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742edo: Difference between revisions

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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>2011-08-08 01:57:20 UTC</tt>.<br>
 
: The original revision id was <tt>244793431</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
742edo is a very strong 19-limit system and a [[zeta peak edo]], and is [[consistency|distinctly consistent]] in the [[21-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] the [[vishnuzma]] and the fortune comma in the 5-limit, [[support]]ing [[vishnu]] and [[fortune]]; [[2401/2400]] in the 7-limit, [[9801/9800]] in the 11-limit, [[4096/4095]], [[6656/6655]], [[10648/10647]] in the 13-limit, [[1701/1700]], [[2058/2057]], [[2601/2600]], [[4914/4913]], [[5832/5831]] in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.
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>
=== 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;white-space: pre-wrap ! important" class="old-revision-html">The //742 equal division// divides the octave into 742 equal parts of  1.617 cents each. It is a very strong 19-limit system and a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak tuning]], and is uniquely [[consistent]] in the 21-limit. It tempers out 2401/2400 in the 7-limit, 9801/9800 in the 11-limit, 4096/4095, 6656/6655, 10648/10647 in the 13-limit, 1701/1700, 2058/2057, 2601/2600, 4914/4913, 5832/5831 in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.</pre></div>
{{Harmonics in equal|742|columns=11}}
<h4>Original HTML content:</h4>
{{Harmonics in equal|742|columns=11|start=12|collapsed=1|title=Approximation of prime harmonics in 742edo (continued)}}
<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;742edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;742 equal division&lt;/em&gt; divides the octave into 742 equal parts of 1.617 cents each. It is a very strong 19-limit system and a &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&gt;zeta peak tuning&lt;/a&gt;, and is uniquely &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt; in the 21-limit. It tempers out 2401/2400 in the 7-limit, 9801/9800 in the 11-limit, 4096/4095, 6656/6655, 10648/10647 in the 13-limit, 1701/1700, 2058/2057, 2601/2600, 4914/4913, 5832/5831 in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== Subsets and supersets ===
Since 742 factors into 2 × 7 × 53, 742edo has subset edos {{EDOs| 2, 7, 14, 53, 106, and 371 }}, of which [[7edo]], [[14edo]] and [[53edo]] are very notable. It supports [[silicon]] ({{nowrap|224 & 518}}) with 14 periods per octave in the 13-limit, and [[iodine]] ({{nowrap|159& 583f}}) with 53 periods per octave in the 17-limit.
 
== 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]] (%)
|-
| 2.3.5
| {{Monzo| 23 6 -14 }}, {{monzo| -84 53 }}
| {{Mapping| 742 1176 1723 }}
| −0.0157
| 0.0555
| 3.43
|-
| 2.3.5.7
| 2401/2400, 14348907/14336000, {{monzo| 23 6 -14 }}
| {{Mapping| 742 1176 1723 2083 }}
| −0.0035
| 0.0525
| 3.24
|-
| 2.3.5.7.11
| 2401/2400, 9801/9800, 172032/171875, 1240029/1239040
| {{Mapping| 742 1176 1723 2083 2567 }}
| −0.0123
| 0.0501
| 3.10
|-
| 2.3.5.7.11.13
| 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325
| {{Mapping| 742 1176 1723 2083 2567 2746 }}
| −0.0302
| 0.0608
| 3.76
|-
| 2.3.5.7.11.13.17
| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655
| {{Mapping| 742 1176 1723 2083 2567 2746 3033 }}
| −0.0317
| 0.0564
| 3.49
|-
| 2.3.5.7.11.13.17.19
| 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211
| {{Mapping| 742 1176 1723 2083 2567 2746 3033 3152 }}
| −0.0295
| 0.0531
| 3.28
|-
| 2.3.5.7.11.13.17.19.23
| 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211
| {{Mapping| 742 1176 1723 2083 2567 2746 3033 3152 3357 }} (742i)
| −0.0468
| 0.0699
| 4.32
|}
* 742et has a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any previous equal temperaments. It is only bettered by [[935edo|935]] in terms of absolute error, and by [[1178edo|1178]] in terms of relative error.
* 742et (742i val) is also notable in the 17- and 23-limit, where it has lower absolute errors than any previous equal temperaments. In the 17-limit it beats [[581edo|581]] and is bettered by [[764edo|764]]; in the 23-limit it beats [[718edo|718]] and is bettered by [[814edo|814]].  
 
=== 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
| 137\742
| 221.563
| 8388608/7381125
| [[Fortune]]
|-
| 1
| 243\742
| 392.992
| 2744/2187
| [[Emmthird]] (7-limit)
|-
| 1
| 303\742
| 490.026
| 896/675
| [[Surmarvelpyth]]
|-
| 2
| 44\742
| 71.159
| 25/24
| [[Vishnu]]
|-
| 14
| 434\742<br>(10\742)
| 701.886<br>(16.173)
| 3/2<br>(105/104)
| [[Silicon]]
|-
| 53
| 239\742<br>(1\742)
| 386.523<br>(1.617)
| 5/4<br>(32805/32768)
| [[Mercator]]
|-
| 53
| 565\742<br>(5\742)
| 913.746<br>(8.086)
| 441/260<br>(196/195)
| [[Iodine]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
== Scales ==
* Silicon[28]: 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43
Retrieved from "https://en.xen.wiki/w/742edo"