292edo: Difference between revisions
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== Theory == | == Theory == | ||
292edo is closely related to [[146edo]], but the [[patent val]]s differ on the mapping for [[3/1|3]]. | 292edo is closely related to [[146edo]], but the [[patent val]]s differ on the mapping for [[3/1|3]]. It [[tempers out]] {{monzo| 3 -18 11 }} (quartonic comma) and {{monzo| 38 -2 -15 }} (luna/hemithirds comma) in the [[5-limit]]; 5120/5103 ([[5120/5103|hemifamity]]), 390625/388962 ([[dimcomp comma|dimcomp]]), 420175/419904 ([[wizma]]), and 4802000/4782969 ([[canousma]]) in the [[7-limit]]; 1375/1372, [[5632/5625]], [[6250/6237]], [[9801/9800]] and [[14641/14580]] in the [[11-limit]]; [[352/351]], [[625/624]], [[847/845]], [[1716/1715]], and [[2080/2079]] in the [[13-limit]]. | ||
It provides the [[optimal patent val]] for the [[undim]] temperament in the 7-, 11-, and 13-limit, and notably [[support]]s [[Hemifamity temperaments #Semiseptiquarter|semiseptiquarter]] and [[semiluna]]. | It provides the [[optimal patent val]] for the [[undim]] temperament in the 7-, 11-, and 13-limit, and notably [[support]]s [[Hemifamity temperaments #Semiseptiquarter|semiseptiquarter]] and [[semiluna]]. | ||
| Line 28: | Line 28: | ||
| {{monzo| 463 -292 }} | | {{monzo| 463 -292 }} | ||
| {{mapping| 292 463 }} | | {{mapping| 292 463 }} | ||
| | | −0.2476 | ||
| 0.2475 | | 0.2475 | ||
| 6.02 | | 6.02 | ||
| Line 35: | Line 35: | ||
| {{monzo| 3 -18 11 }}, {{monzo| 38 -2 -15 }} | | {{monzo| 3 -18 11 }}, {{monzo| 38 -2 -15 }} | ||
| {{mapping| 292 463 678 }} | | {{mapping| 292 463 678 }} | ||
| | | −0.1633 | ||
| 0.2346 | | 0.2346 | ||
| 5.71 | | 5.71 | ||
| Line 42: | Line 42: | ||
| 5120/5103, 390625/388962, 420175/419904 | | 5120/5103, 390625/388962, 420175/419904 | ||
| {{mapping| 292 463 678 820 }} | | {{mapping| 292 463 678 820 }} | ||
| | | −0.2148 | ||
| 0.2219 | | 0.2219 | ||
| 5.40 | | 5.40 | ||
| Line 49: | Line 49: | ||
| 1375/1372, 5120/5103, 5632/5625, 14641/14580 | | 1375/1372, 5120/5103, 5632/5625, 14641/14580 | ||
| {{mapping| 292 463 678 820 1010 }} | | {{mapping| 292 463 678 820 1010 }} | ||
| | | −0.1353 | ||
| 0.2544 | | 0.2544 | ||
| 6.19 | | 6.19 | ||
| Line 56: | Line 56: | ||
| 352/351, 625/624, 847/845, 1716/1715, 14641/14580 | | 352/351, 625/624, 847/845, 1716/1715, 14641/14580 | ||
| {{mapping| 292 463 678 820 1010 1081 }} | | {{mapping| 292 463 678 820 1010 1081 }} | ||
| | | −0.3480 | ||
| 0.2736 | | 0.2736 | ||
| 6.66 | | 6.66 | ||
| Line 63: | Line 63: | ||
| 352/351, 625/624, 715/714, 847/845, 1225/1224, 2025/2023 | | 352/351, 625/624, 715/714, 847/845, 1225/1224, 2025/2023 | ||
| {{mapping| 292 463 678 820 1010 1081 1194 }} | | {{mapping| 292 463 678 820 1010 1081 1194 }} | ||
| | | −0.2376 | ||
| 0.2696 | | 0.2696 | ||
| 6.56 | | 6.56 | ||
| Line 120: | Line 120: | ||
| [[Undim]] | | [[Undim]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
[[Category:Hemifamity]] | [[Category:Hemifamity]] | ||
Revision as of 14:06, 16 January 2025
| ← 291edo | 292edo | 293edo → |
Theory
292edo is closely related to 146edo, but the patent vals differ on the mapping for 3. It tempers out [3 -18 11⟩ (quartonic comma) and [38 -2 -15⟩ (luna/hemithirds comma) in the 5-limit; 5120/5103 (hemifamity), 390625/388962 (dimcomp), 420175/419904 (wizma), and 4802000/4782969 (canousma) in the 7-limit; 1375/1372, 5632/5625, 6250/6237, 9801/9800 and 14641/14580 in the 11-limit; 352/351, 625/624, 847/845, 1716/1715, and 2080/2079 in the 13-limit.
It provides the optimal patent val for the undim temperament in the 7-, 11-, and 13-limit, and notably supports semiseptiquarter and semiluna.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.78 | -0.01 | +1.04 | -0.63 | +1.94 | +1.89 | -1.62 | +0.49 | +1.93 | +1.54 |
| Relative (%) | +0.0 | +19.1 | -0.3 | +25.2 | -15.4 | +47.2 | +46.1 | -39.5 | +12.0 | +47.0 | +37.5 | |
| Steps (reduced) |
292 (0) |
463 (171) |
678 (94) |
820 (236) |
1010 (134) |
1081 (205) |
1194 (26) |
1240 (72) |
1321 (153) |
1419 (251) |
1447 (279) | |
Subsets and supersets
Since 292 factors into 22 × 73, 292edo has subset edos 2, 4, 73, and 146.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [463 -292⟩ | [⟨292 463]] | −0.2476 | 0.2475 | 6.02 |
| 2.3.5 | [3 -18 11⟩, [38 -2 -15⟩ | [⟨292 463 678]] | −0.1633 | 0.2346 | 5.71 |
| 2.3.5.7 | 5120/5103, 390625/388962, 420175/419904 | [⟨292 463 678 820]] | −0.2148 | 0.2219 | 5.40 |
| 2.3.5.7.11 | 1375/1372, 5120/5103, 5632/5625, 14641/14580 | [⟨292 463 678 820 1010]] | −0.1353 | 0.2544 | 6.19 |
| 2.3.5.7.11.13 | 352/351, 625/624, 847/845, 1716/1715, 14641/14580 | [⟨292 463 678 820 1010 1081]] | −0.3480 | 0.2736 | 6.66 |
| 2.3.5.7.11.13.17 | 352/351, 625/624, 715/714, 847/845, 1225/1224, 2025/2023 | [⟨292 463 678 820 1010 1081 1194]] | −0.2376 | 0.2696 | 6.56 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 11\292 | 45.21 | 250/243 | Quartonic (5-limit) |
| 1 | 47\292 | 193.15 | 262144/234375 | Luna |
| 1 | 59\292 | 242.47 | 147/128 | Septiquarter |
| 1 | 111\292 | 456.16 | 125/96 | Qak |
| 2 | 47\292 | 193.15 | 121/108 | Semiluna |
| 2 | 59\292 | 242.47 | 121/105 | Semiseptiquarter |
| 4 | 121\292 | 497.26 | 4/3 | Undim |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct