986edo: Difference between revisions
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 986 factors as {{Factorization|986}}, 986edo has subset edos {{EDOs|1, 2, 17, 29, 34, 58, 493}}. | Since 986 factors as {{Factorization|986}}, 986edo has subset edos {{EDOs|1, 2, 17, 29, 34, 58, 493}}. | ||
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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[Gary]] | | [[Gary]] | ||
|} | |} | ||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
Latest revision as of 13:32, 13 March 2026
| ← 985edo | 986edo | 987edo → |
986 equal divisions of the octave (abbreviated 986edo or 986ed2), also called 986-tone equal temperament (986tet) or 986 equal temperament (986et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 986 equal parts of about 1.22 ¢ each. Each step represents a frequency ratio of 21/986, or the 986th root of 2.
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 |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct