261edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== | 261edo is [[consistent]] in the [[7-odd-limit]] and shares its harmonic 3 with [[29edo]], though with a significant error. Like other small EDO multiples of 29, it tunes the [[mystery]] temperament. A comma basis for the 7-limit is {10976/10935, 15625/15552, 2097152/2083725}. It is also [[enfactoring|enfactored]] in the 2.5.11 subgroup, inheriting it from [[87edo]] which it triples. | ||
261edo's patent val supports and is part of the [[optimal ET sequence]] for the [[diatessic]] temperament in the 11- and 13-limit, [[novemkleismic]] in the 7-limit and the 11-limit. 261ccdee val, {{val|261 414 '''605''' '''732''' '''904'''}} supports the [[superkleismic]] temperament. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|261}} | {{Harmonics in equal|261}} | ||
=== Subsets and supersets === | |||
Since 261 factors as {{Factorization|261}}, 261edo has subset edos {{EDOs|1, 3, 9, 29, 87}}. | |||
[[783edo]], which divides each edostep in three, provides a strong correction for harmonics 3 and 7. | |||
[[Category:Novemkleismic]] | |||
[[Category:Diatessic]] | |||
Latest revision as of 22:35, 21 December 2025
| ← 260edo | 261edo | 262edo → |
261 equal divisions of the octave (abbreviated 261edo or 261ed2), also called 261-tone equal temperament (261tet) or 261 equal temperament (261et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 261 equal parts of about 4.6 ¢ each. Each step represents a frequency ratio of 21/261, or the 261st root of 2.
261edo is consistent in the 7-odd-limit and shares its harmonic 3 with 29edo, though with a significant error. Like other small EDO multiples of 29, it tunes the mystery temperament. A comma basis for the 7-limit is {10976/10935, 15625/15552, 2097152/2083725}. It is also enfactored in the 2.5.11 subgroup, inheriting it from 87edo which it triples.
261edo's patent val supports and is part of the optimal ET sequence for the diatessic temperament in the 11- and 13-limit, novemkleismic in the 7-limit and the 11-limit. 261ccdee val, ⟨261 414 605 732 904] supports the superkleismic temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.49 | -0.11 | +1.29 | -1.61 | +0.41 | +0.85 | +1.39 | +0.79 | +1.34 | -1.82 | +1.61 |
| Relative (%) | +32.5 | -2.3 | +28.0 | -35.0 | +8.8 | +18.5 | +30.2 | +17.2 | +29.1 | -39.5 | +35.0 | |
| Steps (reduced) |
414 (153) |
606 (84) |
733 (211) |
827 (44) |
903 (120) |
966 (183) |
1020 (237) |
1067 (23) |
1109 (65) |
1146 (102) |
1181 (137) | |
Subsets and supersets
Since 261 factors as 32 × 29, 261edo has subset edos 1, 3, 9, 29, 87.
783edo, which divides each edostep in three, provides a strong correction for harmonics 3 and 7.