145edo: Difference between revisions
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== Theory == | == Theory == | ||
145 = 5 × 29, and 145edo shares the same perfect fifth with [[29edo]]. It is generally a sharp-tending system, with prime | 145 = 5 × 29, and 145edo shares the same perfect fifth with [[29edo]]. It is generally a sharp-tending system, with [[prime harmonic]]s 3 to 23 all tuned sharp except for 7, which is slightly flat. It is [[consistent]] to the [[11-odd-limit]], or the no-13 no-15 [[23-odd-limit]], with [[13/7]], [[15/8]] and their [[octave complement]]s being the only intervals going over the line. | ||
As an equal temperament, 145et [[tempering out|tempers out]] [[1600000/1594323]] in the [[5-limit]]; [[4375/4374]] and [[5120/5103]] in the [[7-limit]]; [[441/440]] and [[896/891]] in the 11-limit; [[196/195]], [[352/351]] | As an equal temperament, 145et [[tempering out|tempers out]] [[1600000/1594323]] in the [[5-limit]]; [[4375/4374]] and [[5120/5103]] in the [[7-limit]]; [[441/440]] and [[896/891]] in the 11-limit; [[196/195]], [[352/351]], [[364/363]], [[676/675]], [[847/845]], and [[1001/1000]] in the 13-limit; [[595/594]] in the 17-limit; [[343/342]] and [[476/475]] in the 19-limit. | ||
It is the [[optimal patent val]] for the 11-limit [[mystery]] temperament and the 11-limit rank-3 [[pele]] temperament. It also [[support]]s and provides a good tuning for 13-limit mystery, and because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[ | It is the [[optimal patent val]] for the 11-limit [[mystery]] temperament and the 11-limit rank-3 [[pele]] temperament. It also [[support]]s and provides a good tuning for 13-limit mystery, and because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], and because it tempers out 847/845 it allows the [[cuthbert chords]], making it a very flexible harmonic system. The same is true of [[232edo]], the optimal patent val for 13-limit mystery. | ||
The 145c val provides a tuning for [[magic]] which is nearly identical to the [[POTE tuning]]. | The 145c val provides a tuning for [[magic]] which is nearly identical to the [[POTE tuning]]. | ||
Revision as of 08:02, 27 January 2025
| ← 144edo | 145edo | 146edo → |
Theory
145 = 5 × 29, and 145edo shares the same perfect fifth with 29edo. It is generally a sharp-tending system, with prime harmonics 3 to 23 all tuned sharp except for 7, which is slightly flat. It is consistent to the 11-odd-limit, or the no-13 no-15 23-odd-limit, with 13/7, 15/8 and their octave complements being the only intervals going over the line.
As an equal temperament, 145et tempers out 1600000/1594323 in the 5-limit; 4375/4374 and 5120/5103 in the 7-limit; 441/440 and 896/891 in the 11-limit; 196/195, 352/351, 364/363, 676/675, 847/845, and 1001/1000 in the 13-limit; 595/594 in the 17-limit; 343/342 and 476/475 in the 19-limit.
It is the optimal patent val for the 11-limit mystery temperament and the 11-limit rank-3 pele temperament. It also supports and provides a good tuning for 13-limit mystery, and because it tempers out 441/440 it allows werckismic chords, because it tempers out 196/195 it allows mynucumic chords, because it tempers out 352/351 it allows major minthmic chords, because it tempers out 364/363 it allows minor minthmic chords, and because it tempers out 847/845 it allows the cuthbert chords, making it a very flexible harmonic system. The same is true of 232edo, the optimal patent val for 13-limit mystery.
The 145c val provides a tuning for magic which is nearly identical to the POTE tuning.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +1.49 | +2.65 | -0.55 | +3.16 | +3.61 | +2.63 | +0.42 | +0.69 | -3.37 | -2.97 |
| Relative (%) | +0.0 | +18.0 | +32.0 | -6.6 | +38.2 | +43.6 | +31.8 | +5.1 | +8.4 | -40.7 | -35.8 | |
| Steps (reduced) |
145 (0) |
230 (85) |
337 (47) |
407 (117) |
502 (67) |
537 (102) |
593 (13) |
616 (36) |
656 (76) |
704 (124) |
718 (138) | |
Subsets and supersets
145edo contains 5edo and 29edo as subset edos.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 1600000/1594323, [28 -3 -10⟩ | [⟨145 230 337]] | -0.695 | 0.498 | 6.02 |
| 2.3.5.7 | 4375/4374, 5120/5103, 50421/50000 | [⟨145 230 337 407]] | -0.472 | 0.578 | 6.99 |
| 2.3.5.7.11 | 441/440, 896/891, 3388/3375, 4375/4374 | [⟨145 230 337 407 502]] | -0.561 | 0.547 | 6.61 |
| 2.3.5.7.11.13 | 196/195, 352/351, 364/363, 676/675, 4375/4374 | [⟨145 230 337 407 502 537]] | -0.630 | 0.522 | 6.32 |
| 2.3.5.7.11.13.17 | 196/195, 256/255, 352/351, 364/363, 676/675, 1156/1155 | [⟨145 230 337 407 502 537 593]] | -0.632 | 0.484 | 5.85 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 2\145 | 16.55 | 100/99 | Quincy |
| 1 | 12\145 | 99.31 | 18/17 | Quinticosiennic |
| 1 | 14\145 | 115.86 | 77/72 | Countermiracle |
| 1 | 39\145 | 322.76 | 3087/2560 | Seniority / senator |
| 1 | 41\145 | 339.31 | 128/105 | Amity / catamite |
| 5 | 67\145 (9\145) |
554.48 (74.48) |
11/8 (25/24) |
Trisedodge / countdown |
| 29 | 60\145 (2\145) |
496.55 (16.55) |
4/3 (100/99) |
Mystery |
Scales
Music
- Chromatic piece in magic 16 – magic[16] in 145edo tuning