105edo
| ← 104edo | 105edo | 106edo → |
Theory
105edo is most notable as a tuning of meantone and in particular higher-limit extensions of meantone, such as grosstone and Huygens. 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. 105edo's meantone fifth is nearly identical to the CTE generator for meantone.
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
15-odd-limit interval mappings
The following tables show how 15-odd-limit intervals are represented in 105edo. Prime harmonics are in bold; inconsistent intervals are in italics.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 15/11, 22/15 | 0.192 | 1.7 |
| 7/5, 10/7 | 0.345 | 3.0 |
| 9/5, 10/9 | 0.453 | 4.0 |
| 9/7, 14/9 | 0.798 | 7.0 |
| 13/12, 24/13 | 1.430 | 12.5 |
| 9/8, 16/9 | 1.804 | 15.8 |
| 11/6, 12/11 | 2.066 | 18.1 |
| 5/4, 8/5 | 2.258 | 19.8 |
| 15/8, 16/15 | 2.554 | 22.4 |
| 13/7, 14/13 | 2.584 | 22.6 |
| 7/4, 8/7 | 2.603 | 22.8 |
| 11/8, 16/11 | 2.747 | 24.0 |
| 13/10, 20/13 | 2.929 | 25.6 |
| 13/9, 18/13 | 3.382 | 29.6 |
| 13/11, 22/13 | 3.495 | 30.6 |
| 15/13, 26/15 | 3.688 | 32.3 |
| 7/6, 12/7 | 4.014 | 35.1 |
| 5/3, 6/5 | 4.359 | 38.1 |
| 11/9, 18/11 | 4.551 | 39.8 |
| 3/2, 4/3 | 4.812 | 42.1 |
| 11/10, 20/11 | 5.004 | 43.8 |
| 15/14, 28/15 | 5.157 | 45.1 |
| 13/8, 16/13 | 5.187 | 45.4 |
| 11/7, 14/11 | 5.349 | 46.8 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 15/11, 22/15 | 0.192 | 1.7 |
| 7/5, 10/7 | 0.345 | 3.0 |
| 11/6, 12/11 | 2.066 | 18.1 |
| 5/4, 8/5 | 2.258 | 19.8 |
| 15/8, 16/15 | 2.554 | 22.4 |
| 13/7, 14/13 | 2.584 | 22.6 |
| 7/4, 8/7 | 2.603 | 22.8 |
| 11/8, 16/11 | 2.747 | 24.0 |
| 13/10, 20/13 | 2.929 | 25.6 |
| 3/2, 4/3 | 4.812 | 42.1 |
| 11/10, 20/11 | 5.004 | 43.8 |
| 15/14, 28/15 | 5.157 | 45.1 |
| 13/8, 16/13 | 5.187 | 45.4 |
| 11/7, 14/11 | 5.349 | 46.8 |
| 11/9, 18/11 | 6.878 | 60.2 |
| 5/3, 6/5 | 7.070 | 61.9 |
| 7/6, 12/7 | 7.415 | 64.9 |
| 15/13, 26/15 | 7.741 | 67.7 |
| 13/11, 22/13 | 7.933 | 69.4 |
| 9/8, 16/9 | 9.624 | 84.2 |
| 13/12, 24/13 | 9.999 | 87.5 |
| 9/5, 10/9 | 11.882 | 104.0 |
| 9/7, 14/9 | 12.227 | 107.0 |
| 13/9, 18/13 | 14.811 | 129.6 |
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 |