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253edo: 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:guest|guest]] and made on <tt>2011-04-07 03:18:06 UTC</tt>.<br>
: The original revision id was <tt>217996960</tt>.<br>
: The revision comment was: <tt></tt><br>
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>
<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">=&lt;span style="color: #630080; font-size: 113%;"&gt;253 tone equal temperament&lt;/span&gt;=


253edo divides the octave into 253 steps of 4.743083 cents. It approximates the fifth by 148\253, which is 701.976285 cents, a mere 0.004487 cents sharp. The primes from 5 to 17 are all slightly flat. It tempers out 32805/32768 in the 5-limit; 2401/2400 in the 7-limit; 385/384, 1375/1372 and 4000/3993 in the 11-limit; 325/324, 1575/1573 and 2200/2197 in the 13-limit; 375/374 and 595/594 in the 17-limit. It provides a good tuning for higher-limit [[Schismatic family|sesquiquartififths]] temperament.
== Theory ==
253edo is [[consistent]] to the [[17-odd-limit]], approximating the fifth by 148\253 (0.021284 cents sharper than the just 3/2), and the [[prime harmonic]]s from 5 to 17 are all slightly flat. As an equal temperament, it [[tempering out|tempers out]] [[32805/32768]] in the [[5-limit]]; [[2401/2400]] in the [[7-limit]]; [[385/384]], [[1375/1372]] and [[4000/3993]] in the [[11-limit]]; [[325/324]], [[1575/1573]] and [[2200/2197]] in the [[13-limit]]; [[375/374]] and [[595/594]] in the [[17-limit]]. It provides the [[optimal patent val]] for the [[tertiaschis]] temperament, and a good tuning for the [[sesquiquartififths]] temperament in the higher limits.


**253 tone equal modes**
=== Prime harmonics ===
{{Harmonics in equal|253}}


43 43 19 43 43 43 19: MOS of 5L 2s ([[Superpythagorean]] Tuning)
=== Subsets and supersets ===
41 41 24 41 41 41 24: Meantonic Tuning [[MOS]]
Since 253 factors into 11 × 23, and has subset edos [[11edo]] and [[23edo]]. [[1012edo]] divides 253edo's step size into 4 equal parts and provides a good approximation of the 13-limit.
35 35 35 35 35 35 35 8: MOS of 7L1s (Perfect [[Porcupine-8]] Tuning (Octamonatonic Scale))
 
33 33 33 11 33 33 33 33 11: MOS of 7L 2s (Armodue-Hornbostel (Bright) Tuning)
== Regular temperament properties ==
31 31 31 18 31 31 31 31 18: Armodue-Mesotonic (Mellow) Tuning MOS
{| class="wikitable center-4 center-5 center-6"
26 26 15 26 26 26 15 26 26 26 15: Sensi-11 (or Undecimal Triatonic)</pre></div>
|-
<h4>Original HTML content:</h4>
! rowspan="2" | [[Subgroup]]
<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;253edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x253 tone equal temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="color: #630080; font-size: 113%;"&gt;253 tone equal temperament&lt;/span&gt;&lt;/h1&gt;
! rowspan="2" | [[Comma list]]
&lt;br /&gt;
! rowspan="2" | [[Mapping]]
253edo divides the octave into 253 steps of 4.743083 cents. It approximates the fifth by 148\253, which is 701.976285 cents, a mere 0.004487 cents sharp. The primes from 5 to 17 are all slightly flat. It tempers out 32805/32768 in the 5-limit; 2401/2400 in the 7-limit; 385/384, 1375/1372 and 4000/3993 in the 11-limit; 325/324, 1575/1573 and 2200/2197 in the 13-limit; 375/374 and 595/594 in the 17-limit. It provides a good tuning for higher-limit &lt;a class="wiki_link" href="/Schismatic%20family"&gt;sesquiquartififths&lt;/a&gt; temperament.&lt;br /&gt;
! rowspan="2" | Optimal<br />8ve stretch (¢)
&lt;br /&gt;
! colspan="2" | Tuning error
&lt;strong&gt;253 tone equal modes&lt;/strong&gt;&lt;br /&gt;
|-
&lt;br /&gt;
! [[TE error|Absolute]] (¢)
43 43 19 43 43 43 19: MOS of 5L 2s (&lt;a class="wiki_link" href="/Superpythagorean"&gt;Superpythagorean&lt;/a&gt; Tuning)&lt;br /&gt;
! [[TE simple badness|Relative]] (%)
41 41 24 41 41 41 24: Meantonic Tuning &lt;a class="wiki_link" href="/MOS"&gt;MOS&lt;/a&gt;&lt;br /&gt;
|-
35 35 35 35 35 35 35 8: MOS of 7L1s (Perfect &lt;a class="wiki_link" href="/Porcupine-8"&gt;Porcupine-8&lt;/a&gt; Tuning (Octamonatonic Scale))&lt;br /&gt;
| 2.3
33 33 33 11 33 33 33 33 11: MOS of 7L 2s (Armodue-Hornbostel (Bright) Tuning)&lt;br /&gt;
| {{monzo| 401 -253 }}
31 31 31 18 31 31 31 31 18: Armodue-Mesotonic (Mellow) Tuning MOS&lt;br /&gt;
| {{mapping| 253 401 }}
26 26 15 26 26 26 15 26 26 26 15: Sensi-11 (or Undecimal Triatonic)&lt;/body&gt;&lt;/html&gt;</pre></div>
| −0.007
| 0.007
| 0.14
|-
| 2.3.5
| 32805/32768, {{monzo| -4 -37 27 }}
| {{mapping| 253 401 587 }}
| +0.300
| 0.435
| 9.16
|-
| 2.3.5.7
| 2401/2400, 32805/32768, 390625/387072
| {{mapping| 253 401 587 710 }}
| +0.335
| 0.381
| 8.03
|-
| 2.3.5.7.11
| 385/384, 1375/1372, 4000/3993, 19712/19683
| {{mapping| 253 401 587 710 875 }}
| +0.333
| 0.341
| 7.19
|-
| 2.3.5.7.11.13
| 325/324, 385/384, 1375/1372, 1575/1573, 2200/2197
| {{mapping| 253 401 587 710 875 936 }}
| +0.323
| 0.312
| 6.58
|-
| 2.3.5.7.11.13.17
| 325/324, 375/374, 385/384, 595/594, 1275/1274, 2200/2197
| {{mapping| 253 401 587 710 875 936 1034 }}
| +0.298
| 0.295
| 6.22
|}
 
=== 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
| 35\253
| 166.01
| 11/10
| [[Tertiaschis]]
|-
| 1
| 37\253
| 175.49
| 448/405
| [[Sesquiquartififths]]
|-
| 1
| 105\253
| 498.02
| 4/3
| [[Helmholtz (temperament)|Helmholtz]]
|-
| 1
| 123\253
| 583.40
| 7/5
| [[Cotritone]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Scales ==
* 63 32 63 63 32: One of many [[3L 2s|pentic]] scales available
* 43 43 19 43 43 43 19: [[Helmholtz (temperament)|Helmholtz]][7]
* 41 41 24 41 41 41 24: [[Meantone]][7]
* 35 35 35 35 35 35 35 8: [[Porcupine]][8]
* 33 33 33 11 33 33 33 33 11: [[23edo|"The Hendecapliqued superdiatonic of the Icositriphony"]]
* 31 31 31 18 31 31 31 31 18: [[Mavila]][9]
* 26 26 15 26 26 26 15 26 26 26 15: [[Sensi]][11]
* 20 20 20 11 20 20 20 20 11 20 20 20 20 11: [[11L 3s|Ketradektriatoh scale]]
 
[[Category:3-limit record edos|###]] <!-- 3-digit number -->
[[Category:Tertiaschis]]
Retrieved from "https://en.xen.wiki/w/253edo"