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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|152}}
{{ED intro}}


== 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 [[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 linear temperaments [[amity]], [[grendel]], and [[kwai]], and the 11-limit planar 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, 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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=== 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 ===
Since 152 factors into {{factorisation}}, 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 ==
=== Zeta peak index ===
{{ZPI
| zpi = 965
| steps = 152.052848107925
| step size = 7.89199291517551
| tempered height = 10.468420
| pure height = 7.617532
| integral = 1.593855
| gap = 19.487224
| octave = 1199.58292310668
| consistent = 15
| distinct = 15
}}


== Regular temperament properties ==
== Regular temperament properties ==
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|-
|-
| 2.3
| 2.3
| {{monzo| 241 -152 }}
| {{Monzo| 241 -152 }}
| {{mapping| 152 241 }}
| {{Mapping| 152 241 }}
| −0.213
| −0.213
| 0.213
| 0.213
Line 36: Line 56:
| 2.3.5
| 2.3.5
| 1600000/1594323, {{monzo| 32 -7 -9 }}
| 1600000/1594323, {{monzo| 32 -7 -9 }}
| {{mapping| 152 241 353 }}
| {{Mapping| 152 241 353 }}
| −0.218
| −0.218
| 0.174
| 0.174
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| 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 }}
| {{Mapping| 152 241 353 427 }}
| −0.362
| −0.362
| 0.291
| 0.291
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| 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 }}
| {{Mapping| 152 241 353 427 526 }}
| −0.365
| −0.365
| 0.260
| 0.260
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| 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)
| {{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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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 2
| 2
| 43\152<br />(33\152)
| 43\152<br>(33\152)
| 339.47<br />(260.53)
| 339.47<br>(260.53)
| 243/200<br />(64/55)
| 243/200<br>(64/55)
| [[Hemiamity]]
| [[Hemiamity]]
|-
|-
| 2
| 2
| 55\152<br />(21\152)
| 55\152<br>(21\152)
| 434.21<br />(165.79)
| 434.21<br>(165.79)
| 9/7<br />(11/10)
| 9/7<br>(11/10)
| [[Supers]]
| [[Supers]]
|-
|-
| 4
| 4
| 63\152<br />(13\152)
| 63\152<br>(13\152)
| 497.37<br />(102.63)
| 497.37<br>(102.63)
| 4/3<br />(35/33)
| 4/3<br>(35/33)
| [[Undim]] / [[unlit]]
| [[Undim]] / [[unlit]]
|-
|-
| 8
| 8
| 63\152<br />(6\152)
| 63\152<br>(6\152)
| 497.37<br />(47.37)
| 497.37<br>(47.37)
| 4/3<br />(36/35)
| 4/3<br>(36/35)
| [[Twilight]]
| [[Twilight]]
|-
|-
| 8
| 8
| 74\152<br />(2\152)
| 74\152<br>(2\152)
| 584.21<br />(15.79)
| 584.21<br>(15.79)
| 7/5<br />(126/125)
| 7/5<br>(126/125)
| [[Octoid]] (152f) / [[octopus]] (152)
| [[Octoid]] (152f) / [[octopus]] (152)
|-
|-
| 19
| 19
| 63\152<br />(1\152)
| 63\152<br>(1\152)
| 497.37<br />(7.89)
| 497.37<br>(7.89)
| 4/3<br />(225/224)
| 4/3<br>(225/224)
| [[Enneadecal]]
| [[Enneadecal]]
|-
|-
| 38
| 38
| 63\152<br />(1\152)
| 63\152<br>(1\152)
| 497.37<br />(7.89)
| 497.37<br>(7.89)
| 4/3<br />(225/224)
| 4/3<br>(225/224)
| [[Hemienneadecal]]
| [[Hemienneadecal]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct


== Music ==
== Music ==
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