131edo: Difference between revisions
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131edo is in[[consistent]] to the [[5-odd-limit]] and the error of [[harmonic]] [[3/1|3]] is quite large. However, it is the next [[edo]] after [[81edo]] on the [[Golden meantone|Golden Tone System]] (''[[Das Goldene Tonsystem]]'') of Thorvald Kornerup, using the 131b [[val]]. The [[patent val]] has a fifth sharp by 3.389 cents rather than flat like the meantone fifth; rather than tempering out [[81/80]] it tempers out the [[immunity comma]], 1638400/1594323. In the 7-limit it tempers out [[3125/3087]] and [[245/243]], so that it [[support]]s [[bohpier]]. | 131edo is in[[consistent]] to the [[5-odd-limit]] and the error of [[harmonic]] [[3/1|3]] is quite large. However, it is the next [[edo]] after [[81edo]] on the [[Golden meantone|Golden Tone System]] (''[[Das Goldene Tonsystem]]'') of Thorvald Kornerup, using the 131b [[val]]. The [[patent val]] has a fifth sharp by 3.389 cents rather than flat like the meantone fifth; rather than tempering out [[81/80]] it tempers out the [[immunity comma]], 1638400/1594323. In the 7-limit it tempers out [[3125/3087]] and [[245/243]], so that it [[support]]s [[bohpier]]. | ||
131edo is also notable for having a good approximation to [[natave|acoustic ''e'']], at 189\131, which is a [[semiconvergent]]. This number of steps, 189, is particularly well-factorizable, and divisors of acoustic ''e'' form a sequence of rapidly converging approximations to small rationals. Among these are [[5/4]] (2\9[[edn]] = 42\131), [[19/17]] (1\9edn = 21\131), [[11/10]] (2\21edn = 18\131), [[14/13]] (2\27edn = 14\131), and [[32/31]] (2\63edn = 6\131), with accuracy increasing the smaller the fraction. | 131edo is also notable for having a good approximation to [[natave|acoustic ''e'']], at 189\131, which is a [[semiconvergent]]. This number of steps, 189, is particularly well-factorizable, and divisors of acoustic ''e'' form a sequence of rapidly converging approximations to small rationals. Among these are [[5/4]] (2\9[[EDN|edn]] = 42\131), [[19/17]] (1\9edn = 21\131), [[11/10]] (2\21edn = 18\131), [[14/13]] (2\27edn = 14\131), and [[32/31]] (2\63edn = 6\131), with accuracy increasing the smaller the fraction. | ||
=== Odd harmonics === | === Odd harmonics === | ||
Revision as of 05:06, 15 November 2024
| ← 130edo | 131edo | 132edo → |
Theory
131edo is inconsistent to the 5-odd-limit and the error of harmonic 3 is quite large. However, it is the next edo after 81edo on the Golden Tone System (Das Goldene Tonsystem) of Thorvald Kornerup, using the 131b val. The patent val has a fifth sharp by 3.389 cents rather than flat like the meantone fifth; rather than tempering out 81/80 it tempers out the immunity comma, 1638400/1594323. In the 7-limit it tempers out 3125/3087 and 245/243, so that it supports bohpier.
131edo is also notable for having a good approximation to acoustic e, at 189\131, which is a semiconvergent. This number of steps, 189, is particularly well-factorizable, and divisors of acoustic e form a sequence of rapidly converging approximations to small rationals. Among these are 5/4 (2\9edn = 42\131), 19/17 (1\9edn = 21\131), 11/10 (2\21edn = 18\131), 14/13 (2\27edn = 14\131), and 32/31 (2\63edn = 6\131), with accuracy increasing the smaller the fraction.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.39 | -1.58 | +2.17 | -2.38 | -1.70 | +2.22 | +1.81 | -4.19 | -4.38 | -3.61 | +3.79 | -3.16 | +1.01 | -3.62 | +0.00 |
| Relative (%) | +37.0 | -17.3 | +23.7 | -26.0 | -18.6 | +24.2 | +19.7 | -45.8 | -47.9 | -39.4 | +41.3 | -34.5 | +11.0 | -39.6 | +0.0 | |
| Steps (reduced) |
208 (77) |
304 (42) |
368 (106) |
415 (22) |
453 (60) |
485 (92) |
512 (119) |
535 (11) |
556 (32) |
575 (51) |
593 (69) |
608 (84) |
623 (99) |
636 (112) |
649 (125) | |
Subsets and supersets
131edo is the 32nd prime edo, following 127edo and before 137edo.
Scales
Mos scales
| 33 16 33 33 16 | Pentatonic (comparable with 8edo and 99edo) |
| 23 23 8 23 23 23 8 | Pythagorean tuning (comparable with 17edo) |
| 21 21 13 21 21 21 13 | Meantone tuning (comparable with 50edo) |
| 19 12 19 19 12 19 19 12 | Father Tuning (comparable with 55edo) |
| 18 18 18 18 18 18 18 5 | Porcupine Tuning (comparable with 29edo and 80edo) |
| 17 17 17 6 17 17 17 17 6 | Superdiatonic tuning (comparable with 23edo) |
| 16 16 16 16 16 16 16 16 3 | Bohpier tuning (comparable with 41edo) |
| 13 13 9 13 13 13 9 13 13 13 9 | Sensi-11 Tuning |
| 11 11 11 11 11 5 11 11 11 11 11 11 5 | De Vries 13-tone Tuning |
| 10 10 10 7 10 10 10 10 7 10 10 10 10 7 | Ketradektriatoh Tuning |
| 21 17 21 17 17 21 17 | mohaha7 |
| 4 17 17 17 4 17 17 4 17 17 | mohaha10 |