31 Tone Equal Temperament

Why 31-ET?

31 Tone Equal Temperament (31-ET) is a tuning system that involves 31 equal divisions of the octave. Because it is more difficult to play and learn than the normal 12-tone equal temperament used in most Western music, there need to be compelling reasons to use it if it is ever to attract more than a small niche following.

Here I present the advantages of 31-ET, and lead into comparisons with other commonly advocated microtonal tunings.

Advantages of 31-ET

31-ET vs. Other Microtonal Tunings

The points above are convincing on the grounds that 31-ET is useful, but they do not necessarily convince one that 31-ET is a better choice than other tunings. Here, we consider other tunings that have been more widely used and that, at first glance, might look easier to use because they have a smaller number of divisions of the octave:

There are other tunings possible with similar amounts of divisions of the octave (i.e. greater than 12 but not too much greater than 12), but even there, the best options for matching harmonic intervals create problems with important intervals. 17-ET matches fourths and fifths well, but does not provide an in-tune match for thirds; this is because 34-ET is a division of 17-ET in half, and 34-ET uses odd numbers of steps to create the major and minor thirds.

15-ET is poor overall, and no other equally-tempered tunings less than 41-ET provide a decent match to perfect fourths and fifths.

41-ET, 53-ET, and 72-ET are interesting tuning systems, but they all have so many more intervals to deal with, and their step size is so small that I think 31-ET offers a compelling advantage by virtue of the fact that its step sizes are still reasonably large and it has a more manageable number of intervals.