256edo: Difference between revisions
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256edo is [[enfactoring|enfactored]] in the [[5-limit]] with the same tuning as [[128edo]], and the error of [[harmonic]] [[7/1|7]] leads to in[[consistency]], which is likely one of the reasons this edo attracts little interest. To start with, consider the sharp-tending 256c [[val]] {{val| 256 406 '''595''' 719 886 }}, which [[tempering out|tempers out]] [[2401/2400]], [[3388/3375]], [[5120/5103]], so that it [[support]]s 7-limit [[hemififths]] and [[11-limit]] [[semihemi]]. The [[patent val]] {{val| 256 406 '''594''' 719 886 }} tempers out [[540/539]], 2200/2187, [[4000/3969]], 12005/11979, among others. It is best tuned in the 2.3.7.11 [[subgroup]], in which it is consistent to the [[11-odd-limit]] minus intervals involving [[5/1|5]]. | 256edo is [[enfactoring|enfactored]] in the [[5-limit]] with the same tuning as [[128edo]], and the error of [[harmonic]] [[7/1|7]] leads to in[[consistency]], which is likely one of the reasons this edo attracts little interest. To start with, consider the sharp-tending 256c [[val]] {{val| 256 406 '''595''' 719 886 }}, which [[tempering out|tempers out]] [[2401/2400]], [[3388/3375]], [[5120/5103]], so that it [[support]]s 7-limit [[hemififths]] and [[11-limit]] [[semihemi]]. The [[patent val]] {{val| 256 406 '''594''' 719 886 }} tempers out [[540/539]], 2200/2187, [[4000/3969]], 12005/11979, among others. It is best tuned in the 2.3.7.11 [[subgroup]], in which it is consistent to the [[11-odd-limit]] minus intervals involving [[5/1|5]]. | ||
Latest revision as of 17:11, 20 February 2025
| ← 255edo | 256edo | 257edo → |
256 equal divisions of the octave (abbreviated 256edo or 256ed2), also called 256-tone equal temperament (256tet) or 256 equal temperament (256et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 256 equal parts of about 4.69 ¢ each. Each step represents a frequency ratio of 21/256, or the 256th root of 2.
256edo is enfactored in the 5-limit with the same tuning as 128edo, and the error of harmonic 7 leads to inconsistency, which is likely one of the reasons this edo attracts little interest. To start with, consider the sharp-tending 256c val ⟨256 406 595 719 886], which tempers out 2401/2400, 3388/3375, 5120/5103, so that it supports 7-limit hemififths and 11-limit semihemi. The patent val ⟨256 406 594 719 886] tempers out 540/539, 2200/2187, 4000/3969, 12005/11979, among others. It is best tuned in the 2.3.7.11 subgroup, in which it is consistent to the 11-odd-limit minus intervals involving 5.
In the higher limits, it approximates harmonics 23, 43, and 47 quite accurately.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.17 | -1.94 | +1.49 | +2.34 | +1.81 | -1.47 | -0.77 | -1.83 | -2.20 | -2.03 | -0.15 |
| Relative (%) | +25.0 | -41.4 | +31.7 | +49.9 | +38.6 | -31.3 | -16.4 | -39.0 | -46.9 | -43.3 | -3.2 | |
| Steps (reduced) |
406 (150) |
594 (82) |
719 (207) |
812 (44) |
886 (118) |
947 (179) |
1000 (232) |
1046 (22) |
1087 (63) |
1124 (100) |
1158 (134) | |
Subsets and supersets
Since 256 factors into 28, 256edo has subset edos 2, 4, 8, 16, 32, 64, and 128.