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J, 0 .S. E ,p R S. C lI -I L L r·
.t �

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G- E· R •
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C O U R S E
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Subject: Music
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Lesson LVII.
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VI. Climax and Resistance

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The projection of melody is a mechanical

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trajectory.

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Its kineti� components are balance,

impetus and inertia.

Resistance
produces
impetus,
.
.
.

leading either towards the climax, which is a pt
•••

•

(pitch-time) maximum with respect to primary axis, or
towards the balance.

The -impetus · 1s caused by resistance

which results from rotation.

T�e geometrical projection

of rotation is a circle which· extends itself in time
projection into a cylindrical or Sj.Jher.ical sviral, or
ultimately (through time ex_ter1sion) wave· motion (plane
projection).

The discharge of accumulated centri-

fugal energy is equivalent to a climax.

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The kihetic result 'of rotary motion is

centrifugal energy.

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A heavy object

attached to a string and put into rotary motion about an
axis-point develol,Js cor1siderable energy to move a long
distance, when detached from the string.
Overcoming inertia increases wechanical

\

efficiency (gain of kinetic energy).

Any body set to

move acquires its possible ulti1oate speed in a certain
�eriod of time.

COPYRIGHT 1942
l>SEPH SCHILLINGER
l[W YORK CITY
- . . -··--

The shorter the period from the moment

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2•
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of the application of the initial
,

the moment whEll the

fbrce (impetus) till

bod¥ acquires its uiti.mate speed,

the greater is the mecbanicai efficiency of such motion
.
•
Motion is expressible in the wave amplitude$ and the
projection of kinetic climax is the·maximum amplitude.
Inert matter does not acquire its maxiDJWII amplitude
instantaneously when starting from balance, Just as the
max.imum cawot recede to balance (rest) instantane�usly. ,
This concerns both velocities (frequencies) and
amplitudes.
Mechanical experiences, v1hethe� i.."lstinctive
or intentional, are known to all ty�es of zoological
species and are inherited
and perfected through the
/'f
course of evolution.

I

A gi,own-up animal has a perfect

judgwent of distances and directions and of the amount
of muscular energy necessary to cover them in leaps or
flights, without any theoretical icnowledge of the law of
gravity or mechanics in general.

There is no misjudgment

in the monkey•s flights from tree to tree, or a gazelle
leaving over a creek, or an eagle falling on its prey.
A certa.in amount of intentio1:1al roechanical efficiency
and psycho-physiologic coordination is inherent with
every surviving species of the animal world.

The

relativity of tl1e standards of mechanical efficiency
corresponds to the relativity of reflexes, reactions
and judgments.

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The leaping of a human being over a 14

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foot rod is the highest achievement in the International
Olym}Jics for 1936, and this with the aid of a pole.
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The

mechanical efficienc y of an ordinary flea is fifty times
greater.

A leap of a human being over a rod 50 feet

high would seem supernatural, while the respective leap
of a flea would be below any low standards of efficien�y
(tl1e flea leaps about one hundred times its own size) •
•

The standards of mechanical efficiency vary
wi tl1 ages ar.d places, even among huma11 beings.

They

c.lso vary with different races as v1ell as with different
ages.

The develo�ment of athletic qualities and forms

of locomotion im�ly the raising of the requirements
toward the trends of mechanical efficiency.
Geometrical conception of mechanical apd
bio-mechanical trajectories necessitates the analysis of
the corresJ.101,ding tra.jectories of nervvus imvulses and
muscular reactio11s.

There are corres!-'ondences betv1een

tl1e tv,o, arid the knov;ledge of such correspondences leads
to scientific _µroduct10r1 of excitors (in tr1is case,
esthetic: music in ge:r1eral, or welody in particular)
ca�able of stiwulating the intended reactions.
reflexes

a11d

Simple

react.1.crls !Jl'oject themselves into simple

trajectorial !,atterns; on the other hand excitors having
the forw of sim}'le trajectories stimulate rea.ctior1s of
tl1e corresi.,011ding simplicity.

C

Likevvise, this correspon­

de11ce tc..tCes 1,>lace with the cowplex jJatterns.

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4.

The intensity interdependence betv,een the

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excitor and the reaction was formulated in Weber's
and Fechner•s Psycho-Physiological law.

Thus, both

the configurations (patterns) and the amplitudes

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(intensities) have their corres�ondences between the
excitors and the reactions.

Judgment based on

mechanical experience and mechan ical orientation leads
. higher animals and human bei ngs to certain expectatlons.
In the case of an absolute corresponderice betv,een the
realization of a mechanical �rocess and the expectation,
the resulting reaction is balance (normal satisfaction).
A result above expectation stimulates tl1e ir!teri.sifica­
tion of activity (positive reaction) and at its extreme,

•

ecstasy.

On the other hand, the result of a mechanical

process whi�h is below exj.1ectati0n stimul�tes �assivity
.
(negative reaction) and at its extreme, depression.
The two opposite poles of reactions, led to their
absolute limit, stimulate astonishment, (irratic11al or
zero reaction).
Geometrical projectior1 of the scale of
psychological adject_i ves on a circumference j)roduces
the poles of the two rectangular coordinates (the
diameters of the circle): 1 . normal - absurd;
2. depressing -ecstatic.

Produc ing four new poles on

the intermediate arcs of the circumference through
addition of another pair of rectangular coordinates
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°

to the original pair) we obtain nine poles

(under 45

altogether (including both o

0

°

and 360 ) .

These nine

poles, through the application of the method of
evolving concept series, become expressible in adjectives
standing for the psychological categories.
Scale of psychological categories as
represented through geometFical project�on on a
•

circumference:

The circumference is divided, by the poles
of the coordinates, into 8 arcs, 45° each.

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The

geowetrical poles correspond to the psychological poles.
Arcs represent the transition zones, and the poles -­
their absolute expression.
Zero or 360
90

4

135
45

0

-

0

°

- abnormal

infranormal
- subnormal

- subnatural

(p lease see next page)

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180 ° - normal
270

°-

ultranormal

315 ° - supernatural
225 ° - supernormal

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6.

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Normal
180 9

225° Supernormal

Su bnormaJ 135 °

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Infranormal
90 °

315 ° Supernatural

Subnatural 45 °

Abnormal

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The psychological zones within the above
limits between the adjacent pole.s represent:
The zone around the 0 ° or 360 ° stimulates
astonishment (zero reaction or delayed reaction).
°

The zone around 45

stimulates either pity

or humor.

•

The zone around 90 ° stimulates depression
(pessimism).

•

The zone around 135 ° stimulates the sense
of lyricism (regr,et, melancholy [pleasant sadness,
joyful sadness, controllable, self-imposed sadness])

-

close to positive zone - joy of self-destruction,
self-sacrifice.
. .,

The zone around 180° stimulates the sense
of quiet contemplation, full psychological balance and
satisfaction.
The zone around 225° stimulates the sense
of heroism and admiration.
The zone around 270 ° stimulates the sense
of exaltation, ecstasy and worshipping.
The zone around 315 ° stimulates either the
se11se of fantastic or the sense of fear (unfavorable
surroundings, uncontrollable, unaccountable forces,
fear for existence, struggle for survival).
A discus thrower participating in' the
Olymvics and reaching ·the previous years• record would

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a.
stilllulate the reaction corresponding to 180

point.

The actual reflexes of the spectators would be polite

Throwing beyond the expected range would

applause.

stimulate the reaction corres�onding to the zone
between 180 ° and 225 °, culminating into ultimate
ecstasy when it reached 270 ° (this would be evidenced
by shput1ng, stamping and whistling, the reactions
•

increasing not only in intensity, but in quantity as
well), i.e., the maximum conceivable limit.

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The

clapping reflexes would grow accordingly, from 180°
to 270�

If the disc does not reach the range expected

the reaction would be disappointment, increasing toward
•

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'-...,I

135

°

with the sympathetic spectators, while with the

range reaching only 90° it would lead ultimately to\vard
depression.

The spectators will not applaud when the·

range of disc throwing is near the 90° point.

It 1s

natural to assume that certain groups of spectators,
influenced by their sympathy toward the opponents of
the first d.iscrus thrower, would produce exactly opposite
reactions.

These considerations cover the semicircle

above the. horizon.
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The lower zone, on the negative side, i.e.,
°
between the 0 and ,o

0

,

stimulates the reaction of

laughter and in the case of the discus thrower it would
amount to a range of perhaps onl.y a few yards from his
position after a iong and corresponding preparation to

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9.
a throw.

When the spectators see a husky, muscular

athlete deprived of mechanical efficiency they
unquestionably react to it as seeming decidedly
humorous.
On the positive side of the lower semicircle
between 315° and 360 °, lies the zone of supernatural,
•

where the range of throw of a disc would be beyond any
biomechanical possibility.

For example, if the range

of throw amounted to three miles.

In such cases the

presence of a trick or a supernatural force would be a
necessary ingredient for the logical comprehension of
the phenomenon.

v·

The usual reac.tion would be the

reaction of smile or laughter transforming into aston­
ishment in the direction of the zero point.
The 360° point when reached from the
positive side

YOU ld

amount to the absurd caused by an

impossible mechanical over-effioiency.

Such would be

the case v1hen the disc being thrown would never come
back, or fall anywhere on the ground, vanishing in the
interstellar s�aces and thus overcoming the law of
gravity.
When zero is reached from the negative
side it would mean an impossible mechanical inefficiency.
In the case of a disc thrower· it would happen when the
.
disc. would slip out of the athlete•s hands before he
actually threw it.

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10.
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A tra jectory expressing a mechanically

efficient kinetic process, whether a pendulum or a
musical melody. 11'111 have the mechanical fundamentals
A pendulum carmot start instantaneously

1n comon.

_at its max1mull ampl1tude; neitber can a melody.

A

pendulum cannot stop instantaneously from its maximum
amp litude; neither can a melodf.
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The corresponding

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effects 1n both cases will be either supernatural or
humorous.
The actual quantitative specifications
serl!ing different purposes and expressing different
forms of mechanical efficiency vary with times and
places.

To satisfy any esthetic requirement one l1as

to know th e style in which such requirements have to

V

be carried out, beyond which specifications the entire
kinetic process, whether efficient or not, will be
meaningless.

As the standards vary, the coordinates

on the circle described above change th eir absolute
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positions, i.e., the zero point may move with the
entire system, eit her clockwise or counter-clockwise.
If we would assume, with regard to athletic standards,
18()° to be a limit of certain mechanical operations
when the achievement of the following epoch increases
the quantitative value of normal, placing the point of
norwality to what is 225° on our diagram, the opposite
pole of the coordinate will occupy res�ectively the 45 °

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11.
Referring to music in general and melody

position.

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we find certain standards become old
-in particular,
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fashioned and we begin to feel that though they may
be charming yet they are entirely inadequate for the·
purposes of a more mechanically efficient epoch.

We

feel it in every field concerned with motion, i.e. ,
mechanics.
�

There is a humorous or a pitiful reaction

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toward the 1900 horseless carriage and it becomes
still more humorous where there is an accumulation of
quantities of the symbols of inadequacy, such as the
prerequisites of travel required by a horseless
carriage (dusters, goggles, safety belts).
•

We hav e

exactly the same picture (i.e., if we are people
representing our epoch and not living anachronillms), 1n
melodies composed by a Verdi or a Bellini, where the
mechani.cal efficiency is so low that it makes us smile,
if not laugh.

The same melodies stimulate entirely

different reactions among the octogenarians surviving
in our epoch of 400 miles per hour •
In order to achieve an efficient climax it
is necessary to accumulate energy that would be
effectively discharged into such climax.
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The means for

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accumulating energy, as it was described above, are
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achieved through rotary motion developing centrifugal
energy.

Trajectories ex�ressing musical pitches of

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12.
various frequencies are heard by the.listeners 1n
relation to the entire trajectory.

It is

possible

not only to show the range of frequencies (such as
a form of direct trans.itian from one frequency to
another), but also to show in what way this variation
The portion of a melodic

of frequency was achieved.
trajectory leading

toward the climax, without a

resistance preceding such a climax, does not produce
any dramatic effect.

It is the resistance that makes

the climax appear dramatic.

A portion of melodic

trajectory leading from a climax (maximum amplitude)
towards balance (minimum amplitude) must be performed
in accordance with natural mechanical laws, i.e., it
must contain resistance before it reaches the balance
(compare with pendulum).

excess ot the

The inefficiency or the

forms of resistance (rotary motion)

leads to a mechanical abnormality.

Abnormal melody

stimulates the sense of dissatisfaction or humor.
The forms of resistance leading toward climax acquire
centrifugal form (increasing amplitude).

The forms of

resistance leading toward balance acquire centripetal
form (decreasing amplitude).

The relative period o f

rotary motion an d amplitudes produces various forms
and gradations of resistances.

For example, the

period ot rotation may be long, with the amplitude
reJDa1n1ng constant; or the period of rotation may be
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13 •
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short with rapidly increasing amplitudes.

The period

of rotation way be short with the correspondingly
increasing·amplitude.

The duration of the rotary

. period may be in inverse proportion to the amplitude

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and often the law or squares takes its place •

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14.

Lesson LVIII.
The corresponding forms or resistance as

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applied to melodic trajectories are:
l. Repetition (correspondences: aiming, rotary

motion with infinitesimal amplitudes, affirmation
or the axis level as a starting point).

llusical

rora: repeated attacks of the same pitch discontinued by rests or following each other continuously •

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Pb.Ysical Form

Musical Form
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2. one phase r9tation (corres�ondences: preliminary
contrary motion, initial impulse in archery,
artillery, springboard diving, baseball pitching,

tennis service, etc.)

Musical form: a movement or

a group o f movements in the direction opposite to
the following leap.

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!9sical Form

Physica� Form

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. ' . . . . . . . ... . .

• • __,_ • • • ---a • • ••
-·· --�· • • - •• -··

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This form often acquires more.than one phase
following in one direction which intensifies the

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resistance.

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. . . . . . ---, . . . . . .
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3. Full_p�riodic rotati?B (one or more periods)
A. Constant amplitude (correspondences: rotation around
a stationary point, a top, some�saults - with diving
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and without - lasso, axis and orbit rotation of the
planets, Dervish dances).

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16 •

lilaical Form: mordente, trill, tied tremolo,
groupetto.
Jlusical Form

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B. Variable amplitude (correspondences: giroscope, spiral
motion, tornado, expansion, contraction).

llusical

form: expanding and contracting, simple and compound

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110tion.

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Whereas th e preo1dil11 r.... of ,.1e,111·.t.Ni
require 0ulf, one ot the secondary an•� tbe YU"1able

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· amplitude rotation requires a silliuitaneous oc,llb1nait10D
of two or three second&l'f axes.

ID th1a case 'tbe •Xii

leading towards cli.JDaz or balance will be coii8idered

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fundamen.ta1 and the other au• - oml

17,

Simultaneous combinati ona of two axes:
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(a) Centrifugal (expanding):

Phys,ical Form

Musical Font

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l...,

0

d

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18.
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C•) Calrlpetel (cODtraot:sng)

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Phf1ic&1 l CU

Mlsical Form

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0
b
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Simultan
eous combinati ons o f three ax
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es:
(a) Centrifugal (expanding):
a + 0 + d;

d + O + a
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19 •

(b) Centripetal (contracting):
b + 0 + c;

c + O + b

Physical
-. Form

Musical Form
a + 0 + d

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d + 0 + a

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b + 0 + c

c + 0 + b

,

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20 •
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Lesson LIX.
As the iz\teryal of a pitch level from
the primary axis affects tension (gravity effect
where P.A. is a gravitational field) resistance may
also result from two parallel secondary axes.

The

complementary parallel axis may be placed either aboye
•
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or below the fundamental axis.

The effect of motion

through a pair of parallel axes is that of an extended
trajectory (delayed forced inefficiency).
'

In reality

it is the usual rotary motion only evolving between the
two axis-boundaries.

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The correspondences of such mvti0n are:
raising and falling� zigzag ascending arid desce11ding.
Musical form: revolving around alternately progressine
points (ascending or descending).

(please see next page)

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a
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.......,,)

a.

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--

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C
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24 •

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d.,

,

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J_ -

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,

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0

or

•
,

•

....

•

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•
,

0

•

0

I

The 1, 2 and 3 forms of resistance produce
•

the respective degrees of resistance.

When more than

one form is used in successive portions of melodic
continuity they must follow one another in the
increasing degrees.

The opposite arrangement is

mechanically i nefficient and therefore produces an
effect of weakness.
Resistances lead either toward cl�max or
toward balance.

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Fro■ Balanoe:

ce.r
•

• •

• • •

'--•·

•

•

• • L--.· . --

c.e. .:i[

-

-

- .. --�-.

•

-

•

- -

- -.

'-

- --

•

,.

ee. . m.

Toward Balance:

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Distribution of C1imexes 1n

Melodic Continuity.
The distribution of climaxes in melodic
continuity must be performed with respect to the total
duration of such continuity.

•

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The relative intensity

of climaxes depends both on time and pitch ratios

•

•.

leading toward the respective climaxes.

The natural
•

tendency 1s the expansion of pitoh and the contraction
of time.

These two components mutu ally compensate each

other.
The climactic gain between the two adjacent
climaxes takes place when:
1. The pitch-ratio is increasing and the time-ratio is
constant;
2. The time-ratio is decreasing and the pitch-ratio is
constant.
The climactic gain reaches its mechanical
maximUJD when both forms . are combined (increasing
pitch-ratio and decreasing time-ratio) .
It is practical to save the last effect for
the main climax of the entire melodic continuity and t o
use it only when the extreme exuberance has to be
attained.
As the decreasing time-ratio is charecteristic
of the continuity with a group of climaxes, rhythmic

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28.

material which 110uld appropriately distribute the
climaxes must belong to the decreasi ng series of growth,
such as �•U•NDt'>tion or power series.

Smaller number

values and in inverse correlation serves as material
for the distribution of the pitch ratios for a group
of successive climaxes •

•
•.

This description refers to a traj ectory
moving towards main climax and must be inverted for the

opposite dir ection•
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29 .

Lesson LX,
VII, Superimposition of the Time-RhYthm

on the Secondary Axes
"Beauty" is the resultant of harmonic
relati ons .

In order to obtain a

11

�autitul" (esthetically

efficient) melody it is necessary to establish harmonic
relations between its factorial and fractional rhythm.
This may be achieved by means of a homogeneous series
of factorial-fracti0nal continuity.
durations occurring within the bars

Rhythm of time
lD\lSt

belong to the

same series as the rhythm of the seco ndary axes.
Naturally, there are hybrid melodies ·where fractional
,

and factorial rhythm belong to different series.,

A

homogeneous series is merely an expression of stylistic
consistency.
Melodies wi th structural consistency may be
found nearly in every folk lore, as well as in the works
of composers who synthesized and crystallized the
efforts of their predecessors.

Beethoven crystallized

the melodic style of the "Viennese School", (which at
its time was a revolt agai ns t counterpoint and poly­
phonic writing).

Bach, in his melodic themes, (in many

cases wi th an odd number of bars), crystallized the
efforts of several centuries .,

Mediaeval Cantus Firmus

was the source of his ·toomes.

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30.
Different styles have different evolutionary velocities.

•Jazz" having a veey high one,

(like some o f the specimens of the Alpine flora with
a veey short 11fe-span), has already crystallized its
Examples are numerous and may be found

homogeneity.

•

more in the •swing• playing than in the printed copies
of the songs •

..

After the series has been selected, the
•

actual composition of the fractional continuity may be
accomplished i n two ways:
(1) by using the resultants or the power groups,
(2) by composing freely from the monomial , binomials,
trinomials and quintinomials of a given family,
(see "Evolution of Style in Rhyth m").
An example of composing fractional
continuity 1n ¼ series:
Suppose we have a trinomial of the
Secondary axes, a2T + bT + cT.

In this case 4T ; 16t.
To satisfy 16t we may use r
, or (2+1+1 ) 2 , or any
4
4+3
of their variations, i.e. , the permutations or the
resultants.
A free composition according to (2) may
give results identical with some of the variations.
The groups of the ¼ series are:

,

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31.

•

monomial . . . 4
binomials. • • 3 + 1 and 1 + 3
trinomials . • 2 + 1 + 1, 1 + 2 + 1 and 1 + 1 + 2
the· uniform quadrinomial • . 1 + 1 + l + 1
Deciding upon a2T being (3+1) + (2+1+1),
bT being 1+1+2 and cT being 1+3, we obtain r4+3.

•

By

selecti.ng freely various recurrences of the same
binomial, like 3+1, we obtain: a2T = (3+1) + (3+1),

•

•

bT = 3+1, cT = 3+1, or various recurrences of the same
trinomial ·nith variations, like: a2T = (2+1+1) +
.
+ (2+1+1), bf = 1+2+1, cT = 1+1+2, we obtain groups
•

that are not identical with the resultants or the
power groups.
When a cl imax is desired the maximum time
value wust be pla ced at the corres�onding point of a
Secondary axis (in � at tl1e ez1d, ir1 J?. at the beginning,
in .£ at th e beginning arid in £ at t.l'1e end).

For

instance, if a climax is desired on a2T, it must be
the last term of a rhythmic group of this axi s .
4

In the

•

4 series i· t wou ld be:
,
a2T = (2+1+1) + (1+3)
or (2+1+1) + (1+1+2)
or (2+1+1) + 4 and the like�
To superimpose a fracti onal rhythmic group
on a factorial gro up of the Sec ondary axes, means to

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32.

distribute the points of attack on a pitch trajectory
(the path of a moving point).
Let us assume th�t a gro�p of secondary
axes has been constructed with no reference to any
Placing the

particular logarithmic (tuning) system.

pre-selected fractional group above the axes and

•
•

••

dropping perpendiculars from the points o f attack, we
•

accomplish the distribution of the points of attack

(which become the moments o f attack) along the pitch
trajectory of a hypothetic tuning system •
•

Example

a2T + b'f + ct = (2+1+1) + (1+1+2} + (1+2+1) + (1+3)

·' 0

16T

�

4T
2T+T+T

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l
I I l I r• l• r, t•
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&&$

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,,

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r
r• •1, r
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l

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r

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..

t. -u
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Thus, the red point·s are the moments of attack on
this pitch trajectory.

I

•

•

Here we arrive at the following definition
of melody: melody 1s a resultant traJectorz of the
axis-group moving through the points or attack,
Melody, 1n the academic sense, 1.�. , with sudden pitch
variations within a t1Jning system ., ·is a rectangular
trajectory.

•

Melody ., in the Oriental conception., as

well as in any musical actuality ., is a curvilinear

•

tra jectory, i.e., co ntaining a certain amount of pitch•

sliding.

'

We shall deal with composition of a melody 1n

the academic sense as our musical culture leaves the
bending of a rectangular traj ectory to the instrumental
performer.
•

As the secondary axes form triangles (with
respect to primary axis) , two forms of rectangular
motion through the points of attack are possible :
(1) ascribed (sin phases)
(2) inscribed (cos phases)
Though in c omposing melody a free choice of the two may
take place, in balancing melody at its end on b or c
axes, the ascribed motion produces an incomplete (i.e.,
unbalanced) cadence, while the inscribed motion produces
a 9omplete (i.e. , balanced) one.

The first one is a

device for deviating from balance, i.e;, for accumulating
ter1sion, a stimulus for the new recapi tulation.
Examples of rectangular trajectories evolved
'

through the axes of the previous example:
,.... .

I

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IP?Rlf

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Jlot1on

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,,

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- Ascribed

•
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,

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Example II - Inscribed Motion

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35.

These two potential melodies are
totally different in their pitch progression.

The

usual, commonplace compositicn of pairs varies with
respect to the cadence only.

Such pairs may be

either inscribed or ascribed, but must be identical

'

othe;rwise; the ending of the first one is ascribed,
while the eming of the second - insc.ribed.

..

'

,

•

I

•

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

Lesson LXI.

VIII. Superimposition o f Pitch-Rhythm
(Pitch-Scale) on the Secondary Axes .
Unifo rm time-intervals (du ratio ns) being
ge ometrically pro jected p roduce space-intervals,
(extensio11s) .

Such uniform time seal.es are primary

selective systems v1hen T ::: ra+l · When b f l they
beco me secondary sele c tive systems, (rhythm-scales) .
Unifo rm pitch-intervals of our tuning system

.-------

1
produce lo garithms to the base o f � (semito nes) .

Chromatic scale is the primary selective system o f pitch
in o ur into natio n.

Geo metrical projectio n of sucn scale
Any o ther pitch-

is a uniformity along tne o rdinate.

scale within the same tuning system is � secondary
selective system, (i.e., a derivative of the primary
selective system ) .

•

It is easy to see that a pitch-time traject ory
m oving in either asc ribed o r inscri bed fo rm o f moti on
thro ugh the po ints of intersection of time (abscissa)
and pitch ( ordinate) unifo rmities (primary selective
s,steos) , is (structurally) the simplest form o f melo dy,

..

i.e., a chromatic scale in unif orm rhythm .

I

•

•

2.
Here we arrive at the followi11g definition
of melody: melody is a pitch-time trajectory resulting
from the intersection of the points of intonation

__

.......,_

·
·:

(��tch-units) with the points of attack (time-points)
in a specified axis-system.
When the geometrical points of intersection
do not coincide with the pitch-units of a scale, pitch­

--

units nearest to th e coincidence-points must be used .
Let us superimpose an Aeolian scale
(2+1+2+2+1+2) on the axis-group illustrated in the
preceding chapter.

\

u
,..,,......

Let us assume a2P + bP + cP, i . e . ,

a parallel PT correlation.

And let P

= 5, which in

this case gives a symmetric distributio11.

•

c be the _prims.ry axis.
.f.ii tch £.

.Ql, , bP from .f. to £.

to

Let furtl1er,

Then a2P exterids from
cP from _g_ to £..

a11d

Here is the final cvnstructio11 of tne
•

a.xis grou.1-1:
Scheme of the Points of
Geometrical Intersection.

t: .

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!

1

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.!.
' �

• ---.,.--,.
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• f I ••

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� .. !:.r �
..
f1.
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-ti I•• ..._.__ .J' '
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!/..
:) .,.-...--..r- I t-r
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---L

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'- -, I
-. 'T'
.....
-- '
J..,..

.l..•
.

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II

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3.
This diagram produces a slight deviation
from the description given in the text, due to the
fact that the scale is small enough to give deviations .
•

However, this is not essent�al as further adjustments

-

follow the scale.

The next step is to adjust the �oints of
intersection to the Aeolian scale .

Let us analyze

point by point •

•

If the first point of intersection is £,
the nearest pitch-unit to the second 'point of inter­
section

0
....

on the Aeolian scale is £•

Next, we select

e P as the nearest to the third intersection-point.
f.
The fourth falls exactl y on whicl1 is not in the scale.

r '#
The fifth falls on -

In this case either the

repet ition of £, or g is available.
nearest to g.

-a

Next point is

Through ascribed motion the entire axis

b �.
d and end ori would start on As in inscribed motion, pitch-�evels move

toward the points of intersection; tl1e first pi tch- .
b- axis will be either unit on e�, as the geometrical
f or intersection coincides vii th � ,.
d.
point is nearer tc f

The 11ext intersection­

b- axis
In order tu coruplete -

through inscribed wotion it will be necessary to consider
£ as the last intersection �oint.

C- axis thrcugh the

inscribed motion gives its �oints of intersecticn at

-a '
•

c.
and -

I

•

4.

We shall reconstruct now ��e axis-group
with respect t o the Aeolian scale, as Just described,
This trajectory is

and draw an inscribed tra jectory.

•

the most elementary form of an actual melody.

.

••

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.

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C. nc.

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It would not be difficult to find all
other versions, i.e., the ascribed trajectory and the
trajectories where either axis may ba realized in

0

ascribed or inscribed �otion.

•

•

I

•

- ·-- - - - ---- -- \

5.
Here is a chart of combinatioris :
Axes :

••

a

b

ascribed

ascribed

ascribed

ascribed

ascri bed

inscribed

ascribed

inscribed

ascribed

i11scribed

ascribed

ascribed

inscribed

inscribed

inscribed

inscribed

inscribed

ascribed

i11scribed

ascribed

inscribed

ascribed

inscribed

inscribed

••

There are eight versions altogether.
After obtaining an actual melody, such melody becomes
a subject to scale variation, tonal and geometr ical
ex1,181siu11s and inversions.

For instance, the same

melody in a "blue" scale wo uld sound :
'
•

•

or in a Chinese ( 2+3+2+2) scale (through
the translation of the corresponding degrees) :
(�lease see next page)

•

•

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-. ....

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6•
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_L �

.•.,.

., �·

�

..
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j

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Here an allowance has to be made on the first note
of the last bar, as the VI does not exist in the
Chinese scale, (substituting it by the last degree
of the scale, i.e., V, which is ,!!).

•

'

•

.•

,

-

I

•

-

-

. -

..

-

-

·- . .. . - .

. ...

--

7.
Lesson LXII.
IX. Forms of Trajecto rial Motion
The trajectory obtained above was called

•

"100s t

e lementary for1n of an actual welody 11 because

its form of motion is simple harmonic, (i.e. , scalewi se)

.

•

motion.

According tu Chapter VI, such melody cannot

be too expressive or dramatic.

In order to obtain an

expressive melody it is necessary to build resistances.
This cannot be realized without introducing more
com�lex forms of motion.
We shall present now all the traj ectorial

0

forms with respect to the zero axis.
( 1) Sin wotion with constant awplitude:

•

(2) Cos motion with const&nt amplitude :

(3) Combined sin + cos motiun wi th constar1t awplitude:
•

I

•

--- � -

- ----·--- --

- - ---- ··.. --·· --..,_ ___ -

. .. - ··

-

8.
(4) Cowbined cos + sin motion with constant amplitude :

.c::::>-

.___
.__�"<...:
7�
.7

'
•

( 5) Sin rootio11 \'Ii th increasi11g amplitude :

c:::::::::-,,,

(6) Sin motion with decreasing awplitude:

(7) Sin motion with combined il1c.reasing-decreasing
amplitude:

'I

I

•

-·· -

.

.

. .. •
•

.

(8) Sin motion with combined ciecreasing-increasing

.

amplitude ;
•

••

(9) Cos motion as (5) :

•

(10) Cos motion as (6) :

'

(11) Cos motior1 as (7) :

..
•

•

I

•

10 •
(12) Cos motion as (8) :

•

•

(13) Combined sin + cos motion with co.u1bined

• •

amplitude as (5) :
••

•

,

(14) Combined sin + cos motior.1 with cowbi11ed
amplitude as (6) :

(15) Combined sin + cos motio11 v11. t..1 co1ubir,1ed
amplitude as (7) :

I

•

11.
(16) Combined sin + cos motion with combined
amplitude as (8) :

•.

(17) Combined cos + sin motion with combined
amylitude as (13) :

(18) Combined cos + sin motion with cowbined
amplitude as (14) :

•

(19) Combi11ed cos + sin motion with combir1ed
aruylitude as (15) :

I

•

1 2.

.r

(20) Combined cos + sin motion wi th combi11ed
am.Plitude as (16) :

.

•

These twenty versions are werely variations
.
of the two original forms, l. .e. ' (1) ai\d (5) . Every
.
of the sin a11d every decreasing amplitude
cos l.S
.
l.S G) of the increasing aw�litude •

©

•

Furth er uevelo.t1ment of tr1ese traj ectorial
•

forms may be obtair1ed through ap.iJlicatio11 of the
coefficients of recur·rer1ce of the sir1, tl1e cos an<i t:t1e
growth of aw_pli tudes .

For instar1ce, 3 sin + cos +

+ 2 sin + 2 cos + sin + 3 cos on constant arcvlitud e :

c:'\ .c::\,
,

\J

�V�'\7'CJ

The same case on increasing amplitude:
•

, ...__,
\

•

•

I

•

•

13.
•

All these forms being transformed into

rectangular trajectories, with respect to a definite
intonation (tuning) system, become actual i11tonaticn­
groups, i . e . , melodic forms .

For example, a group�!to

is sin + cos witl1 constant amplitude.

,

IncludiI1g the zero of pitch variation,
•

(absolute zero-axis trajectory) , we have the following
forms of trajectorial motion:
(1) constant �itch trajectories (repeiitioI1 on
extensior1) .

'

(2) sin or cos. trajectories (one phase moti0n ).
(3) combined trajectories (full period wotion or
rotatior1) .
Application of various tr�jectorial forms
to !!., .!2_, £ and £ a.Xes gives the follov1i11g correspon­
All the sin of O remain sin on all other

dences .
•

axes .

All the cos of O rewain cos

011

all other axes.

All the combir1ed forms of O v1ith respect to sin, cos
tl1e
and the co11 stancy of amplitude remain respectively
. .
satne ort all other axes.
be heard.

Zero axis is tr1e only one to

The rest are ruerely hypotl'1etic lines.
Here is an example of the corres�onding

transla tio11s of a curvilinear sin tra jectory into

- -

-

recta11gul&r traj ectories of the o, a, b, c and d
axes .

I

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0

1 l Il j l

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/. •

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,,

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Translations of the cos trajectory:
,

•

-u, r·u

,

•

•

,

•

'

''

:,.

•

,-J

''

•

,

•.

- , ....
... ...
,.....,
'

I

Translations of the combined trajectory:
(1) With continuous tangency!
•
•

•
•
•

•
•

•

•

•

•

•

•

•

•

•,

.
•

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,,
,

........
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--.1✓

--■ollv

I

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. .

16.
(2) With out continuous tangency:

•

(1) may be called revolving trajectories.
(2) may be called crossing trajectorie�.
Devi&tion of a rectangular trajectory from
its corresponding axis signifies inco11s istency and
lowers the esth�ti c value of a melody.
An esthetically efflcient melody must
uisplay, besides consistency, a variety of the forms
of motion.
When a trajectory is controlled by the
two simultaneous axes (fundamental and complementary),
the points of attack may fall on either a.xis according
to the form of alternation.
(please see next vage)

•
•

I

•

-.

•

•

•

16 •

•

Example

I

I

..
..

•

,

......

.... ..

•

...

•

•

•

.......

The form of alternation is subject to distribution,
i.e . , rhythm.
An example of analysis of the trajectorial
motion in J . s . Bach ' s Two-P�_rt Ipver1t�on, l�o . 8:

✓

"

•

'

....
..► .... • - •
.....

,,

,-

•

J

.

:•
•

j

.....

..
�

..

.

"J'
•

I

•

•

•

•

17 .

•

..

The staccato eighths are expressed on the graph as
sixteenths.
This traj ectory has a primary axis
defined by its first, last and two intermediate
The group of the secondary axes is : � + b.

attacks.

The pi tcb a.."l d time ratios are uniform, 1 .e. ;

-

The first attack of b is a c limax .

�PT + bPT .

••

0

'-

The

form of motion on � is sin motion with increasing,
(centrifugal) , amvlitude.

The alternation of the

points of attack on the two conjugated axes is uniform.
The form of motiun on Q is combined (sin + cos) and
has a constant amplitude.
to b .

It is ascribed with respect

The effect of revolving due to the combined
•

•

form vroduces a resistance and delays the balance •

•

This melody v10 uld lose 10ost of it s esthetic value if
the o-axis were eliminated (loss of resistance moving
toward tl1e c limax) , and the b- axis would have one-pha se
motion .
\

•

At this point it would be very advisable
to wake a thorough analysis of the outstanding as well
as the deficient themes taken from the existing music .
This }-lrocedure must follow all the nine cr1apters of
the theory.

A precise statement must be made on each

itew, (reghrding the form

a11d

.
the measurement)
•

Though a theme of any dimensions (durati0n)
•

'

•

I

•

--- --

- .. - . ..

- -

,_..

- -· .. . - -- - -

18.

--

may be constructed to full satisfaction it is more
practical in most cases to compose continuity out
of a short original structure.

Memory is very

limited and the latter will produce an effect of
greater unity •

•

-

After having enough experience in analysis

•

one may start com�osing melodies according to this
theory.

Success depends upon thorough lmowledge of

all the precedi11g material , and a bility · to think •

•

•

•

0

I

•

...

.

-- ..

�

.. .. .....

....

-

,

•

19.
•

Lesson LXIII,
0

X. Composition of Continuity

Melody plotted according to this technique
has the following properties :
l. Permutability of the secondary axes with their
respective melodies in time continuity.

•

2 . Permutability of the individual pitch-units
(preferably through circular permutations)
pertaining to one individual secQndary axis.
3. Geometrical convertibility of the entire melody .
4 . Geometrical convertibility of the portions of
melody pertaining to the individual secondary axes
or any groups thereof.
5 . Tonal expansion of the entire melody.
s. Tonal expansion of the portions o f melody pertaining
to any individual secondary axis or portions thereof.
In this case different axes may appear with dif ferent
coeffic ients of expansion.
7. Combined variati ons of geometrical inversions and
tonal expansions applied to the entire melody.
8 . Combined application of geometrical inversions and
tonal expansions applied to the portions of melody
pertaining to individual secondary axes or any
combinations thereof.

0

In this case different

coefficients of expansi on may be combined with
different geometrical inversions.

I

•

,

... . .. -.. - - - - -·--- ,..

__--

0

- ---·

20.

Continuity may be composed through any
of the abovementioned forms of variation or any
combination thereof.
Here is an example of the quantitative
development of melodic continuity from the original
Let us take a trinomial axial combination,

theme.

•
,

a, b, c.

•

Each of the individual axes has four

geometrical inversions.

Thus, the number of combina­

tions of the three axes being used in identical or
different geometrical inversions · equals 4 3 = 64.
This number refers to one constant E.

If any of the

individual axes appears 1 n three forms of tonal
expansion, the entire quanti ty will be 6 43 = 262,144.
•

The following is a method of indicating a

secondary axis where the geometrical positions and
the coefficients of expansion are specified.

For

example, an axis .!. in position © in the second
•

expansion (E 2 ) may be expressed like this:

A trinomial axial-combination consisting
of a, b and c axes, with specified time and pitch
ratios, and the geo metrical positi ons and coefficients

0

of expansion, assumes the following appearance:

I

•

.. ..

. .-

- - . -- -

--

•

21 .
•

..

Time ratios: 2 + 1 + 1
Pitch ratios: 1 + 2 + 3
Geometrical positions:

@ -,

@, �

Coefficients of expaz1sion: E0 , E 2 , E ,
aP2�0 + bT2P� 2 + cT31W '
·
This method of indication emphasizes not
the axial struc ture alone, but the pitch-units
(intonation) as well.

For example, a melody in its

third displacement, on the axis a, 1.n position @ ,
in the third expansi on, may be expressed as follov,s:

When this method is systematically applied,
the sequence of the dif ferent displacement phases,
with regard to consecutive secon dary axes, may assume
different forms of distribution.

For exawple, it may

start wi th the first phase within the first axis, with
the second phase

\vi. thin

the second &xis, with tl1e

third phase within tre third axis, etc.

It may follow

a rhythm of any resultant or any of the series of
growth·:

Naturally, the rhythm for such variatio11s of motif
• depend upon the numb er of pitc:t1-units wi thin tl1e
motif.

I

•

. . - ---· .. ... ·- ---

---

- .. --·-

-

22 •

•

The ability of producing expressive

--

melodies (themes·) does not make a great composer .
•

The ability of producing an organic continuity out
of original thematic material does.
Going back as far as to the strict style
of counterpoint written to a cantus firmus, we find

.

that composition of continuity is based on uniform

•

factorial periodicity .

The theme regularly appears

in different voices and that keeps the music moving.
In all elementary
homophonic forms,
,

continuity is based on com.Position of biners

(a , + a 2), usually similar structures with different
endings, cons isting of 4 + 4 or 8 + 8 bars.

Next

comes the method of terners, i.e., a , + b + a 2 , intro­
duction of the new material in tl1e center.
The most advanced forms in the past were
offered first by J.S. Bach, who us ed a sequence of
biners i n contractir1g geometrical progressio11s (see
Fugue V, Vol. II, Well Tem�ered Clavichord).

In his

case it meant that a greater overlapping, (stretto
between the the we and the reply), occurred witl1 each
•

fol lowing announcement •
In Beethoven ' s case it meant a co11tinuous
breaking u_p of the original biner.
All tl1ese forms of continuity, (in the
2
past), are rigidly a ttacl1ed to the 2
series.

I •

I

•

23 •
Richard Wagner built his continuity
according to the script, i.e., the operatic libretto.
Though he wrote them himself, and was quite skilful
at it, musical continuity greatly suffered from this
syntactic dominance.

Wagner•s faults were adopted as

virtues by Scriabine and others.

.

•

Literary influence,

toget.�er with linguistic logic and syntactic (propo­
sitional) technique were the factors that delayed,
if not prevented, the sound development of the forms
of musical continuity.

(See the de'firiition of

program music in The Oxford History of Mus ic , Vol. 6,
Page III, which says: "Prograw music is a curious

0

hybrid that is music posing as an unsatisfactory kin d
of poetry").
Forms of musical continuity are purely
quantitative and pertain to motion.

They are bio­

mechanical, i.e., they are forms of grov,th.
they grow norwally, tr1e y survive better.
Darwinism: struggle
the fittest .

When

It is pure

for existence, the survival of

A star-fish is not "just a pretty

pentagon" but an organic form evolved through the
necessity of efficient functioning.
Many an unpretentious melody is appealin g,
i.e., esthetically efficient, due to the fact that
within ��e eight-bar structure certain processes
evolve in a very consistent manner.

It happens quite

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---- -- ------ - -· - --· -- --·

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24.
often that ih soille cases the efficiency of structure
is greater in sinaller portions and smaller in greater
portions and vice-versa.
The bio-mechanical forms are priIDarily
concerned with three basic factors:
(1) Symmetric develo�ment, i.e., the axis-inversion.
(2) The ratio of growth, such· as summation •
••

(3)- Movement with res�ect to tension and release
resulting in balance, i . e . , an arithmetical or
a geometrical mean .

•'

Growth alorig the a.xis of symmetry ( compare
wi th a human body, with its growth along the spinal

....
\

cord) , is a continuity formed by geometrical inver­
.

sions of the original structur e or its portions,
(melody) , along tile primary axis .

The regularity of

recurrence of the different inv ersions is subjected to
rhythm .

Pitch expansions ( tollal and geo metric al) ,
,

combined v,ith their geometrical inversior1s may be used
as components

of musical continuity.
The mos t fluent form of continuity result s

from the symmetric growth along the time-axis .

•

This

is the mos t complete form of continuity as it

.

exemplifies birth, grow th, maturity , decline and
death, all in one process.

To accomplish this it is ,

necess ary to split the origin&l structure i11to a
..

number of elements ( such as

bars or secondary axes) ,

to show these elements in their gradual addition, and

•

•

I

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25.
then in their gradual subtraction.
Suppose we have a three-bar structure
and split it into a, b, c elements*

Gradual

addition of the elements will give : a + ab + abc.
Gradual subtraction of the elements \Vill give:
The cowbination of the two forms --

abc + be + c .

the time-axls on abc.

.

The entir e continuity will

be this:

•

a + ab + abc + be + c
Exau1_ples

'
•

The origir1al structure split il.1to
three elements :
•

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Cor1tinuit y composed thro ugh the time-axis.
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The �rocess of summation. may pertain to
the precedir1g procedure, as well as to factorial
ratios of the secondary axes, or the number of

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26.

individual attacks.

An example o f summation through the
first summation series based on the time axis.

Let

us tak·e an eight-bar structure and split it into
a b c d e f g h elements�

The continuity will have

thls form:
a + ab + abc + abcde + �bcdefgh + defgh + fgh + g h · + h .

•

The entire structure moves acDoss itself through its

•

ovm axis, while time goes on.
The next point is obvious. · Using the
same series for the T of the secondary axes, we
obtain :
T + 2T + 3T + 5T + 8T + 5T + 3T + 2T + f
whatever axis (o, a, b, c or d) each term may represent.
Summation through the number of individual
•

attacks may be found in many melodies.

Take the ·

popular song, "One-Two, Button Your Shoe ", for instance.
The first eight bars give the following summation o f
attacks: 2 + 4 + 6 + 12, i.e. , 2 + 4 = 6, and 6 + 6 = 12.
It means there are four distinct sub-structures, each
containing the number of notes in this part icular
summati on, carried out with an a bsolute precision.
It is important not t0 confuse the rhythm
of attacks wit h the rhythm of durations.
The method o f summation is very flexible,
and with a little initiative one may accomplish a

•

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-· . . ... . . . . .
,

great deal of v ariety.
In the song, "But I Only Rave Eyes for
You", you find the following scheme of attacks;
6 + 9 + 6 + 3.

..

This is an incomplete form of 3 + 6 +

+ 9 + 6 + 3, where the central term is the result of

summation 3 + 6 = 9.
•

At the same time, the central
•

term becomes an axis of time symmetry.
MovEIDent, with respect to tension and

release, resulting in balance may refer to factorial
or fractiona l time-rhythm as well as to the rt1ythm
of the number of individual attacks.

Arithmetical

mean is the most common device in thi s case.
An arithmetical mean is the quotient of
the division of the sum, by the number of elements ..
With two elements, a and b for example, it equals
a+b • Musical intuition has a certain amount of
2
precision, and 1n some cases the se means ca»e out with

a very good approximation.

'•

'I

For example, in the first

3 ¼ bar structure of "Stormy Weather", the first substructure has three attacks, the second

--

seven, and

The exact number fbr the last sub_ 5 ' not 4 . This is a very
structure would be 3+72 good amount of approximati· on, only 20 percent of
the third -- four.

error.

Yet you get a greater satisfaction by adding

one more attack.

Try it by making a triplet out of
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28.
two eighths at the beginning of the third bar.
This procedure 1s adequate mechanically
to : under balancing - overbalancing - balancing, or
- umerbalancing - balancing •
overbalancing
•
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29 .

•

Lesson LXIV.
The following graphs and music serve as
an example of composition of melodic con tinuity.

•

Each example is a comple te musical composition
written for an unaccompanied instrument.
has been greatly negle cte d today.

This art

In the XVII and

A'VIII
Ce nturies, composers possessed enough technique
"
to accomplish such tasks.

J..s. Bach wrote manv

outstanding works, even so11atas, for violir
da gamba alone.

r· v1ola

Today only a very few high ranking

composers lik e Paul Hindemitl1 (Suite for Viola alone)
or Wallingford Riegger, an American, (Suite for Flute
alon e, in seven movements) have dared to write a whole
opus for an unaccompanie d instrument.
The three compositions I offer here are
constructed from th e scales of the First Group.
Each graph represents a theme or iginally plotted .
Musical examples are complete compositions developed
by means of variation.
The notation is as follows:
M -- the entire melody

a, b, c, d -- portions of melody pertainirig
to the respective axes
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30.
a•

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c•

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or
-- parallel binary axes

©,
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@ , @ , @ -- geometrical positions of M
or of the respective axes

Po , P , , P2 , • • • -- permutations of pitch-units
of M or of the respective axes

•

E0 , E,, E 2 , • • • -- tonal expansions of M or of
the res�ective axes
In this form of notation each original
melody (the theme of the composition) appears as

•

•

M @ p0 E0 •
It is advisable to be co11servative in
plarming a complete .i.u€lodic continuity, as ap�lica­
tion of too many variations at a time ( i.e., p, E
and the geometrical positions) incre�ses the

•

complexity of the e11tire compositi on beyond the
listener•s grasp.
f

(please see following pages)

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