253edo: Difference between revisions

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'''253edo''' is the [[EDO|equal division of the octave]] into 253 parts of 4.743083 [[cent]]s each. It 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 a good tuning for higher-limit [[sesquiquartififths]] temperament.
{{Infobox ET
| Prime factorization = 11 × 23
| Step size = 4.74308¢
| Fifth = 148\253 (701.98¢)
| Semitones = 24:19 (113.83¢ : 90.12¢)
| Consistency = 17
}}
{{EDO intro|253}}
 
== 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 a good tuning for higher-limit [[sesquiquartififths]] temperament.


253 = 11 × 23, and has subset edos [[11edo]] and [[23edo]].  
253 = 11 × 23, and has subset edos [[11edo]] and [[23edo]].  
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=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|253}}
{{Harmonics in equal|253}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 401 -253 }}
| [{{val| 253 401 }}]
| -0.007
| 0.007
| 0.14
|-
| 2.3.5
| 32805/32768, {{monzo| -4 -37 27 }}
| [{{val| 253 401 587 }}]
| +0.300
| 0.435
| 9.16
|-
| 2.3.5.7
| 2401/2400, 32805/32768, 390625/387072
| [{{val| 253 401 587 710 }}]
| +0.335
| 0.381
| 8.03
|-
| 2.3.5.7.11
| 385/384, 1375/1372, 4000/3993, 19712/19683
| [{{val| 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
| [{{val| 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
| [{{val| 253 401 587 710 875 936 1034 }}]
| +0.298
| 0.295
| 6.22
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
| 1
| 37\253
| 175.49
| 448/405
| [[Sesquiquartififths]]
|-
| 1
| 105\253
| 498.02
| 4/3
| [[Helmholtz]]
|-
| 1
| 123\253
| 583.40
| 7/5
| [[Cotritone]]
|}


== Scales ==
== Scales ==
* 63 32 63 63 32: [[3L_2s|Pentatonic]]
* 63 32 63 63 32: [[3L 2s|Pentic]]
* 43 43 19 43 43 43 19: [[5L_2s|Pythagorean tuning]]
* 43 43 19 43 43 43 19: [[Helmholtz]][7]
* 41 41 24 41 41 41 24: [[Meantone|Meantonic tuning]]
* 41 41 24 41 41 41 24: [[Meantone]][7]
* 35 35 35 35 35 35 35 8: [[7L_1s|Porcupine tuning]]
* 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"]]
* 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: [[7L_2s|Superdiatonic tuning]] in the way of Mavila
* 31 31 31 18 31 31 31 31 18: [[Mavila]][9]
* 26 26 15 26 26 26 15 26 26 26 15: [[sensi11|Sensi tuning]]
* 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 tuning]]
* 20 20 20 11 20 20 20 20 11 20 20 20 20 11: [[11L 3s|Ketradektriatoh scale]]


[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:Sesquiquartififths]]
[[Category:Sesquiquartififths]]

Revision as of 20:02, 2 April 2022

← 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

Template:EDO intro

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 a good tuning for higher-limit sesquiquartififths temperament.

253 = 11 × 23, and has subset edos 11edo and 23edo.

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)

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 octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 37\253 175.49 448/405 Sesquiquartififths
1 105\253 498.02 4/3 Helmholtz
1 123\253 583.40 7/5 Cotritone

Scales

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