310edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | {{ED intro}} | ||
== Theory == | |||
It inherits its 2.5.7 subgroup from 31edo. It is usable in the 2.5.7.13(.15).17.19 subgroup, being consistent in the tonality diamond of {1, 5, 7, 13, 15, 17, 19}. Note that adding harmonic 15 to the subgroup would be the same as adding harmonic 3, but many ratios of 15 are well represented, especially [[15/13]]. For that reason [[3/2]] is in that diamond, but other intervals of 3 are not. A notable chord using these odds is the 10:13:15 triad, though a much better system of around the same size to utilise it is [[313edo]], which tempers out [[676/675]]. While this system could be interesting, its is nowhere near the best of its size, and for most purposes [[311edo]] is better instead. | |||
It is part of the [[optimal ET sequence]] for the [[31-5-commatic]], [[fantastic]], [[quadrasruta]] and [[wizard]] temperaments. | It is part of the [[optimal ET sequence]] for the [[31-5-commatic]], [[fantastic]], [[quadrasruta]] and [[wizard]] temperaments. | ||
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== Odd harmonics == | == Odd harmonics == | ||
{{Harmonics in equal|310}} | {{Harmonics in equal|310}} | ||
Latest revision as of 17:31, 14 October 2025
| ← 309edo | 310edo | 311edo → |
310 equal divisions of the octave (abbreviated 310edo or 310ed2), also called 310-tone equal temperament (310tet) or 310 equal temperament (310et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 310 equal parts of about 3.87 ¢ each. Each step represents a frequency ratio of 21/310, or the 310th root of 2.
Theory
It inherits its 2.5.7 subgroup from 31edo. It is usable in the 2.5.7.13(.15).17.19 subgroup, being consistent in the tonality diamond of {1, 5, 7, 13, 15, 17, 19}. Note that adding harmonic 15 to the subgroup would be the same as adding harmonic 3, but many ratios of 15 are well represented, especially 15/13. For that reason 3/2 is in that diamond, but other intervals of 3 are not. A notable chord using these odds is the 10:13:15 triad, though a much better system of around the same size to utilise it is 313edo, which tempers out 676/675. While this system could be interesting, its is nowhere near the best of its size, and for most purposes 311edo is better instead.
It is part of the optimal ET sequence for the 31-5-commatic, fantastic, quadrasruta and wizard temperaments.
As a multiple of 10 and 31, it supports many 10th-octave temperaments and 31st-octave temperaments.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.31 | +0.78 | -1.08 | +1.25 | -1.64 | -0.53 | -0.53 | -0.44 | +0.55 | +1.48 | -1.18 |
| Relative (%) | -33.8 | +20.2 | -28.0 | +32.3 | -42.4 | -13.6 | -13.6 | -11.3 | +14.2 | +38.2 | -30.4 | |
| Steps (reduced) |
491 (181) |
720 (100) |
870 (250) |
983 (53) |
1072 (142) |
1147 (217) |
1211 (281) |
1267 (27) |
1317 (77) |
1362 (122) |
1402 (162) | |