187edo: Difference between revisions
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This edo is a bit of a chimera. It has a [[3/2|fifth]] which is 2.9{{c}} flat and a [[5/4|classical major third]] 1.3 cents flat, but the [[7/1|7]], [[11/1|11]] and [[13/1|13]] are more accurate. | == Theory == | ||
This edo is a bit of a chimera. It has a [[3/2|fifth]] which is 2.9{{c}} flat and a [[5/4|classical major third]] 1.3 cents flat, but the [[7/1|7]], [[11/1|11]] and [[13/1|13]] are more accurate. | |||
Using the [[patent val]], the equal temperament [[tempering out|tempers out]] 78732/78125 ([[sensipent comma]]) in the [[5-limit]] and [[225/224]] in the [[7-limit]], providing the [[optimal patent val]] for [[sensei]], which tempers out both. In the [[11-limit]] it tempers out [[441/440]], [[1375/1372]] and [[4000/3993]], and in the [[13-limit]] [[351/350]], [[625/624]], [[1188/1183]], [[1716/1715]] and [[2200/2197]]. | |||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 8: | Line 11: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 187 factors into {{ | Since 187 factors into {{nowrap| 11 × 17 }}, 187edo contains [[11edo]] and [[17edo]] as its subsets. | ||
== Music == | |||
* | ; [[JUMBLE]] | ||
* [https://www.youtube.com/watch?v=vrFU2J25G1Y ''Pillow(33)''] (2024) – a 187edo lullaby, using the [[22L 11s]] [[mos]] | |||
Latest revision as of 10:00, 17 June 2026
| ← 186edo | 187edo | 188edo → |
187 equal divisions of the octave (abbreviated 187edo or 187ed2), also called 187-tone equal temperament (187tet) or 187 equal temperament (187et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 187 equal parts of about 6.42 ¢ each. Each step represents a frequency ratio of 21/187, or the 187th root of 2.
Theory
This edo is a bit of a chimera. It has a fifth which is 2.9 ¢ flat and a classical major third 1.3 cents flat, but the 7, 11 and 13 are more accurate.
Using the patent val, the equal temperament tempers out 78732/78125 (sensipent comma) in the 5-limit and 225/224 in the 7-limit, providing the optimal patent val for sensei, which tempers out both. In the 11-limit it tempers out 441/440, 1375/1372 and 4000/3993, and in the 13-limit 351/350, 625/624, 1188/1183, 1716/1715 and 2200/2197.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.49 | -1.29 | +0.16 | +1.44 | +0.55 | +0.11 | +2.64 | -2.28 | -2.33 | -2.33 | +0.60 |
| Relative (%) | -38.8 | -20.1 | +2.5 | +22.4 | +8.6 | +1.8 | +41.1 | -35.6 | -36.2 | -36.3 | +9.4 | |
| Steps (reduced) |
296 (109) |
434 (60) |
525 (151) |
593 (32) |
647 (86) |
692 (131) |
731 (170) |
764 (16) |
794 (46) |
821 (73) |
846 (98) | |
Subsets and supersets
Since 187 factors into 11 × 17, 187edo contains 11edo and 17edo as its subsets.
Music
- Pillow(33) (2024) – a 187edo lullaby, using the 22L 11s mos