musings: time & the golden matrix - dividing the known PDF Print E-mail
Written by D. Bond   
Wednesday, 26 May 2010 09:27

I was reading an interview given by a very famous, internationally respected composer, who claimed to use high level mathematics in his compositions. I won't identify this composer as I do not have his book in front of me to quote precisely, but let's call him Mr. X [can you guess who it may be?].

Mr. X was described as demonstrating his ideas about rhythm by beating his hand evenly on a tabletop and counting to 12. Mr. X said to his interviewer something like "this is the chromatic scale in time."  I promptly threw the book in a corner! With that simple gaff, this composer, whom I had once greatly admired, crashed and burned in my estimation, and all interest I once had in his formal processes disappeared.

I shouldn't be so harsh, for I made the same error in my one and only 12-tone piece, considering rhythm as additive - the eighth note =1, the quarter =2, the dotted quarter =3, etc. But it should be clear, that a regular pulse, evenly spaced, is no different than the measure of pitch [i.e. 1/1 = 440 cycles/sec or 1/1 = 60 bpm]. If we take a pulse of 60 beats per minute as '1' then '2' is a second voice pulsing at twice the tempo, 120 bpm, '3' 360 bpm, and so on. 1/2 would be a pulse of 30 bpm, etc.

From this it is easy to see that the harmonic matrix applies not only to pitch but also to rhythm. Rhythm is not additive or linear, but is a harmonic relationship. An equally tempered 12-part chromatic scale in rhythm is calculated exactly as it is in pitch, as the twelfth root of 2 times the scalar [1-12] multiplied by the given tempo indication. I challenge anyone to demonstrate these relationships with their hands beating time on a tabletop! Heck, even the computer must approximate this, irrational numbers having no real world representation [yes, we can get arbitrarily close, beyond the limits of perception, but irrational ideals like pi or the square root of 2 only exist in the minds of men, if at all].

Since observing this fact, most of the music I have composed, thankfully heard only by me and my cat Nük [who dislikes it slightly less than I do], uses the harmonic matrix for rhythm as well as pitch relationships. This requires the use of multiple metronomes beating in relation to a specific tempo, either fixed or variable. For example, an accelerando from 60 bpm to 90 bpm with two metronomes in a simple relationship 1/1:3/2 would have one metro beating a 60 bpm and a second beating at 90 bpm maintain their proportional distance as they slide up to 70,105 to 80,120 to 90,135. Translated to pitch this is equivalent to the interval of a perfect fifth sliding up a perfect fifth [3/2^2]. Naturally, more complex rhythmic relationships can be used, and manipulated in a variety of interesting ways.

What we have here is the common way of viewing musical time, as regular flow [velocity], subject to change [acceleration], marked at regular, reoccurring patterns, all against an absolute background. This kind of time seems to start at some point and flows onwards without end, with the rhythmic voices phasing. From here, to create musical form, one may begin the search for rhythmic 'least common multiples', flams, convergences, or go on to calculate densities, topologies, etc. These trivial exercises are left for the enthusiastic reader. :)

A second way to look at time emerges when one considers the absolute background and begins to subdivide. To quote Erik Satie:

          “Before I compose a piece, I walk around it several times, accompanied by myself." 

Here we take a known length of absolute time and create sections, formal structure, and so on up to discreet events through division. If A is some length of time, then there exists B+C=A. There is a proportional relationship between B:C. If A is divided precisely in half, the B:C = 1:1, and the length of each 1/2+1/2=1. If A is divided 2:1 then 2/3+1/3=1, and so on. A matrix M of infinite dimension contains all possible divisions.

M(x,y) = 1-(1/(1+(x/y)))

Here 16*16.

Many will immediately recognize that the golden section, where A=B+C and A:B = B:C, is a special case of this matrix. [Just as the square root of 2, the tri-tone, is a special case never to be found within the harmonic matrix]. Hence I call this matrix the Golden Matrix [in the above, the golden ratio conjugate 1/φ=φ-1=0.61803... is suggested, but it is not difficult to adapt the matrix].

This simple relationship links the harmonic matrix to the golden matrix. Both invert along the diagonal [1:1 2:2 3:3...]. As the harmonic matrix multiplies out to infinity and down to nothing, the Golden matrix similarly divides the known '1' proportionately. Both are enormous fun when used iteratively.

Forgive me for going astray for a while, but may I be permitted to ask: how does one divide an infinite period of time into 2 or any number of parts? how is the frame of reference for absolute time established? isn't it through memory and perception, sensation, mind? so then, what is time? what is distance?


You won't forgive me? I'm not permitted to ask? Oh well then, have you yet caught a glimpse of the eternal, universal harmonic fractal, shimmering as a lump of gold in the river of time?

ah, nvm.


Original research presented courtesy of Bond Institute. Intellectual Property Rights reserved. Further information, publishing, permissions, please contact: This e-mail address is being protected from spambots. You need JavaScript enabled to view it \n




Last Updated on Thursday, 27 May 2010 10:51