986edo: Difference between revisions
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986edo is a good 2.3.7.11 subgroup tuning, but it is in[[consistent]] to the 5-odd-limit and larger due to a high error on the [[5/4|5th harmonic]]. 986edo has an excellent [[11/8|11th harmonic]], being the denominator of a [[convergent]] to log<sub>2</sub>11, after [[949edo|949]] and before [[1935edo|1935]]. In the 2.3.7.11 subgroup, 986edo can be used with optional additions of either [[17/16|17]], [[23/16|23]], [[29/16|29]], or [[31/16|31]]. | 986edo is a good 2.3.7.11 subgroup tuning, but it is in[[consistent]] to the 5-odd-limit and larger due to a high error on the [[5/4|5th harmonic]]. 986edo has an excellent [[11/8|11th harmonic]], being the denominator of a [[convergent]] to log<sub>2</sub>11, after [[949edo|949]] and before [[1935edo|1935]]. In the 2.3.7.11 subgroup, 986edo can be used with optional additions of either [[17/16|17]], [[23/16|23]], [[29/16|29]], or [[31/16|31]]. | ||
In the 2.3.7 subgroup, 986edo tempers out the [[garischisma]], and is a strong tuning for 2.3.7.11-subgroup [[gary]]. It also tempers out | In the 2.3.7 subgroup, 986edo tempers out the [[garischisma]], and is a strong tuning for 2.3.7.11-subgroup [[gary]]. It also tempers out, 131072/130977, 3195731/3188646, 33554432/33480783, 67110351/67108864, and {{monzo|5 4 0 28 -26}} in the 2.3.7.11 subgroup. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{harmonics in equal|986}} | {{harmonics in equal|986}} | ||
Revision as of 17:20, 18 February 2024
| ← 985edo | 986edo | 987edo → |
Theory
986edo is a good 2.3.7.11 subgroup tuning, but it is inconsistent to the 5-odd-limit and larger due to a high error on the 5th harmonic. 986edo has an excellent 11th harmonic, being the denominator of a convergent to log211, after 949 and before 1935. In the 2.3.7.11 subgroup, 986edo can be used with optional additions of either 17, 23, 29, or 31.
In the 2.3.7 subgroup, 986edo tempers out the garischisma, and is a strong tuning for 2.3.7.11-subgroup gary. It also tempers out, 131072/130977, 3195731/3188646, 33554432/33480783, 67110351/67108864, and [5 4 0 28 -26⟩ in the 2.3.7.11 subgroup.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.276 | -0.512 | -0.063 | +0.001 | +0.446 | -0.290 | -0.556 | -0.282 | +0.037 | +0.198 |
| Relative (%) | +0.0 | +22.7 | -42.1 | -5.2 | +0.0 | +36.6 | -23.8 | -45.7 | -23.2 | +3.1 | +16.2 | |
| Steps (reduced) |
986 (0) |
1563 (577) |
2289 (317) |
2768 (796) |
3411 (453) |
3649 (691) |
4030 (86) |
4188 (244) |
4460 (516) |
4790 (846) |
4885 (941) | |
Subsets and supersets
Since 986 factors as 2 × 17 × 29, 986edo has subset edos 1, 2, 17, 29, 34, 58, 493.
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 409\986 | 497.769 | 4/3 | Gary |