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:genewardsmith|genewardsmith]] and made on <tt>2011-04-07 01:06:53 UTC</tt>.<br>
: The original revision id was <tt>217980090</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 (Superpythagorean Tuning)&lt;br /&gt;
! [[TE simple badness|Relative]] (%)
41 41 24 41 41 41 24: Meantonic Tuning MOS&lt;br /&gt;
|-
35 35 35 35 35 35 35 8: MOS of 7L1s (Perfect Porcupine-8 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]]

Latest revision as of 12:47, 12 July 2025

← 252edo 253edo 254edo →
Prime factorization 11 × 23
Step size 4.74308 ¢ 
Fifth 148\253 (701.976 ¢)
(semiconvergent)
Semitones (A1:m2) 24:19 (113.8 ¢ : 90.12 ¢)
Consistency limit 17
Distinct consistency limit 17

253 equal divisions of the octave (abbreviated 253edo or 253ed2), also called 253-tone equal temperament (253tet) or 253 equal temperament (253et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 253 equal parts of about 4.74 ¢ each. Each step represents a frequency ratio of 21/253, or the 253rd root of 2.

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 harmonics from 5 to 17 are all slightly flat. As an equal temperament, 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 the optimal patent val for the tertiaschis temperament, and a good tuning for the sesquiquartififths temperament in the higher limits.

Prime harmonics

Approximation of prime harmonics in 253edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.02 -2.12 -1.24 -1.12 -1.00 -0.61 +1.30 -2.19 -0.33 -1.95
Relative (%) +0.0 +0.4 -44.8 -26.1 -23.6 -21.1 -12.8 +27.4 -46.1 -6.9 -41.2
Steps
(reduced) 		253
(0) 		401
(148) 		587
(81) 		710
(204) 		875
(116) 		936
(177) 		1034
(22) 		1075
(63) 		1144
(132) 		1229
(217) 		1253
(241)

Subsets and supersets

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.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [401 -253⟩ [⟨253 401]] −0.007 0.007 0.14
2.3.5 32805/32768, [-4 -37 27⟩ [⟨253 401 587]] +0.300 0.435 9.16
2.3.5.7 2401/2400, 32805/32768, 390625/387072 [⟨253 401 587 710]] +0.335 0.381 8.03
2.3.5.7.11 385/384, 1375/1372, 4000/3993, 19712/19683 [⟨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 [⟨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 [⟨253 401 587 710 875 936 1034]] +0.298 0.295 6.22

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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
1 123\253 583.40 7/5 Cotritone

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

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