145edo
| ← 144edo | 145edo | 146edo → |
145 equal divisions of the octave (abbreviated 145edo or 145ed2), also called 145-tone equal temperament (145tet) or 145 equal temperament (145et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 145 equal parts of about 8.28 ¢ each. Each step represents a frequency ratio of 21/145, or the 145th root of 2.
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