6.

'

Lesson LXXXV.

SEVE�TH-CHORDS .. S(7)
Diatonic System

A
l
4

. .

1�

-Q

The Third
Inversion

The Second
Inversion

The First
Inversion

Fundamental
Position

t

--.

�

8(7) Seventh­ 'S (�) Fifth­
Sixth Cl1ord
Chord

:i

s(i) Third­
Fourth Chord
-

S(2) Seco11d
Chord

.
A seventh-chord including all inversions

has 24 positions altogether.

The classical system of harmony is based

on the p ospu�ate of resolving seventh: seventh moves
one step_ d9wn_.

'1

•

This postulate provides a medium for

.continuous progression of S(7) as well as establishes
the entire system of diatonic continuity (cycles).

One movement is required to produce C 3 : the

movement of the seventh alone.

wise transformation.

It results in a clock'

•

,
•

\

7.

...

7
-· -·····----

.

-

Two movemen ts are required to produc e

c5:

th e movement of the seventh and of th e fifth one step
d vm
7

It results in a c rosswis e transformation.

•

1

•

5

Three movements are required to produce C1:

the movement of the seventh ., of the fifth and of the

third one step down.

...

transformation.

It results in a counter-clockwise

Taking the chords over two from C 3 we obtain:

I

e7

7�

7
-

j'

--

T

5':>l

3�

✓ l t=\
7
� 5.,1\

This type of mu$iC may be found among

co ntrapuntalists of XVII - XVIII Ce nturies.

Palestrina .,

Bach, Haendel obtained sim ilar results by means o f
suspensions.

Assigning a system of cycles we can produce

•

8.

L

a continuity of S(7).

taken in a n y position.

The starting chord may be

Example: C,. + c, + Cr + C7 + c, + c, +

Cs-

-e- ,,..

• •

•
•

•

•

This continuity being entirely satisfactory

. harmonically may prove, in some cases, unsatisfactory

u

melodically on account of continuous descending in all
voices.

This form, when desirable, may be eliminated

by means of the two devices:

(l) exchange of the common tones

,

(2) octave inversion of the common tones

The same continuity of cycles assumes the

following form:

ffi�-(]I•

!

....

,� I

--

9
�

-.
•

�

,____

-

--

•

�-

&7 J
_
I

j

9.

u

Obviously C 1 does n ot provide common

tones, thus excluding the above devices.

As the continuity of the second type

offers better melodic forms for all voices, it may be

desirable to pre-set certain melodic forms in advance.

For example, it is possible to obtain, by means of
continuous Cs, the following form of descending

through two parallel axes (b) or (d) , as in the music

of Frederick Chopin.

This

may be harmonized as follows:

--------- -------·----- ----- ----

Diatonic C0 becomes a necessity in order to

avoid the excess of saturation typical of the continuity

of S(7) with variable cycles.
•

(J

The principle of moving continuously through

C0 is based on the exchange and inversion of common

tones.

10.
The exchange and inversion of adjacent
'

functions brings the utmost satisfaction.

Neverthe-

less it is not desirable to use the two extreme

functions for such yurpose as the y cause a certain

amount of harshness.

...,

1

�

•

u

An example of continuity of the C0 i

--

-

....,

-

•

•

u

The final form of continuity of S(7)

consists of the mixtures of all cycles (including

C0) based on a rhythmic composition of the coefficients
of recurrence.

11 •

1':----

-

'

•

•

•

12 ..

L

Lesson LXXXVI.

Resolution of 8(7)

Resolution of an 8(7) into an 8(5) in all

•

positions and inversions may be defined as a transi­
tion from four functions t,o three func_tions,

8(5) in the four-part harmony and with a

normal doubling (doubled root) consists of:
1, 1, 5, 5

•

8(7) consists of:

•

1, 3, 5, 7

When a transition occurs, ob viously the

root takes the place of the seventh.

Therefore the

resolution is provided through the motion of 8(7 )

> S(7)

and the substitution of one for the seven, i.e., the
function which would become a seven th in the

continuity of seventh-chords becomes a root-tone when

a resolution is desired.
Example:
C

b

r

7

r::_,

•

� 1

5

> 1

3

) 5

1

) 3

Note: Do not move 8(7)

-

,I

) S(5) in the C0

u

13.
Resolutions in the Diatonic Cycles .
•

•

This case provides an explanation why

.

'

a·tonic triad acquires a tripled root Ein d loses
its fifth.

i

.

(J

-

•

,

I

S,.
�-

�

.

,

c ,,

....
Preparation of S(7)

There are three methods of preparation

of S(7), i.e"", of transition from S(5) to S(7):

0

(1) suspending

(2) d escending
(3) ascending

14.

(__;

The first method is the only one

producing the positive (C3 , Cr , C7 ) cycles.

The methods (2) and (3) are the outcome

of the intrusion of melodic factors into harmony.

They are obviously in confl ict with the nature of

harmony {like the groups with passing chords) as they

produce the negative cycles, which in turn contradict
the postulate of the

resolving seventh universally

observed in classical music •

•

The technique of preparation of the

seventh consists of assigning a certain consonant

function (1, 3, 5) to become a dissonant function (7)
and to either sustain the assigned function of the
8(5) over the bar line or to
ward or upward.

m ove it one step down­

The last two forms of

occur on a weak beat.

a seventh must

Exercise in different positions, inver­

sions and cycles the S(5) --1-➔ S(7) transition.
(1) Suspending:
•

1

7

•

C
•

•

3

7

5

7

15 ..
(2) Descending:

7

(3) Ascending:

,

•

C-7

C-3

(Please see next page)

' .

16.

(1) Suspending

,

,

A

•

J

.3

f.t,

•

•

r�

•

•

•,I

b!'

,,

-

J3

..

-,

I,;' I

___,_..- C.

�•
J�

r

a�
f

rI

rI

r�
�l

0

(2) Descending

•

t

I

'

,,.-, .
,I ,I

L.J '

-;

-61

LII

-

r

-

,

'!

r�

�

·�

-

- •7

,,,
,'

..,

V

.

•

!C

•

•..

(3) Ascending

,.,

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17.
The mixture of the zero, positive and

negative cycles provides the final form of
continuity based on 8(5) and S(7).

For m ore efficient planning of such

continuity use bar lines for the layout.

The

preparation of S (7) may be either positive or

negative; the resolution - al1ivays positive.
Example:

I

I

e7

l!.-.3

•

-

�s-

e.s

C!.o

•

•

•

\

-------------�-----

C

•

I

C,-s--

I

•

18 •
Lesson LXXXVII.
The negative system of tonal cycles may

be used as an independent system.

The negative

system is in reality a geo metrical inversion of the

•

Every principle, rule or regula­

positive system.

tion o f the positive system becomes its own converse
in the

negative.

Chord structures become E
o f the
@

original scale.

E,

®

which forms

Chord progressions are based on
the C- 3•

Clockwise \.transformations

become counterclockwise and vice versa.
Positive

Chord Struetures

Negative

Tonal Cycles:

Negative -

--------➔�-+ Positive
Transformations:
•

---- --.➔ +

2

19.
The postulate of resolving seventh for

the negative system must �be

seventh moves 9ne step up,

read: the negative

The C-� requires the

negative seventh and negative fifth to move one step

up.

The C-1 requires all the

to move up.

tones except the root

This system may be of great advantage

in building up climaxes.

Negative:

Positive:

•
•

F

The root-tone of the negative system is

the seventh of the p ositive and vice-versa.

It is easy to see how the other cycles

would operate.

C-.r

C..r

C

c.

,

20.
If one wishes to read the negative system

as if it were positive, the rules must be changed as
follows:

The

The C-r

,

•

C-3

requires the ascending of 1

The C-7

A. Groups

n

n

"

"
"

" l and 3

n l, 3 and 5

�pecial Applications of S(7)
either
S(7) finds its application' in G•,
"

'

as the first or the last chor d of the group.

The following forms are possible:

8(5) + 8(�) + S(�)
)

8(5) + sc:) + s(i)

<

8(5) + s(!) + 8(2)
(

>

s(7) + s(:) + S(6)
(

8(7) + 8(6) + s(4)
3
)
(
4
8(7) + s(:) + 8(2)
(

)

The cycle between the e,xtreme chords of

G, may be either C0 , or C3 , or c,.
'1

21.

L

Besides Gt

there is a special group

where s(i) is used as a passing chor d .
•

two forms of this group.

(A)

GJ(r)

-

- S(5)(!') + S(i) + 8(7)

There are
or 8(5)
or S (5)

These two forms may be used in one

direction only.

All positions are available.

Rule of voice-leading: bass and one of

the voices of do u. b ling move stepwis e down.
tones sustained.

Common

The cycle between the extreme chords in

the first form is C 3 ; in the second form it is C0 •

••
••
>

•

22.

B. Cadences.
The following applications of S(7) are

commonly known:

(1)

rv,, ..

(2)
(3)

II .,
r

(4)

II"
J

"

"

"

"

"

"

"

"

n

In addition to this the fo�lowing forms
'

may be offered:
(5)

Any of tl1e
previous
forms

"

(6)

Besides these t:t1ere are two ecclesiastic

forms:

(f)
IV (:tr;)

(1)

1,

-

(2)

I5'

- rv©

(Ut")

3

Is-

!5"'

(please see next page)

•

,

23.

.

Ii
-

�

.

-•
•

�

-0

�

-

�

r,>

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�

0

I�

I

I

,_
� 1,�,�,4)

I-,•

-

-

,

•

!>.

,i

II

(/,)

•

1
'.�

-

�

•

J
�

(4,'-, �4)
r
.I

•
:I

•
•

•

0

••

,.

-

�

-

--

--,
�

(1)

t

(5)

{LI)

d:4,.

rJ.

�

•

�

•••

,t;t:i,,,,

·-

.

�

.
-

- t -- - t

•

IS

.

-,

,�

'1!)

•

�

J O S E P H

S C H I L L I N G E R

C O R R E S P O N D E N C E

C O U R S E
Subject: Music

With: Dr. Jerome Gross
Lesson LXXXVIII.

SyPJme,tri!! Z,ero Cycle (C 0 )
Symmetric C0 offers an extraordinary·

versatility on S(7) as seven structures of the latter
have been in use.

If evolving of the forms of \,s (7) would

have been devised scientifically, they would be
obtained in the following order.

Taking c - e - g - bi, (4 + 3 + 3) as

the most common form and producing variations thereof,

we obtain two other forms:

C - e p - g - b \:? (3 + 4 + 3)

and

c - e � - g � - bp ( 3 + 3 + 4)

Taking another form, c - e - g - b

(4 + 3 + 4) , we obtain two other forms:

•

c - e - g

and

- b

(4 + 4 + 3)

c - ep - g - b

(3 + 4 + 4)

•

These two grot1ps of three are distinctly

different but as music has made the use of them for

quite some time our ear does not find it objectionable

any longer to mix all of them in one harmonic continuity.

0
•

.

.

'

•

2.

C - e

-

Besides these

�

•

Sl.X

forms there is a

- gb, - b►lt (3 + 3 + 3 + 3) and might have been

C - e - g ft - b�

(4 + 4 + 4 + 4) if there would not

be an objection t o the fact that c - b

ir

monic octave.

is an enhar­

A continuity on symmetric C 0 of all seven

structures offers-603 0 permutations.

Thus a c - chord

alone can move (without changing its position and

•

without coefficients of recurrence being applied) for
50�0 x 7 = 36,280 chords.

''

A method of selecting the best of the

available progressions must be based on the following

principle: the best progressions on symmetric E0 are
due to i dentig of steps or to contrary motion.
4

•

Example

(1) Identity of Steps:
all semitones

. (2) Contrary motion:

(_)

•

'

The principle of variation of the

chord-structures and their positions remains the

same as in S(5):

Position

Structure

Variable

Constant

Constant

Variable

S(7) in the following table has a dual

system of indications: letter symbols and adjectives.
The adjectives are chosen so that they do not adhere

•

alone.
to the degrees of any scale but to structure
'

'

Thus, such a common adjective as "dominant" ha d to
be sacrificed.

s,

8(7) Table of Structures

S.t

Mtt-
An Example of Continuity in C0 :
Structures: S 3 + S7 + s� + s,
Coefficients (r5+4): 4S 3 + S7 + 3S.., + 2s, + 2S 3 +
+ 3S7 + S 44 + 4S,

U.

'

4.

L
•

,_

•

�

I

r..

I.�

�

--..,

, ...
-

i,, _

b�
'

I

.e-

,.�
,

.,

�

-

..

w�

.
�

•

,r

-

"

+
'y

•
I

. -----

•

--

�

�

-?S
I�

- ---

As in 8(5) , any combination of the forms

of 8(7) by 2, 3, 4, 5, 6 and 7 may be used.
Type III (8Y]Dme_tric),

As in the previous cases when dealing with

symmetrical tonics C0 may be applied either to any of

the tonics or as a continuous change of chord struc­

tures occurring with each tonic.

When structures of 8(5) and 8(7) have to

be specified in one continuity, they must have full

indications :
•

82(5);

s,(7);

82 (7); s3 (7);

S 7 (7)
•

83(5);

s,(5);

S�(5) and

s�(7);

ss(7);

s,(7) ;
•

u

Two Tonics (./2)

As the J2 forms the center of the

octave the progression 1
positive and ./2

l=

) F ) is

.) ../2 (C

) 2 (Ff ) C) is negative.

The system of Two Tonics which was

continuous on S(5) becomes closed on 8(7).
formations correspond t� Ci' •

Trans­

•

-1
•

•

�-

•

Si!!!!

-

>[7)

.,

'
..

,
,

..�

'- �

J-

f

�

S(5")

SC1)

•
•

•

.

�

'

�,

OR�

�

...�

-

S(s-)

• I, '-"

•

.,.
•

Three Tonics (3,!2 )

Continuous system: moves fcur times.

Transformations correspond to C3•
•

To obtain 8(7)

•

after an 8(5) use the position which would corres­
pond to continuous progression of 8(7).

•

..•
,

•

5(7)
....

Example of Conti nuity:

-

•

sCs-)

S(1),

S(s)

S(1)

5{.r}

7.
Four Tonics ( '!/2 )
pond to C3•

�

/

i.
I

Closed system.

S(7) after 8(5) as in Three Tonics.

c.XAM PLE OF C:,QN'flN61 li1'.
.fill

·i

•

,.

�

;:rt,

jt
r

•

,.

•
•
�
r ••

',,.

,

•

,'
-r

�

,-t)

'

¼-· i�

-�
+

�

II" -

11

-

,

Transformations corres­

-

: --

""

•

:"6,·
.-

-

--

'

-s
,.-e

•

,
•

SiX Tonics ( 6J2 )

Continuous system: moves two times*

Transformations correspond to c,.
as :in previous cases-

:s(7) after 8(5)

Both positive and negative

progressions are fully satisfactory.

�

To obtain the

negative progressions read the positive backwards.

I

8.

,.

,�-�
•

I' •
I

�

��

f,(-,}
II

•

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•

�

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.. ,
.I -

S(,)
�-

H•il

S(,J

-e

l

�-

5�)

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-,
i

•

- -

•- •o-

,

�

•6

�

r

!,(7)

�

I ,•

!,(7)
r�

.,..,

i

S{7)

5(5)

--

6
·�

'•

I,·•

-�
,�

5(s)

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9.

Twelve Tonics ( ':/2)
Cl ased sy:stem. Al1 spec:ificatio1)s and

appl1cet1ons as jn S1x Ton1cs.

- <.

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No. 230 Loose Letf 12 Stave Style -Sttndard Punch

- .-

.

•

•

10.
Lesson LXXXIX.

r

it possible

Jixbri9- Five-Par t Harmony

The
to

technique

of continuous 8(7) makes

evolve a hybrid five-part harmony,

where bass is a constant root :tone and the fou r upper
functions assume variable forms of S(7) with respect

•

to

bass •

.eit her on root, or
By placing an S(7)
.
third, or fifth, or �seventh' of the bass root we
•

obt ain all forms of S in five-par t barJQ_ony.

An S(5)

th
t
o
be
ed
wi
t
h
the
addi
t
ion
of
represen
t
1
has
3 . (t he
.
so-called "added sixth ").

Forms of Chords in Hybrid Five-Part (4 + 1} Harmony
The 4
Upper
Part s .

5

3

l

S

13

The
Bass

l

The
Forms of
Tension 8(5)

7

5

9

7

3

11
9

7

1

3

5

1

1

1

S(7)

S(9)

S(ll)

l3

11
9

7

1

S(l3)

It is p.ossible to move con tinuously eit her

form or any of the combina tions of forms in any

11.

C

rhyt hmic form of continuity.

It is impor tant to

realize that the to nal cycles do not correspo nd in

the upper four par t s to the to nal cycles :in the bass

when the

forms of tension are variable.

F or example,

f - a - c - e may be 3 - 5 - 7 - 9 in a DS(9) as well

as 7 - 9 - 11 - 13 in a GS(13).

In such a case a

progression C s- for the bass v-,ith 8(9) -�
> S(l3)

produces C0 for the upper four parts.

The principle of exchange and octave­

•

inversion of the common tones holds true.

Three forms of harmonic continuity will be

used in the followi ng illus trations (these forms o f
co ntinuity are

well).
•

applicable in the four-part harmo ny as

When chord structures acquire greater te nsion

and also when the compensation for the dia t onic

deficiency is requir ed, it is often desirable to use

preselected forms of chord-structures yet moving

diatonicallz.

Such system has a bass belonging to one

defi nite diatonic scale, while the chor d structures
acquire various acciden tal s in or der to pr oduce a
definite so nority.

In the general classificatio n o f

the harmonic progressio ns the latter t ype is known

as diatonic-szmme�ric,

Three Type� 9f H�monic Progressions
I. Diatonic

I I. Diatonic-Symmetric

III. Symmetric

12.
The following examples will be carried

out in all three types of harmonic continuity.

Constant and variable forms of tension will be
offered.

In order to select a desirable form of

structures for the forms of different tension it is

advisable to select a scale first, as such a scale
•

For example, if the

offers all forms of tension.

scale selected is tl - d - e - f:ft - g - a - b Ii,,

•

S (5) = C - e - g - a ;

8(9 )

8(7)

= c - e - g - b � - d;

8(13) = c - b�- - d - r f' - a.

0

8(11 )

Though the- same scale wou ld be ideal for

the progression, it is not impossible and not very

undesirable to use any other scale for the chord­

progressions.

(please

see following pages)
•

u

�ybrid Five-Part Harmony
(Tables an
. d Examples)

(1) Continu,ity 9f S,(5) [moJ1omials]
Scale : c - d - e - _f� - g - a - b'P
Type I .

Type II .

-

•

Type III.

•

,

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13.

14 .

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· ""'

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(ii

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�"'-

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.

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(2) Continuity of p(7) [monomials]
Type I •

•

Type I I .

Type III.

'tt
=-•-··=
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No, 230 Loose Leaf 12 Stave Style -Standard Punch

15.

(3) 9ontipuity_ 9f_ S(9). lmonomials]
Type I •

Type I I .

•

Type III.
,,

�

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No. j30 Loose Leof 12 Stave Style -Standard Punch

16.
(4) Continui;ty of s(,11) [monomials]

Type I .

•

Type II.

Type III.

>

(5) Continuitz of S(lp)_ [mpnomials]
I

17.

-

Type I I .

Type III .

-

...

,

r
\......, ____

Combinations by two (binom.-ials) , three (trinomials),
fo·ur (quadrinomials) and five (quintinomials) may be
=i=-=
s:.!:<
a ---s=im
�=
in
d�
e.:e
.___
· l=:.:!a=r:-...:wz.::a�y'-J•
______ ___ _ ____ _
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18.
Table of Combinations

Arabic numbers in the following tables represenc
Chord Structures (S)

'

Compinations by 2

5 + 9

9 + 13

7 + ll

7 + 13

5 + 11

5 + 13

11 + 13

9 + 11

7 + 9

5 + 7

10 combinations, 2 permutations each
Total: 10 x 2 = 20

•

pombiP?,tions by 3

,

5 + 7 + 9

7 + 9 + 11

5 + 7 + 13

7 + 11 + 13

5 + 7 + 11

5 + 9 + 11

9 + 11 + 13

7 + 9 + 13

5 + 9 + 13

5 + 11 + 13
10 combinations, 6 permutations each

Total: 10 x 6

= 60

•

19.

5 + 7 + 9 + ll

Combinati ons by 4

7 + 9 + 11 + 13

5 + 7 + 9 + 13

5 + 7 + ll + 13
5 + 9 + 11 + 13

5 c ombinations, 24 permutations each

Total: 5 x 24 = 120 .
Combµiations by 5

5 + 7 + 9 + 11 + 13

l combination, 120 permutations

Total: l x 120 = 120
•

All other cases of trinomial, quadri­

nomial, quintinomial and bigger combinations are
treated as c oef'fic ient,s of recurrence.

Example : s> = 28(5) + S (7) + 28(9) =

0

= 8(5) + 8 ( 5) + 8(7) + 8(9) + S(9),

i.e., a quintinomial with two identical pairs.

•

20.

Coefficients of recurrence may be applied to the composition of
continuity consisting of the forms of variable tension.
Examples

Type I .

28 (5) + 8 (9) + 8 ( 13) + 28(7)

..

Type II.

,

Type III.

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J':1:����ff No, 230

Loose Leef 12 Steve Style -Standard Punch

L

Lesson XC.

�i nth-�hords. 8(9)
Diatonic
- System

Ninth-chords in four-part harmony are

used with the root-tone i n, the ba,ss only, tl1us

forming a hybrid fo ur-part harmony [ like S (5) with
the doubled root].

of 3, 7 and 9.

The three upper parts consist

The seven th a nd the ninth are

su bjec t to resolution thro ugh the stepwise downward

motion.

If one fu nction resolves at\,a time it is

always the higher o ne (the ninth) .

o ne fu nctio n at a time produ ces C0 •

A r.esolution of
Other cycles

derive from the simulta neo u s resolu tions of both

f u nctions (the ninth and the seventh) .

No co n­

sec utive 8(9) are possible throu gh this system
[ they alternate with S(7) and S (5) ] .

The reason for resolvi ng the 9th a nd

not the 7th first i n C0 is the latter resu lts i n a

chord-stru cture alien to the usual seven-u nit

diatonic s cales (th� i ntervals in the three upper

voices are fourth� .

u

22.

Positions of 8(9)

As bass remains constant, the three upper

voices are subject to 6 permutations resulting in the
corresponding distributions.

Table of Positions of 8(9)

.e.
•

Resolutions of 8(9)

---\
1

r

cycles only.

Resolut:bons (except C0 ) produce posi tive

C 3 is characteristic of Mozart, Clementi

and others of the same period.

C� (the second resolu­

tion) is the most commonly known, especially with b v

2 :3 .

in the first chord (making a dominant chord of

F-major of it).

contrapuntalists.

c1

is ch aracteristic of Bach and

They achieved such progression

through th e idea of two pairs of voices moving in
thirds in contrary motion.

and f� and add 8(5) g-minor.

Read the last bar with bv
All these cases of

resolution were known to th e classics through melodic

•

man ipulatioris ( contrapuntal heritage) and not through
th e idea of independent structures we call S(9) •
. Preparation of S(9) beaes· a great
''-

There is

similarity with the preparation of 8(7).

even an absolute correspondence in th e cycles with

resp ect to technical proc edures.

The same three m ethods con stitute the

teclmique of preparation (suspending, descending,
ascending).

(1)

Suspending:
•

Table of f>reparations .

=

9

7
•

5

7

9

:3

5

7

Cs
(2) Desceµdi:gg:
3 ..,

;

9

1 -- ;Ji 7
Co

5 -- ,' _ 9

3 - ;. 7
C-3

7- � 9
5 - ,,. 7
C-5'"

24.
(3) fo.scending:

3. ) 9

5-

1- � 7

3_ ➔ 7

•

Prepar�tions of S(9)
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25 .
It follows from the above chart that

some of the pre}Jarations of 8 (9) require an S(5) ,

It is practical
•

some - 8 (7) and some allow both.

to have 8 (5) or 8(7) preparing 8 (9) with the root in
the bass .

The first form of preparation was known

to the classics as �ouble suspension,
Example:

•

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Similar cadence was used in major.

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Example dfi Continuity Coptaining 8(9):

1

•

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1

'I

7

7
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7

.}

_q
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b

7

f

C
•

Homework :

(1) Make complete tables of preparations and
resolutions from

all positions.

(2) Writ e diatonic continuity containing 8(9) .

(3) Make some modal transpositions of the examples
th us obtained.

(4) Write continuity containing S(9) in the second
type (diatonic-symmetric) of harmony .

Select

cbo:Rd-struc tures from the examples of hybrid

I

five-part harmony.
•

•

71'

27 .

L

Lesson XCI.

Ni,nth-Chords, 8(9)
Symmetric Syst em •.

The above described class�cal (prepara­

tion-resolution) t echnique commonly used in the

diatonic system is applicable to the symmetric system
as well.

cycles:

•

and '�-

Symmetric roots correspond to the respective

c, - to ./2;

C3

-

to J./2 and f./2;

c 7 - to 6.f2

With this in view, continuity cor1'sisting of

8(5) , S(7) and 8 (9) and operated through the
classical technique may be offered.

Symmetric C 0 is quit e fruit less when S (9)

alone is used, as the upper three fu nctions (3, 7, 9)

produce an incomplete seventh-chord, the permutations
of which (3 H 7, 3 .( ) 9) sound awkward wit h the
exception of one: 7 H 9.

As 8(9) in the hybrid four-part harmony

is an incomplet e structure (5 is omitted) , the

adjectives may be applied only wit h a certain allowance

for t he 5th.

There are two distinctly different

•
families of S (9) not to be mixed except when lll Co :

(1) The minor sevent h family

'

seventh family
(2) ..The major
•
•

0

•

28 •
•

The minor 7th family includes the

following structures :

•

•

You may attribute to them the following

adjectives in their respective order:
•

large

�
, s,

'

., \t S2 - diminished
,\1s 3 - minor

,'vs.., - small
The major
7th family includes the
•

following structures :

Their respective adjectives are:
, \r s , - major
, � S2

-

augmented I

7� S

-

augmented I I

3

•

These are the only possible forms.

It seems that all combinations of the two

families, except the ones producing consecutive
seventh ( ., t,· S 14 � ➔ 7 -S 1 ;

, r s) < ➔,'1s� ;

7- Sa' � 'J �s, ;

, � s, H 1� s�) , are satisfactory when in C0 • On the
different roots the forms of S (9) must belong to one

family.

Example of C0 Continuity:

•

0

..

Full indication for 8(9) when used in

combinations with S(5) and S(7) :
1 'v s , (9) ;

1 ' s 2 (9) ;

7PS 3 (9) ;

7 � S , (9) ;

7 , s 3 (9) ,
1 q s 2 (9);
Two Tonics ( ./2 ) . The technique corresponds to C s- •
•

..

•

•

I

30.
To resolve the last cl1ord of the

preceding table use position

technique.

(w

of the resolution

Example 0£ Continuity:

-

,

Three Tonics ( 3../2 ) .

The technique corresponds

--

\),J_pf �1 "Pl &\.t :
Awt(vJAt� �r�PS

•

•

•

31.
In order t o acquire a complete under-

standing of the voice-leading in the preceding table
of prog ressions (9 - 6 - 9 - 6 etc. ), reconstruct

mentally an 8(7) instead of an 8(6) .

Then the first

two chords will appear in the following positions:

•

It is clear now th at d� and f� are the

necessary 7 and 9 of the following chord.
•

Example of Co ntinuity:

C.

u

C

The te· chnique corresponds to C 3 •

Four Tonics

-I

,

◄

-I-

•

-I
•

.�

�

,
·-

•-

Progression

Preparation

Resolution

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Example, pf Continuity:
,

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Six Tonics ( 6./2 ) .
Resolution

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The techniqu,e corresponds to c 7 •

Preparation

Progression

7

7

7

•

33.

•

The above consecutive sevenths are

unavoidable with this technique.

The position of every S (9) is based on

the assumption that t11e preceding chord was S(5)
and not S(7) .

Continuity: 8(9) + S(7) + 8(5)

-

•

5

7

7

The negative system which may be obtained

by re ading the above tables in pos ition � is not as
desirable with the se me dia as the positive.

concerns the following '�,12 .

The same

More plastic devices

(general forms of tran sformations) will be offered

later.

Twelve Tonics ( � ) .

•

u

1

The technique corresponds to c 7 •

•

34.

Continuity : 8(9) + S(7) + S(5)

•

•

Homework : Exercises in the

different symmetric

systems containin g 8 ( 5) , 8(7) and 8(9)

with application of different structures

and too C0 betw.een the roots .

•

,

J O S E P H S C H I L L I N G E R
C O R R E S P O N D E N C E C O U R S E
With: Dr. Jerome Gross
Lesson XCII.

Subjec�: Music

Four-Part Harmony (Continuation)
Eleventh-Chords. S(ll)

•

Diatonic System •

•

•

•

Eleventh-chords in four-part harmony are

used with root-tone in the bass only , thus forming a
hybrid four-part harmony [like 8(5) wit h the doubled

root ].

0

The three upper parts consist of 7, 9, 11.

An S(ll) has an advantage over S(9) as the upper

functions .form a complete S ( 5 ) . • All three upper

functions are subject to resolution through the stepwise

dovmv1ard motion.

Resolutions of less than t hree upper

funct ions produce C0 •

this sys t em.

No consecutive S (ll) are possible through

They alternate wit h the other structures�

For the reasons explained in the previous

chapter the C0 resolutions must follow in t he d�rection
of the decreasing func�ions: first 11 must be resolved,

•

then 9, then 7.

When two functions resolve simultane­

ously they are 11 and 9.

chain of resolutions.

An S (ll) allows a continuous

S(ll) 11 �
,

S(9) 9

� 8 (7)

An eleventh-chord through resolution of

the eleventh becomes a ninth-chord; a ninth-chord

through resolution of the ninth becomes an incomplete
4

•
•

seventl1-chord (without a fifth) , or a complete S ( 3 )
as in the correspondin g resolutions of S(9) ; an
incomplete seventh-chord throug h resolution of the

seventh becomes a sixth-chord wit h the - doubJ.ed third.

Positions of S(ll) .

As bass remains constant, the three upper

voices are subject to 6 permutations.

Seventh, ninth

and eleventh form a triad corresponding to a root, a

third and a fifth while the bass is placed one degree

higher.

A c S (ll) has

bass raised one step .

ari appearance of b S (5) with a

•

3.

Resolu_ti on,s of S(ll),

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S{<9 S('} S(:t)

s(,1) 5(1)

•

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As it fo llows from the above table, when

S(ll) resolves into S (9) in C0 , S(9) has its prope�

structural constitution (i.e., 1, 3, 7, 9) .

For the

same reason the c7 -resolution does not appear on this

table, as the structural constituti on of S(9) , into

•

resolve, is 1, 5, 7, 9 and this

which S(ll) would

does not sound satisfactory according to our musical
habits.

.

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S (11

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The above resoluti ons correspond to the

classical resolutions of the triple suspensions.

•

•

•

L

Preparation of S (ll) in the positive

cycles has the cyclic correspondence with the

preparation of S(7) and S( 9) through suspensions.

Nevertheless the manner of reasoning is somewhat

•

different in this

c ase •

As S(ll) has an appearance of an S ( 5)

•
•

with a bass pl.aced one step higher, the most

logical assumption is: take •S ( 5), move its bass one
step u p and this will pro duce an S(ll) of a proper

structural constitution.

In such a case the relation

of th e three s,t ationary upper fm1ctions is C0 •

Being common tones they may be inverted or exchanged.

The first case gives a clue to the prep ara­

tion of oth er cycles ( p ositive and negative as well).
The method of preparation implies merely

the most gradual transformation ( .:::
.., �
-;)
_, or 4:
) ) for the
three upper functions.
To prepare S ( ll) after an 8 ( 5) in C 0 move

all u p per functions down scalewise and leave the

bass stationary (which is the converse of the first
proposition).

(please see next page)

•

5.

Preparations of S(ll)

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When all tones are in common in the

three upper parts it is advisable t o use th e

suspension (over th e bar) me tmd.

When some of the upper par ts move and

some remain stationary ei ther the within th e bar or

•

on may be used •
the over the bar preparati
.

.•
•

Charac teristic progressions and cadences

.

where all forms of tension [from 8(5) to S(ll) ] are
applied:

•

(please see next page)
•

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No. 230 Loose Leaf 12 Stave Style - Standard Punch

•

-

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.,

•

a.
•

Lesson
XClII,
•

· S(11)
Eleventh-Chords,
Symm�t�ic System .

The above described technique of diatonic

progressions containing S(ll) is applicable to the
symmetric system as well.

•

The cyclic correspondence

previously used remains the same.

Thus preparations

of S(ll ) are possible in all systems of the symmetric
roots, while resolutions can be performed
only when
the acting cycle is Cl ( 8../2 and '!/2 ) '-and C.r (./2) .

There is no difficulty wit h any preparation of S(ll)

after a resolution, as the latter aiways consists of

L

1, 3, 5 and therefore may be connected with the

following chord through the usual transformations .

Contrary to 8(9), 8(11) produces a highly

satisfactory C0 , due to the presence of all functions

without gaps in the three upper parts.

As in the nint h-chords, there are two

distinctly different families of S(ll) not to be
mixed except when in C0 •

The distinction becomes even

greater than before and the danger of mixing more

dangerous.

The structural constitution of S(ll)

permits the classification of such structures as
S(5 ) with regard to their ·three upper functions.

9.

Forms of S(11),•
The Major
Seventh Family

Minor
Seve nt h Family
The

,

fJ

•

• "'· J

-

t,,· . '. •- ..J
,r;;�

These are the only possible forms as the

•

diminished in the first group equals ( enh.a.rmonically)
a diminished 8 (9) and the augmented in the second

group equals (enharmonically) the second augmented

8(9) with a fifth and without a third.

•

The selection of better progressions in

C0 for the continu ity of S ( ll ) must be analagous to
the selection of forms ibr S(5).

shall not be used�

Consecutive seventh

10.

Po

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,

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•
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.

Example of C0 Conti nuity .

p c;,

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Full indications for S(ll) when us�d in combinations
with other structures :
7b

_ ___ ___ ___

0

II:

,

,,

Q

1 s2(11);

•

1J s., {11)

The technique corresponds to C s . Clockwise
or counterclockwise transformations for
continuous S(ll).

Resolution

-

•

__ ____ _ ___
9

7�s,
(11);
__;.
_

Two Tonics (J2) .

.L./J

7�S2 (11) ;

S, (ll);

7P s, (11)

Preparation
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,

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p�

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••

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irtll

•
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Continuous S(ll)

Progression

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•

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11 .

You may consider the upper three parts either as 7, 9, 11
in _ and � transformations or as l, 3, 5 with a displaced
Example of Continuity.

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Exam le of Continuit •

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Four Tonics ( �) .

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and � ; transformati·ons only.

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13.

14 .

Homework in the field of S(ll) must correspond to
that of S(9 ) , ut ilizing various structur'es, forms

and progressions .

The transformation technique is

applicable to diatonic and diatonic-symmetric

prog ressions as well.

•

•

0

I

•

15.
Lesson XCIV,

a

Hybrid Four-Part Harmony

•

The general technique of transformations

for - the groups with three fu nctions may be adopted

for the generalization of the forms of voice-leading
.
•

in a hybrid four-part harmony.
perform the

The three u pper parts

tran sformations corresponding to the

grou ps with three functions, and the bass remains
constant.

The following technique is applicable to

any type of harmonic progression (diatonic, diatonic­

C

symmetric, symmetric).

The s pecifications for the

follow ing forms of S are cl1osen with respect to

their sonority.

The ones marked with an asterisk in

unmarked ones ..

The charts of transformations for the

the following tables are less commonly used than the

latter are worked out and you can easily supplement

them for the ones marked wit h the asterisk.
(please see ne�t page)
•

u

16.
Forms of Hybrid Four-Part (3 + 1) Har100ny
The Three

,

upper

parts.

The bass.

.

Forms of

•

tension.

5

5

7

7

9

3

3

.9

3

7

7

9

1

13

5.

3

1

3

1

7

1

1

1

1

1

1

1

*

S ( 5 ) S ( 5)

S (7)

11

13

13

9

11

7

7

1

1

*
*
*
8( 7) S (9 ) 8 (9) 8 (11) 8 ( 13) 8 (13)
'

When the numerals ex pressing the functions in

a group are ident.ical with the nunierals of the following

group, certain forms of transformation, such as constant

abc, have to be eliminated on account of complete parallel­

ism.

When the numerals in the two allied groups are

partly ident i cal so me of the forms (constant a , constant b,
const ant c) give either favorable or unfavorable partial

parallelisms.

Th e partial parallelisms are favorable

when the parallel motioo of functions forms desirable
intervals with th e bass.

They are unfavorable when it

causes con secutive motion of the seventh or ninth with

the bass (consecutive seventh, consecut ive nil.1th) .

As the actual quality of voice-leading

depends on the struct ures o f the two allied chords,

upon completion of all these ch arts in musical notation
you will be able to make your preferential select ion.

•

•

-

•
•

17.
When the numerals in the two allied

groups are either partly or totally different, often

the constant abc transformation becomes the most

favorable form of voice-leading.

There is a natural

compensati on in this case: homogeneous structures are
compensated by heterogeneous transformations and
'

..

heterogeneous structures are compensated by homogeneous

transformations. For ex ample, if the allied groups
both are S(5) the constant abc transformation would

be impossible: 1 � 1, 3

) 3, 5

consecutive octaves and fifths.

) 5,· which gives
''-

On the contrary, when

the functions have different numeral s you acquire the

smoothest voice-1.eading through this particular

transformation .

When two allied groups have different

or partly different numerals for their functions, the
first group becomes the original group and the

following group becomes the Erime group.

When a

transformation between such two gro ups is performed

the prime group in turn becomes the original group for
the next transformation.
The Original
Gro up

a
C

b

The Prime
Group

18.
For example, by co nnecting 8 (5) + 8 (9) +

+ 8 (13) we obtain the following numerals in their
corresponding order:
8(5)

8(9)

l

3

5
.

13

7

7
9

When you co nnect the functions of S (5)

•
•

9

3

S (13)

with the fu nctions of S (9) the first group is the

group.
original group, and the second -- the prime
''
When you connect the functions of 8 (9) with 8 (13)

the functions of 8 ( 9) form the original group, and
the fu nctions of 8 (13) -- the prime gro�p.

Here is a complete table of transforma­

tions.

Fomms of Transformations

in the �omogeneous Groups
•

,- ;!

"

J

a

,.,. \,
b
c
�
a..-+ b
.

b4c
c➔ a

k

...

Const.
a

�

' a"

®

Const.
b
.

a

t;J @

Const.
C

©

@ t) ©

b

c t== ., b

a

)' C

a�a

a

) C

C

)' b

b

b

) b

b

C

➔a

C� C

C

... :,

b➔a

, "j

C

) C

➔b

Const.
abc

a

>b

)a

a

b
C

@

➔a
)b

➔c

19 .
Forms of Transformations in
the Heter9geneous Groups

The Original

The Prime

Group.

Group.

a

..

�

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,:_.,,

le;.

....

'-,//

Const.
a

Const.
b

Const.

Const.
abc

.

a )' b '

'a.➔a'

a ) b•

a-4-

b➔c•

c -"7 b f b ➔ C '

b ➔b '

c -+ a '

b ➔ a ' c --+- b '

c➔a '

•

u

b•

b

C

ci

a� a '

•

a➔ c '

C

b -t a •

c➔c'

b )'b'
c· �c •

20 •
•

Lesson XCV.

Here are all the comb:wations for the

two allied gro ups talcen, applied to all forms of
tension.

Binomi_?l Combir12.tions pf_ the Origi�l

•

and the Prime Groups.
0

.

•

S (5) (
S (5) �
S (5)(
S(5) �

➔ 8 (7)
) S (9)

) S (ll)
f 8 (13)

S (7) ( ) S (9)

8 (7) (
8 (7) (

), S (ll)

) 8 ( 13 )

S (9) ( ) S (11)

S (ll)�

78 ( 13 )

S (9) � 1 8 ( 13)
'

10 Combi nations, 2 permutations each.

Total number of cases: 10 x 2 = 20 •

•

Table of transformations for the twenty

binomials consisting of one original and one prime group.

Each S tension is represented in this table by one

.

structure only.

The sequence of the forms of transforma-

tions in this table remains the same for all cases:
(1)

f� ;

(2) :' � ;

(5) Const. c ;

•
•

u

(3) Const. a ;

(6) Const. abc •

(4) Const. b;

•

21.

•

3
5

S(5)

) S(7)

)5

1

,

1

➔3

1

➔1

1� 5

1

)7

3

)r 3

)7

3

)5

3

➔3

3

3

➔• 5

5

�5

5

�3

5

�7

➔5

5

>3

5

�7

➔7

)3

..

8 (7)

•

>- 5

3

➔3

3

)1

5� 5

5� 3

5

)1

5

)- 3

7

7

1

7

➔5

7

➔5

1 >9

1� 7

1

)3

3� 3

3

)7

B -4 3

34 5

3

5 -4 5•

5� l

➔1

7

7' 3

➔ 8 (5)

') 1
.), 3

3

➔

I

8(5)

1-

➔ 8 (9)

)7

1� 9

1

:3� 9

3� 3

3 -4, 9

3

5

➔7

5

5� 7

5

➔3

➔3

➔7

►3

-8 ( 9)

)1

3

>l

7

)5

9 ---4 3

9

7

➔3

9

·3

7� 5

7

9

➔1

)5

5

➔9

3

➔3

5 ••➔ 9

➔ 8 ( 5)

3

)3

•

➔5

>3
�l

1--..:, 1
9

➔5

3 =• )- 1 .
7--r 3
9

➔5

•

22.

8 ( 5)

1
-

1➔11

1

3➔11

3

➔9

1� 9
3� 7

3 7> 9

54 9

5

➔7

5 ➔11

5 ➔ 11

7➔ 3

7➔1

)9

1�11

1

5➔11

3� 7
➔9

➔7

5

5

➔ S (ll)

)- 7

)7

.

•

8 (11)
7

➔3

9➔ 5

,,

)1

7� 5

7➔ 1

9

9 ➔5

)l

11➔3

ll-t3

8(5)

1
-

), 9

1➔13

l

', 7

➔ 8(5)
7➔ 5

9➔3

9➔1

➔ 8 (13)
1➔13

1

)9

1

), 7

)7

3

)9

3

') 7

3 ➔13

3

}9

3

5 -) 7

5

�9

5

5

-:, 1

5 ➔13

8 (13)
•

7 ➔5

7---jl

7 )5

)5

9�1

9-45

9�3

13 -4 3

13➔3

13➔ 1

13 ➔ 1

5___:;.13

.

► 8(5)

7 ➔3
9

11 )• 5

11➔1 11➔5

�13

�9

9➔3

7

)3

9➔
.. 1

13-t5

7

)1

9

)3

13-+ 5

23.
.

S (7 )

➔ 8 (9)

5

)7

3

>9

3

)3

3 --4 9

3� 7

3

)3

5

)3

5

)3

5

➔9

5

-, 1

5 -.+ 3

5

>7

7

�3

7_._.::, 7

7 • �7

7

13

7

�9

7--r 9

3� 7

3�5

3�3

7, ) 3

7� 5

- � S (7)

6(9)
� -4 5

3

)7

5

)7

7

)3

t7� 7

7

)5

�3

9

➔5

9�5

9

➔3

9

u

)3

9- ➔ 7

9➔ 7

•

➔ S ( ll )

S (7)
3

)'!)

5 �ll

...'

➔7

>11

)7

5� 7

5➔11

5----t 9

5 ---j- 7

5➔ 9

7

7

7

7

➔*l

7➔11

3

➔

9

:, 9

, (11 )

7�5

7

9,7

9� 3

-t 3
l
l

)7

11�5

3

)11

➔7

3

►9

3

)7

➔ 8 (7)

7➔3

7� 7

7➔ 5

7➔ 3

',7

9� 5

9

9 )5

11--+ 5

11 ➔ 3

9

➔3

ll-r7

11➔7

24.

S (7 )
•

)9

3 )13

'7

5 ➔13

5 -4 7

5 ➔13

7 --..::, 7
.

7

_.,

)9

7

>7
)9

➔ S(13 )
3➔13

3� 9

3

)7

5➔ 9

5

>7

5➔ 9

7

7 ➔13

7�13

)7

•

•
•

S (13)

•

7➔ 5

7

)7

9 -4 7

9

)3

L3 ➔ 3

13➔ 5

.

� S(7)

7� 3

7

}7

7

)5

7

)3

>7

9

)5

9

�3

9

;> 5

9

tl.3 ➔ 5

S (9)

13➔3

13 ➔7

13 ➔7

� S(ll)

➔7

·➔ 9

3 ➔ 11

3➔ 7

3� 11

3�9

3

'➔11

7

)7

r7➔11

7� 9

7 )7

9 �7

9 -4 9

9 ----r 9

9 ) 7

9➔ 11

7 , ;>- 9
9 ➔ 11

7 )9

7� 7

7�3

9 ---f 7

9

9

S (ll)
7➔ 7.

9 ➔9
Ll4 3

1➔ 9

9 -,\ 3
11 ➔7

7 -4- 3

9➔9
Ll➔7

➔ S(9)

1 1 :,)-3

➔3

11➔ 9

>7

11➔ 9

•

•

25.

➔ S ( 13)

5 (9)

l3

), 9

3

)13

�

)7

3�13

3� 9

3--"t 7

7� 9

7

)7

7--r 9

7 �13

7� 7

7 ➔13

-.:, 7

9 ---) 9

9 --+ 9

9

e➔ 7

9--tl3

9➔13

7�
' 7

7--,. 3

13➔ 9

13 ➔ 9

.•
S (13) � S (9)

•

7

)7

7

)9

7

)'3

7

9

}9

9

). 3

9-4 9

9

134 7

1 347

13-4 3

l3---t 3

S (ll)

7

9

►9

7➔13

7

)13

94 7

9 ➔13

11� 7

.,'

u

)9

9➔11
13 ➔ 7

11➔ 9

7411
9

)7

13➔ 9

)9

➔7

9➔ 3

9 -4 7

➔ 8 (13)
7➔13

7

)- 7

1 1➔9

9➔ 9

7___,::,. 9
9� 7

9

)9

11 ,1

11➔13

11➔13

S ( l3)

) S (11)
7 )11

7

)9

7 )7

)9

9

)7

9

)7

.

7-4 7

9 ➔11
13 ➔9

9

1347

13➔11

➔9

13➔11

S ( 5) �
> S ( 7)
.

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Please complete all tables.
Make addi tional tables for; 8(5) �
� s(5); S(7) - > S(7) ;
S(9) ➔ S(9 ) ; S(ll)· ➔ S(ll);
S(13)

) S(13).

==
==
====
---======= = ================= ====================================
•· �•·=:::-IIP,o.

USIC1.•o1

BRAND

No. 230 Loose Leaf 12 Stave Style-Standard Punch

•

28.

It is easy to work out all cases in

musical notation applying
tonal eye les.

each case to all three

As in the p�evious cases, continuity

may be composed in all three types of harmony

(diatonic, diatonic-symmetric and symmetric ) .

Struc tures of different tension may be selected for
the composition of continuity.

Different individual

styles depend upon the coefficients of recurrence

applied to the structures of different tension.
'
.
The first of the follo,ving two examples
o f continuity is produced through the stru c tures

of constant form and tension [ S (l3) ]., and the

•

second -- illustrates continuity of variable forms

,

and tensions distributed through r3+2•

(please see next page)

u

29.

Continuity of Groups_ with I�entical Fun��ions
Type TT.

z-

,,,-.,J

�

•

Continuity of Groups with Different Functions
28(9) + 8(7)�+ 8(13) + 2S(ll);
Typ& III.
'J'J
4

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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
Lesson XCVI.

C O U R S E
Subject: Mup ic

Generalization of Symmet�ic Progressions
illL•

•

The forms of symmetric progressions here­

tofore used in this course of Harmony were based on
monomial symmetry of the u.niform intervals of an

octave.

In order t.o obtain various mixtures

{binomials, trinomials and polynomials) of the

original forms of symmetry within an octave, it is

necessary to establish a gen,eral nomenclature for all
intervals of an octave.

As all intervals are special

cases of the twelve-rold symmetry, any diatonic form

may be cons�dered a special case of symmetry as well.

The system of enumeration of intervals may

follow the upward or dovmward direction from an y

established axis point.

As both directio ns include

all intervals (which means both positive and negative

tonal cycles), the matter or preference must be deter­
mined by the quan titative predominance of the type of
intervals gen er·ally used.

It seems that the descending

system is more practical, as smaller numbers express

2.
the positive steps �n three and four tonics, and the
negative -- on six and twelve tonics.

In the following exposition . the descending

system will be used exclusively.

•

This does not prevent

you from using the ascending system.

Scales of Intervals within one Octave Range :

•

.

•

•

Descending System:
C� C

=

0

C � C =

c -...
) b = l

c ---J
/ bV = 2
C ---j' a

=

c� g = 5
c ---,> ff= 6

c -�
> e

=

=

Two Tonics: 6 + 6

C

)

C

) e- = 4

C

>

=

f

5

f = 6

7

7

c -�
> g =

8

C -+
/""a� = 8

10

c-...
) a

=

c� b

=

9

c) b'v = 10

c -�
) d�= 11

) C1 ;:::
C...

) d� = l

c --) f

c -➔) e� = 9
c -�
) d

0

c

3

c ---,> a� = 4

c� f =

Ascending System:

12

C

---j'

I

C

11

:: 12

Monomials

Three Tonic s : 4 + 4 + 4 or 8 + 8 + 8

Four Tonics:

3 + 3 + 5 + 3 or 9 + 9 + 9 + 9

Six Tonics: 2 + 2 + 2 + 2 + 2 + 2 o r 10 + 10 + 10 + 10 +
10 + 10

Twelve Tonics: 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
or ll+ll+ll+ll+ll+ll+ll+ll+ll+ll+ll+ll

3.
Thus, each constant systempf tonics

•

•

becomes a form of monomial period icity of a cert ain

pitch-interval, expressible in the form of a constant

number-value, which in turn expresses the quantity of

semitones from the preceding pitch-unit .
:

In the light of t h is system the problem of

mixing the various tonics (or any interval-steps in

general) becomes reduced to the process of composing

binomials, trinomials or any more extended groups
(such 89 rhythmic resultants, their modificat ions

through permutations and powers, series of growth),
i.e. , to the rhythmic distribution of steps.

u

The vitality of such groups, i.e. , the

quantity of their recurrence until the completion of

their cycle, depends upon the divisibility-properties
of the sums of their interval-quantities.

The total

sum of all number-values expressing the intervals

becomes a divisor to 12, or any multiple thereof.

signifies the motion of a certain group through an

This

octave (or octaves).

For example, a binomial 3 + 2 has 12

recurrences until it completes its cycle, as 3 + 2 = 5,

and the smallest mult iple of 12, divisible by 5 is 60.
This is true of all prime numbers being used as
divisors.

•

I

C - A -

I

y-

�·-

E -� - B -

r

- c*- B

I

B - Gf- F..- D"'- C t- A:f"-

,E ►- C - �� - G -

•

E- D -

- E

C'

The above property makes the mixtures of

three m:id fo ur tonics very desirable when a long

harmonic span is necessary without a need
' of the
.'-

variety of steps.

The process of division serves as a testing

tool of the vitality of compound symmetric groups.

Two tonics close after two cycles, as
6 + 6 = 12, or 12
6 = 2;

r�+ closes after one cycle, as 3 + 1 + 2 +
3
+ 2 + 1 + 3 = 12, and
= l;

i�

- r5 + closes after three cycles, as 4 + 1 +
4
= 3.
+ 3 + 2 + 2 + 3 + 1 + 4 = 20, and

�g

A greater variety without deviating from

a given style may be achieved by means of permutations

of the members of a group.

For example, a group witµ

a short span may be revital ized through permutations:

•

5.
( 3+1+2) + ( 3+2+1) + (2+3+1) + ( 1+3+2) + (1+2+3) + (2+1+3)
�-- ·- E � - D ► - C - B ,- G - F:ft- F� - D - C
o r .. C - A - G•- .....
f

The selection of number values is left to

the composer' s dis cretion.

•

If he wants to obtain a

tonic-dominant character of classical music,
the onl y
•
'

thing he needs is the excess o f the val ue 5.

Anyone equipped with this method can dodge

the extremities by a cautious selection o f the
•

coefficients of recurrence.,.

For instance, in order to

produce the style o f progressions which lies so mewhere

between Wagner and Ravel it is necessary to have the
5, the 3, and the 10 in a certain proportion, l ike

C - A - F - '.ltt

C*- D�-

C - A

�
- E - F

etc.

Naturally, the selection of the ten sions

and the forms of structures in definite proportions

is as important as the selection o f the forms o f

progressions whe n a certain definite style must be­

produced.

y
6.

On the other hand, this method of fers a

woriderful pastime, as one can produce chord pro­
gressions from any number combinations.

Thus, a

telephone directory becomes a source of il1spiration.
Example

Columbus 5 - 7573

•

5 + 7 + 5 + 7 + 3 is equivalent to

•

C - G - C - G - C - A.

''

This progression closes after 4 cycles:
•

C - G - C - G - C - A - E - A - E - A - rW-

#- C ...- F ,.- Cf:- Ftt- qf'- A�- D�- A•- D1t"- C
F
,
When zeros occur in a number-combination

they represent zero-steps, i.e*, zero cycles (C0) .
'

Then the form of tension, the structure or the
position of a chord has to be ch anged.

(please see next page)

7.
Example of Contiµuity:
Progression :

e,

-

-

�

�

f/"
.:1-

ft�

-

- -

- ----

----

-

-

-

"""�

*
I

-

r5 +3

1

(,,

-

8.

•

•

Lesson XCVII.

Applicat ion of,,the G_eneralized Symmetric
Progressions to Modulation
•
• •

The rhythm of chord progressior1s expressed

in number-values may serve the purpose of transition

from one key to anoth. er.

This procedure can be

approached in two ways: (1) the connect ion concerns

the tonic chords of the preceding and the following

•

key; and (2) any chord of the preceding key, in its

relatior1 to any cl'1 ord of the following, key.

The last

case requires movement thro ugh diatonic cycles in
both the preceding and the following key.

The technique of performing modulat ions,

based on the rhythm of symmetric progressions, consists

of two steps: (1) th e detection of the number-value

expressing the interval between the two chords, where

•

su ch connection must be established; (2) compos ition
of a rhythmic group from the numeral expressing the
interval between the abovementioned ch ords�

For

example, if one wants to perform a modulation by

means of symmetric progressions from the chord C

(which may or may not be in the key of C) to the
•

chord E�(wh ich may or may not be in the key of -/),

the first procedure t o perform is to compose rh ythm
from the interval 9.

The knowledge of the Theory of

•

Rhythm offers many ways of composing such groups :
composition of binomials, trinomials or larger

groups from the original number, or any permutations

thereof.

The quantity of the terms in a grou.p will

define the number of chords for the modulatory trans i­
tion.

Breaking up number 9 into binom ials, we obtain :

8 + 1, 7 + 2, 6 + 3, 5 + 4, and their rec iprocals.

When a binomial is used in th is sense, the two chords
are connected through one intermed iate chord.
'example, taking 5 + 4 we acquire: C - G - E.�

For
If

more chords are desired any other rhythmic group may
For exa.mple, 4 + 1 + 4,

be devised from number 9.

which will give C - A�- G - E' , i.e. , two intermed iate
chords.

When a number-value expressing the interval

between the two chords to be connected through modula­

tion is a small number, it is necessary to add the
invariant 12.

This places the same p itch-unit (or

the root of the chord) into a different octave, with­

•

out changing its intonation.

For example, if a

modulation from a chord of C to the chord of B� is

•

required, such addition becomes very desirable .
C

) B \,

=

2

B�- ) Br= 12

u

12 + 2 = 14

•

•

10.
Some possible rhythms derived from the value 14:
7 + 7

5 + 2 + 2 + 5

=

=

C - F - BP

C - G - F - E

\,

- B t:,

In cases like this rhythmic resu lta11ts may be used
as well, providing the necessary cr1anges are made.

r4 +3 = 3 + 1 + 2 + 2 + 1 + 3
•

•

Readjustment:

3 + 1 + 2 + 2 + 1 + 3 + 2 = C - A - A\, - F·4'- F � - E V- C - BP
Or:

r + = 3 + 2 + 1 + 3 + 1 + 2 + 3
5 3
Readjustment:

3 + 2 + 1 + 2 + 1 + � + 3 = C - A - G - F�- E - E�- D�- B�
Thus, all these procedures guarantee the appearance of

the desirable B V point.

When a modulation of still greater extension

is required, the invariant of addition becomes 24, 36,

or even a higher multiple of 12, from which rhythmic

groups may be composed.

Many persons engaged in the work of

arranging find this type of transition more effective

than the modulations proper.

Naturally, the selection

of the structures of different tension and form may be

made according

to the requirements of the general

style of harmony used in a particular arrangement.

11.
The best modulations will result from the symmetry
that may be detected in a given piece of music.

Even when tonic-dominant progression is characteristic
of harmonic continuity, this method may be used with
success, as it simply requires the composition of a

rhythmic group, where the original value is 5 .

In

this seemingly limited case there is still a choice

of steps: 4 + l; 3 + 2; 2 + 3; 1 + 4 •
•

Examples of �odulations

Through Symmetric Gr!)'ups

(1) Key of C

to Key;bf E V; 1 = 9
9

Symmetric Group: 1 + 3 + 1 + 3 + 1 (r 3 of 9 series)

•

u

•

12.
(2) Key of C to Key of E �

Chords to be Connected: D -- BV ;
4 + 12

= 16

Symmetric Group: r

•

u

3 + l
=
4+3
+

1=

4;

+ 2 + 1 + l + 1 +

1 + 2 + 1 + 3

13 ..
Lesson• XCVIII.
•
•

Chro matic Sys tem of Harmony

The basis of th is system is transformation

of diatonic chordal functions into chroma tic ch ordal
functions and back in to diatonic.

Chromat ic cor1tinu ity

evolved fro m th is basis emphasizes various phenomena

of harmony wh ich do not confine t o diatonic or
symmetric sys tems.

The usually known modula tions are

but a special case of the chromatic sys tem .

Chord

progress ions usually known as "alien � ,chord pro­

gressions find their exhaus tive explanation in this
system.

•

Wagner was the first composer to manipulate

in tu itively with this type of harmonic con ti nuity.

Not

having an y theoretical basic principle of handling su ch
progressions, Wagner of ten wro te th em in an enharmoni­
cally confusing way.

(J. S. Bach made an unsuccessful

a ttemp t to move in chromatic sys tems.

See 111Nell

Tempered Clavich ord� - Vol. I, Fugue 6 - bar 16) .

It

is necessary, for analy tical purposes, to rewri te su ch

music in proper notation, i.e. , chroma tically and not

enharmonically.

A more consis tent notation of

chromatic continui ty may be fou nd among th e followers

of Wagnerian harmony, su ch as Borodin and Rimsky­
Korsakov.

The chrom atic system of harmonic con tinu ity

l4 .

is based on progressions of chromati
. c groups.

Every

chromatic group consists of three chords, which

express - the following mechanical process: balance tension - release.

These three moments correspond

to the diatonic - chromatic - diatonic transformation.

A chromatic group may consist of one or more simul­
taneous operations.

Such operations are alterations

of diatonic to11es into chromatic tones, by raising or

lowering them.

The initial diator1ic tone of a

chromatic group retains its name, while being altered,
'

and changes it during the moment of release.

The two forms of chromatic operations are :

(1)
(2)
•

In application to musical names it may

become, for ins tance, g - / . _ a or g - g P - f.
steps are always semitones.

Suc·h

At such moment of release,

in a chromatic group, a new chordal function (and in

some cases the same) becomes the starting point of the

next chromatic group, thus evolving into an infinite

chromatic continuity.

u

appearance :
•

Such c ontinuity acquires ·the following

•
I

,

15.
d - ch - d

d - ch - d

d - ch - d

etc.

Chromatic continuity in such form off ers a very

practical bar distribution by placing two chords in
a bar.

•

Such distribution places the release on the

dovwnbeat and sounds satisfactory to our ear, probably

due to the habit of hearing them in such distribution •

•

As in the diaton ic progressions, the

. r the resolutio� of chordal
commonness of to11es, o

functions, or as in the symmetric progressions the

become the stimuli of motion, likewise

symmetric roots

•

in the chromatic progressions such stimuli are the

chromatic alterations of the diatonic tones.

Besides the form of continuity of

chroma tic groups offered in the preceding diagram,

two other for ms are possible.

Thus, the latter do not

necessarily require the technique of the chromatic

system.

The first of these forms of continuity

produces an overlapping, over one term:
(1)

d - ch - d

d - ch - d

d - ch - d

i.e.,

the second part produces the first term of a chromatic
group, while the first one produces the second term.

16.
(2)

d - ch - d
d - ch - d

i.e., two or more parts of harmony coincide in

their transformation in time, though the form of

transformation may be different in eahh part.

Any chord acquiring . a chromatic alteration

becomes more intense than the corresponding form of
tension, without it.

If the middle term of a

chromatic group has to be intensj,fied, the follovving

forms of tension may constitute a chromatic group:
S ( 5)

S ( 7)

8 ( 5)

S (7)

S ( 5)

8 ( 7)

S ( 5)
8(7)

8 ( 7)

S (7)

8 ( 7)

S ( 7)

The only combination which is u ndesirable,

as it produces an effect of weakness, is when the

middle term is S(5) .

Operations in a given chroma tic group

correspond to a group of chordal functions wh ich may

be assigned to any form of al terations.

As for

technical reasons the 4-part harmony is limited to
8(5) and S (7) forms, with th eir inversions, all

transformations of func tions. in the chromatic group

u
.,.,

•

deal with the four lower functions (9, 11 and 13

are excluded) •

•
•

•

'

17.
Numerical Table of Transformations
for the Cpromatic Groups.•

•

•

•

1-1-1

3-3-3

5-5-5

7-7-7

1-1-3

3-3-1

5-5-1

7-7-1

1-3-1

3-1-3

5-1-5

7-1-7

3-1-1

1-3-3

1-5-5

1-7-7

1-1-5

3-3-5

5-5-3

7-7-3

1-5-1

3-5-3

5-3-5

7-3-7

5-1-1

5-3-3

3-5-5

1-1-7

3-3-7

5-5-7

7-7-5

1-7-1

3-7-3

5-7-5

7-5-7

7-1-1

7-3-3

7-5-5

5-7-7

l-�-5

1-3-7

1-5-7

3-5-7

1-5-3

1-7-3

1-7-5

3-7-5

5-1-3

7-1-3

7-1-5

7-3-5

3-1-5

3-1-7

5-1-7

5-3-7

3-5-1

3-7-1

5-7-1

5-7-3

5-3-1

7-3-1

7-5-1

7-5-3

..

''

3-7-7

Some of these combinations must be

excl uded because of the adherence of the Seventh to
the classical system of voice-leading (descending

• ••

resolution) •
••
•

18.
The preceding table offers 16 different

versions for each starting func tion (1, 3, 5, 7) .

In addition t o th.is, any middle chord of a chromatic

group may assume one of the seven forms of S(7), and

any of the last ch ords of a chromatic group -- either

four f orms of S(5) or seven forms of S(7) .

Thus,

each starting_ point offers either 28 or 49 forms.

The to tal number of starting points for one
equals 16 .

function

These quant i t ies mus t be mult iplied by 16

in order to show the total number of �ases.
'

28

X

16 - 448

49 X 16 - 784

This applies to one initial function only, and as any

group may start with either of the fo ur functions,
the total quan t it y is 4 (784 + 448) = 4, 928.

A number

of these cases eventually excludes themselves on

accoun t of the abovementioned limitation s caused b y
the traditional voice-leading .

The actual realization of chromatic

•

groups must b e performed from the two fundamental

bases: the major and the minor .

The concept of . a

harmonic basis expresses any three ad jacen t �hordal
functions, such as:

•

5

7

9

11

13

3

5

7

9

11

1

3

5

7

9

•

19.
Due t o practical lim itations this course
5

of Harmony will deal with the first (3) basis only .

The terms major and minor correspond to the structural
co nstitut ion in the usual sense: major

4 + 3, and

All fundamental chromatic operations

minor = 3 + 4.

are derived from these two bases.
•

=

Major Basis

Minor Basis

•

These six forms of chromatic operations (3 from each

basis) are used independently.

Chromatic operations

available from the maj or basis are: raising of the

root-tone, lowering of the th ird, raising of the fifth.
They are the oppos ite from the minor basis.
(please see following pages)

--•

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..

Try to find the remaining cases through

the table of transformations of the chordal functions.
•

Please remember that the classical system of voice­

leading must be carried out through chromatic continuity.

A Seventh either descends or remains (as in traditional
cadences) ; it may even go up one semitone, due to the
chord structure, yet it positively must retain its

0

original name, like d - d*.

22.

Through the selection of different chromatic

groups (which may be used with coefficients of recurrence)

a ch romatic continuity may be composed.

With the amount of explanation offered so

far, every last chord of the preceding group (and

therefore the first chord of the following group) must

be major or minor,

•

as the operations from other bases

will be explained in the following lesson •

Example of Chroma tic_Conj:;�nui ty:

--

u

-•• --- --- -=---=---�--- � -- -...---·-- ---- -�- ..------- -++-

23.,
Lesson XCIX.
Operations from S 3 (5) and S�(5) bases
As 3 of S3 (5) is identical with 3 ·of

S 1 (5) , the fun�amental operations correspond to

S , (5) •

They are :

(1) raising of l

(2) lowering of 3

Function 5 does not participate in the
'

fundamental operat-io11S , as it is already altered.

As the form of the middle coord is pre-selected, the
fifth requires rectification in many cases though it

retains its name.

acceptable.

All forms of doublings are

As 3 of Sq(5) is identical with 3 of

S2 (5) , the fundamental operations correspond to

S2 ( 5) .

They are:

(1) lowering of l

(2) raising of 3

•

Fifth does not participate in the fund.a-

mental operations, but may be rectified.,
Figure I.
0

Operat,ions from an augmented basis •
(please see next page)

•

•

u

24.

1

-----

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fl-·

�

•
'.
4

-

I •�

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-

i

3

•

-

--

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....

IJ ,
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r
(.] D
,

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V '-"
r1 rI
•

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r

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+

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---

r

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f
.
::
Ir

-

I
�
•.

I/

.

1

-

0
•

I�

"-

-

l"I,

I

25.
Figure III,

Chroma�i9 continuity inclu?ing all pases.

/

--· - •
•

r.--.u----------------------------------------s-

(__;
., .

Chromatic Alteration of the Seventh.

Due to the classical tendency of a

downward resolution of the seventh, chromatic altera­
tions follow the same direct ion.

Lowering of the

seventh (both major and minor) can be carried out
from all forms of 8(7) .

If the seventh is minor, it

is more practical to have it as shanp or natural, as
lo1;vering of the flat produ ces a double-flat.

operate from a diminished seventh.

-

Do not

'

Figur e IV,

Examples of opepations from �pe Seventh.

I�
�

,
,
I
_....

I
i
•
j: -

r,,

••

••

•

�� .
- •]I

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• :�
,:·
_
�
,•
�-'

-

-

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.

•

I

-

.-

17

A

,� _,,I

,..--,

•

-

. _,.,' ,. -a:- F
- ',J. ,

-

r-•

�

-

•I

�

I
k; '
..
�ii?
1 -J
-S.7
- r •

•

lll rf

"

�

•

•

We can incor porate n ow all the single

operations into the final form of chromatic continuity.
Figure V.

Oper atiops from l, 31 5 and 7.
All bases.

u

/

-~

rf
.

0

.

J O S E P H

S C H I L L I N G E R

C O R R E S P O N D E N C E
With: Dr. Jerome Gross
Lesson c,

C O U R S E
Subject,: Musio

Parallel Double Chromatics

(Double chromatic operations)

••

•

•

•

Parallel double chromatics occur when

an opp osite
fundamental operatio�s �re performed from
•
base.

In such a case the rectification of the third

is required.

If, for example, we decide to lower 1

of s , (5) basis, it becomes necessary to alter 3 to its

proper basis, i.e. to lower in this case.

We shall consider the alterations of 1

and 5 as funda mental and the correction of 3 as comple­
mentary chroma ti cs.
operations.

s, (5) basis
Fundamental

The f ollowing table represents all
Figure VI,.

Parallel Double Chromatics.
s2 (5 )

Fundamental

Complementary

Complementary

Fundamental

F'\lndamen tal

Complementary

Complementary
•

2.

Fundamental chromatics represent the

middle term of a complete chromatic group, whereas

the complemen tary chroma tics do n ot necessarily perform

the conclusive movement designated by their alterations.
Thus, the scheme of chromatic groups for

the parallel double chromatics appears as follows:
•

(fundamental)

(1)

•

(complement ary)

•

(2)

(fundamental)
(complement ary)
For example, if c - c�- b� is a fundamental

operation, the

oomplementary chromatic is: e - eV .

oomplementary chromatic e� does not necessarily move

int o d.

The

It may remain or even move upward, depending on

the chordal funct ion assigned t o it .

The same is t rue of the ascending chromatics.

If o - c - d is the fundamental operation, the complementary chromatic is e p- e.

The complementary chromatic

e does not necessarily move into f.

It may remain or

even move downward, depending on the chordal function
assigned to it .

The assignment of chordal functions must be

performed for the two simultaneous operations :. funda­

m ental and complementary.

It is practical to designate

the ascending alterations as:
Bescending -- as: 7 or 5.
3
5

3
·1

or

5,
3

and the

This protects harmonic continuity from a

wrong direction and sometimes from an excess of

•

accidentals.

This remark refers to the middle term

of a chromatic group.

Figure VII,

Examples
= of Doub le Parallel Chromatics,
a

G

(please see next page)
•

•

u

'
•

•

4.

Fl� m
•

-·-------�

- -·-------

---==�

By assigning the opposite bases, we can

obtain double parallel chromatics at any desirable

place of chroma.tic continuity.

Figwe VII I .

Con�inuity of Double ?arallel Chromatics •

•

Double parallel chromatics are the

quintessence of chromatic style in harmony.

It

created the unmistakable charac ter of Wagner and the
post-Wagnerian music .

While the a n al ysis of Borodin,

Rimsky-Korsakov, Frank and Delius does not present

any difficulties for the analyst familiar with this

theory, the music of Wagner often requires transcribing

into chromatic notation .

One of the progressions

typical of the later Wagner ' s period (we find much of

it in "Parsifal") is :

-

Being transcribed into chromatic

notati on it acquires the following appearance:

•

be

•

"\ I

I
"

, 3

.-

✓� -

•

•

I

-·l;, �

(1 lr -

7

-

This corresponds to

,

�

s , (5) basis:

There are many instances when double
'

parallel chromatics are evolved on a bas�s of passing

chromatic tones.

u

They are abundant in the music of

R i.msky-Korsakov , Borodin and, lately, became very

common in the American popular and show songs ("Cuban

Long Song", "The Man I Love").

The source of passing

chromatic tones, t he technique of which we shal l

discuss later, is more Chopin than Wagner or the

post-Wagnerians.

u

7.

u

Lesson CI.

Tri£l� and Quadruple

Parallel Chromatics

Triple parallel chromatics occur when 1

is raised in S� (5) basis.

This, being the fu nda­

mental operati on, requires the correction of the

third (3f) and of the fifth (5f) .

alterations become

5
3

7

or

5

1

u

The triple

3

Fig-qr� IX.

•

Triple parallel ch_romatics

- -·-- ------------........

==�·�==-=:::-=-=n-�_o:·.

I
1!�Qti��t&._
==·
=_t_�-�-���-.:_
=_
����==-�o�=�_t2�===-�9ta
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•

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u

...

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a
.-, :s -·-.

r

.-

A

s
'
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..,

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•

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-ti Q- 2-_

-- · -

c..- -

-

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.
• _q,.
....
---.

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g�

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-•

.

------41-

,�
·�

.:-

;;..
-

J
I-

-.. -' I

,
_.

8.
Quadruple parallel chromatics occur

when 1 is raised in s 1 (7) basis [diminished seven th­
chord].

This requires the alterat ion of all

remaining func tions, i .e. 3 :#', 5 -t' and 7�

This is

the only in terpretation satisfying the cases of

chroma tic parallel motion of the dimi.nished seventh•

chords.

See Beethoven 's Piano Sonata No. 7 Largo

(bar 20 from the end and the following 5 bars in

relation to the ad jacent harmonic coutext).

•

•

Such a

continuous chain of quadruple parallelisms truces place
'-

when the same operation is performed several times in
suc cession.

u

As chromatic syst em is limited to four

functions (1, 3, 5, 7), quadruple parallel chr omatics
remain wi th their original assignmen ts (while being
altered).

Figpre X .

Quadruple
Parallel Chroma tics
•

- .,,.
•

u

•

u

By combining all forms of chr omatic

oper ati ons, i.e. single, double, triple and

quadruple, we obtai n the final form of mixed
chromatic cor1tinuity.

figure XI••

Continuity of Mixed Chromatic Oper ations

-- ---

•

�-- --------- -..r::===�-, ---·-- --

Enh�rmonic Tr�atmept of the Chromatic Sys�em

By r eversing the original directions of

chromatic operations we more than double the original

resources of the chr oma.tic system.

0

Enharmonic treatment of chr omat ic groups

•

10.
consists of substituting r aising for lowering a n d

vice-ver sa.

a group and
third term.

Tbis changes the original direction of
brings to new p oints of release in its

The following formula express�s all

conditions n ecessa ry for the enha rmonic treatment.

.•

(1)

-4
� x = y 'v
X
�z

•

(2)

� x'v : y-$-- _,

z

(1 , 3,

5,

7)

(1, 3, 5, 7)

Progression s of this kind are character­

istic of post-Wagnerian composers (Borodin 's "Prince
Igor ", Rimsky-Korsakov' s ''Coq D•Or" and "Khova n ­

schina11 ) .�)

Figure XII._

�xa mples of enharmonic treatment
of the chr omatic system.

•

(please see follov,ing pages)

-

,

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12.

(Figure XII, cont . )

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13 .
In cases of double and triple chromatics,

all or some of the altered functions can be en har­

monized.

Figur e XIII.

Enharmonic treatment of double and
tr iEle
cb,,r,oma tics,
'

•

.
•

-n
--·

,
�:

("
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1· 1

14.

u

Lesson CJ:I.

Overlapp ing Chromatic Groups,
Overlapping groups produ ce a highly

saturated form of chromatic continuity.

The altera­

tions in the two overlapping groups may be e ither

both ascend ing, or both descending, or one of the

•

groups can

be ascending, wh ile the other descending •

The choice of ascending and descending groups depends

•

on the possibilities presente� by the ,precedin,g groups
during the moment of alteration.

G

groups •is:

The general form of overlapping chromatic
d - ch - d

d - ch - d

This scheme, being applied to ascending

and descending alterations, offers 4 variants.

,

(2)

x� �
X

�y

�Ay

X� X

CJ

,

15 .

u

x,.A

(3)

·� y

x

x�
xV

"--.»y

(4)
•

•

Thus, parallel as well as �ontrary forms

are possible.

Each of the mutually overlapping groups

has a single chromatic operation.

Figure
• XIV1

Examples of overlapping chromatic grpups •

- ,.,

-

(_)

-

•

r•

r-.

..,

a

-l

-- -

-,�

(;}

-r•
-s • -- - -

-6

-

I•

-

•

.-

rI

••

-

., ••

,..-:,,

",-:,,

-

�

.

-

••

.

�

,--y

.

-

r•

--

-

•r

.

,�

I �
•
•

,

'

- --

16.

The sequence in which such groups can

be constructed is as follows:

In the first e�ple of Figure XIV

(and similar procedure refers to all cases) we

write the f irst ch ord first:

,.
�·

�

•

•

'

I

•

•

one voice.
bass:

The next step is to make operations in
In this example it was chosen in the
.

':t,...•

c,.'•

I

<•
..

:i

The next step is to construct the

middle chord of this group: (1�

was assumed to

remain 1, which gave the C� seventh-chord) :

•

I

•

.

,

-

a

17.
The next step is to estimate the

possibilities of other voices with re•gard to
chromatic alterat ions.

The b --)
• b � step permits to cons truct

a chord which necessitates the inclusion of d and

bP .
g

Another possibility might have been to produce

; g�, which would also permit the use of d in

the bass ..

See the sec·ond example of Figure

XIV.

=tt='
The third possibility might have been the step e-�
) e,

•

in the alto voice, which also permits �he use of d .

➔ e�or g-_.
> g�would be possible,
Even steps like e --•
though the latter require an augmen ted S (7) , i.e.

0

(reading upward) d - g � - e ' - b�.

-Continuity

Figure

xv.

of Overlapping Chromatic Groups,

Fl � ll

•

0

,

Lesson CIII,

a

18.

Coinciding Chromatic Groups .
0

The ,technique of evolving �oinciding

chromatic
groups is quite different from all the
•

chromatic techniques previously described.

It 1s

more similar to the technique of passi!lg chromatic

•
•

tones, at which we shall arrive later •

Coinciding chroma.tic groups are evolved

as a form of contrary motion in two voices being a

doubling of the chord, with which the group begmns.

The general form of a coinciding chromatic

group is:

d - ch - d
d - ch - d

Contrary directi ons of the chromatic

operations can be either outward or inward:

(1)

(2)

u

X

x� y
�
�-

19.

u

The assignment of the two remaining

functions in the middle chord of a coinciding group

can be performed by sonority, i.e. enharmonically.
For instance, in a group

•

-.

the
b

c*

c'f

interval can be read enharmoni�ally, i.e. as

in which case it becomes

7

l

or

9
3

etc.

It is easy

t hen to find the two remaining func tions, like 3 and
5.

Thus, we can construct a chord c� - e - g - b.
As coinciding chromatics result from

d.oublings, it is very important t o have full control
of the variable doublings t echnique.

Thus the

doubling of 1, 3, 5 and also 7 (major or minor) must

be used intetionally in all forms and inversions of

S(5) and S(7 ) .
the doubled 7.

The

la tter, naturally, for obtaining

Figure XVI.

Examples of Coinciding Chromatic Groups.

(Notation of ct1romatic operations as in all other

u

forms of ctiromatic groups) .

(please see next page )

20.

(_) - - -------- - -;-----�-----.
-

©

Fl4 M

.______.,u___----111----------=-==-

==::::it:=======z:2::=jj:

- - -'-----------------

21.

,

It is important to take into consideration,

while executing the co inciding chromatic gro ups, that
the first procedure is to establish the chromatic
operations

-

"

,l ,, ••

..

•

•

.
,�

-.

• I•

.
•

-

�

·�

•

•

and the second procedure is
'' to add the

two missing functio·ns.

l

�

.,

.
�

,�

After performing this, the final step is

to assign the functions in the last chord of the

group.

•

••
""I•

••

...

.

..

..

•

••

All coincidip.g groups _are reversible,

Whe11 moving from an octave inward by semitones, the

•

I

22.

u

last term of the group produces
-- a minor sixth.

When

moving outward from unison or octave, the last term
of the group produces a nyijor third,

It is important to take these considera­

tions into account while evolving a continuity of

co i nciding chromatic groups.,

.

•

my such grou p can

start from any two vo ices produ cing (vertically) a

unison, an octave, a major third or a minor sixth.
The following are all movements and

directions with respect to c.

-

23.

0

(4)

Fifillre XVII.

Contiµuity of Co,inciding Chromatic Gro:ups.
•

•

•

-

•

u

All techniques of chroma tic harmony can

be utilized in the mixed forms of chromatic

continuity •
•

'·

0

,

•
•