263edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|263}}
{{EDO intro|263}}
==Theory==
263et tempers out 393216/390625 (Würschmidt comma) and |50 -33 1> in the 5-limit. Using the patent val, it tempers out 4375/4374, 50421/50000, and 458752/455625 in the 7-limit; 441/440, 3388/3375, 16384/16335, and 26411/26244 in the 11-limit; 364/363, 2080/2079, 2197/2187, and 3584/3575 in the 13-limit; 595/594, 833/832, 936/935, and 1156/1155 in the 17-limit.


Using the 263d val, it tempers out 5120/5103, 16875/16807, and 1959552/1953125 in the 7-limit; 540/539, 1375/1372, 16384/16335, and 43923/43750 in the 11-limit; 351/350, 1001/1000, 1573/1568, 2197/2187, and 4225/4224 in the 13-limit.  
== Theory ==
263et [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and {{monzo| 50 -33 1 }} in the 5-limit. Using the [[patent val]], it tempers out [[4375/4374]], [[50421/50000]], and 458752/455625 in the 7-limit; [[441/440]], [[3388/3375]], [[16384/16335]], and 26411/26244 in the 11-limit; [[364/363]], [[2080/2079]], [[2197/2187]], and 3584/3575 in the 13-limit; [[595/594]], [[833/832]], [[936/935]], and [[1156/1155]] in the 17-limit.  


Using the 263df val, it tempers out 352/351, 640/637, 729/728, and 3584/3575 in the 13-limit.
Using the 263d val, it tempers out [[5120/5103]], [[16875/16807]], and 1959552/1953125 in the 7-limit; [[540/539]], 1375/1372, 16384/16335, and 43923/43750 in the 11-limit; [[351/350]], [[1001/1000]], [[1573/1568]], 2197/2187, and [[4225/4224]] in the 13-limit.
===Prime harmonics===
 
Using the 263df val, it tempers out [[352/351]], [[640/637]], [[729/728]], and 3584/3575 in the 13-limit.
 
Finally, it is accurate for the 17th harmonic, as the denominator of a convergent to log<sub>2</sub>17, after [[80edo|80]] and before [[343edo|343]].
 
=== Prime harmonics ===
{{Harmonics in equal|263}}
{{Harmonics in equal|263}}
===Subsets and supersets===
 
263edo is the 56th [[prime EDO]]. It is accurate for the 17th harmonic, as the denominator of a convergent to log<sub>2</sub>17, after [[80edo|80]] and before [[343edo|343]].
=== Subsets and supersets ===
==Regular temperament properties==
263edo is the 56th [[prime edo]].  
 
== 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|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|417 -263}}
| {{monzo| 417 -263 }}
|{{val|263 417}}
| {{val| 263 417 }}
| -0.2229
| -0.2229
| 0.2229
| 0.2229
| 4.89
| 4.89
|-
|-
|2.3.5
| 2.3.5
|393216/390625, {{monzo|50 -33 1}}
| 393216/390625, {{monzo| 50 -33 1 }}
|{{val|263 417 611}}
| {{val| 263 417 611 }}
| -0.3666
| -0.3666
| 0.2728
| 0.2728
| 5.98
| 5.98
|}
|}
=== 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
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|40\263
| 40\263
|182.51
| 182.51
|10/9
| 10/9
|[[Minortone]]
| [[Minortone]]
|-
|-
|1
| 1
|85\263
| 85\263
|387.83
| 387.83
|5/4
| 5/4
|[[Würschmidt]]
| [[Würschmidt]]
|}
|}
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
[[Category:Prime EDO]]

Revision as of 04:58, 11 March 2024

← 262edo 263edo 264edo →
Prime factorization 263 (prime)
Step size 4.56274 ¢ 
Fifth 154\263 (702.662 ¢)
Semitones (A1:m2) 26:19 (118.6 ¢ : 86.69 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

Theory

263et tempers out 393216/390625 (würschmidt comma) and [50 -33 1⟩ in the 5-limit. Using the patent val, it tempers out 4375/4374, 50421/50000, and 458752/455625 in the 7-limit; 441/440, 3388/3375, 16384/16335, and 26411/26244 in the 11-limit; 364/363, 2080/2079, 2197/2187, and 3584/3575 in the 13-limit; 595/594, 833/832, 936/935, and 1156/1155 in the 17-limit.

Using the 263d val, it tempers out 5120/5103, 16875/16807, and 1959552/1953125 in the 7-limit; 540/539, 1375/1372, 16384/16335, and 43923/43750 in the 11-limit; 351/350, 1001/1000, 1573/1568, 2197/2187, and 4225/4224 in the 13-limit.

Using the 263df val, it tempers out 352/351, 640/637, 729/728, and 3584/3575 in the 13-limit.

Finally, it is accurate for the 17th harmonic, as the denominator of a convergent to log217, after 80 and before 343.

Prime harmonics

Approximation of prime harmonics in 263edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.71 +1.52 -1.53 +0.77 -0.98 -0.01 -0.94 +1.38 +1.60 +0.21
Relative (%) +0.0 +15.5 +33.3 -33.4 +16.9 -21.6 -0.3 -20.5 +30.3 +35.1 +4.6
Steps
(reduced) 		263
(0) 		417
(154) 		611
(85) 		738
(212) 		910
(121) 		973
(184) 		1075
(23) 		1117
(65) 		1190
(138) 		1278
(226) 		1303
(251)

Subsets and supersets

263edo is the 56th prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [417 -263⟩ ⟨263 417] -0.2229 0.2229 4.89
2.3.5 393216/390625, [50 -33 1⟩ ⟨263 417 611] -0.3666 0.2728 5.98

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 40\263 182.51 10/9 Minortone
1 85\263 387.83 5/4 Würschmidt

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

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