105edo: Difference between revisions
Cleanup and -factual errors (127edo seems to be the last meantone pval edo) |
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=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|105}} | {{Harmonics in equal|105}} | ||
=== Intervals === | |||
[[Table of 105edo intervals]] | |||
=== Miscellany === | === Miscellany === | ||
Revision as of 17:50, 31 March 2023
| ← 104edo | 105edo | 106edo → |
105edo is the equal division of the octave into 105 equal parts of 11.429 cents each.
Theory
105edo is most notable as a tuning of meantone and in particular higher-limit extensions of meantone. It tempers out 81/80 in the 5-limit; 81/80, 126/125 and hence 225/224 in the 7-limit; 99/98, 176/175 and 441/440 in the 11-limit; and if we want to push that far, 144/143 in the 13-limit. This is the sharper fifth mapping of 11-limit meantone (aka huygens rather than meanpop), for which it gives the optimal patent val, and provides a good tuning for the 13-limit extension, though 74edo is in that case the optimal patent val.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.81 | +2.26 | +2.60 | +1.80 | -2.75 | +5.19 | -2.55 | -2.10 | -0.37 | -2.21 | +0.30 |
| Relative (%) | -42.1 | +19.8 | +22.8 | +15.8 | -24.0 | +45.4 | -22.4 | -18.4 | -3.2 | -19.3 | +2.6 | |
| Steps (reduced) |
166 (61) |
244 (34) |
295 (85) |
333 (18) |
363 (48) |
389 (74) |
410 (95) |
429 (9) |
446 (26) |
461 (41) |
475 (55) | |
Intervals
Miscellany
105 is fairly composite, being the product 3 × 5 × 7 of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.
Scales
Since 105edo has a step of 11.429 cents, it also allows one to use its mos scales as circulating temperaments, which it is the first triangular edo to do[clarification needed].
| Tones | Pattern | L:s |
|---|---|---|
| 5 | 5edo | equal |
| 6 | 3L 3s | 18:17 |
| 7 | 7edo | equal |
| 8 | 1L 7s | 14:13 |
| 9 | 6L 3s | 12:11 |
| 10 | 5L 5s | 11:10 |
| 11 | 6L 5s | 10:9 |
| 12 | 9L 3s | 9:8 |
| 13 | 1L 12s | |
| 14 | 7L 7s | 8:7 |
| 15 | 15edo | equal |
| 16 | 9L 7s | 7:6 |
| 17 | 3L 14s | |
| 18 | 15L 3s | 6:5 |
| 19 | 10L 9s | |
| 20 | 5L 15s | |
| 21 | 21edo | equal |
| 22 | 17L 5s | 5:4 |
| 23 | 13L 10s | |
| 24 | 9L 15s | |
| 25 | 5L 20s | |
| 26 | 1L 25s | |
| 27 | 24L 3s | 4:3 |
| 28 | 21L 7s | |
| 29 | 18L 11s | |
| 30 | 15L 15s | |
| 31 | 12L 19s | |
| 32 | 9L 23s | |
| 33 | 6L 27s | |
| 34 | 3L 31s | |
| 35 | 35edo | equal |
| 36 | 33L 3s | 3:2 |
| 37 | 31L 6s | |
| 38 | 29L 9s | |
| 39 | 27L 12s | |
| 40 | 25L 15s | |
| 41 | 23L 18s | |
| 42 | 21L 21s | |
| 43 | 19L 24s | |
| 44 | 17L 27s | |
| 45 | 15L 30s | |
| 46 | 13L 33s | |
| 47 | 11L 36s | |
| 48 | 9L 39s | |
| 49 | 7L 42s | |
| 50 | 5L 45s | |
| 51 | 3L 48s | |
| 52 | 1L 51s | |
| 53 | 52L 1s | 2:1 |
| 54 | 51L 3s | |
| 55 | 50L 5s | |
| 56 | 49L 7s | |
| 57 | 48L 9s | |
| 58 | 47L 11s | |
| 59 | 46L 13s | |
| 60 | 45L 15s | |
| 61 | 44L 17s | |
| 62 | 43L 19s | |
| 63 | 42L 21s | |
| 64 | 41L 23s | |
| 65 | 40L 25s | |
| 66 | 39L 27s | |
| 67 | 38L 29s | |
| 68 | 37L 31s | |
| 69 | 36L 33s | |
| 70 | 35L 35s | |
| 71 | 34L 37s | |
| 72 | 33L 39s | |
| 73 | 32L 41s | |
| 74 | 31L 43s | |
| 75 | 30L 45s | |
| 76 | 29L 47s | |
| 77 | 28L 49s | |
| 78 | 27L 51s | |
| 79 | 26L 53s | |
| 80 | 25L 55s | |
| 81 | 24L 57s | |
| 82 | 23L 59s | |
| 83 | 22L 61s |