 7a
Forms of the distribution of coefficients:
7 = 1+6; 2+5; 3+4 .

A = a + 6b; 6a + b .
7 1 = 5040 =
61
720

7

Each of the above 2 permutations of the first form
of distribution ha s 7 general permutations .
Total: 7 • 2 = 14
A = 2a + 5b;
B?

=

5a + 2b .
71
2 1 51

=

5040 = 21
2 • 120

Each of the above 2 permutations of the second form
of distribution has 21 general permutations.
Total:
A =

3a + 4b;
P.7 -

21 • 2 = 42
4a + 3b -

•

= 5040
31 41
6 • 24

71

35

Each of the above 2 permutations of the third form
of distribution bas 35 general permutations.
The total number of cases: A = 7a
General permutations: 14

+ 42 + 70 =

126

=

42

Circular permutations: 7 •6-

V

0

13.
A = 8a

Forms of the di.stribution of coefficients :
8 = 1+7; 2+6; 3+5; 4+4 A = a + 7b; 7a + b .
8 1 = 40,320 = 8

p =
a

5, 040

71

Each of the above 2 permutations of the first
form of distribution has 8 general permutations.
Total : 8 • 2 = 16

•

A

=

6a + 2b .

2a + 6b;
p

8

= �o,320 = 28
2 •720
21 6 1
a1

=

Each of the abov e 2 permutations of the second
form of distribution has 28 general permutations.
Total : 28 • 2 = 56
A = 3a

+

5b ;

P8 _
-

5a + 3b .

8!
= 40,920 = 56
3I 5 l
6 • 120

Each of the above 2 permutations of the third
form of distribution has 56 general permut ations.
56•2 = a12

Total:
A = 4a + 4b
P8

=

81
_ 40,320 _ 70
4 1 4 1 - 24 • 24 -

The above invariant (fourth) form of distribution
has 70 general permutati ons.

0

14 ..
The total number of cases : A = 8a
General permutations: 16 + 66 + 112 + 70 = 254
Circular permutati ons :

8•7

= 56

A = 12a

Forms of the distribution of coefficients:
12 = l+ll; 2+10; 3+9; 4+8; 5+7; 6 6
+
A = a + llb; lla + b .

= 12 1 = 47� 1991� 600 = 12
39 ,916 , 800
11 !
Each of the above 2 permutations of the first
form of distribution has 12 general permutat ions.
Total: 12 • 2 = 24
A = 2a + lO b;

lOa + 2b .
_ 479,001.i 600
121
21 101 - �- 3, 628, 800

= 66

Each of the above 2 permutations of the second
form of distribution has 66 gen eral permutations.
Total : 66•2 = 132
A = 3a + 9b;

9a + 3b .

P,l = 311 291 I

= 479,091,600 = 220
6 • 362, 880

Each of the abov e 2 permutations of the third
form of distribution has 220 general permutations.
•

•

Total: 220•2 = 440

0

15 .
A = 4a + Sb;

_
-

8a + 4b ,

12 1 _ 479,0,01,600
4 1 8 1 - 24 •40,320

=:

495

Each of the abov e 2 permutations of the
. fourth form of distribution has 495 general permutati ons .
Total : 495 • 2 = 990

A = 5a + 7b; 7a + 5b .
12 1

= 479,001J 600
120 • 5 , 040
51 71

= 792

Each of the ab_ove 2 permutations of the
fifth form of distribution has 792 general permutations.
Total:

792• 2 = 1584

A = 6a + 6b

00 = 924
479,001,6
_
_
12
1
°
p,�
720 •720
- 61 61 The above invariant (sixth) form of
distribution has 924 general permutations.
The total number of 6ases : A = 12a
General permut ations: 24 + 132 + 440 + 990 +

+ 158 4 + 924 = 4094.

Circular permutations:

12 • 11 = 132

0

16.
Lesson CLXXXII.
Figure I I .
The interval of octave can be changed to
any other interval.

For tl1e groups with more than 6

attacks o�ly circular. per�utati9ps are included.
(please see pages 17-22)

figure I I I .
Examples of the polynomial atta cK-groups
(coefficients of recurrence) .
(please see page 22)

•

•

0

A = a

Figµre I I .
A = 3a : 2a+b; a+2b.

A = 2a; 2 forms

2 combinations,

17.

3 permutations each. Total 2•3 = 6
A � 4a: 3a+b; 2a+2b; a+3b
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No. 1. Loose Leaf

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23.
Lesson CLXXXIII.
Instrumental forms ofS = �E
Material:
(a) coupled melody:

M (

P II
) '
PI

(b) harmonic forms of two-unit scales;
(c) two-part harmony;
(d) two-parts of any harmony.

•

Pr
Prr '

I = a:

I = ab, ba: permutations of the higher orders.
Coeffic ients of recurren c e : 2a+b; a+2b; . . •

• • • ma + n b.

Figure IV.
(please see pages 24-29)

Individual attacks emphasizing one or two
parts can be combined into one attack-group of any
desirable form.

Example :
I (S

= 2p) :

bbbb b
bb bbb
bb
aa ; aaaa ; aaa aa ; aa a aa ; • • •
b

Figyre V.
(please see page 30)

0

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31.
Lesson CLXXXIV.
(• = 3p)
General classification of I S
(Ta ble of the combinations of attacks for
a, b and c)

A = a ; 2a ; 3a ; 4a ; 5a ; 6a ; 7a ; 8a ; 12a.
The following is a complete table of all forms
of IS
( = 3p).

It includes all the combinations end

per�utations for 2, 3, 4, 5, 6, 7, 8 and 12 sequent
attacks�
(1) I = ap (one part, one attack).

Three invariant form s : a or b or c .

A = a p , 2ap, . . • map .

This is equivalent to I (S = p).

(2) I = a2p (one attack to a part, tv,o sequent parts)
Three invariant forms : ab, ac, be •
E
a ch inva riant form produce s 2 attacks and has
2 permuta ti o ns.
This is equivalent to I ( S

= 2p).

Further combinations of ab, ac, be are not
necessary as it corresponds to the form s of (3) .

(3) I = a3pt (one attack to a part, three sequent
parts).
One invariant form : abc.
The invariant form produces 3 attacks and has
S permutations:
a bc, a c b, cab, bac, bca, cba .

•

0

All other attack-groups (A = 3 + n) develop
from this source by means of the coefficients
of recur rence.
Figure V"J..

I(S

= 3p) : attack-groups for one simultaneous p.
(please see page 33)
Development of attack-groups by means of the
4

•

coefficients of recurrence.
A = 4a; 2a+b+c; a+2b+c; a+b+2c.

C.

41
P'i = 2 t = �4 = 12
Each of the above 3 permutations of the coefficients
has 12 general permutations.

Total in general permutations: 12 •3 = 36
Total in circular permutations: 4 • 3 = 12
A = 5a.
Forms of the distribution of coefficients :
5 = 2+2+1 and 5 = 1+1+3

A = 2a+2b+c ; 2a+b+2c; a+2b+2c
p _ 51
5" - 2 1 21

120 = 30
2•2

.
Each of the 3 permutations of the first form
of distribution has 30 general permutations.
30•3 = 90.

Total:

•

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Figur� VI.

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0

34 .
A = a+b+3c; a+3b+c ; 3a+b+c .
P,r

5 ! = 120 = 20
6
31

=

Each of the above 3 permutations of the second
form of distribution has 20 general permutations.
Total: 20•3 = 60.
The total number of cases: A = 5a.
General permutations : 90 + 60 = 150
Circular permutations : 5 • 6 = 30

A = 6a.
Forms of the distribution of coefficient s :
6 = 1+1+4 ;

A = a+b+4c;
PLV =

1+2+3;

2+2+2.

a+4b+c;

4a+b+c.

6 1 = 720 = 30.
41
24

Each of the above 3 permut ations of the first
form of distribution has 30 general permutations.
Total: 30•3 = 90
A = a+2b+3c ;

a+3b+2c ;

2a+3b+c ;

3a+2b+c.

3a+b+2c ;

2a+b+3c ;

Each of the a bove 6 permutations of the second
form of distribution has 60 general permutations .
Total: 60 · 6 = 360.

u

0

A = 2a+2b+2c.

The third form of distribution (invariant)
has 90 general permutat ions .
The total number of cases: A = 6a.
General permutations : 90 + 360 + 90 = 540.
Circular permutations : 18 + 36 + 6 =

60 .

A = 7a.
Forms of the distribution of coefficients:
7 = 1+1+5;
A = a+b+5c;
•

1+2+4;

a+5b+o;

2+2+3;

3+3+1

5a+b+c .

7 1 = 5040 = 42
120
1 - 51

p

_

Each of the above 3 permut ations of the first
form of distribution has 42 general permutat ions .
Total: 42 • 3 = 126.
A = a+2b+4c;

a+4b+2c ;

2a+4b+o ;

4a+2b+c.

P1 =

4a+b+2c;

2a+b+4c;

7 ! = 5040 = 105
2 ! 41
2 • 24

Each of the above 6 permutations o f the second
form of distribution has 105 general permutations.
Tot al: 105 • 6 = 630
A = 2a+2b+3c ;
p =
7

2a+3b+2c;

7_,
! -- =
..,._.__,;,.

2! 31 21

3a+2b+2c .

= 210

0

36.
Each of the above 3 permutations of the third
form of distribution has 210 general permutations.
Total: 210 •3 = 6 30
A = 3a+3b+c;
P.., =
,

71

31 31

3a+b+3c ;
=

a+3b+3c .

5040 = 14
0
6·6

Each of the above 3 permutations of the fourth
form of distribution has 140 general permutations.
•

•

Total : 140 •3 = 420
The total number of cases: A = 7a
General permutations: 126 + 630 + 630 + 420
Circular permutations:

21

= 1806

+ 42 + 21 + 21 = 10 5

A = Sa.
Forms of the distribution of coefficients:
8

=

1+1+6 ; 1+2+5; 1+3+4 ; �+2+4; 2+3+3

A = a+b+6c;
_

PS -

a+6b+c;

6a+b�c.

8 1 _ 40,320 _ 56
61 720 -

Each of the above 3 permutation.s o f the first
form of distribution has 56 general permutations.
Total : 56•3 = 168
A = a+2b+5c;

a+5b+2c;

5a+b+2c;

2a+b+5c;

2a+5b+c;

5a+2b+c .

8!
21 5 !

=- �0, 320
2 • 120

= 168

Each of the above 6 permutations of the second
form of distribution has 168 general permutations.
Total : 168 •6

= 1008

0

37.

A = a+3b+4c;

a+4b+3c;

4a+b+3c ;

3a+b+4c ;

3a+4b+c;

4a+3b+c.

= 4 0� 320 = 280
6 • 24

81
P.I _ 31 4 1

F.ach of the above 6 permutations of the third
form of distribution has
Total :

A = 2a+ 2b+4c;
•

280
28 0 •6

2a+4b+2c;

81
21 2I 41

Po• =

general permutations.

= 168 0

4a+2b+2c

=
= 42 0• ,2 520
420
• 24

Each of the above 3 permutations of the fourth
form of distribution has 420 general permutations .
Total : 4 20 • 3

A = 2a+3b+3c ;
p =
g

3a+ 2b+3c ;

= 1260

3a+3b+2c

81
= �0,320
2 •6 • 6
21 3 1 31

= 560

Each of the above 3 permutations of the fi fth
form of distribution has 56 0 general permutations.
Total: 560 • 3 = 1680
The total number of cases : A = Sa
General permut ations: 168 + 1008 + 168 0 + 126 0 +
+ 1680 = 5796
Circular permut ations :

24

+ 48 + 48 + 24 + 24 = 168

A = 12a.
Forms of the distribution of coefficients :

u

8 = 1+1+10 ; 1+2+9; 1+3+8 ; 1+4+7; 1+5+6 ;
2+3+7;

2+4+6 ; 2+5+5;

3+3+6 ;

3+4+5;

2+2+8 ;

4+4+4.

0

38 .
a+b+lOc ;

A =
p

, i.

_
-

•

a+lOb+c ;

lOa+b+c

121 _ 4791p011 600
3,628,800
101 -

= 132

Each of the above 3 permutations of the first
form of distribution has 132 general permutations.
Total: 132 •3 = 396

•

A = a+2b+9c ;

a+9b+2c;

9a+b+2c;

2a+b+9c;

2a+9b+c;

9a+2b+c.

p

_
,:a. -

121
21 9 1

= 479,ppl,§OO = 660
2 • 362,880

Each of the above 6 permuta tions· of the second

form of d istribu tion has 660 general permutations .
Total : 660 • 6

= 3960

A = a+3b+8c ;

a+8b+3c;

8a+b+3c;

3a+b+8c;

3a+8b+c;

8a+3b+c.

12 1 = 47�,001,6p�
6 •40,320
31 81

= 1980

Each of the abov e 6 permutations of the third
£orm of distribution bas 1980 general permutations.
Total : 1980 • 6 = 11,880

A = a+4b+7c;

a+7b+4c;

7a+b+4c;

4a+b+7c;

4a+7b+c ;

7a+4b+c.

p'"

=

12 1

41 7 I

=

479,001,600
24•5 , 040

= 3960

Each of the above 6 permutations of the fo urth

u

form of distribution bas 3960 general permutations.
Total: 3960•6

= 23,760

0

39 ..
A = a+5b+6c;

a+6b+5a;

6a+b+5c;

5a+b+6c;

5a+6b+c;

6a+5b+c.

=

121

51 61

479,001,600 = 5544
120•720

Each of the above 6 permutations of the fifth
form of distribution has 5544 general permutations .,
Total: 5544• 6
A = 2a+2b+8c;

=

2a+8b+2c;

121
2 1 2 1 81

32,264
8a+2b+2c

= 479,001, 600 = 2970
2• 2 • 40,320

Each of the above 3 permutations o� the sixth
form of distribution has 2970 general permutations.
Total: 2970 • 3 = 8910

A = 2a+3b+7c;

2a+7b+3c ;

7a+2b+3c ;

3a+2b+7c;

3a+7b+2c ;

7a+3b+2c.

P,,_,

121
479,001,600 = 7920
=
= 2 1 31 7 1
2 • 6 • 5, 040

Each of tr1e above 6 permutations of the seventh
form of distribution has 7920 general permutations.
Total : 7920•6

= 47,520

A = 2a+4b+6c;

2a+6b+4e;

6a+2b+4c;

4a+2b+6c;

4a+6b+2c;

6a+4b+2c .

12 1

21 41 6 1

= 479,00�,600 = 1386
2 • 24 • 720

Each of the above 6 permutations of the eighth
form of distribution has 1386 general permutations .
Total: 1386•6

=

8316

0

40.

A = 2a+5b+5c;

5a+2b+5c;

121

2 1 51 51

5a+5b+2c .

=
= 479,001,600
16 ' 632
2 •120 • 120

Each of the above 3 permutations of the ninth
form of distribution bas 16,632 general permutations.
Total: 16 , 632 •3
A = 3a+3b+6c;
=
P,4

3a+6b+3c;

12 1
31 3! 6 1

= 49,896

6a+3b+3c

_ 479,001,600
6 • 6 • 720
0

-

= 18,480

Each of the above 3 permutations Qf the tenth
form of distribution has 18,480 general permutation s .
Total: 18,480•3

= 55,440

A = 3a+4b+5c;

3a+5b+4c;

5a+3b+4c;

4a+3b+5c;

4a+5b+3c ;

5a+4b+3c.

P.·�

=

12 1
= 479,oo�,spo
6 • 24 •120
31 41 51

= 27,720

Each of the above 6 permutations of the eleventh
form of distr ibution has 27,720 general permutations .,
Total: 27, 720 · 6

= 166,320

A = 4a+4b+4c
479
1,6
00 = 34, 650
12
1
,00
,
=
=
P,:2..
4 1 4 1 41
24•24•24
The twelfth form of distribution (invariant) has
34,650 general permutations.
The total number of case s : A = 12a.
General permutations: 396 + 3960 + 11,880 + 23,760 +
+ 3�64 + 8910 + 47,520 + 8316 + 49,896 + 55,440 + 166,320 +
+ 34 , 6 50 = 443, 3 12.
Circular permut ations : 36 + 72 + 72 + 72 + 72 + 36 +
+ 72 + 72 + 36 + 36 + 72 + 12 = 660.

0

Lesson CLXXXV .
A � 4a; 2a+b+c ; a+2b+c; a+b+2c

Figure VII .

-.

�otal in ge�eral p�mutations: 12+12+1.2 = 36
Total in circular permutations: 4+4+4 =

A = 5a; 2a+2b+c; 2a+b+2c ; a+2b+2c

-

•J

- -

"I

'(I I I I••

•1

- I'I
I :1 I

J -

••

.

!"l .,. II.I"...II

.l

"I I I

.

!] I

12

..

)l �I I• I

••
-

·•I I -i IfI

.

c:=========

-

Total in general permutations; 30+30+30 = 90
Total jn cjrc11Jar perrm1tatiaos: 5±5+5 = 15
A = 5a; a+b+3c; a+3b+c; 3a+b +c

,.

�
,,

0

- -

-

-

--•

"J

�

.-.

�

• f'.

-

-

'"'

( 'I I 1'11 �I

-

•
•• •I• • •

•

.
• •
". I• ■III•

-

. 11"111 I I 1•111111 1 11.111
1

Total in general permutations: 20+20+20 =
Total in circular permutations: 5+5+5 = 15
No. t. l.ooae Lear

0

The entire total for 5 attacks: in general permutations: 150

0

42.

1n circular permutations: 30
A = 6a; a+b+4c ; a+4b+c; a+b+4c
•

•

Total in general permutations: 30•3 = 90
Total in circular permutations; 6•3 = 18

A = 6a ; a+2b+3c ; a+3b+2c ; 3a+b+2c; 2a+b+3c ; 2a+3b+c ; 3a+2b+c
•

•

•

•

,

- 'IIIJI I II I
Total in general permutations: 60• 6 = 360

I. I

�
�
.......... ­

fl

I I t:a

•

I-. I II"• 1•1II I 11

Total in circular permut ations: 6 • 6 = 36

No. 1. Loose Leaf

KIN
•&as e·w,y. "· Y.

0

43.

A = 6a; 2a+2b+2c
•

The entire total for 6 attacks: in general permutations: 540
1n circ11Jar perm11tations: 60

A = 7a; a+b+5e; a+5b+c; 5a+b+c

•

0

Total in general permutations: 42•3 = 126
Total in circular permutations: 7 • 3 = 21

A = 7a; a+2b+4c ; a+4b+2c ; 4a+b+2c; 2a+b+4c; 2a +4b + c; 4a+2b+c
•

�
-.;;i.� -

..........

No. t . l.oose Leaf

KIN

159& s·way. N. Y.

0

44.
IA,,

-- -

, ------:c-·....-

-

'

IIJ.;. •

-

1-1

I •••- f 1 -l • 11 _ 1

I

_
-_
_
_
-_
_
_
-_
_
_
-_
_
_
-_
_
_
_
_
_
_
_
_
_
_
-_
_
_
_
_
_
_
-_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
-_
_
-_
_
_
-_
_
_
-_
•

Total in general permutations : 105•6 = 630
Total in circular permutations: 1•6 = 42
A = 7a; 2a+2b+3c; 2a+3b+2o; 3a+2b+2o

.,

-

l

.

-

:,,III

"' "�-I lJ•

•
( ("" '1 1,.IJI

-l

• IIIJI

Total in general perm utations: 210• 3 = 630
Total i•n circ11Jar permutatj otls: 7•3 = 2l
A = 7a ; 3a+3b+c; 3a+b+3c; a+3b+3c

•

Total in general permutation s : 140•3 = 420
Total in circ11Jer pernn1tations: 7•5 = 21
The entire total for 7 attacks :

in general permutations: 1806

in circular permutat ions : 105
No. 1. Loose Leaf

KIN

°

lti9S 'll •�y. If.Y.

0

A = Sa; a+b+6c ; a+6b+c; 6a+b+c

45.
•

'�

-

-

..

I ••

t•I1 I-III

'
•

•• •,

..

Total in general permutat ions : 56•3
Total in circular permutations : 8 • 3

,,

A

-- Ba; a+2b+5c; a+5b+2c; 5a+b+2c; 2a+b+5c; 2a+5b+c ; 5a+2b+c

.,

r

= 168
= 24

�. -

11

••

--- -

--

. ".

'

•;a

I II_J_

Ll&.o.l!fl I I Ill■

I ajI III

•
ll I

•
••
I 1· 11•■1
I
II [

- I I 1 11111! ■ l11fl
Ill

•

�... ....�.

Total in general permutations: 168•6 = 1008
Total in circ ular permutations: 8•6 = 48
No. 1 . Loose Leaf

"''"

auo

169S 11·way. N. Y.

0

•

46.

A. :::

8a; a+3b+4c; a+4b+3c; 4a+b+3c;
3a+b+4c; 3a+4b+c; 4a+3b+c
�

'
•

J

'4

•

..

1

.'.

,,

.

.,

-,8

,,

,

n1 ...,.,.,,, I� - �1 ---'1l"'\'I"'\-

-

,,

.,,
ij

•

i--":l"'\1J � •Tl c:,.y-r� I rt.T�

-

-

-

I" II·,:.,l"'l 1

•
•

"
·.

-

•

•

Total in general permutations :
Total in circular permutations:

/

I•

1680
48

A = Ba; 2a+2b+4c; 2a+4b+2o; 4a+2b+2c

--

..,- I�('!.JI f"\( -. I"'I
,-LJ -

11I

-'

1111 I 111,

•

�

0

·,. I - •
'"I 1,, I
I •"
- r, , ,s
Total in general permutat ions : 420 •3 = 1260
Total in circular permutations: 8 • 3 = 24

No. t. Loose Leaf

KIN

°

169$ 8 way. N. Y.

0

47.

A = Ba ; 2a+3b+3c; 3a+2b+3c; 3a+3b+2c

-

.ti•'-I

•1 1::111
••

I .J. III

I JI'.I
I _J• II •I.a •

Total in general permutations : 560 • 3 = 1680
Total in circ11Ja.r perro,1tat1ons: 8•3 = 24
The entire total for 8 attacks: in general permutations: 5796
in circular permutations : 168
A � 12a; a +b+lOc; a+lOb+c; lOa+b+c

r-

u
:.::::

A

r,

�
'

A

I '. .I;,,,: LI 1.1II

• ...
I
Total in general permutations : 132•3 = 396
Tota l in circular permutations; 12•3 = 36

-.

=

I• -

I

•

•

12a ; a+2b+9c ; a+9b+2c; 9a+b+2c;
2a+b+9c; 2a+9b+c; 9a+2b+c
•

•

0
No. 1 . Loose Leaf

0

48.

0
•

•

______--_
___--_
-_
-_
-_
_
_
-_
______
-_
_
-_
_
__"'"----- --------- _--_
-_
_____
-_
_
-_
-_
-___
-_
-_
_____

Total in general permutations : 660•6 = 3960
Total in circular permutat ions: 1 2 • 6 = 72
A = 12a ; a+3b+8c; a +8b+3c; 8a+b+3c;
3a+b+8c; 3a+8b+c; 8a+3b+c

•

•

,

....
•

I

.,, I

I (

I

.....,

,,,.

11

,

Total in general permutations : 1980•6 = 11 ,880
Total in circular per.mutations :

72
No. t. Loose Lear

KIN

1$9, tfway. N. Y.

0

49.

A = 12a; a+4b+7c; a+7b+4c; 7a+b+4c;
4a+b+7c; 4a+7b+c; 7a+4b+c

-

�

---

• .. I

.

.

I

,,

,..

• ,

. -I

•

I• - II I':'III • ..
•

-

.... I•. T_I1 •

•

J'aI'• I

-

• , �,· u11..- ,. ,..1r1-

•

Total in general permut ations: 3960 .6 = 23, 760
Total in circular permutations : 12 •6 - 72
A = 12a; a+5b+6c; a+6b+5c; 6a+b+5c;
5a+b+6c; 5a+6b+c; 6a+5b+c
•

•

•

,-

- -- -

-

'

IrI

•

I

••

'

- r•
' .,..,,1....e

-

.•

•• •,

0
No. t. Loose Leaf

"'"

169f',

uND
s·••Y· N. Y.

0

50 •
•

Total in general permutations : 5544•6 = 32, 264
Total in circular permutations : 12•6 = 72
A = 12a; 2a+2b+8c ; 2a+8b+2c; 8a+2b+2c

Total in general permutations : 2970 • 3 = 8910
Total in circ ular permutat ions : 12•3 = 36
A = 12a; 2a+3b+7c;
3a+2b+7c·

2a+7b+3c;
3a+7b+2c ·

7a+2b+3c;
7a+3b+2c
•

•

•

Total in general permutat ions : 7920•6 = 47, 520
Total in circular permutations : 12 • 6 = 72
No. 1. Loose Lear

KIN

■uo

169� e'way. ti. Y.

0

A = 12a; 2a+4b+6c;
4a+2b+6c;

51.

2a+6b+4c; 6a+2b+4c;
4a+6b+2c; 6a:t:4b±2c

'
-

•

-

I

•

- l■I I
-·

l 11 l�)JI

Total in general permut ations : 1386 •6 = 8316
Total in circular permut ations: 12 • 6 = 72
A �

12a; 2a+5b+5c; 5a+2b+5c; 5a+5b+2c

--

a

II[ ■•

r.

I ..-•-I ll

I

1""1 -11 1

..

Total in general permutations: 1�632 • 3
Total in circular permutations: 12 • 3

•

•

•

= 49,896
= 36

No. t. l.ooae Leaf

Kll«
169$

s·•�y. N. Y.

0

0

-

52.

A = 12a; 3a+3b+6c; 3a+6b+3c; 6a+3b+3c

•

--

�

t

I

..

'

•

J

r,...,.,

-

lz r•-I

•JI

•

'

•

,-nTJCt

•

Total in general permutations : 18480 • 3 = 55,440
Total in circular permutations : 12 • 3
36
A =

12a ; 3a+4b+5c ;
4a+3b+5c;

3a+5b+4c;
4a+5b+3c ;

5a+3b+4c ;
5a+4b+3c

•

0

•

•

�

,-

I

•

•

•.I.

•

•

•

Total in general permutations: 27 , 720 • 6 = 166,320
Total in circular permutations: 12·8 = 72

0
__ .,.,,...._

� ,,

No. 1. Looae Leaf

0

53.
n I ,..

- II:

The entire total for 12 attacks� in general permutations : 443, 312
in circular permuta tions: 660

I

0