31 Tone Equal Temperament

41 Tone Equal Temperament

41 Tone Equal Temperament (41-ET) is a microtonal tuning that is finer than 31 tone equal temperament. 41-ET has numerous advantages over 31-ET, but it also has the downside of being more complex and difficult to implement, as well as a few other downsides of intonation for certain intervals. Overall, 41-ET is one of the best tunings in its neighborhood in terms of matches to natural harmonics: it is the smallest equal temperament which matches all overtones up through 16 while being off by less than 10 cents (one-tenth a 12-ET half-step).

I see 41-ET as a natural alternative to 34 equal temperament, and a tuning that combines some of the best features of both 31-ET and 34-ET.

I have only minimally explored 41-ET, and probably will not turn to it until I feel like I have exhausted 31-ET or reached its limitations, which may never happen.

Strengths or Advantages of 41-ET over 31-ET

Weaknesses of 41-ET

Relative to 31-ET and other tunings

What does 41-ET add beyond 31-ET?

41-ET has an additional 10 intervals relative to 31-ET; these can be seen as five intervals and their inversions. They are:

A note on the whole tones

The major tone in 41-ET is nearly perfectly in-tune, but the minor tone is significantly narrower (almost 7 cents) than its true interval. The difference is audible by most people, and produces the net effect that the major and minor tones are more different from each other in 41-ET than they are in just intonation. This actually makes the intervals easier to distinguish in this tuning. As the major tone is more important harmonically, it is beneficial than the major tone is the more in-tune of the two intervals.

On thirds and the 5th harmonic

41-ET matches the 5th harmonic well, but the match is the most out-of-tune (off by about 5.83 cents) of all the harmonics up through 11.

The situation for major and minor thirds in 41-ET is less-than-ideal and poses some slight drawbacks for this tuning relative to 31-ET. Although both of these intervals are close matches to their true intervals (much closer than 12-ET), the major third is narrow and the minor third is wide, making these intervals a little harder to hear, and even farther from their 12-ET approximations than they are in 31-ET. Furthermore, both of these intervals are less in tune in 41-ET than in 31-ET. Fortunately though, both intervals are still close to be harmonically useful, much more in tune than in 12-ET, and probably indistinguishable from their true intervals to some listeners.

The matches to septimal minor and major thirds, the undecimal neutral third, and the tridecimal minor third are all very close and their differences from just intervals are probably imperceptible to most listeners.

On the 7th harmonic

41-ET provides a very good match to the 7th harmonic, only slightly off from 31-ET; for practical purposes the match is probably imperceptibly close for most people. The matches to ratios 8:7, 7:6, and 7:5 are all very close. In contrast to 31-ET, 41-ET does a very good job of matching 9:7 as well. There's also a slight improvement to 14:7, the interval relating the 7th to 11th harmonics, and perhaps more importantly, in 41-ET, this interval is distinguished from 9:7, whereas in 31-ET the two are lumped together.

On the 11th harmonic

41-ET handles the 11th harmonic rather differently from 31-ET. Unlike 31-ET, 41-ET distinguishes between the greater (11:10) and lesser (12:11) neutral seconds, although it lumps the greater interval in with the minor tone (10:9), and does not match it particularly well. One effect of this is that you cannot divide a minor third in half. Two lesser neutral seconds add up to a tridecimal minor third, matching the 13:12:11 harmonics.

I personally find this interval math harder to wrap my mind around than the way 31-ET works. The way Arab music and other traditional musics utilizing the 11th harmonic have usually been notated does not distinguish between greater or lesser neutral seconds. Given that neither system perfectly matches the natural harmonics and just intonation intervals, I'd rather choose to go with the simpler one, as it still seems quite complex.

On the 13th harmonic

41-ET is the smallest tuning that handles the 13th harmonic at all, while also handling all smaller harmonics. Although 17-ET, 34-ET, and 36-ET all match the 13th harmonic more closely, they don't come close to all the smaller harmonics.

Only some of the intervals involving the 13th harmonic are well-matched. The best matches include:

These matches are close enough to actually utilize the harmonic.

External resources