152edo: Difference between revisions

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


== Theory ==
== Theory ==
152et is a strong 11-limit system, with the 3, 5, 7, and 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 [[Mirkwai comma|16875/16807]] in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], [[4000/3993]], [[5632/5625]] and [[9801/9800]] in the 11-limit.  
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 [[patent val]] tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1001/1000]], [[1573/1568]], and [[4096/4095]]. The 152f val tempers out [[352/351]], [[625/624]], [[640/637]], [[729/728]], [[847/845]], [[1188/1183]], [[1575/1573]], [[1716/1715]] and [[2080/2079]].  
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]].  


It provides the [[optimal patent val]] for the 11-limit [[grendel]] and [[kwai]] linear temperaments, the 13-limit rank-2 temperament [[Ragismic microtemperaments #Octoid-Octopus|octopus]], the 11-limit planar temperament [[laka]], and the rank-5 temperament tempering out 169/168.  
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 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 ==
=== 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 ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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|-
|-
| 2.3
| 2.3
| {{monzo| 241 -152 }}
| {{Monzo| 241 -152 }}
| [{{val| 152 241 }}]
| {{Mapping| 152 241 }}
| -0.213
| −0.213
| 0.213
| 0.213
| 2.70
| 2.70
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| 2.3.5
| 2.3.5
| 1600000/1594323, {{monzo| 32 -7 -9 }}
| 1600000/1594323, {{monzo| 32 -7 -9 }}
| [{{val| 152 241 353 }}]
| {{Mapping| 152 241 353 }}
| -0.218
| −0.218
| 0.174
| 0.174
| 2.21
| 2.21
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| 2.3.5.7
| 2.3.5.7
| 4375/4374, 5120/5103, 16875/16807
| 4375/4374, 5120/5103, 16875/16807
| [{{val| 152 241 353 427 }}]
| {{Mapping| 152 241 353 427 }}
| -0.362
| −0.362
| 0.291
| 0.291
| 3.69
| 3.69
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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
| [{{val| 152 241 353 427 526 }}]
| {{Mapping| 152 241 353 427 526 }}
| -0.365
| −0.365
| 0.260
| 0.260
| 3.30
| 3.30
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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
| [{{val| 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]].
* 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 [[270edo|270]] do we find a better equal temperament that does better in either of those subgroups.


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+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<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 560.53
| 560.53
| 242/175
| 242/175
| [[Whoosh]] / [[whoops]]
| [[Whoops]]
|-
|-
| 2
| 2
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| 55.26
| 55.26
| 33/32
| 33/32
| [[Biscapade]]
| [[Septisuperfourth]]
|-
|-
| 2
| 2
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| [[Hemienneadecal]]
| [[Hemienneadecal]]
|}
|}
<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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* "athlete's feet" from ''razorblade tiddlywinks'' (2023) – [https://open.spotify.com/track/32c34U3syZDMAJkBzgh2pd Spotify] | [https://birdshitestalactite.bandcamp.com/track/athletes-feet Bandcamp] | [https://www.youtube.com/watch?v=lXqVaVn3SrA YouTube]
* "athlete's feet" from ''razorblade tiddlywinks'' (2023) – [https://open.spotify.com/track/32c34U3syZDMAJkBzgh2pd Spotify] | [https://birdshitestalactite.bandcamp.com/track/athletes-feet Bandcamp] | [https://www.youtube.com/watch?v=lXqVaVn3SrA YouTube]


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Amity]]
[[Category:Grendel]]
[[Category:Grendel]]
[[Category:Kwai]]
[[Category:Kwai]]
[[Category:Laka]]
[[Category:Laka]]
[[Category:Listen]]
[[Category:Listen]]

Latest revision as of 13:18, 24 March 2025

← 151edo 152edo 153edo →
Prime factorization 23 × 19
Step size 7.89474 ¢ 
Fifth 89\152 (702.632 ¢)
Semitones (A1:m2) 15:11 (118.4 ¢ : 86.84 ¢)
Consistency limit 11
Distinct consistency limit 11

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

Approximation of prime harmonics in 152edo
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

Table of rank-2 temperaments by generator
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

Music

birdshite stalactite
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