105edo: Difference between revisions
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== Theory == | |||
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone, such as [[grosstone]] and [[huygens]]. It [[tempering out|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 (a.k.a. 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 tuning|CTE generator]] for meantone. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|105}} | |||
=== Subsets and supersets === | |||
105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35. | |||
As such, the val [105 165 245 294], which is contorted in 2.n for each prime n in the subgroup, may be used to extend the concept of 21edo's 5-limit harmony to the 7-limit, producing an independent dimension for each prime. | |||
== Intervals == | |||
{{Main|Table of 105edo intervals}} | |||
=== 15-odd-limit interval mappings === | |||
{{Q-odd-limit intervals|105}} | |||
== Instruments == | |||
=== Lumatone === | |||
The [[lumatone]] can be used to play 105edo. For key mappings, see: [[Lumatone mapping for 105edo]]. | |||
[[Category:105edo| ]] <!-- main article --> | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | |||
[[Category:Huygens]] | |||
[[Category:Meantone]] | |||
Latest revision as of 19:44, 30 April 2025
| ← 104edo | 105edo | 106edo → |
105 equal divisions of the octave (abbreviated 105edo or 105ed2), also called 105-tone equal temperament (105tet) or 105 equal temperament (105et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 105 equal parts of about 11.4 ¢ each. Each step represents a frequency ratio of 21/105, or the 105th root of 2.
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 (a.k.a. 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) | |
Subsets and supersets
105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35.
As such, the val [105 165 245 294], which is contorted in 2.n for each prime n in the subgroup, may be used to extend the concept of 21edo's 5-limit harmony to the 7-limit, producing an independent dimension for each prime.
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 |
Instruments
Lumatone
The lumatone can be used to play 105edo. For key mappings, see: Lumatone mapping for 105edo.