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935edo: Difference between revisions

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**Imported revision 267776178 - Original comment: **
 
 
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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>2011-10-23 23:34:35 UTC</tt>.<br>
 
: The original revision id was <tt>267776178</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
935edo is a very strong [[23-limit]] system, and is [[consistent]] through to the [[27-odd-limit]]. It does reasonably well in the higher limits, though the sharply tuned [[11/1|11]] and [[23/1|23]] and the flatly tuned [[29/1|29]] and [[31/1|31]] create inconsistencies together, those being [[29/22]], [[29/23]], [[31/22]], [[31/23]] and their [[octave complement]]s; it is otherwise consistent to the [[39-odd-limit]]. It is a [[zeta peak edo]].  
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>
As an equal temperament, it [[tempering out|tempers out]] the {{monzo| 39 -29 3 }} ([[alphatricot comma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), and {{monzo| 91 -12 -31 }} (astro comma) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; [[117649/117612]], [[151263/151250]], [[161280/161051]] in the [[11-limit]]; [[2080/2079]], [[4096/4095]], [[4225/4224]] in the [[13-limit]]; [[2058/2057]], [[2500/2499]], [[4914/4913]] in the [[17-limit]]; [[2432/2431]], [[3136/3135]], [[3250/3249]], [[4200/4199]] in the 19-limit; and [[2025/2024]], [[2300/2299]], [[2646/2645]] among others in the 23-limit.
<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 935 equal division divides the octave into 935 parts of 1.283 cents each. It is a very strong 23-limit system, and distinctly consistent through to the 27 odd limit. It is also a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak tuning]]. In the 5-limit it tempers out the tricot comma |39 -29 3&gt;, the septendecima, |-52 -17 34&gt;, and astro, |91 -12 -31&gt;. In the 7-limit it tempers out 4375/4374 and 52734375/52706752, in the 11-limit 161280/161051 and 117649/117612, and in the 13-limit 2080/2079, 4096/4095 and 4225/4224.</pre></div>
 
<h4>Original HTML content:</h4>
=== Prime harmonics ===
<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;935edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 935 equal division divides the octave into 935 parts of 1.283 cents each. It is a very strong 23-limit system, and distinctly consistent through to the 27 odd limit. It is also a &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&gt;zeta peak tuning&lt;/a&gt;. In the 5-limit it tempers out the tricot comma |39 -29 3&amp;gt;, the septendecima, |-52 -17 34&amp;gt;, and astro, |91 -12 -31&amp;gt;. In the 7-limit it tempers out 4375/4374 and 52734375/52706752, in the 11-limit 161280/161051 and 117649/117612, and in the 13-limit 2080/2079, 4096/4095 and 4225/4224.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Harmonics in equal|935|columns=11}}
{{Harmonics in equal|935|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 935edo (continued)}}
 
=== Subsets and supersets ===
Since 935 factors into primes as {{nowrap| 5 × 11 × 17 }}, 935edo has subset edos {{EDOs| 5, 11, 17, 55, 85, and 187 }}.
 
== 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| 1482 -935 }}
| {{Mapping| 935 1482 }}
| −0.0243
| 0.0243
| 1.89
|-
| 2.3.5
| {{Monzo| 39 -29 3 }}, {{monzo| -52 -17 34 }}
| {{Mapping| 935 1482 2171 }}
| −0.0157
| 0.0233
| 1.82
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 36 -5 0 -10 }}
| {{Mapping| 935 1482 2171 2625 }}
| −0.0259
| 0.0268
| 2.08
|-
| 2.3.5.7.11
| 4375/4374, 117649/117612, 131072/130977, 161280/161051
| {{Mapping| 935 1482 2171 2625 3235 }}
| −0.0527
| 0.0588
| 4.58
|-
| 2.3.5.7.11.13
| 2080/2079, 4096/4095, 4375/4374, 78125/78078, 117649/117612
| {{Mapping| 935 1482 2171 2625 3235 3460 }}
| −0.0490
| 0.0543
| 4.23
|-
| 2.3.5.7.11.13.17
| 2058/2057, 2080/2079, 2500/2499, 4096/4095, 4375/4374, 4914/4913
| {{Mapping| 935 1482 2171 2625 3235 3460 3822 }}
| −0.0520
| 0.0508
| 3.96
|-
| 2.3.5.7.11.13.17.19
| 2058/2057, 2080/2079, 2432/2431, 2500/2499, 3136/3135, 4375/4374, 4914/4913
| {{Mapping| 935 1482 2171 2625 3235 3460 3822 3972 }}
| −0.0526
| 0.0475
| 3.70
|-
| 2.3.5.7.11.13.17.19.23
| 2025/2024, 2058/2057, 2080/2079, 2300/2299, 2432/2431, 2500/2499, 2646/2645, 4375/4374
| {{Mapping| 935 1482 2171 2625 3235 3460 3822 3972 4230 }}
| −0.0616
| 0.0515
| 4.01
|}
* 935et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19- and 23-limit. It is the first to beat [[764edo|764]] in the 13-limit, [[814edo|814]] in the 17- and 23-limit, and [[742edo|742]] in the 19-limit, only to be bettered by [[954edo|954h]] in all 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
| 103\935
| 132.19
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
| 1
| 339\935
| 435.08
| 9/7
| [[Supermajor (temperament)|Supermajor]]
|-
| 1
| 442\935
| 567.27
| 104/75
| [[Alphatrillium]]
|-
| 17
| 194\935<br>(26\935)
| 248.98<br>(33.37)
| {{Monzo| -23 5 9 -2 }}<br>(100352/98415)
| [[Chlorine]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms|minimal form]] in parentheses if distinct
Retrieved from "https://en.xen.wiki/w/935edo"