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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:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-08 14:32:17 UTC</tt>.<br>
 
: The original revision id was <tt>601731906</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
665edo is best known for its unfathomably accurate [[3/2|fifth]], only 0.00011 cents compressed. 665edo is the denominator of a convergent to log<sub>2</sub>3, after [[41edo]], [[53edo]], and [[306edo]], and before [[15601edo]], and is the member of this series with the highest 3-2 [[Telicity #k-Strong Telicity|telicity ''k''-strength]] before being finally surpassed in this regard by [[190537edo]].  
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>
However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out|tempers out]] the [[satanic comma]], {{monzo| -1054 665 }} in the 3-limit; the [[enneadeca]], {{monzo| -14 -19 19 }}, and the [[monzisma]], {{monzo| 54 -37 2 }} in the 5-limit; the [[ragisma]], 4375/4374, the [[meter]], 703125/702464, and {{monzo| 36 -5 0 -10 }} in the 7-limit; [[4000/3993]], 46656/46585, [[131072/130977]] and 151263/151250 in the 11-limit, providing the optimal patent val for the 11-limit [[brahmagupta]] temperament. In the 13-limit, it tempers out [[1575/1573]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]; since it tempers out 1575/1573, the nicola, it [[support]]s nicolic tempering and hence the [[nicolic chords]], for which it provides an excellent tuning. In the 17-limit it tempers out [[1156/1155]], [[1275/1274]], [[2058/2057]], [[2500/2499]] and [[5832/5831]]; in the 19-limit it tempers out 969/968, [[1445/1444]], 2432/2431, 3136/3135, 3250/3249, and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184, and 2737/2736.
<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 //665 equal temperament// divides the octave into 665 equal parts of 1.80451 cents each. It is best known for its extremely accurate fifth, only 0.0001 cents flat. 665edo is the denominator of a convergent to log2(3), after [[41edo]], [[53edo]] and [[306edo]], and before [[15601edo]]. However, it also provides the [[optimal patent val]] for the rank four temperament tempering out 4000/3993. In the 3-limit it tempers out the 'satanic' comma, |-1054 665&gt;; in the 5-limit the enneadeca, |-14 -19 19&gt; and the monzisma, |54 -37 2&gt;; in the 7-limit, 4375/4374 and the meter, | -11 2 7 -3 &gt;; in the 11-limit, 4000/3993 and | 17 -5 0 -2 -1 &gt; and | -1 2 -4 5 -2 &gt;, providing the optimal patent val for 11-limit [[Ragismic microtemperaments#Brahmagupta|brahmagupta temperament]]. In the 13-limit it tempers out 1575/1573, 2080/2079, 4096/4095 and 4225/4224; since it tempers out 1575/1573, the nicola, it supports nicolic tempering and hence the [[nicolic tetrad]], for which it provides an excellent tuning.</pre></div>
 
<h4>Original HTML content:</h4>
665edo provides relatively great approximations for the 7-limit intervals and harmonics 13, 17, 19, and 23, with minuscule absolute error. It is considered as an excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is consistent in the [[27-odd-limit]], and in the full 27-odd-limit, the only inconsistencies are [[11/10]], [[25/22]], [[15/11]], [[17/11]], [[23/22]], and their [[octave complement]]s. 665edo provides relatively poor approximations for intervals of 11, with two mappings possible for the [[11/8]] fourth: a sharp one from the [[patent val]], and a flat one from the 665e val. Using the 665e val, [[41503/41472]], 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-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;665edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;665 equal temperament&lt;/em&gt; divides the octave into 665 equal parts of 1.80451 cents each. It is best known for its extremely accurate fifth, only 0.0001 cents flat. 665edo is the denominator of a convergent to log2(3), after &lt;a class="wiki_link" href="/41edo"&gt;41edo&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt; and &lt;a class="wiki_link" href="/306edo"&gt;306edo&lt;/a&gt;, and before &lt;a class="wiki_link" href="/15601edo"&gt;15601edo&lt;/a&gt;. However, it also provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for the rank four temperament tempering out 4000/3993. In the 3-limit it tempers out the 'satanic' comma, |-1054 665&amp;gt;; in the 5-limit the enneadeca, |-14 -19 19&amp;gt; and the monzisma, |54 -37 2&amp;gt;; in the 7-limit, 4375/4374 and the meter, | -11 2 7 -3 &amp;gt;; in the 11-limit, 4000/3993 and | 17 -5 0 -2 -1 &amp;gt; and | -1 2 -4 5 -2 &amp;gt;, providing the optimal patent val for 11-limit &lt;a class="wiki_link" href="/Ragismic%20microtemperaments#Brahmagupta"&gt;brahmagupta temperament&lt;/a&gt;. In the 13-limit it tempers out 1575/1573, 2080/2079, 4096/4095 and 4225/4224; since it tempers out 1575/1573, the nicola, it supports nicolic tempering and hence the &lt;a class="wiki_link" href="/nicolic%20tetrad"&gt;nicolic tetrad&lt;/a&gt;, for which it provides an excellent tuning.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== Prime harmonics ===
{{Harmonics in equal|665}}
 
=== Subsets and supersets ===
Since 665 factors into {{factorisation|665}}, 665edo has subset edos {{EDOs| 5, 7, 19, 35, 95, and 133 }}. One step of 665edo has been proposed as an [[interval size measure]], called a '''Delfi unit'''. A Delfi unit is exactly 48 imps ([[31920edo|48\31920]]).  
 
[[1330edo]], which doubles 665edo, provides a good correction of the harmonic 11.  
 
[[7315edo]], which undecuples 665edo, is the last 3-2 telic multiple, and fully consistent to the [[27-odd-limit]] and almost the [[31-odd-limit]].
 
=== Miscellany ===
A [[maximal evenness]] scale deriving from the {{nowrap|118 &amp; 665}} temperament, known as [[vavoom]], can also theoretically serve as a calendar leap week cycle corresponding to a year length of {{nowrap| 365d 5h 48m 37{{frac|17|19}}s}}, about 7 seconds shorter than the average length of the tropical year today. Given the excellence of both 118 and 665 in 5-limit, this is a great point of intersection of solar calendar leap rules and just intonation-based temperaments.
 
== Intervals ==
See [[Table of 665edo intervals]].
 
== 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| -1054 665 }}
| {{Mapping| 665 1054 }}
| +0.0000
| 0.0000
| 0.00
|-
| 2.3.5
| {{Monzo| -14 -19 19 }}, {{monzo| 54 -37 2 }}
| {{Mapping| 665 1054 1544 }}
| +0.0213
| 0.0301
| 1.67
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| 36 -5 0 -10 }}
| {{Mapping| 665 1054 1544 1867 }}
| −0.0015
| 0.0474
| 2.63
|-
| 2.3.5.7.11
| 4000/3993, 4375/4374, 117649/117612, 131072/130977
| {{Mapping| 665 1054 1544 1867 2301 }}
| −0.0511
| 0.1078
| 5.97
|-
| 2.3.5.7.11.13
| 1575/1573, 2080/2079, 4096/4095, 4375/4374, 31250/31213
| {{Mapping| 665 1054 1544 1867 2301 2461 }}
| −0.0594
| 0.1002
| 5.55
|}
 
=== 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
| 62\665
| 111.88
| 16/15
| [[Vavoom]]
|-
| 1
| 138\665
| 249.02
| {{Monzo| -26 18 -1 }}
| [[Monzismic]]
|-
| 7
| 288\665<br>(3\665)
| 519.70<br>(5.41)
| 27/20<br>(325/324)
| [[Brahmagupta]]
|-
| 19
| 276\665<br>(4\665)
| 498.05<br>(7.21)
| 4/3<br>(225//224)
| [[Enneadecal]]
|}
<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:3-limit record edos|###]] <!-- 3-digit number -->
[[Category:Satanic]]
[[Category:Wizardharry]]
[[Category:Monzismic]]
Retrieved from "https://en.xen.wiki/w/665edo"