For about
the last twenty years I have been intensively studying music
from its mathematical approach using the *Schillinger
System of Musical Composition* and Joseph Schillinger's
*Mathematical Basis of the Arts* as a guide. I have
deciphered his great work and want to share some of my
findings.

The fundamental
components of music, viz., scales, cadences, triads,
sevenths, etc., have been transformed into their coherent
natural geometric form, in color and in 3rd dimension. This
enables us to visually analyze these musical structures as
well as their mirror images, thus dissolving the inherent
bewilderment we all encounter in our music studies. It also
accelerates the learning process by a thousand fold by
enabling us to see the general overall mechanics of music.

Currently we are
precluded and forbidden from expressing geometrics because
of the manner in which we read, write and describe music.
For example if we were to ask a dozen composers to compose a
horizontal or vertical line, a specific triangle, a
tetrahedron, a square, a pyramid, a sea shell or a pine cone
we would get a dozen different compositions for each of
these simple forms. Not so with visual music because it
enables these blind architects to see these forms and
compose the structures accordingly.

Tradition from the days
of the Just Intonation tuning system still has us counting
to 8; C being 1, D being 2, E, 3, etc. Since the 1700s,
however, we have been using the Equal Temperment system
where C is zero; C sharp is 1, D is 2, E flat is 3, etc...B
is 11 and C octave is 12 and not 8.

Music has remained in
the dark, without geometric form, because we still refer to
C as 1 instead of zero. Geometry begins with 0, not 1. With
C as 0, coherent visual form ensues. The twelve notes in our
primary selective system are used because 12 is the most
versatile number; 12 is the smallest number with the most
divisors.

The 12 notes in the
primary selective system are placed on the 12 numbers on the
clock: middle C is zero (midnight), C sharp(D flat) is 1
o'clock, etc. The C octave in the treble is +12, high noon,
or 360 degrees . The C octave in the bass is -12, or
yesterday noon!

With C as zero,
Schillinger categorizes music into two general forms: **
symmetric and diatonic**.

####
Symmetric

The Symmetric category
therefore is 12/1 or one 12 note chromatic scale. 12/2 is
two six note whole tone scales. 12/3 yields three four note
diminished scales. 12/4 yields four three note augmented
scales. 12/6 gives us six two note flat fifth scales.

The Symmetric category
therefore may visually express a circle or the spokes of a
wagon wheel, a hexagon, chicken wire, snowflakes, the
benzene ring, squares, equilateral triangles. The flat 5th
scale may represent a satellite circling the earth or an
electron orbital, vertical lines, horizontal lines, 30
degree and 60 degree lines, hemispheres, orange slices, a
baton (stationery or thrown into the air as a majorette
would during a parade). The mirror images or manifolds as
Schillinger calls them are alike in this category. A later
correspondence will describe the wave mechanics of these
polar, non polar and planar enantiomeres geared towards
science from fact learned in music.

####
Diatonic

In the remaining general
category, the Diatonic, we find these manifolds radically
different than in the Symmetric category. This is where the
real essence of music is. Most of us know our diatonic
scales, cadences and 7ths, but their manifolds are least
understood if ever thought of at all. The enclosed diagrams
illustrates the principles of the manifolds. Simply, if we
place both thumbs on 0 (middle C) and play the C diatonic
Ionian scale with the right hand, and using the same
intervals in the left hand, it will reflect C's Phrygian
mode. This case the intervals are 0 2 2 1 2 2 2 1. C's
Phrygian is thus in the key of A flat, starting at C.

Let us digress a minute
and discuss the key principle in modern scientific
navigation which is identical to Schillinger's manifolds.
Navigation is an activity we take for granted in our day to
day lives. There are two methods; meets and bounds and grid
coordinates. In music, navigation is currently in the meets
and bounds system. Since Schillinger was a scientist and
navigated his music theories by grid coordinates, permit me
to offer a comparative example.

About 1948 an invention
appeared which revolutionized navigation. It ranks second
only to the invention of the compass and is known as the
Visual Omni Range. Its concept is similar to Schillinger's
manifolds. The principle is simple: two beacons or discs
rotating in opposite directions, each emitting a radio
signal at a certain frequency. When both beacons start at 0
or North they are "in phase". As they rotate in opposite
directions they become out of phase by 30 degrees, 60
degrees ,...180 degrees, etc., until the cycle is complete.
The receiver in the aircraft measures the phase displacement
and registers what magnetic direction the aircraft is in
relation to the VOR. Two such devices permit triangulation
and will pinpoint the exact location. Virtually all the
navigation used today uses the VOR. Schillinger developed
this same principle to navigate in music some 30 years
before it was formulated for the VOR. Once this general
principle is grasped more complex manifold signals readily
fall into place.

We see the logic of this
for scales, diads, triads and sevenths. Cadences are no
exception. Cadence is fundamental to music and there are two
general forms: Classic and Modern. We also find two general
forms of mathematics which deal with mechanics: classical
mechanics and quantum mechanics. Until about 1900 classic
cadences predominated in music as did classical mechanics in
mathematics. Each era is confined within the limits of its
mathematical or musical systems. In the classical mechanics,
Euclidian and Newtonian systems prevail, triangular
measurement dominating this system. In classical music,
harmonic triads prevail. After 1900 quantum theory envelopes
all science with a 4th dimensional parameter being added to
the classical system. In music the RE SOL DO LA (D-, G7, CM,
A-) 4th dimensional cadence tones appeared in music;
particularly in Scriabin, Youmans, Gershwin. As in science
from that time on (1900-1925) all things entered the quantum
era.

Once the mechanics of
music is seen as a whole we see and grasp this overview as
we would view a plastic model of an internal combustion
engine or a clock. Once the concept of the component parts
are understood it makes things easier to comprehend. Perhaps
all of us agree that our knowledge of the 7ths is
proportional to our understanding of music. "Sevenths are
where it`s at" is a truism. As pianists we review our
scales, cadences and 7ths, but when the manifolds are
included with each of them in turn, they cause a profound
intellectual and physical improvement. Each entity has its
unique shape and position; each is special, so much more so
now that visual perception is the new parameter.

The cornerstone of the
Schillinger System is his Theory of Rhythm. Once the process
of generating rhythm patterns is understood, Schillinger`s
Encyclopedia of Rhythms proves to be an invaluable aid. It
is easy to read and comprehend and affords a wealth of
information. The 2 volume Schillinger System of Musical
Composition is very difficult and requires years of study,
but with the coordinated simultaneous study of the
Encyclopedia of Rhythms and cross reference to his
Mathematical Basis of the Arts it begins to make sense. The
manifolds are as critical to understanding Schillinger as
are the rhythm generators. From these comes quadrant
rotation, composition from geometric projection and infinite
series. Step by step, this system is interesting and most
valuable.

Harry Lyden