152edo: Difference between revisions
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152edo is a strong [[11-limit]] system, with the [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[11/1|11]] slightly sharp. It [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 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]]. | 152edo is a strong [[11-limit]] system, with the [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[11/1|11]] slightly sharp. It [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 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]], [[support]]ing 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]]. | It has two reasonable mappings for [[13/1|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]], [[support]]ing 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]]. | ||
[[Paul Erlich]] has suggested that 152edo could be considered a sort of [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3038.html#3041 universal tuning]. | [[Paul Erlich]] has suggested that 152edo could be considered a sort of [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3038.html#3041 universal tuning]. | ||
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=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|152}} | {{Harmonics in equal|152}} | ||
=== 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 [[stretched and compressed tuning|compressing the octave]] is acceptable. [[241edt]] is a great example for this. | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
Revision as of 13:00, 24 March 2025
| ← 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 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.
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 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 |
- 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