# Alternative Numerical Representation of Pitch

I'm aware of a number of different ways of representing pitch as numbers (including one I developed for use in my own software) but am interested if there are others.

The most widespread is probably MIDI (where C4 is 60 and each semitone up/down adds/substracts 1 from the value).

The MIDI note value system conflates enharmonics (which is problematic for a lot of the work I'm interested in and pretty much a deal breaker) and can really only handle a 12eq but it does have the advantage that both pitch class and octave are incorporated in one number.

I've written a lot of code that uses my own circle of fifths system http://jtauber.com/blog/2007/12/13/numerical_representation_of_pitch/ assigning D to 0 and +1 means a fifth above and -1 a fifth below. This allows for distinct enharmonic spelling, and easy interval arithmetic (which makes it easy to give the correct answer to questions like: what's the augmented fourth above D♯?) and can easily test for enharmonic equivalence via a modulo operation. It can, however, extend beyond the 35 note names for non-12eq tunings. One disadvantage is it only represents pitch class so you need a second number to indicate octave.

Hewlett's Base-40 system http://www.ccarh.org/publications/reprints/base40/ is very clever in supporting distinct enharmonic spelling plus representing pitch class and octave in a single number in such a way that allows easy interval arithmetic. The only disadvantage I can see is it does assume 35 note names and so has a 12-note octave assumption built in.

Hewlett's page (linked to above) does mention work by Clements and Zimmerman.

Is Hewlett's Base-40 the dominant one used in computational musicology or are there other systems (that maintain interval invariance and enharmonic spelling differentiation) in use, and, if so, what are they?

• I don't know if this will help you, but have you heard of pitch set theory? It's good for abstracting relationships between a series of pitches, but does not assign absolute numbers. en.wikipedia.org/wiki/Set_theory_(music) – cotroxell May 20 '11 at 12:35
• @James Tauber: The answer to "Are there other systems in use in computational musicology?" is presumably yes. Are you looking for something in particular? For example: A system that provides greater advantages than the one you're using, with an explanation of it? If so, please add that information to the question; I don't see a real question here otherwise. – delete me May 20 '11 at 13:12
• Just represent the pitch using the frequency in Hertz. Problem solved. :) At least, as a physicist, this is what I would do. – Noldorin May 20 '11 at 19:11
• @Matthew Read: I've edited the final paragraph to make it clearer I'm interested in what other systems are in use – James Tauber May 21 '11 at 2:12
• @Noldorin: joking aside, frequency isn't really that different than the MIDI note value. In fact, if you assume 12-eq and allow for non-integer MIDI note values, they are homeomorphic. Systems like Hewlett's and mine maintain note naming for us people that care that A♯ is not B♭ :-) – James Tauber May 21 '11 at 2:19