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 provides close matches to most common musical intervals, including perfect fifths and fourths, major and minor thirds. The major third is exceptionally in-tune, and the minor thirds fare better than in standard 12-tone tuning.
• In addition to the intervals familiar to western music, 31-ET also matches other musical intervals, including those using the 7th and 11th harmonics. This means:
  • 31-ET can accomodate all the musical intervals used in Arab Music.
  • 31-ET also allows you to play intervals involving the 7th harmonic, allowing a harmonic seventh chord (like in Barbershop Quartet music), and also appearing as the "blue notes" common to American blues and Jazz, originating in traditional African music.
  • 31-ET more closely matches some of the musical intervals that appear in Indian classical music but are not easily accommodated by Western scales.
• 31-ET can accomodate the basic major and minor scales, and associated chords, used in most western music. It can thus be used to play most existing music.
• The individual steps of the 31-ET scale are smaller than the standard 12-tone chromatic scale by a factor of less than 1/3rd: they represent a fine-tuning of that scale that is small enough to have a better match to a wide range of musical intervals, and yet large enough that they can be heard as distinct notes by the untrained ear.

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:

• 19 Tone Equal Temperament - This tuning is similar to 31-ET in many respects, but has numerous shortcomings.
• 22 Tone Equal Temperament - This interesting tuning warrants mention; it is arguably quite xenharmonic as it lacks a whole tone.
• Quarter Tones or 24 Tone Equal Temperament - Quarter tones are convenient to work with for people familiar with 12-ET; this page explains why 31-ET offers enough advantages over quarter tones to be worth the added learning curve.
• 34 Tone Equal Temperament - For how close in number of steps this tuning is to 31-ET, it is very different. It is difficult to notate and poorly matches the 7th and 11th harmonics, but better matches the familiar intervals of Western music.
• 41 Tone Equal Temperament - I consider this tuning to be the next logical step after 31-ET; it improves on the intonation of most intervals, but is more unwieldly and difficult to notate.

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.