145edo: Difference between revisions
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=== Prime harmonics === | === Prime harmonics === | ||
{{ | {{Harmonics in equal|145|intervals=prime|columns=11}} | ||
== 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 62: | Line 62: | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per | ! Periods<br>per 8ve | ||
! Generator<br>( | ! Generator<br>(Reduced) | ||
! Cents<br>( | ! Cents<br>(Reduced) | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 77: | Line 77: | ||
| 12\145 | | 12\145 | ||
| 99.31 | | 99.31 | ||
| | | 18/17 | ||
| [[Quinticosiennic]] | | [[Quinticosiennic]] | ||
|- | |- | ||
| Line 84: | Line 84: | ||
| 115.86 | | 115.86 | ||
| 77/72 | | 77/72 | ||
| [[ | | [[Countermiracle]] | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 90: | Line 90: | ||
| 322.76 | | 322.76 | ||
| 3087/2560 | | 3087/2560 | ||
| [[ | | [[Seniority]] / senator | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 96: | Line 96: | ||
| 339.31 | | 339.31 | ||
| 128/105 | | 128/105 | ||
| [[Amity]] | | [[Amity]] / catamite | ||
|- | |- | ||
| 5 | | 5 | ||
Revision as of 11:01, 5 December 2022
| ← 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
| 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) | |
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 (Reduced) |
Cents (Reduced) |
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 |