Alternative Numerical Representation of Pitch

Question

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?

Answer 1

There are many others. Some piano technicians identify the keys on a piano by number, from 1 = the lowest A to 88 = the highest C.

In electrical engineering, the most common way is to use Hertz which is the number of cycles per second. In standard pitch, A is 440. A disadvantage of this system is that it is not linear, but you can convert it to a linear scale by taking the logarithm. (What the piano technicians use is the log to base 22.5 of this number, if my math is correct.) An advantage is that almost-exact values can be given for notes in different tuning systems and for enharmonics. A disadvantage is that they can only be almost-exact because most of the numbers are irrational.

Harry Partch in his book Genesis of a Music formulated a system of ratios starting at a low G which is 1/1. His system depends on instruments that are tuned using only rational numbers so that the intervals relate more closely to the harmonic series. The D a fifth above low G is 3/2.

There are also systems that are not purely numerical. There is a very commonly-used one in which the lowest C on the piano keyboard is CCC, the next is CC, the next is C, then c, then c', then c'' etc. Another one labels the low C as C1, the next as C2 etc.

Early computer-based musical typology systems had only punched cards for input, so they needed a system. This was for printed music, not sound generation. One way is to indicate the staff line and note value; another is pitch class, octave and note value like "C5Q".