152edo: Difference between revisions

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== 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 [[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].

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 == Theory ==
-edo 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]].
+edo 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 [[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].