Prime-Number Meter and Celestial Motion

 
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Aurora’s Dance

Prime-Number Meter and Celestial Motion
 

The most conspicuous feature of Aurora’s Dance is its 7/8 meter. Not until the piece was finished did I, in hindsight, name it for the aurora borealis, as something about it had the numinous quality of outer space. Later, when I heard Bobby McFerrin’s composition, Stars, I understood why. That piece was written in 13/8.

Paul Hindemith wrote:

Even in a series of sounds identical in every respect recurring at uniform time-intervals, the ear tends to hear regular groupings … we can understand such a series as subdivided into groups of two or three beats.1

And Paul Creston wrote:

Psychologists and writers on rhythm have elaborated ad nauseam on the tendency of the mind to group into two’s, three’s, etc. any series of regular, periodic sounds … and it is only natural that this tendency has been apotheosized in musical meter.2

One kind of rhythmic classification conceptualizes meter as binary or ternary depending on how the pulse is subdivided, and another kind of classification designates meter as bipartite or tripartite depending on how the pulses divide a measure. Therefore, regardless the complexity, it appears to be true that on a visceral level, the human organism perceives ordered movement through time as a composite of rudimentary, irreducible building blocks of which there are two kingdoms: one that beats in two and one that beats in three. You cannot waltz to a marching drum. So what does this have to do with space?

There is another type of meter, a kind of irregular subdivision, that results when the kingdoms are combined. In “regular” meters, a measure is naturally divided into pulses of equal duration, and if a pulse which beats in two throughout a measure is subdivided into a triplet, its duration (the total value of the triplet) still remains the same. All beats are resolvable to one kingdom and we feel a continuous, uniform rhythmic flow. However, in meters whose numerators are prime numbers greater than 3, i.e., 5, 7, 11 and 13…, a measure must necessarily contain at least two unequal pulses — pulses of differing duration — one being even and one being odd. Since the odd pulse can only be divided by three or reduced to two plus three, we are always left with a two-beat group and a three-beat group. The pulses cannot be resolved and remain forever discrete. While the math is simple, the result for the listener can be strong. As we move with the music, measure by measure, we continuously experience a rhythmic shift between the rudimentary, primal groups. We feel two separate, but integrated motions, and need not be aware of it.

We look to nature as a source of artistic understanding. Man, and certainly his music, emanate from nature, are a part of nature, and to dismiss the connection would be futile. The moon looms in the sky each night whether thought about or not. Its phases change. The earth has days, nights, and seasons. Throughout the entire universe there flows a ubiquitous stream, encompassing atom to distant galaxy, from which we are derived: a stream of simultaneous and synchronized rotational and forward movement. Electrons, moons and planets spin about their axes while orbiting larger central bodies (nuclei, planets and suns) which are also spinning and moving through space. And our solar system, along with billions of other stars contained within the spiraling arms of the Milky Way, revolves around a galactic center which, relative to the universe at large, moves at a velocity of 600 km/sec and completes a single rotation in approximately 200 million years.3 When radio telescopes were invented and new galaxies discovered, it was seen that they, too, have spiraling arms and rotate as they move through space. We walk straight on earth, but ride the teacups into the universe.

Carl Sagan wrote, “We are a way for the Cosmos to know itself.” We need not understand celestial motion to know how to walk, or to love music, but we have the ability to think and wonder, and so we do. The integration of simultaneous rotational and forward movement is a common denominator of each and every one of the heavenly bodies – a duality which underlies and defines their motion. Music written in prime number meter also holds a duality: the regular, bi-rhythmic oscillatory motion between rudimentary two-beat and three-beat pulse groups. When we say we are “moved” by music, this has literal meaning. Music comes from the depths of the living; it is an expression of, and also evokes, the movement of energy within us. Because prime number meter has the peculiarity of generating a continuously alternating pulse, it causes us to be literally moved, like the celestial bodies, in two directions at once.

An intuitive idea, i.e., based on individual perception, must face the test of reality. It would be nonsensical to think that all compositions written in prime-number meter have an intended celestial theme, or that all music having celestial themes is written in prime-number meter. But that does not invalidate a connection (because: some people may not feel it; there are other qualities of space a composer may write about; there are other devices, such as the whole-tone scale, a composer may use; and on … ). So what is the evidence that a connection exists, is sensed by others — that it is real? I will go through some examples, and you can listen for yourself.

Bobby McFerrin used 13/8 meter to write Stars (recording posted above). In that piece, the rhythmic shift is especially pronounced due to the relatively large difference in duration between the five-beat and eight-beat metric groups (as opposed to a more subtle shift between the two-beat and three-beat groups in 5/4 meter).

In The Planets, Gustav Holst began and ended his 7-movement suite with Mars and Neptune, respectively. Both are written in 5/4 meter.

Mars


Neptune




Day of the Comet, from George Crumb’s Zeitgeist, is written in 5/16 meter.

The first movement of Paul Hindemith’s symphony, Die Harmonie der Welt (The Harmony of the Universe), is comprised of alternating 5/4 and 4/4 measures. The subject of this symphony, originally written as an opera, was Johannes Kepler himself, the 17th century astronomer and musician. In the liner notes accompanying this CD, Philip Borg-Wheeler wrote:

Kepler’s obsession with the harmony of the spheres … [h]is attempts to relate mathematical proportions in the structure of musical sounds to proportions in the orbits of the planets was a theme which Hindemith found very attractive … Die Harmonie der Welt, completed in 1957, is the consummation of his artistic and philosophical beliefs.4

The most powerful example I have heard of the expression of celestial themes through prime number meter occurs in Olivier Messiaen’s duo piano work, Visions de l’Amen. Of the second movement, entitled Amen of the Stars, and of the Ringed Planet, Messiaen wrote:

A wild and brutal dance. Stars, suns and Saturn, surrounded by its multicoloured rings, spin crazily. The second piano introduces the theme of the dance of the planets, the five introductory notes of which form the essential part of the movement … the second piano gives diversity to the five notes heard at the beginning of the theme by means of changes in rhythm and abrupt leaps in register.5 [Emphasis added]

In this movement, the “five introductory notes” are written as a dimeter (a frame of two measures that comprises the meter) having the time signatures 3/16 and 2/4, containing three sixteenth notes and two quarter notes, respectively. Together, these measures have the same duration (3/16 + 8/16), but not the same accents, as an 11/16 monometer (which is felt as 5/16 + 6/16). Thus, the “essential part” of Messiaen’s planetary theme not only is an exemplar of the above prime number principle, but further exploits its power (accentuates the feeling of rhythmic shift) by contracting the odd pulse to a short three-beat group, expanding the even pulse to the equivalent of a long eight-beat group, and, by use of the bar line, placing unequivocal primary accents on both of them. Messiaen split the musical atom, whether consciously or not, and the result is potent.

Michael Gerber’s improvisation in Aurora’s Dance is a realization in music of what I’ve tried to say with words. Underneath spatial whole-tone based melodies played with his right hand, Mike surprisingly maintains the 7/8 ostinato base line with his left hand throughout his entire solo up until the ending. In doing this, he not only provides orientation in open harmonic territory, but more importantly, he relentlessly holds us balanced, without retreat, in the quintessential duality of celestial motion. Only towards the very end of his solo, where he uses light, pointalistic staccato and arpeggiated ribbon-like sheets of sound to evoke sensations of pulsation and streaming of energy, does he abandon it.

Once while speaking about the conveyance of emotion through musical improvisation, David Liebman said to me, “I don’t play feelings. I accept the feeling and play the music.” Therein lies the epiphany shaping his solo in Aurora’s Dance. Impelled by the underlying bi-rhythmic current, David’s solo generally consists of a series of phrases or sections, each containing one or more central notes around which he plays harmonic variations, textural nuances and passing tones. As the solo develops, these central notes gradually rise in pitch up to a point, where a sharp melodic drop then occurs. Following the drop, the central pitches again rise, but at a steeper rate, until, after a final ascent, he reaches an apical note and breaks into the melody of the composition. The solo feels alive! A look at the central notes6 reveals fascinating proportions: their pitches not only rise successively, but the intervals between them continuously expand (min 3rd, maj 3rd, perfect 4th and perfect 5th). If we transpose this into physics, equating pitch with distance, the melodic movement up to this point is increasing in momentum; the central pitches are moving faster with time. Then, following the melodic drop, the expansions between the intervals occur at greater rates (the intervals are: min 7th, octave, two octaves plus a 2nd, three octaves plus a 2nd). There is an increase in acceleration. In other words, the contraction is followed by an increase in force and energy which propels the next expansion.7 The overall shape of this solo could just as well be the sonic equivalent of the movement of the auroral lights the first time I saw them. Glowing through a cold, far northern nighttime sky, they contracted and expanded with increasing intensities, pushing upward like plasmatic ribbons streaming toward a central dome. Both realms of perception, sound and light, led to similar feelings within me.

It would be painful to imagine that David was calculating intervallic distances, or that Mike was contemplating existential duality, as they each played their solos. Nor is it surprising to find a pervading continuity of natural law within musical expression. Rather, it is a revelatory and exhilarating affirmation of our oneness, both physically and creatively, with the cosmic ocean.


1.  Paul Hindemith, Elementary Training for Musicians, Schott & Co., Ltd., Second Edition, 1949, p. 93.
2.  Paul Creston, Principles of Rhythm, Library of Congress Catalogue Card Number 64-15438, p. 17.
3.  Charles Konia, M.D., “The Rotation of Spiral Galaxies,” Journal of Orgonomy, Vol. 19, No. 2, Orgonomic Publications, Inc., 1986.
4.  Sound recording liner notes. Philip Borg-Wheeler, Hindemith, Symphony Die Harmonie Der Welt, Chandos Records Ltd., Chan 9217.
5.  Sound recording liner notes. Olivier Messiaen, Olivier Messiaen Visions de l’Amen 3, Montaigne Auvidis, Reinbert de Leeuw Edition 5, MO 782050.
6.  The central notes, pitch drops and important related intermediary notes are: e flat’g flat’b flat’e flat’’b flat’’ – (drop to) b flat – (up to original starting note) e flat’d flat’’ – (drop to) f’f’’ – (drop to) bd flat’’’ – (drop to) b flat – (up to a starting note) b – ascent to d flat’’’’.
7.  This is based upon the following fundamental physical equations: momentum (p) = mass (m) x velocity (v) = m x distance (d)/time (t); acceleration (a) = v/t = d/t2; and force (F) = m x a.
 
Reference: Wilhelm Reich, Cosmic Superimposition, Farrar, Straus and Giroux, New York, 1973

Re-written from the original version contained in:
Souls & Masters: Thoughts and Notes by the Artists
Aurora’s Dance and this Essay: © Rhoda Averbach 1997
Aurora’s Dance is from the album, Souls & Masters

 

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