111edo: Difference between revisions
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Adopt template: EDO intro; cleanup; clarify the title row of the rank-2 temp table; -redundant categories |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|111}} | |||
== Theory == | == Theory == | ||
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest | 111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With [[harmonic]]s 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. | ||
It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. | It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. | ||
In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. | In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. | ||
It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31&80 temperament, and [[buzzard]], the 53&58 temperament. | It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 & 80 temperament, and [[buzzard]], the 53 & 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 17: | Line 16: | ||
== 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|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 28: | Line 27: | ||
| 2.3 | | 2.3 | ||
| {{monzo| 176 -111 }} | | {{monzo| 176 -111 }} | ||
| | | {{mapping| 111 176 }} | ||
| -0.236 | | -0.236 | ||
| 0.236 | | 0.236 | ||
| Line 35: | Line 34: | ||
| 2.3.5 | | 2.3.5 | ||
| 78732/78125, 67108864/66430125 | | 78732/78125, 67108864/66430125 | ||
| | | {{mapping| 111 176 258 }} | ||
| -0.570 | | -0.570 | ||
| 0.510 | | 0.510 | ||
| Line 42: | Line 41: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 1728/1715, 3136/3125, 5120/5103 | | 1728/1715, 3136/3125, 5120/5103 | ||
| | | {{mapping| 111 176 258 312 }} | ||
| -0.797 | | -0.797 | ||
| 0.591 | | 0.591 | ||
| Line 49: | Line 48: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 176/175, 540/539, 1331/1323, 5120/5103 | | 176/175, 540/539, 1331/1323, 5120/5103 | ||
| | | {{mapping| 111 176 258 312 384 }} | ||
| -0.639 | | -0.639 | ||
| 0.615 | | 0.615 | ||
| Line 56: | Line 55: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 176/175, 351/350, 540/539, 676/675, 1331/1323 | | 176/175, 351/350, 540/539, 676/675, 1331/1323 | ||
| | | {{mapping| 111 176 258 312 384 411 }} | ||
| -0.655 | | -0.655 | ||
| 0.562 | | 0.562 | ||
| Line 63: | Line 62: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714 | | 176/175, 256/255, 351/350, 442/441, 540/539, 715/714 | ||
| | | {{mapping| 111 176 258 312 384 411 454 }} | ||
| -0.672 | | -0.672 | ||
| 0.523 | | 0.523 | ||
| Line 70: | Line 69: | ||
| 2.3.5.7.11.13.17.19 | | 2.3.5.7.11.13.17.19 | ||
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475 | | 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475 | ||
| | | {{mapping| 111 176 258 312 384 411 454 472 }} | ||
| -0.740 | | -0.740 | ||
| 0.521 | | 0.521 | ||
| Line 82: | Line 81: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated | ! Associated<br>Ratio* | ||
! Temperament | ! Temperament | ||
|- | |- | ||
| Line 207: | Line 206: | ||
| [[Misty]] | | [[Misty]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== Music == | == Music == | ||
* [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 | ; [[Gene Ward Smith]] | ||
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning | |||
[[Category:Buzzard]] | [[Category:Buzzard]] | ||
[[Category:Semisept]] | [[Category:Semisept]] | ||
Revision as of 08:57, 24 October 2023
| ← 110edo | 111edo | 112edo → |
Theory
111edo is consistent through to the 21-odd-limit, and is the smallest edo uniquely consistent through the 15-odd-limit, marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to 37edo, with which it shares the mappings for 5, 7, 11, and 13.
It is also significant for lower limits, especially in terms of what it tempers out in its patent val; for example, it tempers out 176/175 and gives an excellent optimal patent val for the corresponding 11-limit rank-4 temperament.
In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the quartisma.
It is a particularly good tuning for the 11- or 13-limit versions of semisept, the 31 & 80 temperament, and buzzard, the 53 & 58 temperament. Gene Ward Smith's trio in #Music section is in guanyin temperament, the planar temperament tempering out 176/175 and 540/539, for which 111 also provides the optimal patent val.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.75 | +2.88 | +4.15 | +0.03 | +2.72 | +3.15 | +5.19 | -1.25 | -2.55 | +0.91 |
| Relative (%) | +0.0 | +6.9 | +26.6 | +38.4 | +0.3 | +25.1 | +29.2 | +48.0 | -11.5 | -23.6 | +8.4 | |
| Steps (reduced) |
111 (0) |
176 (65) |
258 (36) |
312 (90) |
384 (51) |
411 (78) |
454 (10) |
472 (28) |
502 (58) |
539 (95) |
550 (106) | |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [176 -111⟩ | [⟨111 176]] | -0.236 | 0.236 | 2.18 |
| 2.3.5 | 78732/78125, 67108864/66430125 | [⟨111 176 258]] | -0.570 | 0.510 | 4.72 |
| 2.3.5.7 | 1728/1715, 3136/3125, 5120/5103 | [⟨111 176 258 312]] | -0.797 | 0.591 | 5.47 |
| 2.3.5.7.11 | 176/175, 540/539, 1331/1323, 5120/5103 | [⟨111 176 258 312 384]] | -0.639 | 0.615 | 5.69 |
| 2.3.5.7.11.13 | 176/175, 351/350, 540/539, 676/675, 1331/1323 | [⟨111 176 258 312 384 411]] | -0.655 | 0.562 | 5.21 |
| 2.3.5.7.11.13.17 | 176/175, 256/255, 351/350, 442/441, 540/539, 715/714 | [⟨111 176 258 312 384 411 454]] | -0.672 | 0.523 | 4.84 |
| 2.3.5.7.11.13.17.19 | 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475 | [⟨111 176 258 312 384 411 454 472]] | -0.740 | 0.521 | 4.83 |
Rank-2 temperaments
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed.
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperament |
|---|---|---|---|---|
| 1 | 11\111 | 118.92 | 15/14 | Subsedia |
| 1 | 13\111 | 140.54 | 13/12 | Quanic |
| 1 | 14\111 | 151.35 | 12/11 | Browser |
| 1 | 16\111 | 172.97 | 400/363 | Undetrita |
| 1 | 20\111 | 216.22 | 17/15 | Tremka |
| 1 | 23\111 | 248.65 | 15/13 | Hemikwai |
| 1 | 31\111 | 335.14 | 17/14 | Cohemimabila |
| 1 | 35\111 | 378.38 | 56/45 | Subpental |
| 1 | 41\111 | 443.24 | 162/125 | Sensipent / warrior |
| 1 | 43\111 | 464.86 | 17/13 | Semisept |
| 1 | 44\111 | 475.68 | 21/16 | Vulture / buzzard |
| 1 | 46\111 | 497.30 | 4/3 | Kwai |
| 1 | 49\111 | 529.73 | 19/14 | Tuskaloosa |
| 1 | 55\111 | 594.59 | 55/39 | Gaster |
| 3 | 7\111 | 75.68 | 24/23 | Terture |
| 3 | 12\111 | 129.73 | 14/13 | Trimabila |
| 3 | 13\111 | 140.54 | 243/224 | Septichrome |
| 3 | 17\111 | 183.55 | 10/9 | Mirkat |
| 3 | 23\111 (14\111) |
248.65 (151.35) |
231/200 (12/11) |
Hemimist |
| 3 | 46\111 (9\111) |
497.30 (97.30) |
4/3 (18/17~19/18) |
Misty |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Music
- Trio for SoftSaturn, NebulaSing and TromBonehead (archived 2010) – SoundCloud | details | play – guanyin[22] in 111edo tuning