152edo: Difference between revisions
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== Theory == | == Theory == | ||
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 [[harmonic]]s [[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 [[13/1|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]]. | 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 [[integer limit|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 [[17/1|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 [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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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 152 factors into 2<sup>3</sup> × 19, 152edo has subset edos {{EDOs| 2, 4, 8, 19, 38, 76 }}. | Since 152 factors into primes as {{nowrap| 2<sup>3</sup> × 19 }}, 152edo has subset edos {{EDOs| 2, 4, 8, 19, 38, 76 }}. | ||
== Approximation to JI == | == Approximation to JI == | ||
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|- | |- | ||
| 2.3 | | 2.3 | ||
| {{ | | {{Monzo| 241 -152 }} | ||
| {{ | | {{Mapping| 152 241 }} | ||
| −0.213 | | −0.213 | ||
| 0.213 | | 0.213 | ||
| Line 54: | Line 56: | ||
| 2.3.5 | | 2.3.5 | ||
| 1600000/1594323, {{monzo| 32 -7 -9 }} | | 1600000/1594323, {{monzo| 32 -7 -9 }} | ||
| {{ | | {{Mapping| 152 241 353 }} | ||
| −0.218 | | −0.218 | ||
| 0.174 | | 0.174 | ||
| Line 61: | Line 63: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 4375/4374, 5120/5103, 16875/16807 | | 4375/4374, 5120/5103, 16875/16807 | ||
| {{ | | {{Mapping| 152 241 353 427 }} | ||
| −0.362 | | −0.362 | ||
| 0.291 | | 0.291 | ||
| Line 68: | Line 70: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 540/539, 1375/1372, 4000/3993, 5120/5103 | | 540/539, 1375/1372, 4000/3993, 5120/5103 | ||
| {{ | | {{Mapping| 152 241 353 427 526 }} | ||
| −0.365 | | −0.365 | ||
| 0.260 | | 0.260 | ||
| Line 75: | Line 77: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 352/351, 540/539, 625/624, 729/728, 1575/1573 | | 352/351, 540/539, 625/624, 729/728, 1575/1573 | ||
| {{ | | {{Mapping| 152 241 353 427 526 563 }} (152f) | ||
| −0.494 | | −0.494 | ||
| 0.373 | | 0.373 | ||
| 4.73 | | 4.73 | ||
|- | |||
| 2.3.5.7.11.13.19 | |||
| 352/351, 400/399, 495/494, 540/539, 625/624, 1331/1330 | |||
| {{Mapping| 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 | |||
| {{Mapping| 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 [[130edo|130]] and is superseded by [[224edo|224]]. In the 19- and 23-limit it is the first to beat [[140edo|140]] and is superseded by [[159edo|159]]. | * 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 [[130edo|130]] and is superseded by [[224edo|224]]. In the 19- and 23-limit it is the first to beat [[140edo|140]] and is superseded by [[159edo|159]]. | ||
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| [[Hemienneadecal]] | | [[Hemienneadecal]] | ||
|} | |} | ||
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[ | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct | ||
== Music == | == Music == | ||