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369edo: Difference between revisions

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**Imported revision 248804365 - Original comment: **
 
Countritonic -> ragitritonic
 
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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>2011-08-27 14:21:19 UTC</tt>.<br>
 
: The original revision id was <tt>248804365</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
369edo shares its [[3/2|perfect fifth]] with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp.  
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 [[escapade comma]] and the [[ennealimma]] in the 5-limit; [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament; in the 11-limit, [[4000/3993]], [[5632/5625]] and [[16384/16335]], so that it supports [[escapade]] in the [[2.3.5.11 subgroup]] and in fact provides the [[optimal patent val]]. It also provides the optimal patent val for the 11-limit {{nowrap| 152 & 217 }} temperament (an escapade extension), the {{nowrap| 130 & 239 }} temperament (a weak escapade extension), and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as semiporwellic, the no-7 subgroup version thereof.
<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 369 equal division divides the octave into 369 equal parts of 3.252 cents each. It tempers out 2401/2400 and 4375/4374 in the 7-limit, so that it supports ennealimmal temperament; in the 11-limit, 4000/3993 and 5632/5625. 369 provides the [[optimal patent val]] for both the 11-limit 21&amp;109 temperament and the 65&amp;152 temperament. 369 factors as 3^2 * 41.</pre></div>
 
<h4>Original HTML content:</h4>
Extension to the 13-limit is viable by the 369f val, tempering out [[1575/1573]], [[2080/2079]], [[2200/2197]], and 3584/3575. The [[Tenney–Euclidean tuning|TE-optimal tuning]] of this temperament is [[consistent]] in the 15-integer-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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;369edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 369 equal division divides the octave into 369 equal parts of 3.252 cents each. It tempers out 2401/2400 and 4375/4374 in the 7-limit, so that it supports ennealimmal temperament; in the 11-limit, 4000/3993 and 5632/5625. 369 provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for both the 11-limit 21&amp;amp;109 temperament and the 65&amp;amp;152 temperament. 369 factors as 3^2 * 41.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== Prime harmonics ===
{{Harmonics in equal|369|columns=11}}
 
=== Subsets and supersets ===
Since 369 factors into primes as {{nowrap| 3<sup>2</sup> × 41 }}, 369edo has subset edos {{EDOs| 3, 9, 41, and 123 }}.
 
== 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| 32 -7 -9 }}, {{monzo| 1 -27 18 }}
| {{Mapping| 369 585 857 }}
| −0.1991
| 0.1409
| 4.33
|-
| 2.3.5.7
| 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }}
| {{Mapping| 369 585 857 1036 }}
| −0.1743
| 0.1294
| 3.98
|-
| 2.3.5.7.11
| 2401/2400, 4000/3993, 4375/4374, 5632/5625
| {{Mapping| 369 585 857 1036 1277 }}
| −0.2277
| 0.1576
| 4.85
|-
| 2.3.5.7.11.13
| 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575
| {{Mapping| 369 585 857 1036 1277 1366 }} (369f)
| −0.2685
| 0.1703
| 5.24
|}
 
=== 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
| 17\369
| 55.28
| 33/32
| [[Escapade]]
|-
| 1
| 172\369
| 559.35
| 864/625
| [[Tritriple]] (5-limit)
|-
| 1
| 181\369
| 588.62
| 128/91
| [[Ragitritonic]]
|-
| 9
| 77\369<br>(5\369)
| 250.41<br>(16.26)
| 140/121<br>(100/99)
| [[Ennealimmapine]]
|-
| 9
| 97\369<br>(15\369)
| 315.45<br>(48.78)
| 6/5<br>(36/35)
| [[Ennealimmal]] / enneabiotic
|-
| 9
| 68\369<br>(14\369)
| 221.14<br>(45.53)
| 25/22<br>(77/75)
| [[Quadraennealimmal]]
|-
| 41
| 55\369<br>(1\369)
| 178.86<br>(3.25)
| 567/512<br>(352/351)
| [[Hemicountercomp]]
|}
<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
 
[[Category:Semiporwellismic]]
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