253edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|253}}
{{ED intro}}


== Theory ==
== 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. 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.
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 = 11 × 23, and has subset edos [[11edo]] and [[23edo]].
=== Prime harmonics ===
{{Harmonics in equal|253}}


=== Prime harmonics ===
=== Subsets and supersets ===
{{Harmonics in equal|253|columns=11}}
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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>Stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 2.3
| 2.3
| {{monzo| 401 -253 }}
| {{monzo| 401 -253 }}
| [{{val| 253 401 }}]
| {{mapping| 253 401 }}
| -0.007
| −0.007
| 0.007
| 0.007
| 0.14
| 0.14
Line 30: Line 32:
| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| -4 -37 27 }}
| 32805/32768, {{monzo| -4 -37 27 }}
| [{{val| 253 401 587 }}]
| {{mapping| 253 401 587 }}
| +0.300
| +0.300
| 0.435
| 0.435
Line 37: Line 39:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 32805/32768, 390625/387072
| 2401/2400, 32805/32768, 390625/387072
| [{{val| 253 401 587 710 }}]
| {{mapping| 253 401 587 710 }}
| +0.335
| +0.335
| 0.381
| 0.381
Line 44: Line 46:
| 2.3.5.7.11
| 2.3.5.7.11
| 385/384, 1375/1372, 4000/3993, 19712/19683
| 385/384, 1375/1372, 4000/3993, 19712/19683
| [{{val| 253 401 587 710 875 }}]
| {{mapping| 253 401 587 710 875 }}
| +0.333
| +0.333
| 0.341
| 0.341
Line 51: Line 53:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 325/324, 385/384, 1375/1372, 1575/1573, 2200/2197
| 325/324, 385/384, 1375/1372, 1575/1573, 2200/2197
| [{{val| 253 401 587 710 875 936 }}]
| {{mapping| 253 401 587 710 875 936 }}
| +0.323
| +0.323
| 0.312
| 0.312
Line 58: Line 60:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 325/324, 375/374, 385/384, 595/594, 1275/1274, 2200/2197
| 325/324, 375/374, 385/384, 595/594, 1275/1274, 2200/2197
| [{{val| 253 401 587 710 875 936 1034 }}]
| {{mapping| 253 401 587 710 875 936 1034 }}
| +0.298
| +0.298
| 0.295
| 0.295
Line 66: Line 68:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per Octave
|-
! Generator<br>(Reduced)
! Periods<br />per 8ve
! Cents<br>(Reduced)
! Generator*
! Associated<br>Ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
Line 89: Line 92:
| 498.02
| 498.02
| 4/3
| 4/3
| [[Helmholtz]]
| [[Helmholtz (temperament)|Helmholtz]]
|-
|-
| 1
| 1
Line 97: Line 100:
| [[Cotritone]]
| [[Cotritone]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
* 63 32 63 63 32: [[3L 2s|Pentic]]
* 63 32 63 63 32: One of many [[3L 2s|pentic]] scales available
* 43 43 19 43 43 43 19: [[Helmholtz]][7]
* 43 43 19 43 43 43 19: [[Helmholtz (temperament)|Helmholtz]][7]
* 41 41 24 41 41 41 24: [[Meantone]][7]
* 41 41 24 41 41 41 24: [[Meantone]][7]
* 35 35 35 35 35 35 35 8: [[Porcupine]][8]
* 35 35 35 35 35 35 35 8: [[Porcupine]][8]
Line 108: Line 112:
* 20 20 20 11 20 20 20 20 11 20 20 20 20 11: [[11L 3s|Ketradektriatoh scale]]
* 20 20 20 11 20 20 20 20 11 20 20 20 20 11: [[11L 3s|Ketradektriatoh scale]]


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:3-limit record edos|###]] <!-- 3-digit number -->
[[Category:Tertiaschis]]
[[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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