365edo/Eliora's approach
Theory
As 365 is the number of days in the common year, and there is a way to implement such a fact into music. An octave stretch of -0.796 cents would compress 365edo to an interesting intepretation: the pure 2/1 would represent 365.24219edo, which is the length of solar days in a tropical year. Therefore, each step is equal to 3.2854936 cents, and there are common octaves consisting of 365 steps and leap octaves consisting of 366 steps. The additional step does not amount to a JI mapping on its own, but rather resets the stretched octave to match the pure octave. Given that a standard gamut of music suitable for human hearing uses 8 octaves, this means 2 of them would be leap.
Using a val with a selected octave stretch
In 23-limit, 365eeffgghiii val's octave stretch of -0.79428 cents is very close to the target value, and makes 2/1 correspond to 365.241917 days, or 365 days 5h 48m 21.7s, which is only about 20 seconds short of the tropical year in the present era.
In the 7-limit, the val is the same as patent val for 365edo, and hence it supports the hemiwürschmidt temperament just like the unstretched pure-octave 365edo does.
Mapping used
| Prime Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 |
|---|---|---|---|---|---|---|---|---|---|
| Map | 365 | 579 | 848 | 1025 | 1264 | 1352 | 1493 | 1551 | 1653 |
| Patent val 365edo | 365 | 579 | 848 | 1025 | 1263 | 1351 | 1492 | 1550 | 1651 |
| Reduced | 365 | 214 | 118 | 295 | 169 | 257 | 33 | 91 | 193 |
Mappings different from the patent val highlighted in bold.
Using a uniform map
It is possible to take interval approximations directly in 365.24219edo.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.80 | +0.34 | -0.22 | -1.20 | +1.54 | +1.46 | +0.28 | +1.57 | -0.64 | -1.12 | -1.58 |
| Relative (%) | -24.2 | +10.5 | -6.6 | -36.4 | +47.0 | +44.3 | +8.6 | +47.8 | -19.6 | -34.0 | -48.2 | |
| Steps (reduced) |
365 (365) |
579 (213.75781) |
848 (117.51562) |
1025 (294.51562) |
1264 (168.27343) |
1352 (256.27343) |
1493 (32.03124) |
1552 (91.03124) |
1652 (191.03124) |
1774 (313.03124) |
1809 (348.03124) | |
One might take note that for harmonics 19 and 23, the uniform map features a distinct mapping from the val with the best octave stretching. Overall, the uniform map performs quite poorly in lower limits, though it provides good tunings for the 2.3.5.17.23 subgroup.
Regular temperament properties
365eeffgghiii val is used.
- 11-limit commas: {896/891, 6144/6125, 6250/6237, 532400/531441}
- 23-limit commas: {256/255, 300/299, 352/351, 456/455, 896/891, 1225/1224, 3136/3125, 13608/13585}
Table of intervals
| Step | Note name | Interval name | Associated ratio* |
|---|---|---|---|
| 0 | January 1 | Prime, unison | 1/1 |
| 32 | February 1 | Dodecaphonic semitone | |
| 33 | February 2 | Septendecimal semitone | 17/16 |
| 59 | February 28 | ||
| 59 II | February 29 | inserted once every 4 octaves | accumulating octave stretch |
| 60 | March 1 | Dodecaphonic major second | |
| 63 | March 4 | Classical major second, meantone | 9/8 |
| 91 | April 1 | Undevicesimal minor third, dodecaphonic minor third | 19/16 |
| 118 | April 28 | Classical major third | 5/4 |
| 169 | June 18 | Undecimal superfourth | 11/8 |
| 182 | July 1 | Dodecaphonic tritone | |
| 193 | July 12 | Vicesimotertial lalala | 23/16 |
| 214 | August 2 | Perfect fifth | 3/2 |
| 257 | September 14 | Tridecimal neutral sixth | 13/8 |
| 295 | October 22 | Harmonic seventh | 7/4 |
| 365 | January 1 | Octave | 2/1 |
* Based on 365eeffgghiii val