Intonation

4 October 2020

Intonat.c is a little program I wrote already in May 1993, and slightly improved recently, keeping the output essentially the same. The source code can also be downloaded as a text file here. It is a simple command line program that works on MSDos (the original platform of development), Windows command line, Linux and other Unixes.

However, there is no apparent need to ever compile and run it, because the only output it can ever produce, I already publish here: the full output, and a reduced version produced by using the command line option -53.

The full version includes a division of the octave into 19, 31, and 43 equal parts. The reduced output only has divisions into 12, 24 and 53 equal parts, which hopefully makes the listing less confusing and less overwhelming.

The reason I looked at this program again, more than 27 years after writing the first version, was in the musical experiments for my article A flat is not g sharp [...], and more specifically, its technical elaboration.

So what do we see in the program’s output? Each line describes a musical interval, first the prime and continuing all the way to the octave. The lines are sorted by ascending pitch, in other words by interval width relative to the prime.

The leftmost column shows the ratio of the interval’s frequencies, as a decimal number. For example, the just fifth has note frequencies at the ration of 3:2, or 1.500000 decimal. The next column to the right shows the interval expressed in cents, which is a logarithmic scale such that an octave is 1200 cents, and 100 cents is one semitone distance in equidistant temperament.

The next two columns show either a

frequency ratio: For example, the aforementioned just fifth, which is 701.955 cents wide, has a frequency ratio of 3 to 2, for example of a note ‘e’ at 660 hertz and a note ‘a’ of 440 Hz; or an
octave division: Very close to that interval of 701.955 cents, we see the equidistant interval of 7 semitones, which is exactly 700 cents wide by definition. It is indicated as “7/12 octave”, because it has exactly 7 parts, of one 12th of an octave each. Also close by, we see the interval of thirty-one 53rds of an octave, indicated as “31/53 octave”, at 701.887 cents.

What is interesting in the list, is that it shows how well 53 EDO fits to just tunings and equidistant tunings as well. Just look at the near equivalents of ratios 256:243 (four fifty-thirds of an octave), 16:15 (5/53 of an octave), 10:9 (8/53), 9:8 (9/53), 32:27 (13/53), 6:5 (14/53), 5:4 (17/53), 81:64 (18/53), 4:3 (22/53), etc.