152edo
| ← 151edo | 152edo | 153edo → |
152 equal divisions of the octave (abbreviated 152edo or 152ed2), also called 152-tone equal temperament (152tet) or 152 equal temperament (152et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 152 equal parts of about 7.89 ¢ each. Each step represents a frequency ratio of 21/152, or the 152nd root of 2.
Theory
152edo is a strong 11-limit system, with the harmonics 3, 5, 7, and 11 slightly sharp. It tempers out 1600000/1594323 (amity comma) and [32 -7 -9⟩ (escapade comma) in the 5-limit; 4375/4374, 5120/5103, 6144/6125 and 16875/16807 in the 7-limit; 540/539, 1375/1372, 3025/3024, 4000/3993, 5632/5625 and 9801/9800 in the 11-limit. It provides the optimal patent val for the 11-limit rank-2 temperaments amity, grendel, and kwai, and the 11-limit rank-3 temperament laka.
It has two reasonable mappings for 13, with the 152f val scoring much better. The 152f val tempers out 352/351, 625/624, 640/637, 729/728, 847/845, 1188/1183, 1575/1573, 1716/1715 and 2080/2079, supporting and giving an excellent tuning for amity, kwai, and laka. The optimal tuning of this temperament is consistent in the 15-integer-limit. The patent val tempers out 169/168, 325/324, 351/350, 364/363, 1001/1000, 1573/1568, and 4096/4095, providing the optimal patent val for the 13-limit rank-5 temperament tempering out 169/168, as well as some further temperaments thereof, such as octopus.
Extending it beyond the 13-limit can be tricky, as the approximated harmonic 17 is almost 1/3-edostep flat of just, which does not blend well with the sharp tendency from the lower harmonics. The 152fg val in turn gives you an alternative that is more than 2/3-edostep sharp. However, if we skip prime 17 altogether, we can treat 152edo as a no-17 23-limit system with the 152f val, where it is strong and almost consistent to the no-17 23-odd-limit with the sole exception of 13/8 and its octave complement. It tempers out 400/399 and 495/494 in the 19-limit and 300/299, 484/483 and 576/575 in the 23-limit.
Paul Erlich has suggested that 152edo could be considered a sort of universal tuning.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.68 | +0.53 | +2.23 | +1.31 | -3.69 | -2.32 | +2.49 | +3.30 | -3.26 | -0.30 |
| Relative (%) | +0.0 | +8.6 | +6.7 | +28.2 | +16.6 | -46.7 | -29.4 | +31.5 | +41.9 | -41.3 | -3.8 | |
| Steps (reduced) |
152 (0) |
241 (89) |
353 (49) |
427 (123) |
526 (70) |
562 (106) |
621 (13) |
646 (38) |
688 (80) |
738 (130) |
753 (145) | |
Octave stretch
152edo's approximated harmonics 3, 5, 7, 11 can all be improved, and moreover the approximated harmonic 13 can be brought to consistency, if slightly compressing the octave is acceptable. 241edt is a great example for this.
Subsets and supersets
Since 152 factors into primes as 23 × 19, 152edo has subset edos 2, 4, 8, 19, 38, 76.
Approximation to JI
Zeta peak index
| Tuning | Strength | Octave (cents) | Integer limit | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per 8ve |
Step size (cents) |
Height | Integral | Gap | Size | Stretch | Consistent | Distinct | |
| Tempered | Pure | |||||||||
| 965zpi | 152.052848 | 7.891993 | 10.46842 | 7.617532 | 1.593855 | 19.487224 | 1199.582923 | −0.417077 | 15 | 15 |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [241 -152⟩ | [⟨152 241]] | −0.213 | 0.213 | 2.70 |
| 2.3.5 | 1600000/1594323, [32 -7 -9⟩ | [⟨152 241 353]] | −0.218 | 0.174 | 2.21 |
| 2.3.5.7 | 4375/4374, 5120/5103, 16875/16807 | [⟨152 241 353 427]] | −0.362 | 0.291 | 3.69 |
| 2.3.5.7.11 | 540/539, 1375/1372, 4000/3993, 5120/5103 | [⟨152 241 353 427 526]] | −0.365 | 0.260 | 3.30 |
| 2.3.5.7.11.13 | 352/351, 540/539, 625/624, 729/728, 1575/1573 | [⟨152 241 353 427 526 563]] (152f) | −0.494 | 0.373 | 4.73 |
| 2.3.5.7.11.13.19 | 352/351, 400/399, 495/494, 540/539, 625/624, 1331/1330 | [⟨152 241 353 427 526 563 646]] (152f) | −0.507 | 0.347 | 4.40 |
| 2.3.5.7.11.13.19.23 | 300/299, 352/351, 400/399, 484/483, 495/494, 540/539, 576/575 | [⟨152 241 353 427 526 563 646 688]] (152f) | −0.535 | 0.333 | 4.22 |
- 152et (152fg val) has lower absolute errors in the 11-, 19-, and 23-limit than any previous equal temperaments. In the 11-limit it is the first to beat 130 and is superseded by 224. In the 19- and 23-limit it is the first to beat 140 and is superseded by 159.
- It is best at the no-17 19- and 23-limit, in which it has lower relative errors than any previous equal temperaments. Not until 270 do we find a better equal temperament that does better in either of those subgroups.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 7\152 | 55.26 | 33/32 | Escapade / alphaquarter |
| 1 | 31\152 | 244.74 | 15/13 | Subsemifourth |
| 1 | 39\152 | 307.89 | 3200/2673 | Familia |
| 1 | 43\152 | 339.47 | 243/200 | Amity |
| 1 | 49\152 | 386.84 | 5/4 | Grendel |
| 1 | 63\152 | 497.37 | 4/3 | Kwai |
| 1 | 71\152 | 560.53 | 242/175 | Whoops |
| 2 | 7\152 | 55.26 | 33/32 | Septisuperfourth |
| 2 | 9\152 | 71.05 | 25/24 | Vishnu / acyuta (152f) / ananta (152) |
| 2 | 43\152 (33\152) |
339.47 (260.53) |
243/200 (64/55) |
Hemiamity |
| 2 | 55\152 (21\152) |
434.21 (165.79) |
9/7 (11/10) |
Supers |
| 4 | 63\152 (13\152) |
497.37 (102.63) |
4/3 (35/33) |
Undim / unlit |
| 8 | 63\152 (6\152) |
497.37 (47.37) |
4/3 (36/35) |
Twilight |
| 8 | 74\152 (2\152) |
584.21 (15.79) |
7/5 (126/125) |
Octoid (152f) / octopus (152) |
| 19 | 63\152 (1\152) |
497.37 (7.89) |
4/3 (225/224) |
Enneadecal |
| 38 | 63\152 (1\152) |
497.37 (7.89) |
4/3 (225/224) |
Hemienneadecal |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct