•

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: Mus,ic

With: Dr. Jerome Gross
Lesson CXXX.

MELODIZATION OF HARMONY.
•

Composition of melody W?-th harmonic ?�coru­

pan���pt can be accomplished either by correlating the

•

melody with a chord progression or by composing the
melody to such a progression.

While the tirst �rocedure

is more commonly known, and attempts have been made

even to develop a theory to thls effect, the second
procedure has brought forth music of unsurpassed
harmonic expressiveness ,.

Many composers, particularly

the operatic ones (and among them Wagner) indulged in
composing the melodic parts to harmonic progressions.
So far as this theory is concerned, .the

technique of harmonization of melody can only be
developed if the opposite is known.

If melody can be

expressed in t�rms of harmony, i.e. as a sequence of

chordal functions and their tension, then a scientific
and universal metl1od for the harmonizaticn of melody

can be formulated by reversal of the system of operations.
The process of composing melody to chord

progressio11s thus becomes tl'1e melodiza_tion of 11.§1.rmony.

"

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2.
Though such a word cannot be found in the English

\ dictionaries of today, we can be certain it will be

there very soon , as the discovery of a new technique
necessitates the introduction of a new operational
concept.

This �heog of Melodization will be applied
$

•

to harmonic progressions satisfy ing the definition of

•
•

the Sp�cial Tl1eon: of J-Iarmony:.

•

According to this

definition all chord-structures are based on E, of the

names
seven-unit scales contai ning seven musiGal
. without
'-

identical intonations.

Thus any pitch unit of melody

can only be one of the seven functions: 1, 3, 5, 7, 9,
11, 13.

C

These seven functions produce the man ifold

which we call the scale of tension.

By arrangi ng the

scale of tension in a circular fashion, we obtain two

harmonic directions: the clockwise and the counterclockwise.

•

Clockwise functioning of the consecutive

pitch-units of a melo�y neces§�tat�s the p9si�ive
•

..

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3.

form of =tonal cycles.

= •

r

Counterclockwise functioning of the

pitch-units of a mel99¥ necessitates
consecutive
-

the negativ.e form of ,tonal 9ycles ..

Assuming that all pitch-units of a

melody are _stationag and_ identi.cal, and therefore

.•
-

could be any pitch-unit that is stationary, we shall
By assigning the clockwise

choose c as such a unit.

functioning to such a unit, we obtain the positive
form of harmonic progressions.
Melody:

Chords:

1

3

5

7

9

11

13

1

c + c + c + c + c + c+ c + c

C + A+ F + D + B + G + E + C

By reading the above progression backwards,

we obtain the negative form.

Omission of certain chordal functions for

the consecutive pitch-units of the melody will result

in the change of cycles but not of the direction.
Melody:
Chords:

Likewise:

Melody:

C

Chords:

1
C

+

5·

9

C + C

+

13

3

7

C + C + C

11
1
+ C+ C

C + F + B + E + A +D + G + C
➔
l

7

13

C + C + C

5

+C

11
3
9
1
+ C + C + C +C

C + D + E + F + G + A + B + C

Cs-

c,

0

0

It follows from the above reasoning that

every �hord has seven for�s of melodiz?tion, as
1, 3, 5, 7, 9, 11 or 13 can be, added to it.

The reduction of the scale of tension

decreases this quantity respectively.

We shall consider all the reduced forms

.
•

of the scale of tension to be the ranges of tension .

When each chord is melodized by one attack (or one
pitch-unit) the range of tension can be entirely

under control.

The minimum range of tension possible

can be acquired by assigning only one chordal function
to appear in the melody.

Let us assume that such a

function is the root-tone of the chord.

Then if

harmony consists of three parts, melody will sound

like the bass of progressions of 8(5) const.
For example:

Melody:
Chords:

2c1

+

c,

2c 7

c + f + b + g + c + d + e + • • •

C + F + B + G + C + D + E + • • •
Figure I.

•

C

+ Cr +

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0

•

5.

C

It is easy to see that the pattern of

•

melody in such a case is conditioned by the cycles

through which the chords move.

The predominance of

c7 produces scalewise steps or leaps of the seventh.

Other cycles influence the melodic pattern accordingly.
Now, if we assign any other chordal

.

•

•

function (still one for the entire progression) , the

resulting melodic pattern does not change, but the
form of tension does.•

This time we shall take the seven th to

melodize the same chord progression.
Figure ):I._

,.,
•

-•

1

•

1

•

-------•

The different ranges of tension produce

different types (styles) of melodization.

Music

progresses cloc kwise through the scale of tension.
A narrow range, confined to lower

functions produces more archaic or more conservative

C

styles.

The resulting melodization may suggest Haydn

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6.

or other early forms (in wost cases sueh styles

later become trivial) .

Whereas a narrow range

confined to higher furictions results in melodization

suggesting stylistically Debussy or Ravel.

The

intermediate form may produce Wagner, Frank, Delius.

Wrien the entire scale is used as a range of tension,

the resulting melodization becomes highly flexible
in its expression.

•

•

•

'

•

•
•

C

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•

7.

C

Lesson CXXXI.

•

I. Diatonic Melodization

As it follows from the preceding exposi­

tion, any chordal function can partici�ate in melo­

dization.

The only necessary step which follows is

the assignment
of chordal functions for melodization
•

•
•

with regard to actual chord-structures.

-- as H.

We shall express melody as Mand harmony

In terms of attacks, one pitch-unit

M
assigned for melodization of one chord_' becomes H = 1 .

Under such conditions it is possible theoretically to

u

evolve seven forms of melodization.

For example, a C- chord can be melodized

by c (l) , e (3) , g (5), b (7) , d (9) , f (ll) and a (13).
Figure III.

•

•

•

•

•

.. .

I

0

-

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•

a.
It is easy to see that the majority of

pitch-units of M are satisfactory.

Two of them

(d and f), however, do not result in a satisfactory

melodization.

The reason for the latter is that -

high functions, without the support by the

immediately preceding function in harmony, are not

acceptable .

.

•

Likewise, the presence of lower

functions in the melodization of high-tension cl:1ords

•
•

satisfactory
The 13 is fully
'

is equally inacceptable.

as melodization of 8(5) because by sonority it

converts an 8(5) into 8(7) .

Now we can construct tbe table of

L

melodization for the fifth voice above four-part·

harmony, where both melody and harmony are diatonic.

•

Figure IV.

Table I:
M

7, 13
5
1

s

S(5)

9, 13X)
.

I'=

1.

5, 11, 13

5, 13

5, 11

5, 9
13

7

9

11

13

5

7

9

9

11

3

7

7

7

1

1

1

1

8(7)

8(9)

S(ll)

3

1

S(l3)

.

•

•

0

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9.

It follows from the above table that:

(1) classical and hybrid four-part harmony can be
used for the diatonic melodization;

(2) all chordal tones actually participating in the
chord as well a s the functions designated as M
can be used for the diatonic ruelodization;

•
•

•

(3) by diatonic melodization we shall mean the

participation of pitch units of one diatonic

scale and fnom which the chord-progression is

evolved;

(4) the use of 13 in 8(7) is acceptable when the root
of the chord is in th e-bass (i.e. do not use
inversions);

(5) the alternative in selection of functions for the
melodization of 8(13) is due to t\vo forms of

structures covered by the branch of hybrid four­

part harmony.

Assuming that there are on the average

•

about five practical pitch-units (functions) for the

melodization of each chord through the form

M=1
,
H

the number of possible melodizations of one harmonic
cor1tinuit·y (under such conditions) equals 5 to the

power, the exponent of which represents the number of

chords.

Thus a progression consisting of 8 chords

produces 5

L)

B

= 390,625 melodizations.
•

•

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10.
The two fundamental factors in deter­

mining the quality and the character of melodization
are:

(a) the range of tension;

(b) the melodic pattern (i.e. the axial
combination of melodic structure)

The int -erest may be concentrated on

either one or on both.

Attack-interference patterns

add interest to melodization.

•

In the following examples�' R represents

the range of tension, A -- the axial combination.

All the following examples can be played in any

•

system of accidentals •

Figure V..

Examples of Diatonic Melodization

M iI

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1

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Les.son CXXXII.
The increase of the number of attacks

necessitates a slight remodeling of Table I (Fig. IV).
Any higher funct_ion can be supported. by the

,immediately 12receding function of immediately:
preq_edi:ng rank.

For instance, 9 can be used for melodi-

••

zation of 8(5) providing it is immediately preceded
by 7, and the root of 8(5) is in the bass (the

•

necessary condition for the support o� 9).

same reason 11 can be used for

For the

melodization of

8(7) if preceded by 9 and when 8(7) has a root in the

bass.

Figure VI.

•

Table II:

M
H

-

2, 3, 4, • • •

Additions to Table I:
7

➔9

9� 11

5

7
5

3

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1

1

8(5)

8(7)

.

0

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14.

•

Fig�e VII.

Examples of Diatonic Melodization.
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16.
mov_ing int9 chordal tones, actually present in the
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associated with Mozart, Chopin, Schumann, Chaikovsky

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•

Figur e VIII •

Examples of Diatonic Melodization.
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J O S E P H

S C H I L L I N G E R

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

C O U R S E
Subject : Music

With : Dr. Jerome Gross
Lesson CXXXIV.

Composition of the Attack-Groups
of Melody.

.

•

In all the previous forms of melodization,

the attack-group of M was constant.

Any assumed

quantity of attack per chord (H) was ca�ried out
consistently .

The monomial attack group (A) in all

cases was an integer remaining constant throughout
�.

L

This monomial form of an attack-group can be

f

as
expressed
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one to infinity) .

= A, where A can be any integer (from

Now we arrive at binomial attack-groups
M expressed
be
can
This
.
for the melork•
as
�
2H .-

= A , + A 2 , i. e. melody covering two successive cl:1or ds
consists of two different ·attack-groups .
For insta11ce :

(1)

-! H -M

2a +. a ;

(3) . 2H = 5a + 3a;
.II

as :

. '.

.

(2)

-H

:H = 3a 1r 2a;
= a + 8a;

•

• • •

These expressions can be further deciphered

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(1 )

= 2a + a;

(3)

=
The

(2 )

5a + 3a;

M + M
H ' R2

=

3a + 2a;
a + Ba; • • •

(4)

main significance of a b inomial attack­

group is the introduction of contrast between the two
successive por tions of M.

The greater the c ontrast

required, the greater the difference between the two
number-values of a binomial .

This proposition can be

reversed into the following : the cont�ast between the
two terms o f a binomial decreases when �heir values

approach equality.
Thus,

than

M

2H

3a + 6a;

=

2H

2a + 6a;

2a + 6a is more contrasting than

3a + 6a is more contrasting than the least

c ontrasting 5a +. 6a .

obtain a monomial as :
that

= a + 6a is more contrasting

)(

With fur ther balancing we

M

M
+
H2
H,

M
1I - 6a .

=

6a + 6a which means

If permutation takes place in a binomial

attack-group, it results in the second order binomial

attack group.
•

For instance :

: = 4a + 2a;
H

this becomes :

4H

M =

MH,

in the course of H ) = 4H,

+M +
H2

MHa, +-�!H

'I

= 4a

•

+ 2a

+

2a + 4a.

0

0

•

3.
The above described method of binomial

attack-groups is true of any polynomials.
are sub ject to permutations •

Examples of trinomial attack-groups :

•

•
•

(1)

M
3H

(2)

M

3H

(3)

(4)

0

The latter

= 3a + 2a + a ;

14 + M + M = 3a + 2a + a ;
H2
Ha
If,

-

4a + a + 3a;

-HM ,

+ H2 + H3

a + 2a + 4a;
3H

-HM,

M
M
+ - + Ha · = a + 2a + 4a;
Ha

M

M

3H

=

M

M

-

4a + a + 3a;

M
M + 14 = 3a + 5a + 8a.
+
3a + 5a + 8 a ; H'
H2
H3
Examples of polynomial attack groups

based on the resultant.s of interference:

M _ M
6H - H,

M + M - 3a + a + 2 +
+ a
H,- H•
+

14
+ M
+
Hr
Ii.

= 2a + a + a + a + · a + a + 2a.

2a + a + 3a.

,

0

0

•

4.

(3)

r

9+8
M

16H

.•
Ba + a + 7a + 2a + 6a + 3a + 5a + 4a +

+ 4a + 5a + 3a + 6a + 2a + 7a + a + 8a .
The effect produced b y such composition

of attacks as ( 3) is that of counterbalancing the

original binomial: it starts with excessive animation
•

over H , (8a) and complete lack of it over Hz (a) ; it

f ollows into the state closest to b alance, after which

the counterbalancing begins, ultimatel� reaching its

converse:

a + 8a.

In all cases of ra +b the maximum
animation takes place at the beginning and at the

end.

When the opposite effect is desirable (minimum

animation at the beginning and at the end) use the

permutation of binomials (which is possible when the

number of terms in the polynomial is even) .

For instance: (3) can be transformed into

M = a + 8a + 2a + 7a + 3a + 6a + 4� + 5a + 5a +
16H
+ 4a + 6a + 3a + 7a + 2a + 8a + a .
In addition to resultants, involution ·

(power) groups as well as various series of variable

velocities (natural harmonic series, arithmetical and
geometrical progressions, summation series) can be
used as the forms of attack-gro ups.

•

0

0

For instance: ( 2 + 1) 2

:

( 1 + 3) 2

:

=
�
4

4a + 2a + 2a + a ;

:H =

a + 3a + 3a + 9a ;

•

2a + 3a + 5a + 8a + 13a .
For the time being we shall use the

simplest durati on-equivalents of attacks, as this

•

•

subject is a matter of further analytical investigation
(which will follow in the next lesson) ;
Figure xy.

Examples of Diatonic Melodization with

Variable Quantity of Attacks of M over H:
M
-

=

H

A var.

(please see following pages)

•

•

'

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0

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liiii

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(Fig. XV, cont .. )
I

•

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•

Ties in the above examples were added after the

completion of melodization.

(

'-.,...

•

•

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0

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a.

•

Lesson CXXXV.
Composition of Durations for the
Attack-Groups of Melo�y.

Composition of durations for the attaclt­

groups of melody can be accomplished by means of

technique previously defined as Evolution of Style

Every attack-group, monomial, binomial,

in Rhythm.

trinomiaJ.., quintinomial, etc. can be expressed
through the different series.

binomial of

•

3
3
4
4

For instance, a

series is 2 + 1 or its '' converse; a

binomial of

-

binomial of

8 series is 5 + 3 or its converse.

series is 3 + 1 or its converse; a

8

Likewise a trinomial of

4
4

•

•

series is

2 + 1 + 1 or one of its permutations; a trinomial of

f

is 4 + l + 1 or one of its permutations; and the

trinomial of

•

p�rmutations .

8
8

series is 3 + 3 + 2 or one of its

Selection of durations for the attack­

groups through the different series permits to

translate a piece of music fro,m one rhythmic style
into another.

When a choice is to be made as to the

f orm of a binomial or a trinomial, the form of balance
(unbala nci ng, balancing) becomes the decisive factor.

..

•

0

0

Thus, out of the two binomials 3 + 1

and 1 + 3, the fir st is more suitable at t he

beginning of m elody and the second -- at the end.
•
4 series:
2 + 1 + 1
In the case of a trinomial in 4
at the beginning, l + 2 + l somewhere about the
center and l + l + 2 at the end.

Likewise, in

series: 3 + 3 + 2 at the beginning, 3 + 2 + 3 about

.
•

the center and 2 + 3 + 3 at the end •

Four at tacks

can be achieved by splitting one of the tefms of a

•

as a
Splitting of the terms serves
''-

trinomial.

general technique for acquiring more terms for low
determinants.

0

•

Examples of composition
• of durations

for the attack-groups o f melody where each term
of an attack-group corresponds to one cl1ord:

I

=

A.
l

A�

= a + b + c + d + e ;

Ao

=

a + b ;

A�

=

As = a + b + c ;

a

A )" = a + 2a + 3a + oa + 3a + 2a + a
Series:

3

3

T = 3H , + (2+l) H2 + (l+l+l) H3 + (1 + 1 + 1 + !. + !) H 't' +
2
2
2
2

u

•

0

0

10.

5

4

J.

d J jJJ

fl J D l J J J J

j ,
•

--.
U-1 t.1-1

Series :

i

Yv1 T

V1 •

¼

+ (1+1+2)H5 + (1+3)H� + •4H�.
i
4

0

S er1.e
. s:

6
6

T = 6H, + (5+1)H� + (4+l+l)H3 + (1+1+2+l+l) H� +
+ (1+1+4) Hs + (1+5)H� + 6H7 •

(Waltz or Mazurka)

Li

q'

0

0

11 •
S eries
. :

8
8

•

+ (5+3)H2 + (3+3+2) H 3 + ( 2+1+2+1+2)H" +
,
+ (2+3+3) ff( + (3+5)H_ + 8H 7 •

T = 8H

(Foxtrot, Rhumba, Charleston)

� i, 1 •

•

,....

1 · v1 ,

1 v 1 -v i

1 1

..--... ,,....._

u,

•
•

,--

1 · -v 't

0

•

Figure zyr .
•

.,

�. �., SERlE)
9

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(Fig. XVI, cont . )

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13 •
The final and most refined technique

of coordination of the attack and the duration­

groups takes place when the attack-groups are
c onstructed independently of T.
interference between the

As this causes

attaek and the duration­

groups, the duration of the individual chords is not

conformed to bars or their simplest subdivisions •
•

We shall take a simple case for our

illustration •

Let us cl1oose A

+

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

= r5+4 == 4a + a + 3a +
'

= 20 a .

Let us execute the dur ations as T

As T in this case has lOa and A has 20a, the

=

interference is very simple.

a A
a

1 (20)
2 (10)

= 16t •2 - 32t .
Let T" = at, then :

Hence, T •

NT"

-

32 -a 4

The duration of each consecutive H equals

the sum of dur ations during the time of attacks

corresponding to such an H. ,

Thus, H , corresponding to 4a, the dura tions

of which cons titute 3t + t + 2t + t, will last 7t.

Likewise the next chord, i . e . H 2 will last t as at

)

0

0

V
•

14 .,

this p oint melodization consists of one attack, and

that attack c orresponds to one unit of duration.

Here is the final so luti on of the case.

(1)

(2)

a
a

Ji = i + 1. + 2 + £. + £. + Q + 1. + ! =
1
l
1
l
1
l
H
1
1

T

M) _

= 4aH, + aH 2 + 3aH 3 + 2aH� + 2aHr +

T H

1+1+2 + L+3 ) + ( 3+1
+ !.
+
l
4
4
4;

+

2+1+1 +
4

+

1 + 1+2+1+3) =[(3t+t+2t+t ) H
, +
1
7
7t

2t) H + ( t+3t ) H J +
+ (i ) H2 + (t+t+
�
3
4t
4t
t

•

t +t
3 t
+ [{ !� ) H,r + (2 4i ) H" + {½) H ?

+

M

8
8

�

1 . r1

( t+2t;t +3t) H g ]

n

.11 J

r.r .

J . rJ
•

n

rJ

+

J P l.

0

0

15 .
•

'

u

Figur e XVII,.

•

....
,

1�,·=t
•

.,,
-

•

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L,, ✓

-

•
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1
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16.

u

Lesson CXXXVI .
Direct Compos iti on of Durations p9rr,elating
M�lody and Harm9ny,

Time-rhythm of both melody and harmony can

be set simultaneously by means of a proporti9p�te

distribution of §ur ations for a constap� guantity of
IIC

attacks of

I.

This can be achieved by synchronizing a

polynomial (consisting of the corresponding number of

•

terms, representing at·tacks) with its square, or the
square of a polynomial with its cube ,· etc .

For instance, we would like to have 4

attacks per chord in the style of durations of the

!

series.

Let us take a quadrinomial: 3 + 1 + 2 + 2

and square it.
(3+1+2+2) 2

T (M) .

=

(9+3+$+6) + (3+1+2+2) + (6+2+4+4) +

+ (6+2+4+4)

The above distributive square represents

The T (H) is the or iginal quadrinomial,

synchronized with the
8 (3+1+2+2)

distributive square:

= 24 + 8 + 16 + 16

Thus we obtain:

T

M
H

=

9t + 3t + 6t + 6t +
24t

3t + t + 2t + 2t
8t

+

0

0

17 .
6t + 2t + 4t + 4t
+
16t

+

n...,..1 '

-

M

6t + 2t + 4t + 4t
''fst •

8
8

Ir' ....,
.

_J
-

J . f> l J

-

C

M

•

J.

•\

0

-

J. i

_J-

C.

•

FigPr� XVIII.

=·t1

,
-.

•

"\.

..f'I

-JI
-· ,
�

�-

,. r

8
"

.
•

-

•

. .,.

-

:..,,_z

-

�
�

-

•

II

""

.
...
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3;

r
-

-

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a

�

�
-

I•

•

$=_
�

[

0
I

.

-

C

...
..

-

-

�'

'

Likewise, synchronization of the

distributive square with the distributive cube can

be used for melodizat ion of harm ony.
I.

0

The group of the

square furnishes durations for the chords and the gr oup
,
•

/

0

0

18.
of tjle cube furnishes durations for the melody.
T
T

M
2+1+1 3
H . = 4 2+1+1 2

=

8t+4t+4t
4t+2t+2t +
+
16t
8t

2t+t+t
+ 4t+2t+2t + 4t+2t+2t
+
8t
l8t
4t
2t+t+t +
4t

+
••

melody: M

4t+2t+2t
8t

+ 2t+t+t
+
4t

This produces harmony:

H

-......

M

d

2t+t+t •
4t

= 9H, and

= 27a, with constant 3 attacks per chord .

M

8
8

+

-

�

L J

J fl l fJ

d l J

9 4

0

H

•

Figure XIX·,
(please see next page)
•

•
•

q 4

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For greater contrast in the quantity · o r

at tacks between M and H ) , use the synchronized

first power group for H' ) and the distributive
cube for

•

M•

In addition to distributive powers,

coefficients of durat ion can be used.

.

•

For instance:

- (3+1+2+J.+l+�_:t-;l.j-2+1+3) + (3+1+2+1;+-1+l+.1+2+1+3)
6+2+4+2+2+2+2+4+2+6

.C

)

0

0

21.
Lesson CXXXVII,
Chromatic Variation of the Diatonic
Melodization,

It is mor e expedient to obtain a chromatic

melody to diatonic chor d progr essions by using two

successive operations:

.

•

(1) Diatonic Melodization of Harm9ny

(2) Chromatization of Diatonic Melody

The first is fully cove�ed by the

preceding techniques.

The second (chromatization) can b e

accomplished by means of passing or auxiliary
chromatic tones.

The most pr actical way to per form

• this r hythmically is by means of split-�nit_groups
(see "Theory of Rhythm": Var iations ) .

This does not

c hange the character of dur ations (wit h respect to
their style) but

merely incr eases the degree of

animation of melody.

Figur e XX •

Example of the phromatization of
Diatonic M.elody,

(please see next page)

0

0

22.

(Etg. XX)

r

1> 1 ?.10N \t ME.lOJ>\ l.A"llON -

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I

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23.

u

The r: (13) FapJ1lies,

(Intr oduc tion to Symmetric Melodization)
Each style of symmetric harmonic continuity

(Type II, III and the generalized) is governed by the

2 (13) families.

Pure styles are controlled by any

one � (13) , while hybrid styles are

based usually on

two, and seldom as many as three, � (13) .

The c omplete manifold of � (13) fa milies

corresponds to the 36 seven unit pitch scales which

c ontain the seven names of non-identi�al pitches.
The � (13) are the first expansion (E,) of such

scales.

We shall classify all forms by associating

1, 3, 5 and 7 as the lower structure [as S(7) ] with

9 , 11 and 13 as the upper structure [as S ( 5) ] ,

eliminating all enharmonic coincidences, as well as
all adjacent thirds which do r:i not satisfy i = 3 and
i = 4.

These limitatior1s are necessitated by

the scope of the Special Theory of Harmony.
Figu;re XXI.
.
Complete Table of � (13)
(please see next page)

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

Subject: Music

Lesson CXXXVIII,

pymmetric. }.\elod,ization of Harmony:

.

•

Symmetric melodization provides the

composer with resources particularly suitable for

equal temperament ( '1:/2 ) .

Whereas in .the diatonic

system s ome chord-structures , particular·ly o f a high

tension, produce harsh sounding harmonies, in the

r

symmetric system both the chord-struc tures and the

intonations of melody are entirely under control and

are subject to choice.

The teclmique of symmetric

melodizati�n makes it pos sible to surpass the refine­
ments of Debussy and Ravel.

And, whereas it took any

important c omposer many years to crystallize his

original style, thi s technique of melodization offers

36 styles to choose from when one
a time.

Y

(13) is used at

The amount of p ossible styles grows enormously

with the introduction of blends based on two L (13) .
Then the number of styles becomes 36 2
Likewise by blend.ing three

Y (13) ,

limit of mixing, we acquire 36 3

=

=

1296.

which is a reasonable

46,656 style s .

It is cor rect to admit that only about 4-

0

0

2.
of the 36 master-structures have been explored to

any extent, the rest being virgin territory packed

with most expressive resources of melody and harmony .
In offering the following technique, I

shall use symmetric progressions of type II, III and

the generalized form in four and in five part harmony.

•

The main difference between the four and the five
parts is density .

For massive accompaniments use

five and for lighter ones use four-part harmony .

When all substructures [S(5)
, 8 (7) , S (9) ,
'

S ( ll) ] derive from one master-structure [

L

(13) ] ,

they adopt all intona tions of that master-structure.

The easiest way to acquire - a quick orientation in any

r

(13) is to prepare a chromatic table of such a

matter-struc ture.

Ta.king <£. (13) XIII from

Figure XXI, we obtain the following table of trans­

pos itions .

Figure_ X,XII...
•

•

•

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Such a table is very helpful, as all

intonations for both melody and harmony can be

found for any symmetric progression.
Each

L (13)

being E, of a seven-unit

scale corresponds to E0 of the same scale.

The rest of the procedure of melodizat ion

.

•

is based on the same principle of tension as in the.

diatonic melodization.

The functions added to

respective tens ions of chords are the most desir,able

•

ones as axes of the melody.

Thus the axis of the

melody above 8(5) in four-part harmony is either 7 or

u

13.

Actually such a choice creates polymodality, as
0

S(5) d0 serves as an accompaniment to �elody which
is db or d� respectively.

It is pGlymodality that

makes such music more expressive.

The following is the table of melodic

axes for the respective structures in four and five­

part harmony.

more than one.

In some cases there is a choice o f
Some of the forms are admitted

because there has been practical use of them already.
For example, 8(5) in five-parts with melodic axis
on d , (= 9) .

It is interesting to note that

L

(13)

XIII is used most of all, and that it is the most

ob vious master-structure, as it consis ts of a large

G

8(7) and

a major 8(5) .

)

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Figure XXIII .

Table of Melodic .Axes in Relation to
the Tension of H.

Master-Structure : � (13) XII I.

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Lesson CXXXIX,
•
•

Using this "f: (13) we shall melodize

a generalized symmetric progres sion in four parts
.
M
in ff = a .
Figure XXIV.
--

Theme: 2 + 2 + 2 + l; tension: 8 (5) + 28(7) +

•

+ 8 ( 9 ) + 2S ( 13)

"2:" (13)-f XIII

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a

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Figure XXV•

Theme: Type I I :
� (13) : XIII

C ::: 2C.s-

I3

S

\I

+ C-7 + 2C3 + C-s-

Il

tensi on: 8(5) + 8 ( 7) + 2S(9) + S (ll)

M = a
ff

(please see next page)

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(Fig. XXV)
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With more than one attack of M per H',

the quality of transit ion in melody, during the chord

changes, becomes more and more noticeable .

In melodizing each H with. more than one

attack of M, it becomes neces sarr �o perform modu�a ­

tions in melogy.

Such mod1.1Ja tions are equivalent

to polytonal-unimodal and poly:tonal-po_lymodal transi­
tions.

The technique for this based on common tones,

chromatic alterat ions or identical motifs is provided
in the Theory_ of Pitch Scales (The First Group).
ExB.!Dples or SY!Pmetr�c
Melodization.

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being more stationary than it would be desirable in

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structures for one harmonic continuity, it is

desirable to alter either• the lower part of the•

_(_1_ 3 ), i . e . 1, 3, 5, 7 or the uppe� part of it,

L

i .e. 9, 11, 13, wit hout altering the lower.

Let us produce a mixed style of master-

•

structures, confining the latter to

After su c h a sele ction

(13) XVII.

L

L

(13) XIV and
•

J.S

made, the

master structures become simply: �, and Y- •
Now in devising the style we inusb r esort to the
''
coefficients of re cur rence, as the predominance of

•

2,

one � over another is the chief stylistic character­

istic.

scheme :
M

n

Let us assume the following recurrence-

2 L,

+Y2. .

= a + 4a .·;

¼ serie� of T.

= 2S(9) + 8(13) .
Figure x;xvrr.
(please see next page)

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12.
Lesson CXL.

Chro��tic Variation of the Sy�metr�c
Melod :Lzation .

Any melody evolved by means of symmetric

melodization can be co11verted into chromatic type by
means of passing and auxiliary chromatic tones.

Such

chromatic tones do not belong to the master-structure .
Rhythmic treatment of dur ations must be performed by

•

means of split-unit groups.

Fig ure XXVIII.

Example of ChJ'omati� Vapiation of

=·

the SY!D�etr�c Melodizatipn.

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13.

All rhythmic devices such as composition

of attack and duration-groups are applicable to all

forms of symmetric melodization.

Chromatic Mel odiza�ion of Harmony
Chromatic Mel odizat ion of Harmony serves

the purpose of melodizing all forms of chromatic
.
•

continuity.

This includes: chromatic syst.em,

!Do.dulation, enharmonics, altered chords and also

•

hybrid p.armonic con.tinu,ity,

As a consequence, it is

applicable to all forms of symmetric progressions,
but by th is we have nothing to gain as symmetric

melodizat ion is a more general technique .

There are two fundamental forms of

•

chromatic melodization.

One of them produces

melodies of either chromatic. type, or of extensivefy

�hr oma,t ized type .

Another produces melodies o f

purely diatopic t ype .

The first technique consists of

as auxiliary
anticipatipg cho.rdal tones and usipg them
•

tones .

In a sequence R, + H2 + Ha + • • •

the

chordal tones of H2 are the auxiliaries and the

chordal tones of H ' are chordal tones while this
chord sounds .

In the next chord (H 2 ) the chordal

tones of H 3 are the auxiliaries and the chordal tones
of H 2 are chordal to nes wh ile this chord sounds.

This

0

0

14.
procedure can be extended ad infinitum.

As all the disturbing p itch-units are

•

harmonically justified as soon as the next chord

appears, the listener is not aware that nearly all

chromatic units of the octave are used against each

chromatic group, especially when there is a sufficient
number of attacks of .M against H •

Auxiliary tones must be written in a

proper manner, i.e. by raising the lower (ascending)

•

auxiliary and by lowering the upper (�escending)

auxiliary, even if they have a different appearance
in notation of the following chord.

L

Figure XXIX.

•

elodization by_ M
eans
Example of Cpromatic M
of .Anticipated Chord�l Topes.•

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Lesson CXLI,

u

The second technique is based on the

method of construc ting a guan61tative scale .

Such

a scale can be evolved by a p11rely ,statistical

method, Whereas it is not obvious even to the most
0

discriminating ear, it is easy to find by plain

addition the quantity in which each chromatic pitch-

.
•

unit appears during the course of harmonic co11tinuity •
In order to find a quantitative scale it

is necessary to write out a full chromatic scale

from ap.;y note (I do it usu.ally from . .£) .

·,

The next procedure is to add all the c-

pitches in a given harmonic progr ession (doubled

-

tones to be c ounted as on e and enharmonics to be
included).

Then all the c� - pitches , d - pitches,

etc . , until we sum up the enti re cliromati c scale.

This produces a quantitative analysis of a full chromatic

scale .

Now by eliminating some of the units which have

lower marks, we obtain a guantitat�ye (diatonic) scale.
If .there is one unit having highest mark,

it should become the root-tone of the scale and,
possibly, the axis of the future melody.

If there are

more than one U:Qits having highest mark, it is up to
the composer to assign or1e of th;,em as an axis.
In the chromati c progression of

0

0

0

17.
Fig._ XXIX, the quantitative analysis of the
chromatic scale appears as follows.
Figure XXX.

-

,_

4

•

•
•

By excluding all values below 4, we

obtain the following nine-unit scale 'v,ith the·

root-tone on � (maximum value) .

•

Figure �I._

If such a scale still appears to be

too chromatic, further exclusion of · the lov,er marks
may reduce it to fewer units.

By exc�uding all the marks below 5 (in

this case) it will reduce the scale to five units

. and give it a purely diatonic appearance.

0

0

18.
figJJ!' e WII,
•

The next procedure is the actual

•

melodizabion, which is to be performed according

to the diatonic tecrmique.

•

By this method, the

tones which quantitatively predonl'inat� during the

course of chromatic continuity (an d which, affec·t

0

us as such physiologically, i.e. as excitations)

become the �its some of wh,igh sat�sfy every chord

and attribute a great stylistic unity to the entire
J

•

•

product of melodization.

The quantity of attacks of M against H

largely depends on the possibilities of melodization.
Fig ure XXXIII.
4

C

Example of C hromatic Melo�izatio� by

means of 24ant� tative piatonic S�ale
(please see next page)

I

0

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19.

(Fig . XXXIII)
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u

The two techniques of chromatic

melodization can be combined in sequence.

This

results in contrasting groups of diatonic and of
The quantity of H covered by

c hromatic nature.

one method can be specified by means of the
coefficients of recurrence.

For example: 2H di + H ch.

•

•

Figure XJQCIV.

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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 EN C E
With: Dr. Jerome Gross
Lesson CXLII.

C O U R S E
Subject: Music_

HARMONIZATION OF MELODY

The usual approach to harmonization of

melody is entirely superficial when the very fact

of finding a "suitable" harmonization seems to solve
the problem in its entirety.

Looking baqk at the

'

music which has already been written, we find quite

a diversity of

styles of harmonization.

In some

cases melody has a predominantly diatonic character

w hile chords seem to form a chromatic progression,

and others when meiody has a predominantly chromatic

'

character while the accompanying harmony is entirely
diatonic.

Operatic works by Rimsky-Korsakov and

Borodin may serve as an illustration of the first typ�,

and music by C hopin, Schumann and Liszt, of the
second type.

This brings up the question of

systematic classification of the styles of harmonization
By a pure method of combinations we

arrive at the following forms of harmonization :

(1) Diatonic harmonization of a diatonic melody.

(2) Chromatic harmonization of a diatonic melody.
(3) Symmetric harmonization of a diato nic melody •

•

•

0

0

2.

(4) Symmetric harmonizati on of a symmetric melody.

(5) Chromatic harmonization of a symmetric melody.

(6) Diatonic harmonizati on of a symmetric melody .

(7) Chromatic harm onization of a chromatic melody.

•

.

•

(8) Diatonic harmonization of a chromatic melody •

(9) Symmetric harmonization of a chromatic melody.
In addi tion to this, various hybrids may

be formed intentionally, · and they do exist in the
music written on an intuitive basi s .

The necessity

of handling the hybrid forms of harmonic continuity,

which is inevitable not only in p opular dance music ,

but frequently in music of composers who are considered
"great" and "classical", for the purpose of arranging

o r transcribing such music, requ ires a thor o�gh know­

ledge of all pure, as well as hybrid, forms of

harm onizati on.

1. Diatonic harmonization of a
diatonic melody:

There are two

fundamental procedures

required for t� above method of harmonization:

(a) The distribution of the quantity of attacks in

melody and harmony, i . e . the quantity of attacks of
melody harmonized by one' chord, or the qu�nt ity of
chords harmonizing one attack in melody.
(b) Selec�i on of the range of tension • .

CJ
•

0

0

3.

attacks.

Let us take a melody consisting of 12

Such a melody may be harmonized by 12

different chords, each attack in the melody acquiring
its individual chord.

It may offer as well two attacks

of a melody harmonized with one chord�

In this case

6 different chords will constitute the harmonic progression.

Further, eacl1 3 attacks of a melody may

acquire a chord, thus requiring 4 chords thr oogh the

entire melody.

•

Proceeding in a similar fashion one

may ultimately arrive at one chord harmonizing the
entire melody.

''

This is pos sible because no pitch-unit

in a diatonic scale may exceed the function of 13th,

and will merely require an 11th chord for harmonization,

u

in order to support the 13th as an extreme function in

a melody where all the remaining units of the scale may
be present as

melody.

well.

Let us take, for example, the following

Figµre I._

•

•

0

0

4.
In order to harmonize this melody with

12 different c hords it is necessary to assign each

pitc h-unit of the melody to a cl1ord.

Such an

assignment is based o n a selection of the range of

tension.

Let us suppose that we limit our range

o f tension from the 5th to the 13th.

Having a

consider able choice in the assignment of pitch-units

as chordal functions we will give preference to t hose

•

forming a positive cycle.

Examples of assignment of the above

melody:
M
H

T

R ange of tension : 5 -- 13

1

Figure II.

A.

?

7

q

7

7

• •

•

•

•
•
.

.
•

•

u

•

•••

I

0

0

-------.._.---

5.

B.
1

,

,

1

s

S'

5

.

•

•

u

In assigning 2 attacks in the melody

against 1 chord, it is necessary to conceive the 2
ad jacent pitches in a scheme · of chordal functions
(thi�ds in this case) .

Thus, the f irst 2 units,

a + b, have to be translated into

: , which may

assume the following assignments:

13

a

9

11

b

3

5

7

C

9

lI

13

3

5

Likewise, C + d transforms itself into:

•

d

7

The next two units produce:

5
3

7

9

5 - 7

11
9

13
11

0

0

•

The next two units produce :

d

5

7

9

11

b

3

5

7

9

9

11

3

5

7

g

9

11

13

a

3

5.

7

The next two units produce :
The next two units produce:

6.

13
11

13

This group of ass.ignments offers quite a

variety of harmonizati ons, even 1vvith the preservation
of the pos itive system of progressions.
Figur e III..

Range of tension: 3 -- 13

.�

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0

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•

7.

Lesson CXLIII.
Assigning every 3 pitch-units of the

melody to one chord, and distributing them thr ough
the scheme of chordal functions, we acquire the
follov,ing table ..

•

M
H -

•

•

Range of tension : 1 - 13

3

Figure IV.

•

A•

-·� n

-,

'

'•

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13

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fi

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0

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8.

c.

M - 4
ii -

Range of tensi on: 1 -- 13
•

•

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II 1

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Range of tension: 1 -- 13

�

II 1
11

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r• .

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0

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18.
4. Symmetric harm on�zatiop of �
symmetri� melody:

There is a very small probab·ility that

melodies composed from symmetric scales outside of

this method have been in existence , as the conception
of symmetric scal es it self is unknovm to the musical

world.

.
•

The problem of harmonization of melodies

composed from symmetric scales requir es , therefore ,

•

As it ha s been

the existence of such melodie s ..

of symmetric
explained in the third and fourth grou P.
'pitch scales, melodie s can be composed through

permutation of pitch-units in the s ectional scales

u

(each starting with a new tonic) .

After the complete

melodic form is achieved the fir1al step corisists of

superimposition of the rhythm of durations on such a

continuity of melodic forms ..

Let us take a scale

based on 12 tonics where each sectional scale has a
structure 3 + 4 and limit the entire scale t o the
•

first 3 tonics.

As

scales of the 12 tonic system

have a wide range expanse it is desirable, in many
cases , to re duce the range by means of octave­

contraction.

Figµre XI..

•

L,

•

•

•

•

•

•

•

•

►

•

,

•

•

•

•

• •
• •

-

0

0

The next step is to select a melodic

form based on circular permutations of pitcr1-units
in the above scale and the rhythmic form based on

synchro nization of 2 + 1 and (2 + 1) 2 •

I

u

Figure XI.I •
201#:

fJ.J

•
t

e.
•

•

By superimposing the rh ythm of durations

on melodic form we obtain an interference as the

number of attacks i n th e melodic form is 9, and t he

number of attacks in the rhythmic form is 6.

Thus,

melodic form will ap_pear twice and rhythmic form

three times.

Figure XIII .

Composit i on of Melodic Continuity

Melodic form consists of 9 attacks

-69 -_ -23

Rh ythmic form consists of 6 attacks
Melodic Continuity
•

(please see next page)

0

0

•

20 .

(Fig . XIII)

..

•

s

• •
•
..:..:.....,.--,-----i,,,---'-·--,ta;.....;...
· • -----·

1�

•

r
IL----::�---

a

T

n

..,,.
C

18
s
$■•"""•••-------=---------

•

•

•

In the above melody the sequence of

chords will be assigned to each tonic .
'
\......I

Thus, the

first sectional scale emphasizes 13t, the second -5t, the third -- 13t, the second recurrence o f

the first -- 5t, the second recurrence of the

S

•

second -- 13t, the second recurrence of the third -St, and an axis ( = 18t) is added for completion.

0

0

'
21 ..
Lesson
CXLV,
•
Here are tv,o methods of symmetric

harmonization of melodies constructed on symmetric

pitch scales.

The fir st provides an extraordinary

variety of devices while the second is limited to a

;

considerably smaller number of harmonizations .

A. The first method assigns the importa.tlt

tones (all pitch-units in this case) of a sectional
scale to be the three upper functions of a

L (13}

adding the remaining functions dovmward- through any
The first sectional scale in

desirable selection�

the als)ve melody has three pitch-units (c, e P, g)

which we shall originally conceive as 13 - 11 - 9,
downwards.

The continuation of this chord downwards

will require pitch-units of the following nam es:
a, f, d, b.

In the following L°(13) a certain

structure is offered as a special case of many other

possible

L.

Figur e XIV.

� 13

..
I

...

10

0

0
"
p 0

0

0

•

22 .

The upper three functions of the

chord (red ink) may produce their own chord in

harmony.

Thus, the functions 9 - 11 - 13 of the

L may actually become 1 - 3 - 5 .

All pitch-units

of melody and harmony are iden tical in this case.
(See Figur e XV - A) .

By assigning the same three

pitch-units as 3 - 5 - 7 we have to add one
function down. (See Figure XV - B) •

A.11 further assignments of thl3 three

•
•

pitch-units, namely 5 - 7 - 9, 7 - 9 � 11,

9 - 11 - 13,

11 - 13 - 1,

13 - 1 - 3

are the

c , �, e, f, g, respectively, on Figure XV.

This

Figure offers a complete transposition of all the
assignments through the three tonics employed in

•

the melody •

Figure XV.
(please see next page)

•

I

0

0

Figure x:y_.

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24.

As Figure XV exhausts all the possi­

bilities under the given group of chords it is

possible to exhaus t all the forms of harmonization

for the given melody through various forms of

constant and variable assignment of functions �

As

the melody consists of 3 groups, the sequence of

.

•

chords with regard to these 3 groups can be read
directly from Figure XV, and the letters on

Figure XVI represent the respective bars of

Figure XV in such a fashion that the f�rst letter

refers to the first group of the melody, the sec ond
t o the second, and the third to the third.

0

Figure· XVI.

•

•

aaa

bbb

CCC

ddd

ggg

fff

eee

a ab aba baa

cca cac ace

eea eae aee

gg11 gag agg

aad ada daa

ccd cElc dee

eec ece cee

ggc geg egg

aac aca caa

ccb cbc bee

aae aea eaa

cce cec ecc

aag aga gaa

eeg cge gee

aaf afa faa

eef cfc fee

eeb ebe · bee
eed ede o.ee
eef efe fee
eeg ege gee
•

G
•

ggb gbg bgg
ggd gdg dgg
gge geg egg

ggf gfg fgg

0

0

25 •

•

G-

bba bab abb

bbc bcb ebb
bbd bdb dbb

bbe beb ebb

bbf bfb fbb
bbg bgb gbb

ffd fdf dff

ddf dfd fdd
ddg dgd gdd

abe

bcf

cdg

cef

acd

bdf

cfg

acf

bef'

ace
acg

ade

adf
adg
aef

bde

bdg

def

deg

cdf

beg

abg

u

dde ded edd

ddc dcd cdd

cde

abf

�

ffb fbf bff

bcb

bee

ffa faf aff

ddb dbd bdd

abc

abd

•

dda dad add

ffc fcf cff

'

ffe fef eff
ffg fgf gff

efg

•

dfg

ceg

beg
bfg

•

/

aeg

afg
•

The total number of possible harmoniza­

tions to be derived from Figure XVI is as follows :
7 cases on constant te.nsion: aaa, bbb, etc .

•

18 x 7 =

0

0

26.

= 126 cases on a tension that is constant for 2 of
the three groups.

35 x 6 = 210 cases with variable

tension for all 3 groups.

Thus, the total number

of harmonizations for the meloey o ffered is

7 + 126 + 210 = 343.
B.

The second method is based on a

random selectior1 of a ""i:""(13) based entirely on the

preference with regard to sonority.

•

As any � (13)

has definite substructures and often 'in limited

quantities, the possibilities of harmonization are

less varied than through the first method.

If one

selects L (13) w ith b� and f4{ on a c scale (see

Figure XVII) the possibilities of accommodating a

sectional scale 3 + 4 (minor triad) becomes limited

•

to only tv10 assignme11ts, namely, 5 - 7 - 9 and
13 - 1 - 3.

Figur� XVII.
l:" (13)

•

••

•

•

•

•
•

•

Retransposing these functions to the

melody assigned for harmonization we obtain the

following results.

0

0

I

27.
Figure XVI�_I.

C - �R.o.Jf>
•

•
•

•

•

(J../

As it follows from this figure, each

sectional scale of the melody permits only two
versions of chords.

Thus, by a constant or

variable assignment of the t,,o possible versions,

a complete table of possible harmonizations is

obtained .,
•

Figure •XIX •
aaa

bbb

aab

bl:>a

aba

baa
•

'

bab
abb

Thus, the total number of possible

harmonizations amounts t o a.

0

0

- - - - - -- - --

- --

-- -

.

-

-

�

.

-

-

28.

In tbe cases where sectional scales

are too complete, the assignment of only certain
For

tones as chordal functions is necessary.

example, in the following

scale based on 3 tonics

and 5-unit sectional scales, it is sufficient to

assign the wh ite notes as chordal functions, then

in the m�lody derived from such a scale, black

notes become the auxiliary

•

Figur� XX.

•

.
•

aid passing tones .

,

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In some symmetrical scales the structm·�

of individual sectional scales is such that the

sonority of certain pitch-units does not conform to
the structures of special harmony (i.e. harmony of
thirds).

Some of the units of such sectional scales

may be disturbing, and though they may fit as

passing tones in some other chord structures than
the ones emphasized by special harmony, the y

•

•

decidedly do not fit as passing ton es · in many r:,_ (i.�.
In such a case each pitch-unit in such sectional

scale of a compound symmetric scale must be assigned

•

•

0

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•

J..

either as a chordal function or an auxiliar y tone
with a definite direction.

These pair s, i.e.

the chor dal tone and its auxiliary tcne, ar e
dir ectional units r

In composing melodic for ms from the

scales containing dir ectional units it is necessary

to permute·.. · the dir ecticnal units and not the

individual pitch-unit s .

After all the units ar e

assigned the above described procedur e of harmoniza­

•

tion (the second method ) may b e applied.
Figure XXJ .

•
•

•

• -. .. .
,..

•
The arr ows on the above figur e lead

fr om an auxiliary tone to a chordal function.

•

0

0

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•

J O S E P H

S C H I L L I N G E R

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

C O U R S E
Subject: Music

With:,.Dr, Jerome Gross

Lesson CXLVI.

5 . Chromatic harmonization of a

.

•

Chromatic harmonization of a symmetric

•

melody is based on the same principle a·s chromatic
harmonization of a diatonic melody (see Form 2,
page 1 of Lesson CXLII) .

The proc�dure consists of

inserting passing and auxiliary cllromatic tones into

symmetric harmonic continuity .

As a result of such

insertion of passing or auxiliary chromatic tones

altered chor ds may be formed as independent forms .

This type of harmonization may sound as

either chromatic continuity or symmetric continuity

with passing chromatic tone s to the listeners.

•

(please see next page)

,,

•

0

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If you find that certain passing or

auxiliary tones in the above example sound unsatis­
The greater the

factory, you may eliminate them.

allowance given for altered chords, the greater the

number of possibilities for the chromatic character
of symmebt'ic harmonic co11tinultty.

6. Diatonic harmonization of a

.
•

Melodies constructed from sy mmetric

•

scales cannot be harmonized by a pure diatonic

continuity .

The style that has diatonic characteristics

is in reality a hybrid of diatopic progression�

stmmetr,ically conn�cted.

0

This type of harmonization

is possible when melody evolved within the scope of an

individual sectional scale can b€ harmonized by several

chords belonging to one key.

The relationship of

symmetric sectional scales defines the form of

symmetric connecti ons between the diatonic portions
of harmonic continuity.

The diatonic portions of

harmonization are conformed to one key.

Symmetrical

tonics do not necessarily represent the root chords of
a key .

For example , a note c in a melody scale may

be 1, 3, 5 , etc. of any ch ord.

In most cases of the

music of the past such harmonizations usual ly pertained

u

to identical motifs in symmetric arrangement, as in

0

0

the first announcement of a theme by the celli in

Wagner 's Overture to "Tannheuser", where identical
motifs are arranged· through '!/2, and the diatonic

portions appear as follows: the first in B minor
making a progression IV - I - V - III.

The following

sections are exact transpositions through the

,

'!/2,

i.e. they appear in D minor and F minor, respectively.

•

Figur e XXIII .

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In the following example of harmonization

the melody is based on a symmetric scale with three

pitch-units (2 + 1) connected through

3

,/2.

Figure •XXIV .

•

•

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0

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•

6.

Each bar comprises one sectional scale

As there are many

utilizing the melodic form abcb.

ways of harmonizing such a motif, here is one of them
producing C0 + c, + C t for each group, and all the

following groups are identical rep roduc tions- of the

original group connected through

•

Figure XXV.

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Music by Rimsky-Korsakov, Borodin and

Moussorgsky is abundant with such forms .of harmoni­

zation.

In order to transform the above

harmonization into a chroma tic on.e , all that is

necessary is to insert passing an d auxiliary

chromatic tones.

Diatonic harmoniza t ion of symmetric

melodies not composed on the sequence of identical
motifs where different portions pertaining to
•

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0

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7.

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individual sectional scales- are c onnected

symmetrically is possible as well.

The latter

form is not as obvious and may seem somewhat
incoherent to the ordinary listener.

.

•

•

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•

_/

0

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8.
Lesson CXLVII.
7. Chromatic harmonization of

•

a chromatic melo?y:

A melody which can be harmonized

chromatically must be a chromatic melody consisting

of long durations ..

be assigned to a chromatic operation in a chromatic

.

•

group of harmony .

•

•

Each group of three units must

The usual sequence d - ch - d

refers to every three not es, if the middle note is
a chromatic alteration.

'

Thus , in the 'following

melody the chromatic groups of harmony will be
�

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as signed as follows:

Group 1 : C - cf- d

•

Group 2: d

- d*- e

Group 3 : a - a'- g

Group 4 : g - g-f- a
Group 5: a - a*- b

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The process of harmonization �f a

chromati c melody chromati cally, consists of two

pr o cedures after the pit ch-units have been assigned
to some number combinations.

As our technique of

chr omati c harmony deals with 4-part harmony, the

melody must become one of the four parts.

Let us

assign the chromatic groups to the above melody as

.

•

follows:

•

Group 1 :

1 - 1 - 1

1 - 1 -

Group 2 :

Group 3 :

5 - 5 - 3

Group 5:

l - 1 - 1

Group 4: 3

•

- l - l

In group 3, a� is a lower ed fifth •
.f"
In group 5, a is a r aised r oot tone. The

following example r epresents the abov e melody in

a 4-part setting .

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10.

·v

The final procedure of chromatic

harmonization of a chro1na.tic melody consists of

isolating the melody, placing it above harmony and

melodizing the remaining 3-part harmony with an

additional voice ..

This additional voice is

devised according to the fundamental forms of
melodization, i.e. it may double any of the

.

•

functions present i n tl1e chord, or add the function
next in rank.

•

In the following example .'the notes in
parenthesis represent such added voice. The

functions of this voice are:
g

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11.
8 . Diatonic harmonization of a
chro�atic �elody:

A chromati c melody may be diatoni cally

harmonized when it has a considerable degree of

animation ( short durations).

In such case some of

the to nes are chordal functions and some be come

.
•
•

aux iliary or passing chromatic tones.

The principle

of assigning the fun ctions which are supposed to be
diatonic, must take place in this case.

The following example is th� melody

which was used as an illustration in the preceding

paragraph and only used in its most animated form.

lJ

Figyre XXIX.

By assigning

C - 5

\

we a c quire F chord.

d - 13
a - 5
In the next bar , by assigning
we obtain D
e - 9
g - 1
chord. By assigning
we obt_ain G chord, and by
a - 9

•

assigning b 1 - 5 we obtain B and E chords .

Thus,

the entire melody can be placed into a certain ·

I

0

0

12.

0

desirable key (C major in this case). The units
�
�
.
a and c in the second bar are auxiliary tones to

•

the third and fif'th respectively of the G chord.

The entire harmonization has a Phrygian character.
Figure XXX.

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same melody .

-

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Another example of harmonization of the
By assigning the following functions

we obtain another harmonization :
C -

e - 13

5

d - 13

•

g - 5

Figur e XXXI.
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13 ..
9. Symme�tic harmonization of a

(J

a

cpromatic �elody:

Symmetric harm onization of a chr omatic

melody is used for tne melodie� of long durations •

In most cases each pitch-unit of a melody has to be

harmonized by a different chord.

-.

The advantage o f

the symmetric method of harmonization is tha. t if a

melody is partly diatonic there is an opportunity of
using one c�ord against more than one pitch-unit of
a melody.

Any symmetric harmonizati on� as in the
·cases above, must be based on a preselected

u

L ��.

Let us assign the following � (13) and

use it for the harmonization of melody utilized in
the previ ous exampl.es ..

The important considerations

in the following procedure are variation of tension
.
and utilization of enharm onics as partic ipants o f

r (13) (a

',

SUjp.ements an equivalent of g

-:f

13th of a B chord) •

(please see next page)

0

for the

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14 .

Figure XXXII.
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