(This article originally appeared in the January 1980 issue of Contemporary Keyboard magazine and is here presented in honor of Schillinger's 121st birthday.)
THE HORIZONS OF MUSIC have expanded immeasurably during the twentieth century—but this expansion has been due more to changes in practice by the artists themselves than to any major theoretical advances. Several theorists have, however, made significant contributions. Arnold Schoenberg's serial techniques have of course had tremendous impact. Heinrich Schenker's analysis procedures are of major importance, but they are useful more for understanding existing works than for creating new ones. A third Sch, Joseph Schillinger, is perhaps less widely known than these two, but his extensive and systematic explorations have played a significant role in the musical development of thousands of musicians, including such giants as George Gershwin and John Coltrane. The Schillinger system is useful primarily for the composer, since it shows how combinatorial processes familiar to mathematicians can be applied to pitch and rhythm to produce new musical ideas. What follows is a brief look at Schillinger's life and his ideas.
Joseph Schillinger (1895-1943) was a mathematician, an inventor, an artist, and a photographer, as well as a composer. He spent the first 33 years of his life in Russia, where he was a highly regarded composer and educator; he was at one time the head of the music department in the Board of Education for the Ukraine and a professor in the State Institute for the History of the Arts in Leningrad. In addition, he organized and directed the first jazz orchestra in Russia; their first concert opened with a lecture by Schillinger on "The Jazz Band And Music Of The Future." He spoke of jazz as the music of the masses and of its role in rejuvenating music.
Schillinger wrote 22 extended compositions and numerous shorter ones. His major compositions for the piano are the Five Pieces, Op. 12, and the Excentriade, Op. 14. His major orchestral works are the Symphonic Rhapsody, which was selected by the State Committee in Russia as the best work composed during the first ten years of the Soviet Union, and March Of The Orient, which was premiered by the Cleveland Orchestra. He also composed the first piece for the Theremin, the First Airphonic Suite, which was performed by Leon Theremin with the Cleveland Orchestra.
Schillinger was invited to America in 1928 by the American Society for Cultural Relations with Russia to lecture on contemporary Russian music. Although he was considered one of the leading young Russian composers, he felt that his primary mission in life was to make his theories of music known. Once he had come to America, most of his teaching was carried on privately or through correspondence courses. In fact, both of his major treatises, The Schillinger System Of Musical Composition and The Mathematical Basis Of The Arts, were published posthumously.
The most famous of Schillinger's pupils was probably George Gershwin. Gershwin, whose lifelong interest in serious compositional study may have reflected his feelings of inadequacy at having come up through Tin Pan Alley, also studied briefly with such figures as Henry Cowell and Wallingford Riegger, and at one time or another expressed interest in studying with Schoenberg, Ravel, and Nadia Boulanger, among others, but never got around to it. The story that Gershwin studied with Schillinger for 4 1/2 solid years cannot be verified, but it seems to be true that the pedagogue's influence on the tunesmith's orchestration, including the orchestration of Porgy And Bess, was considerable. Schillinger's other students included Benny Goodman, Tommy Dorsey, and Glenn Miller, as well as a number of film composers. A second generation of Schillinger disciples, who learned his ideas from his books rather than through lessons with the man himself, includes Eubie Blake and saxophonists Sonny Rollins and John Coltrane.
Critics of Schillinger's ideas usually focus on their mathematical nature. There is a widespread belief that anything artistic which has truck with math will come out sounding mechanical and contrived. Adherents of this view conveniently overlook the fact that conventional music theory is mostly math, too. Another criticism is that the Schillinger system replaces the intuitional aspects of musical composition with purely intellectual ones. But again, as with conventional theory, the intuition comes in knowing when, where, and how to apply the intellectual tools. Misunderstandings have also grown from Schillinger's use of unconventional terminology and graph notation. There are even people who think that the system is designed exclusively for the creation of background and incidental music, because it has been used by many movie composers.
It would be impossible to discuss all of Schillinger's ideas in a magazine article, so I will concentrate on introducing a few basic ideas, which might be useful for improvising pianists and composers in expanding their tonal and rhythmic conceptions at the keyboard.
It was Schillinger's intention to set out the total riches of the tempered scale and all of the rhythmic possibilities within our Western system of rhythmic notation. He went about this in a logical, mathematical way so as not to overlook a single scale or rhythmic pattern. Nicolas Slonimsky, author of the Thesaurus Of Scales And Melodic Patterns, wrote that "Schillinger has done for music what Mendeleyev did for chemistry; he has provided an exhaustive periodic chart of all its elements, making possible the discovery of those that are not now known." Elliott Carter wrote that "Within the covers of these two volumes [The Schillinger System] one finds the most comprehensive tabulation of musical elements, devices, and procedures that probably has ever been made."
Instead of using traditional terminology to refer to the intervals between pitches, Schillinger adopted a simplified numerical system in which a half-step is designated as 1, a whole step is 2, a minor third is 3, a major third 4, and so on. When this system is used, the process of generating new pitch groupings (which could be played sequentially as scales or simultaneously as chords) becomes much more intuitively obvious. For example, the table below shows all of the possible three-note groupings within the octave (that is, when the outer two notes of the group are a major seventh or less from one another).
1+1 2+1 3+1 4+1 5+1 6+1 7+1 8+1 9+1 10+1 1+2 2+2 3+2 4+2 5+2 6+2 7+2 8+2 9+2 1+3 2+3 3+3 4+3 5+3 6+3 7+3 8+3 1+4 2+4 3+4 4+4 5+4 6+4 7+4 1+5 2+5 3+5 4+5 5+5 6+5 1+6 2+6 3+6 4+6 5+6 1+7 2+7 3+7 4+7 1+8 2+8 3+8 1+9 2+9 1+10
To find 3+2 on the keyboard, pick a starting note, then count up three half-steps to find the middle note and two more to find the top note. Starting on C you would arrive at the three-note grouping C-Eb-F. Starting on any other note would give you a transposed version of the same relationship.
The same mathematical process can be used to find the four-note scales, of which there are 165 within the octave. Major and minor scales belong to the family of seven-note scales. There are 462 possible seven-note scales within the octave! Think about that the next time you're practicing scales.
This same approach allows you to extend beyond the octave and derive scales based on series of numbers found in mathematics and nature. For example, the Fibonacci series (a series of numbers which can be seen in many organic forms in nature) could be used to generate a scale. The Fibonacci series is generated by adding each pair of consecutive numbers in the series to get the following number. Beginning with two 1's, we get 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13, and so on, so the Fibonacci series begins 1-1-2-3-5-8-13-21-34-55.... This would generate the following scale:
Symmetrical scales can be generated by dividing the octave into two equal parts (6+6), three equal parts (4+4+4), four equal parts (3+3+3+3), or six equal parts (2+2+2+2+2+2):
By filling in the spaces between these points with fragments of identical construction, we can generate many symmetrical scales. The diminished scale is one such scale that many musicians are familiar with. It is divided into four equal parts, 3+3+3+3, and each of these parts is divided 2+1. Starting on C, this gives us:
I recently composed a song, which used a symmetrical scale divided into three equal sections in which I divided each of the segments into 1+2+1. Starting on B, I arrived at the following scale:
It's also possible to divide the keyboard into identical segments of 8 or 9 half-steps, with whatever subdivision of notes you prefer within the segments. This will give you a scale that repeats its pattern every two or three octaves, but one in which the octaves adjacent to one another may have entirely different sets of notes in them. Here is an example of a three-octave 9+9+9+9 scale:
Schillinger handles rhythm in the same expansive and methodical fashion that he uses for pitch. Beginning with quarter-note rhythms in 4/4, we derive all the possible combinations that add to 4:
1+1+1+1 2+2 1+1+2 3+1 1+2+1 1+3 2+1+1 4
Of course, any of these numbers could represent either a note or a rest. The same process will allow us to discover all the possible rhythms for bars of any length, divided into whatever units (eighths, triplets, sixteenths) we prefer.
One of Schillinger's most important rhythmic ideas is that of combining two regular rhythmic patterns, such as 4 beats and 5 beats, and deriving a resultant rhythm by combining them.
Schillinger's theories cover not only scales and rhythms but harmony, counterpoint, instrumental forms, dynamics, and orchestration. Any one of these areas may provide the basis for what Schillinger terms the major component of a composition. Applying numerical terminology to intervals, rhythms, harmonies, and other aspects of music makes visible a greater unity and cohesiveness between one of these elements and another.
The way I use Schillinger's ideas is by selecting some component, such as an unusual pitch scale or rhythm, which interests me, and using it as the basis of a composition. Applying the Fibonacci series or some other natural mathematical progression to either pitch or rhythm provides a musical framework that in many cases has supplied the necessary musical blueprint for creating a composition that expressed a particular musical feeling. By limiting my choice of musical materials, I was saved from sitting down to compose and facing that vague white noise of the musical infinite. Instead, I felt a remarkable freedom as other musical ideas began to grow from the original ones that I had selected.
Schillinger's is the first non-restrictive-school of music theory. In his "Overture" to The Schillinger System, Henry Cowell wrote: "The Schillinger system makes a positive approach to the theory of musical composition by offering possibilities for choice and development by the student, instead of the rules hedged round with prohibitions, limitations, and exceptions which have characterized conventional studies." In addition to his insights into musical structure, Schillinger had visionary ideas about electronic instruments, humanity, and the synthesis of the arts that go beyond the scope of this article. The book Joseph Schillinger: A Memoir By His Wife, by Frances Schillinger, gives a better picture of this remarkable man.
Joseph Schillinger's contribution to music has been to make us aware of the wealth of musical resources available to us. There is a universe of sounds as yet untapped within the equal-tempered scale. To close, let us listen as Schillinger himself addresses this issue in his book Kaleidophone: "The better your sight, the more stars you see in the sky. But an unarmed eye, no matter how perfect, has its limitations. Our ancestors lived in a limited world. Ours is so great that our minds can hardly absorb what our eyes can see through a modern telescope. It is about time to get off the uncontrollable magic flying carpet of strained imagination and take a rocket, which will bring the stars closer to us.... While the scope of intonations in the music we have been so proud of for several centuries is confined to but a very few scales and chords, here, due to a mathematical method, are inexhaustible resources of raw material. Whatever your ambitions and aspirations, and whether you are contemplating a long journey or just want to take a walk around the corner—don't forget your glasses!"
Gil Goldstein is a jazz pianist whose recording credits include work with guitarist Pat Martino, percussionist Ray Barretto, bassist Eddie Gomez, and harmonica virtuoso Toots Thielemans. A former student of classical pianist Ivan Davis, Goldstein is presently working on a doctoral degree in primary music education.