53 Tone Equal Temperament
53 tone equal temperament (53-ET) is a tuning that is substantially more fine-tuned than 31-ET. 53-ET is the next finer equal temperament after 41 equal temperament that improves on the match of fifths and fourths to their true intervals in just intonation.
The step size is about 22.64 cents, an interval wide enough for nearly all listeners to hear, but narrow enough that the notes sound very close to each other. This step size is between 1/4th and 1/5th the size of a half-step in 12-ET, and is wider than one half the step size of 31-ET.
I have not composed in 53-ET and do not plan to devote much energy to doing so in the forseeable future. This is because 31-ET is much more manageable, and yet is completely new territory for me and offers possibilities that seem inexhaustible by a single composer in a single lifetime. Perhaps some day others will find 53-ET worth exploring in a way that I have not.
Strengths of 53-ET
- Many intervals are so close as to be nearly perfect - These intervals include perfect fourths and fifths, major and minor thirds, major and minor whole tones. These intervals are so close to their true intervals that the difference is imperceptible to even most trained listeners.
- Many more intervals are still well matched - All intervals involving the 7th harmonic are still well-matched, with the largest deviation being the septimal tritones, which are 6.17 cents off (26 steps of the scale is wider than the 7:5 interval). All intervals involving the 11th harmonic are also closely matched, with the largest deviation being the 11:8 tritone, which is 7.92 cents too narrow in 53-ET. All intervals involving the 13th and 15th harmonic are even closely matched!
- No problematic intervals - 53-ET is remarkable in how all the scale degrees correspond to intervals for which there are useful relationships with natural harmonics, and the possibility of providing harmonic context for the intervals. This contrasts with 31-ET which has one problematic interval.
Challenges with 53-ET
- 53-ET is devilishly difficult to notate - This tuning runs into the problem of distinguishing between major and minor tones, and also, the major tone takes up 9 steps of the scale, creating the need for a huge number of accidentals with few enharmonic equivalences.
- Poor match to the 11th harmonic - 53-ET's closest match to the 11th harmonic is 7.92 cents off. This creates a variety of problems: for how many divisions of the octave and intervals there are, 53-ET still fails to distinguish the greater (11:10) and lesser (12:11) neutral seconds. All of the intervals involving the 11th harmonic and lower harmonics are off by at least 6.51 cents; the 14:11 interval is off by 9.96 cents. Although a few intervals are improved on over 31-ET and 41-ET, many become worse, particularly 11:9 and 12:11.
- Many of the intervals require high harmonics to interpret them - Although, as noted above, 53-ET has no truly problematic interval, it does have many that require use of high harmonics. The 23-step interval is an extreme case, with the smallest match being to the 19:14 interval. Many intervals require using the 13th and 17th harmonics too.