316edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
While not highly accurate for its size, 316et is the point where a few important temperaments meet, and is | While not highly accurate for its size, 316et is the point where a few important temperaments meet, and is [[consistency|distinctly consistent]] in the [[11-odd-limit]]. It [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }}, the [[undim comma]], {{monzo| 41 -20 -4 }}, and the [[maquila comma]], {{monzo| 49 -6 -17 }} in the 5-limit; [[3136/3125]], [[4375/4374]], [[10976/10935]] in the 7-limit; [[3025/3024]], [[3388/3375]], [[9801/9800]], and [[14641/14580]] in the 11-limit; and using the [[patent val]], [[1716/1715]], [[2080/2079]], [[2197/2187]], [[4096/4095]], [[4225/4224]], [[6656/6655]], and [[10648/10647]] in the 13-limit; notably supporting [[Abigail]] and [[semiparakleismic]]. | ||
It provides the [[optimal patent val]] for the rank-4 temperament tempering out 3388/3375, and [[triglav]], which also tempers out 3025/3024. | It provides the [[optimal patent val]] for the rank-4 temperament tempering out 3388/3375, and [[triglav]], which also tempers out 3025/3024. | ||
| Line 11: | Line 11: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
316 factors into | 316 factors into {{factorisation|316}}, with subset edos {{EDOs| 2, 4, 79, and 158 }}. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 26: | Line 27: | ||
| 2.3 | | 2.3 | ||
| {{monzo| 501 -316 }} | | {{monzo| 501 -316 }} | ||
| | | {{mapping| 316 501 }} | ||
| | | −0.182 | ||
| 0.182 | | 0.182 | ||
| 4.79 | | 4.79 | ||
| Line 33: | Line 34: | ||
| 2.3.5 | | 2.3.5 | ||
| {{monzo| 8 14 -13 }}, {{monzo| 41 -20 -4 }} | | {{monzo| 8 14 -13 }}, {{monzo| 41 -20 -4 }} | ||
| | | {{mapping| 316 501 734 }} | ||
| | | −0.269 | ||
| 0.193 | | 0.193 | ||
| 5.08 | | 5.08 | ||
| Line 40: | Line 41: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 3136/3125, 4375/4374, {{monzo| -26 -1 1 9 }} | | 3136/3125, 4375/4374, {{monzo| -26 -1 1 9 }} | ||
| | | {{mapping| 316 501 734 887 }} | ||
| | | −0.160 | ||
| 0.252 | | 0.252 | ||
| 6.64 | | 6.64 | ||
| Line 47: | Line 48: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 3025/3024, 3136/3125, 4375/4374, 131072/130977 | | 3025/3024, 3136/3125, 4375/4374, 131072/130977 | ||
| | | {{mapping| 316 501 734 887 1093 }} | ||
| | | −0.088 | ||
| 0.267 | | 0.267 | ||
| 7.04 | | 7.04 | ||
|- | |- | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 1716/1715, 2080/2079, 3025/3024, 3136/3125 | | 1716/1715, 2080/2079, 2197/2187, 3025/3024, 3136/3125 | ||
| | | {{mapping| 316 501 734 887 1093 1169 }} | ||
| | | −0.016 | ||
| 0.293 | | 0.293 | ||
| 7.72 | | 7.72 | ||
| Line 62: | Line 63: | ||
=== 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 | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br> | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 92: | Line 94: | ||
| 512/375 | | 512/375 | ||
| [[Maquila]] (5-limit) | | [[Maquila]] (5-limit) | ||
|- | |||
| 1 | |||
| 149\316 | |||
| 565.82 | |||
| 18/13 | |||
| [[Threedic]] | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 106: | Line 114: | ||
|- | |- | ||
| 2 | | 2 | ||
| 83\316<br>(75\316) | | 83\316<br />(75\316) | ||
| 315.19<br>(284.81) | | 315.19<br />(284.81) | ||
| 6/5<br>(33/28) | | 6/5<br />(33/28) | ||
| [[Semiparakleismic]] | | [[Semiparakleismic]] | ||
|- | |- | ||
| 4 | | 4 | ||
| 131\316<br>(27\316) | | 131\316<br />(27\316) | ||
| 497.47<br>(102.53) | | 497.47<br />(102.53) | ||
| 4/3<br>(35/33) | | 4/3<br />(35/33) | ||
| [[ | | [[Unlit]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
[[Category:Parakleismic]] | [[Category:Parakleismic]] | ||
[[Category:Semiparakleismic]] | [[Category:Semiparakleismic]] | ||
[[Category:Triglav]] | [[Category:Triglav]] | ||
Latest revision as of 22:50, 20 February 2025
| ← 315edo | 316edo | 317edo → |
316 equal divisions of the octave (abbreviated 316edo or 316ed2), also called 316-tone equal temperament (316tet) or 316 equal temperament (316et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 316 equal parts of about 3.8 ¢ each. Each step represents a frequency ratio of 21/316, or the 316th root of 2.
Theory
While not highly accurate for its size, 316et is the point where a few important temperaments meet, and is distinctly consistent in the 11-odd-limit. It tempers out the parakleisma, [8 14 -13⟩, the undim comma, [41 -20 -4⟩, and the maquila comma, [49 -6 -17⟩ in the 5-limit; 3136/3125, 4375/4374, 10976/10935 in the 7-limit; 3025/3024, 3388/3375, 9801/9800, and 14641/14580 in the 11-limit; and using the patent val, 1716/1715, 2080/2079, 2197/2187, 4096/4095, 4225/4224, 6656/6655, and 10648/10647 in the 13-limit; notably supporting Abigail and semiparakleismic.
It provides the optimal patent val for the rank-4 temperament tempering out 3388/3375, and triglav, which also tempers out 3025/3024.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.58 | +1.03 | -0.47 | -0.69 | -1.29 | +1.37 | -1.31 | -1.69 | -0.46 | +1.80 |
| Relative (%) | +0.0 | +15.2 | +27.1 | -12.4 | -18.0 | -33.9 | +36.2 | -34.5 | -44.6 | -12.2 | +47.4 | |
| Steps (reduced) |
316 (0) |
501 (185) |
734 (102) |
887 (255) |
1093 (145) |
1169 (221) |
1292 (28) |
1342 (78) |
1429 (165) |
1535 (271) |
1566 (302) | |
Subsets and supersets
316 factors into 22 × 79, with subset edos 2, 4, 79, and 158.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [501 -316⟩ | [⟨316 501]] | −0.182 | 0.182 | 4.79 |
| 2.3.5 | [8 14 -13⟩, [41 -20 -4⟩ | [⟨316 501 734]] | −0.269 | 0.193 | 5.08 |
| 2.3.5.7 | 3136/3125, 4375/4374, [-26 -1 1 9⟩ | [⟨316 501 734 887]] | −0.160 | 0.252 | 6.64 |
| 2.3.5.7.11 | 3025/3024, 3136/3125, 4375/4374, 131072/130977 | [⟨316 501 734 887 1093]] | −0.088 | 0.267 | 7.04 |
| 2.3.5.7.11.13 | 1716/1715, 2080/2079, 2197/2187, 3025/3024, 3136/3125 | [⟨316 501 734 887 1093 1169]] | −0.016 | 0.293 | 7.72 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 51\316 | 193.67 | 28/25 | Didacus |
| 1 | 83\316 | 315.19 | 6/5 | Parakleismic (7-limit) |
| 1 | 84\316 | 322.78 | 3087/2560 | Seniority |
| 1 | 141\316 | 535.44 | 512/375 | Maquila (5-limit) |
| 1 | 149\316 | 565.82 | 18/13 | Threedic |
| 1 | 155\316 | 588.61 | 45927/32768 | Countritonic (7-limit) |
| 2 | 55\316 | 208.86 | 44/39 | Abigail |
| 2 | 83\316 (75\316) |
315.19 (284.81) |
6/5 (33/28) |
Semiparakleismic |
| 4 | 131\316 (27\316) |
497.47 (102.53) |
4/3 (35/33) |
Unlit |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct