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1600edo: Difference between revisions

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**Imported revision 556762361 - 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>2015-08-16 13:50:19 UTC</tt>.<br>
 
: The original revision id was <tt>556762361</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
1600edo is a very strong 37-limit system, being [[consistency|distinctly consistent]] in the [[37-odd-limit]] with a smaller [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than anything else with this property until [[4501edo|4501]]. It is also the first division past [[311edo|311]] with a lower 43-limit relative error.  
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>
In the 5-limit, it supports [[kwazy]]. In the 11-limit, it supports the rank-3 temperament [[thor]]. In higher limits, it tempers out [[12376/12375]] in the 17-limit and due to being consistent higher than 33-odd-limit it enables the essentially tempered [[flashmic chords]].
<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 **1600** equal division divides the octave into 1600 equal parts of exactly 0.75 cents each. It is a very strong 37-limit system, being distinctly consistent in the 37-limit with a smaller [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] than anything else with this property until [[4501edo|4501]]. It is also the first division past [[311edo|311]] with a lower 43-limit relative error. One step of it is the [[relative cent]] for [[16edo|16]].</pre></div>
 
<h4>Original HTML content:</h4>
=== Odd 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;1600edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;strong&gt;1600&lt;/strong&gt; equal division divides the octave into 1600 equal parts of exactly 0.75 cents each. It is a very strong 37-limit system, being distinctly consistent in the 37-limit with a smaller &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&gt;relative error&lt;/a&gt; than anything else with this property until &lt;a class="wiki_link" href="/4501edo"&gt;4501&lt;/a&gt;. It is also the first division past &lt;a class="wiki_link" href="/311edo"&gt;311&lt;/a&gt; with a lower 43-limit relative error. One step of it is the &lt;a class="wiki_link" href="/relative%20cent"&gt;relative cent&lt;/a&gt; for &lt;a class="wiki_link" href="/16edo"&gt;16&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Harmonics in equal|1600}}
 
=== Subsets and supersets ===
Since 1600 factors into {{factorization|1600}}, 1600edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, and 800 }}.
 
One step of it is the [[relative cent]] for [[16edo|16]]. Its high divisibility, high consistency limit, and compatibility with the decimal system make it a candidate for interval size measure. One step of 1600edo is already used as a measure called ''śata'' in the context of 16edo [[Armodue theory]].
 
== 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| -53 10 16 }}, {{monzo| 26 -75 40 }}
| {{Mapping| 1600 2536 3715 }}
| −0.0003
| 0.0228
| 3.04
|-
| 2.3.5.7
| 4375/4374, {{monzo| 36 -5 0 -10 }}, {{monzo| -17 5 16 -10 }}
| {{Mapping| 1600 2536 3715 4492 }}
| −0.0157
| 0.0332
| 4.43
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, {{monzo| 24 -1 -5 0 1 }}, {{monzo| 15 1 7 -8 -3 }}
| {{Mapping| 1600 2536 3715 4492 5535 }}
| −0.0172
| 0.0329
| 4.39
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 823875/823543
| {{Mapping| 1600 2536 3715 4492 5535 5921 }}
| −0.0087
| 0.0356
| 4.75
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 14875/14872, 63888/63869
| {{Mapping| 1600 2536 3715 4492 5535 5921 6540 }}
| −0.0163
| 0.0331
| 4.41
|}
 
=== 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
|-
| 2
| 217\1600
| 162.75
| 1125/1024
| [[Crazy]]
|-
| 32
| 23\1600
| 17.25
| ?
| [[Dam]] / [[dike]] / [[polder]]
|-
| 32
| 121\1600<br>(21/1600)
| 90.75<br>(15.75)
| 48828125/46294416<br>(?)
| [[Windrose]]
|-
| 32
| 357\1600<br>(7\1600)
| 267.75<br>(5.25)
| 245/143<br>(?)
| [[Germanium]]
|-
| 80
| 629\1600<br>(9\1600)
| 471.75<br>(6.75)
| 130/99<br>(?)
| [[Tetraicosic]]
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
Retrieved from "https://en.xen.wiki/w/1600edo"