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 ~~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].

Line 11:

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=== Prime harmonics ===

=== 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 ===

Since 152 factors into 23 × 19, 152edo has subset edos {{EDOs| 2, 4, 8, 19, 38, 76 }}.

Since 152 factors into primes as {{nowrap| 23 × 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 ==

{~~{comma basis begin}}~~

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

| {{~~monzo~~| 241 -152 }}

| {{Monzo| 241 -152 }}

| {{~~mapping~~| 152 241 }}

| {{Mapping| 152 241 }}

| ~~−0~~.213

| −0.213

| 0.213

| 2.70

Line 27:

Line 56:

| 2.3.5

| 1600000/1594323, {{monzo| 32 -7 -9 }}

| {{~~mapping~~| 152 241 353 }}

| {{Mapping| 152 241 353 }}

| ~~−0~~.218

| −0.218

| 0.174

| 2.21

Line 34:

Line 63:

| 2.3.5.7

| 4375/4374, 5120/5103, 16875/16807

| {{~~mapping~~| 152 241 353 427 }}

| {{Mapping| 152 241 353 427 }}

| ~~−0~~.362

| −0.362

| 0.291

| 3.69

Line 41:

Line 70:

| 2.3.5.7.11

| 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

| 3.30

Line 48:

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| 2.3.5.7.11.13

| 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

| 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 ~~begin}}~~

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

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|-

| 2

| 43\152 (33\152)

| 43\152 (33\152)

| 339.47 (260.53)

| 339.47 (260.53)

| 243/200 (64/55)

| 243/200 (64/55)

| [[Hemiamity]]

|-

| 2

| 55\152 (21\152)

| 55\152 (21\152)

| 434.21 (165.79)

| 434.21 (165.79)

| 9/7 (11/10)

| 9/7 (11/10)

| [[Supers]]

|-

| 4

| 63\152 (13\152)

| 63\152 (13\152)

| 497.37 (102.63)

| 497.37 (102.63)

| 4/3 (35/33)

| 4/3 (35/33)

| [[Undim]] / [[unlit]]

|-

| 8

| 63\152 (6\152)

| 63\152 (6\152)

| 497.37 (47.37)

| 497.37 (47.37)

| 4/3 (36/35)

| 4/3 (36/35)

| [[Twilight]]

|-

| 8

| 74\152 (2\152)

| 74\152 (2\152)

| 584.21 (15.79)

| 584.21 (15.79)

| 7/5 (126/125)

| 7/5 (126/125)

| [[Octoid]] (152f) / [[octopus]] (152)

|-

| 19

| 63\152 (1\152)

| 63\152 (1\152)

| 497.37 (7.89)

| 497.37 (7.89)

| 4/3 (225/224)

| 4/3 (225/224)

| [[Enneadecal]]

|-

| 38

| 63\152 (1\152)

| 63\152 (1\152)

| 497.37 (7.89)

| 497.37 (7.89)

| 4/3 (225/224)

| 4/3 (225/224)

| [[Hemienneadecal]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct

== Music ==

; [[birdshite stalactite]]

* "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:Amity]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|152}}
+{{ED intro}}
 == 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 linear temperaments [[amity]], [[grendel]], and [[kwai]], and the 11-limit planar 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, 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].
@@ Line 11: / Line 13: @@
 === Prime harmonics ===
 {{Harmonics in equal|152}}
+=== Octave stretch ===
+edo'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 ===
-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 ==
-{{comma basis begin}}
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
-| {{monzo| 241 -152 }}
+| {{Monzo| 241 -152 }}
-| {{mapping| 152 241 }}
+| {{Mapping| 152 241 }}
-| &minus;0.213
+| −0.213
 | 0.213
 | 2.70
@@ Line 27: / Line 56: @@
 | 2.3.5
 | 1600000/1594323, {{monzo| 32 -7 -9 }}
-| {{mapping| 152 241 353 }}
+| {{Mapping| 152 241 353 }}
-| &minus;0.218
+| −0.218
 | 0.174
 | 2.21
@@ Line 34: / Line 63: @@
 | 2.3.5.7
 | 4375/4374, 5120/5103, 16875/16807
-| {{mapping| 152 241 353 427 }}
+| {{Mapping| 152 241 353 427 }}
-| &minus;0.362
+| −0.362
 | 0.291
 | 3.69
@@ Line 41: / Line 70: @@
 | 2.3.5.7.11
 | 540/539, 1375/1372, 4000/3993, 5120/5103
-| {{mapping| 152 241 353 427 526 }}
+| {{Mapping| 152 241 353 427 526 }}
-| &minus;0.365
+| −0.365
 | 0.260
 | 3.30
@@ Line 48: / Line 77: @@
 | 2.3.5.7.11.13
 | 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)
-| &minus;0.494
+| −0.494
 | 0.373
 | 4.73
-{{comma basis end}}
+|-
+| 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 begin}}
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br>per 8ve
+! Generator*
+! Cents*
+! Associated<br>ratio*
+! Temperaments
 |-
 | 1
@@ Line 114: / Line 164: @@
 |-
 | 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]]
 |-
 | 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]]
 |-
 | 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]]
 |-
 | 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]]
 |-
 | 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)
 |-
 | 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]]
 |-
 | 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]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
 == Music ==
 ; [[birdshite stalactite]]
-* "athlete's feet" from ''razorblade tiddlywinks'' (2023) &ndash; [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:Amity]]