149edo
| ← 148edo | 149edo | 150edo → |
149 equal divisions of the octave (abbreviated 149edo or 149ed2), also called 149-tone equal temperament (149tet) or 149 equal temperament (149et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 149 equal parts of about 8.05 ¢ each. Each step represents a frequency ratio of 21/149, or the 149th root of 2.
Theory
149edo is the smallest division which is uniquely consistent through the 17-odd-limit. It has a general flat tendency, with the fifth 1.28 cents flat, but the major third is a quarter of a cent sharp. In the 5-limit it tempers out the sensipent comma, 78732/78125; in the 7-limit, 1029/1024, 3136/3125 and 19683/19600; in the 11-limit 385/384 and 441/440; in the 13-limit 351/350 and 676/675; in the 17-limit 273/272 and 561/560; in the 19-limit 286/285 and 343/342. It provides the optimal patent val for 7-, 11-, 13-, and 17-limit heinz temperament and the rank-3 temperament ominous in the 13- and 17-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.28 | +0.26 | -2.38 | -3.67 | -2.94 | -0.26 | +0.47 | -0.09 | +1.30 | -1.41 |
| Relative (%) | +0.0 | -15.9 | +3.3 | -29.6 | -45.5 | -36.6 | -3.2 | +5.9 | -1.1 | +16.1 | -17.5 | |
| Steps (reduced) |
149 (0) |
236 (87) |
346 (48) |
418 (120) |
515 (68) |
551 (104) |
609 (13) |
633 (37) |
674 (78) |
724 (128) |
738 (142) | |
Subsets and supersets
149edo is the 35th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-236 149⟩ | [⟨149 236]] | +0.405 | 0.405 | 5.03 |
| 2.3.5 | 78732/78125, [-34 20 1⟩ | [⟨149 236 346]] | +0.232 | 0.411 | 5.11 |
| 2.3.5.7 | 1029/1024, 3136/3125, 19683/19600 | [⟨149 236 346 418]] | +0.386 | 0.445 | 5.53 |
| 2.3.5.7.11 | 385/384, 441/440, 3136/3125, 19683/19600 | [⟨149 236 346 418 515]] | +0.521 | 0.481 | 5.97 |
| 2.3.5.7.11.13 | 351/350, 385/384, 441/440, 676/675, 847/845 | [⟨149 236 346 418 515 551]] | +0.567 | 0.451 | 5.60 |
| 2.3.5.7.11.13.17 | 273/272, 351/350, 385/384, 441/440, 676/675, 847/845 | [⟨149 236 346 418 515 551 609]] | +0.495 | 0.453 | 5.62 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 3\149 | 24.16 | 686/675 | Sengagen |
| 1 | 16\149 | 128.86 | 14/13 | Tertiathirds |
| 1 | 18\149 | 144.97 | 49/45 | Swetneus |
| 1 | 24\149 | 193.29 | 28/25 | Hemithirds |
| 1 | 29\149 | 233.56 | 8/7 | Slendric |
| 1 | 47\149 | 378.52 | 56/45 | Subpental |
| 1 | 55\149 | 442.95 | 162/125 | Sensipent |
| 1 | 57\149 | 459.06 | 125/96 | Majvam |
| 1 | 60\149 | 483.22 | 45/34 | Hemiseven |
| 1 | 61\149 | 491.28 | 3645/2744 | Fifthplus |
| 1 | 68\149 | 547.65 | 11/8 | Heinz |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct