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) | |
Octave stretch
145edo's approximated harmonics 3, 5, 11, 13, 17, 19, and 23 can be improved at the cost of a little worse 7, and moreover the approximated harmonic 13 can be brought to consistency, if slightly compressing the octave is acceptable. 375ed6 is about at the sweet spot for this.
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 |
| 2.3.5.7.11.13.17.19 | 196/195, 256/255, 343/342, 352/351, 361/360, 364/363, 476/475 | [⟨145 230 337 407 502 537 593 616]] | -0.565 | 0.486 | 5.87 |
| 2.3.5.7.11.13.17.19.23 | 196/195, 256/255, 276/275, 352/351, 361/360, 364/363, 460/459, 476/475 | [⟨145 230 337 407 502 537 593 616 656]] | -0.519 | 0.476 | 5.75 |
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