145edo: Difference between revisions
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| Fifth = 85\145 (703.45¢) (→ [[29edo|17\29]]) | | Fifth = 85\145 (703.45¢) (→ [[29edo|17\29]]) | ||
| Major 2nd = 25\145 (206.90¢) | | Major 2nd = 25\145 (206.90¢) | ||
| | | Semitones = 15:10 (124.14¢ : 82.76¢) | ||
| | | Consistency = 11 | ||
}} | }} | ||
The '''145 equal divisions of the octave''' ('''145edo''') or '''145(-tone) equal temperament''' ('''145tet''', '''145et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 145 [[equal]] parts of 8.28 [[cent]]s each. | |||
The '''145 equal divisions of the octave''' ( | |||
== Theory == | == Theory == | ||
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]] and [[364/363]] in the 13-limit; [[595/594]] in the 17-limit; [[343/342]] and [[476/475]] in the 19-limit. | |||
It | 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 [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], and because it tempers out 847/845 it allows the [[cuthbert triad]], 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]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
Revision as of 11:59, 24 October 2021
| ← 144edo | 145edo | 146edo → |
The 145 equal divisions of the octave (145edo) or 145(-tone) equal temperament (145tet, 145et) when viewed from a regular temperament perspective, is the tuning system derived by dividing the octave into 145 equal parts of 8.28 cents each.
Theory
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 and 364/363 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 minthmic chords, because it tempers out 364/363 it allows gentle chords, and because it tempers out 847/845 it allows the cuthbert triad, 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
Script error: No such module "primes_in_edo".
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 octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 2\145 | 16.55 | 100/99 | Quincy |
| 1 | 12\145 | 99.31 | 35/33, 18/17 | Quinticosiennic |
| 1 | 14\145 | 115.86 | 77/72 | Mercy / countermiracle |
| 1 | 39\145 | 322.76 | 3087/2560 | Senior / seniority |
| 1 | 41\145 | 339.31 | 128/105 | Amity |
| 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 |