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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-09-24 05:00:08 UTC</tt>.<br>
: The original revision id was <tt>453635564</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The //152 equal division// divides the octave into 152 equally sized parts of 7.895 cents each. It is a strong 11-limit system, with the 3, 5, 7, and 11 slightly sharp. It provides the [[optimal patent val]] for the 11-limit [[Mirkwai clan|grendel]] and [[Mirkwai clan|kwai]] linear temperaments, the 13-limit rank two temperament [[Ragismic microtemperaments#Octoid-Octopus|octopus]], the planar temperament [[Hemifamity family|laka]], and the rank five temperament tempering out 169/168. It tempers out 1600000/1594323, the amity comma, in the 5-limit; 4375/4374, 5120/5103, 6144/6125 and 16875/15807 in the 7-limit; 540/539, 1375/1372, 4000/3993, 5632/5625 and 9801/9800 in the 11-limit, and 169/168, 325/324, 351/350, 364/363, 1001/1000, and 4096/4095 in the 13-limit.


152 = 8 * 19, with divisors 2, 4, 8, 19, 38, 76.</pre></div>
== Theory ==
<h4>Original HTML content:</h4>
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]].
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;152edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;152 equal division&lt;/em&gt; divides the octave into 152 equally sized parts of 7.895 cents each. It is a strong 11-limit system, with the 3, 5, 7, and 11 slightly sharp. It provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for the 11-limit &lt;a class="wiki_link" href="/Mirkwai%20clan"&gt;grendel&lt;/a&gt; and &lt;a class="wiki_link" href="/Mirkwai%20clan"&gt;kwai&lt;/a&gt; linear temperaments, the 13-limit rank two temperament &lt;a class="wiki_link" href="/Ragismic%20microtemperaments#Octoid-Octopus"&gt;octopus&lt;/a&gt;, the planar temperament &lt;a class="wiki_link" href="/Hemifamity%20family"&gt;laka&lt;/a&gt;, and the rank five temperament tempering out 169/168. It tempers out 1600000/1594323, the amity comma, in the 5-limit; 4375/4374, 5120/5103, 6144/6125 and 16875/15807 in the 7-limit; 540/539, 1375/1372, 4000/3993, 5632/5625 and 9801/9800 in the 11-limit, and 169/168, 325/324, 351/350, 364/363, 1001/1000, and 4096/4095 in the 13-limit.&lt;br /&gt;
 
&lt;br /&gt;
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]].
152 = 8 * 19, with divisors 2, 4, 8, 19, 38, 76.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
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].
 
=== Prime harmonics ===
{{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 ===
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 ==
{| 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 }}
| {{Mapping| 152 241 }}
| −0.213
| 0.213
| 2.70
|-
| 2.3.5
| 1600000/1594323, {{monzo| 32 -7 -9 }}
| {{Mapping| 152 241 353 }}
| −0.218
| 0.174
| 2.21
|-
| 2.3.5.7
| 4375/4374, 5120/5103, 16875/16807
| {{Mapping| 152 241 353 427 }}
| −0.362
| 0.291
| 3.69
|-
| 2.3.5.7.11
| 540/539, 1375/1372, 4000/3993, 5120/5103
| {{Mapping| 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
| {{Mapping| 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
| {{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 ===
{| 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
| 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<br>(33\152)
| 339.47<br>(260.53)
| 243/200<br>(64/55)
| [[Hemiamity]]
|-
| 2
| 55\152<br>(21\152)
| 434.21<br>(165.79)
| 9/7<br>(11/10)
| [[Supers]]
|-
| 4
| 63\152<br>(13\152)
| 497.37<br>(102.63)
| 4/3<br>(35/33)
| [[Undim]] / [[unlit]]
|-
| 8
| 63\152<br>(6\152)
| 497.37<br>(47.37)
| 4/3<br>(36/35)
| [[Twilight]]
|-
| 8
| 74\152<br>(2\152)
| 584.21<br>(15.79)
| 7/5<br>(126/125)
| [[Octoid]] (152f) / [[octopus]] (152)
|-
| 19
| 63\152<br>(1\152)
| 497.37<br>(7.89)
| 4/3<br>(225/224)
| [[Enneadecal]]
|-
| 38
| 63\152<br>(1\152)
| 497.37<br>(7.89)
| 4/3<br>(225/224)
| [[Hemienneadecal]]
|}
<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]
 
[[Category:Amity]]
[[Category:Grendel]]
[[Category:Kwai]]
[[Category:Laka]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/152edo"