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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.
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.
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.
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