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digital harmony

musical harmony as ternary computing

This is a theory, but it works in practise, and yields surprisingly specific and rich musical results.

We know that binary numbers represent the presence or absence of powers of two:

1910 = 100112 
  = [[(1)×24]+[(0)×23]+[(0)×22]+[(1)× 21]+[(1)×20]
  = 16 + 2 + 1

We might also use binary numbers to represent the presence or absence of powers of three:

8510 = 100113 
  = [[(1)×34]+[(0)×33]+[(0)×32]+[(1)×31]+[(1)×30]
  = 81 + 3 + 1

Using threes in these positions is a step toward ternary computing, using digits 0, 1, and 2.

It also turns out to give a very fruitful representation of musical scales. Twos, in that scheme, turn out to be redundant — or probabalistic — when all is reduced to a single octave. So the system is surprisingly complete. It can be used to analyze audio and music notation, or to create music dynamically.

Circle of Fifths: 3n/2x

Instead of thinking of these powers of three as a sum (or dot product), as in a number system, we could think of them as populating a collection:

10011 = [(1)×34],[(1)×31],[(1)×30]

For musical purposes, it is useful to think of this as a collection of multiples of a base freqency:

10011 = [(freq)×34],[(freq)×31],[(freq)×30]

The circle of fifths is just such a sequence of twelve consecutive powers of three, from 30 to 311. Thus, if the above set of numbers were applied to a base frequency of middle C (as ‘freq’, above), the ones would correspond to C, G, E — a major triad. The sum of the waves at these frequencies sounds like this:

A major triad as powers of three, octave reduced.

The ratios of the complete circle of fifths pattern (starting at F) look like this in a single octave:

F:  3^0/2^0,
C:  3^1/2^1,
G:  3^2/2^3,
D:  3^3/2^4,
A:  3^4/2^6,
E:  3^5/2^7,
B:  3^6/2^9,
F#: 3^7/2^11,
C#: 3^8/2^12,
G#: 3^9/2^14,
Eb: 3^10/2^15,
Bb: 3^11/2^17


Our conventional musical notes correspond to powers of three, divided by whatever power of two is necessary to bring the frequency down into the the right octave. Twos define register: if their exponent changes, their octave changes.

In practical reality, this register change (which doesn’t change the name-identity of a note) may be important. It seems a sort of trivial reduction, but in the end the matter of register may help us identify the separateness of tones within a chord. The overtones of a pitch played on a real instrument multiply quickly beyond the first octave, but if there is a big spike within an octave of a perceived tone, it’s likely that it is the result of a secondary source.

Back to binary

In logical terms, then, we can derive the following: a tonal ‘note’ in the system is defined as the presence or absence of a power of three (with presence/absence marked by 0/1). A power of 2 changes the octave, but not the identity, of the note.

We can thus represent a group of notes as a single binary number, showing the presence or absence of the power of three relative to some core frequency.

Using this system, then, we can represent a C Major scale like this:

The above pattern serves as a kind of identity matrix, through which all other combinations of notes can be filtered, re-organized, and re-conceived.

Two things to notice: 1) this is a rearrangement of the white and black keys on the piano and 2) the order is reversed here to match the piano, ascending left to right. (C Major would be 000001111111 if the numbers went up from left to to right.) I’ve put the highest-exponent-value note at the right; numbers would increase to the left…

Maybe that was a mistake from musical habit.

For finding mathematical patterns (and there are many to find), it might be best to consider it the other way around…

The Audio End

We only come across the chromatic scale insofar as we consider powers of three to be musically meaningful. And it is fair to say that it is meaningful: the ratio of 3:2 lives powerfully in real-world pitches. Here it is on a violin. The lower band is C and neighboring pitches. The upper band is G, the frequency at a ratio of 3:2:

A violin playing a middle C

On Intonation and Tuning

There’s no point insisting on the finer points of intonation at this moment. In fact, I am confident in saying, as a highly competent professional violinist, that intonation is a moving target, with ever-shifting possibilities for ‘rightness’.

It is possible, for example, to tune dominant seventh chords to the overtone series, with a low major third, and a very low seventh. It sounds good – it’s not ‘out of tune’ – but it’s quite static, and not terribly useful for polyphony. The harmonic series functions most powerfully in the area of timbre, giving focus and color rather than dynamism and note-predictions. Chords can function (and can be suggestively tuned) far more flexibly in this binary system, where consonance is defined not only by the harmonic series, but also, simultaneously, by the entropy and patterning of its powers of three.

Reasons

I write and pursue this musical system for two reasons.

  1. The presence of a large amount of conventional harmony at the root of the computer suggests a durable representation in the digital age. What can show up in binary can continue to exist digitally, because it is maximally efficient in the medium/mechanism.
  2. This system suggests a kind of predictive computing, as I will suggest in upcoming posts. Perhaps it can serve as a simple, tangible example of the basic processes of networks, memory, and even backpropagation. Evidence of its usefulness can be seen in the physical disturbance that every harmony change can cause a player of an instrument.

That is to say, it seems to be real enough.

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