316edo: Difference between revisions
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+commas and misc. cleanup |
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== 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. | ||
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! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 121: | Line 121: | ||
| 497.47<br>(102.53) | | 497.47<br>(102.53) | ||
| 4/3<br>(35/33) | | 4/3<br>(35/33) | ||
| [[ | | [[Unlit]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[Category:Parakleismic]] | [[Category:Parakleismic]] | ||
[[Category:Semiparakleismic]] | [[Category:Semiparakleismic]] | ||
[[Category:Triglav]] | [[Category:Triglav]] | ||
Revision as of 19:06, 11 December 2023
| ← 315edo | 316edo | 317edo → |
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 it is distinct