935edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
ArrowHead294 (talk | contribs)
15,213 edits
mNo edit summary
 
(3 intermediate revisions by 2 users not shown)
Line 3: Line 3:


== Theory ==
== Theory ==
935edo is a very strong 23-limit system, and is [[consistency|distinctly consistent]] through to the [[27-odd-limit]]. It is also a [[zeta peak edo]]. It [[tempering out|tempers out]] the {{monzo| 39 -29 3 }} ([[tricot comma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), and {{monzo| 91 -12 -31 }} (astro) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; 161280/161051 and 117649/117612 in the 11-limit; and [[2080/2079]], [[4096/4095]] and [[4225/4224]] in the 13-limit.  
935edo is a very strong [[23-limit]] system, and is [[consistent]] through to the [[27-odd-limit]]. It does reasonably well in the higher limits, though the sharply tuned [[11/1|11]] and [[23/1|23]] and the flatly tuned [[29/1|29]] and [[31/1|31]] create inconsistencies together, those being [[29/22]], [[29/23]], [[31/22]], [[31/23]] and their [[octave complement]]s; it is otherwise consistent to the [[39-odd-limit]]. It is a [[zeta peak edo]].  
 
As an equal temperament, it [[tempering out|tempers out]] the {{monzo| 39 -29 3 }} ([[alphatricot comma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), and {{monzo| 91 -12 -31 }} (astro comma) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; [[117649/117612]], [[151263/151250]], [[161280/161051]] in the [[11-limit]]; [[2080/2079]], [[4096/4095]], [[4225/4224]] in the [[13-limit]]; [[2058/2057]], [[2500/2499]], [[4914/4913]] in the [[17-limit]]; [[2432/2431]], [[3136/3135]], [[3250/3249]], [[4200/4199]] in the 19-limit; and [[2025/2024]], [[2300/2299]], [[2646/2645]] among others in the 23-limit.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|935}}
{{Harmonics in equal|935|columns=11}}
{{Harmonics in equal|935|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 935edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 935 factors into {{factorization|935}}, 935edo has subset edos {{EDOs| 5, 11, 17, 55, 85, and 187 }}.
Since 935 factors into primes as {{nowrap| 5 × 11 × 17 }}, 935edo has subset edos {{EDOs| 5, 11, 17, 55, 85, and 187 }}.


== Regular temperament properties ==
== Regular temperament properties ==
935et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19- and 23-limit. It is the first to beat [[764edo|764]] in the 13-limit, [[814edo|814]] in the 17- and 23-limit, and [[742edo|742]] in the 19-limit, only to be bettered by [[954edo|954h]] in all of those subgroups.
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 1482 -935 }}
| {{Mapping| 935 1482 }}
| −0.0243
| 0.0243
| 1.89
|-
| 2.3.5
| {{Monzo| 39 -29 3 }}, {{monzo| -52 -17 34 }}
| {{Mapping| 935 1482 2171 }}
| −0.0157
| 0.0233
| 1.82
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 36 -5 0 -10 }}
| {{Mapping| 935 1482 2171 2625 }}
| −0.0259
| 0.0268
| 2.08
|-
| 2.3.5.7.11
| 4375/4374, 117649/117612, 131072/130977, 161280/161051
| {{Mapping| 935 1482 2171 2625 3235 }}
| −0.0527
| 0.0588
| 4.58
|-
| 2.3.5.7.11.13
| 2080/2079, 4096/4095, 4375/4374, 78125/78078, 117649/117612
| {{Mapping| 935 1482 2171 2625 3235 3460 }}
| −0.0490
| 0.0543
| 4.23
|-
| 2.3.5.7.11.13.17
| 2058/2057, 2080/2079, 2500/2499, 4096/4095, 4375/4374, 4914/4913
| {{Mapping| 935 1482 2171 2625 3235 3460 3822 }}
| −0.0520
| 0.0508
| 3.96
|-
| 2.3.5.7.11.13.17.19
| 2058/2057, 2080/2079, 2432/2431, 2500/2499, 3136/3135, 4375/4374, 4914/4913
| {{Mapping| 935 1482 2171 2625 3235 3460 3822 3972 }}
| −0.0526
| 0.0475
| 3.70
|-
| 2.3.5.7.11.13.17.19.23
| 2025/2024, 2058/2057, 2080/2079, 2300/2299, 2432/2431, 2500/2499, 2646/2645, 4375/4374
| {{Mapping| 935 1482 2171 2625 3235 3460 3822 3972 4230 }}
| −0.0616
| 0.0515
| 4.01
|}
* 935et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19- and 23-limit. It is the first to beat [[764edo|764]] in the 13-limit, [[814edo|814]] in the 17- and 23-limit, and [[742edo|742]] in the 19-limit, only to be bettered by [[954edo|954h]] in all of those subgroups.
 
=== 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
| 103\935
| 132.19
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
| 1
| 339\935
| 435.08
| 9/7
| [[Supermajor (temperament)|Supermajor]]
|-
| 1
| 442\935
| 567.27
| 104/75
| [[Alphatrillium]]
|-
| 17
| 194\935<br>(26\935)
| 248.98<br>(33.37)
| {{Monzo| -23 5 9 -2 }}<br>(100352/98415)
| [[Chlorine]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms|minimal form]] in parentheses if distinct

Latest revision as of 12:07, 4 June 2026

← 934edo 935edo 936edo →
Prime factorization 5 × 11 × 17
Step size 1.28342 ¢ 
Fifth 547\935 (702.032 ¢)
Semitones (A1:m2) 89:70 (114.2 ¢ : 89.84 ¢)
Consistency limit 27
Distinct consistency limit 27

935 equal divisions of the octave (abbreviated 935edo or 935ed2), also called 935-tone equal temperament (935tet) or 935 equal temperament (935et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 935 equal parts of about 1.28 ¢ each. Each step represents a frequency ratio of 21/935, or the 935th root of 2.

Theory

935edo is a very strong 23-limit system, and is consistent through to the 27-odd-limit. It does reasonably well in the higher limits, though the sharply tuned 11 and 23 and the flatly tuned 29 and 31 create inconsistencies together, those being 29/22, 29/23, 31/22, 31/23 and their octave complements; it is otherwise consistent to the 39-odd-limit. It is a zeta peak edo.

As an equal temperament, it tempers out the [39 -29 3⟩ (alphatricot comma), [-52 -17 34⟩ (septendecima), and [91 -12 -31⟩ (astro comma) in the 5-limit; 4375/4374 and 52734375/52706752 in the 7-limit; 117649/117612, 151263/151250, 161280/161051 in the 11-limit; 2080/2079, 4096/4095, 4225/4224 in the 13-limit; 2058/2057, 2500/2499, 4914/4913 in the 17-limit; 2432/2431, 3136/3135, 3250/3249, 4200/4199 in the 19-limit; and 2025/2024, 2300/2299, 2646/2645 among others in the 23-limit.

Prime harmonics

Approximation of prime harmonics in 935edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.077 -0.004 +0.158 +0.554 +0.114 +0.285 +0.241 +0.603 -0.272 -0.223
Relative (%) +0.0 +6.0 -0.3 +12.3 +43.1 +8.9 +22.2 +18.8 +47.0 -21.2 -17.4
Steps
(reduced) 		935
(0) 		1482
(547) 		2171
(301) 		2625
(755) 		3235
(430) 		3460
(655) 		3822
(82) 		3972
(232) 		4230
(490) 		4542
(802) 		4632
(892)
Approximation of prime harmonics in 935edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) +0.207 -0.399 +0.568 +0.590 +0.506 -0.348 -0.307 +0.265 -0.017 -0.624 -0.045
Relative (%) +16.1 -31.1 +44.2 +45.9 +39.4 -27.1 -23.9 +20.7 -1.4 -48.6 -3.5
Steps
(reduced) 		4871
(196) 		5009
(334) 		5074
(399) 		5194
(519) 		5356
(681) 		5500
(825) 		5545
(870) 		5672
(62) 		5750
(140) 		5787
(177) 		5894
(284)

Subsets and supersets

Since 935 factors into primes as 5 × 11 × 17, 935edo has subset edos 5, 11, 17, 55, 85, and 187.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [1482 -935⟩ [⟨935 1482]] −0.0243 0.0243 1.89
2.3.5 [39 -29 3⟩, [-52 -17 34⟩ [⟨935 1482 2171]] −0.0157 0.0233 1.82
2.3.5.7 4375/4374, 52734375/52706752, [36 -5 0 -10⟩ [⟨935 1482 2171 2625]] −0.0259 0.0268 2.08
2.3.5.7.11 4375/4374, 117649/117612, 131072/130977, 161280/161051 [⟨935 1482 2171 2625 3235]] −0.0527 0.0588 4.58
2.3.5.7.11.13 2080/2079, 4096/4095, 4375/4374, 78125/78078, 117649/117612 [⟨935 1482 2171 2625 3235 3460]] −0.0490 0.0543 4.23
2.3.5.7.11.13.17 2058/2057, 2080/2079, 2500/2499, 4096/4095, 4375/4374, 4914/4913 [⟨935 1482 2171 2625 3235 3460 3822]] −0.0520 0.0508 3.96
2.3.5.7.11.13.17.19 2058/2057, 2080/2079, 2432/2431, 2500/2499, 3136/3135, 4375/4374, 4914/4913 [⟨935 1482 2171 2625 3235 3460 3822 3972]] −0.0526 0.0475 3.70
2.3.5.7.11.13.17.19.23 2025/2024, 2058/2057, 2080/2079, 2300/2299, 2432/2431, 2500/2499, 2646/2645, 4375/4374 [⟨935 1482 2171 2625 3235 3460 3822 3972 4230]] −0.0616 0.0515 4.01
  • 935et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19- and 23-limit. It is the first to beat 764 in the 13-limit, 814 in the 17- and 23-limit, and 742 in the 19-limit, only to be bettered by 954h in all of those subgroups.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 103\935 132.19 [-38 5 13⟩ Astro
1 339\935 435.08 9/7 Supermajor
1 442\935 567.27 104/75 Alphatrillium
17 194\935
(26\935) 		248.98
(33.37) 		[-23 5 9 -2⟩
(100352/98415) 		Chlorine

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

Retrieved from "https://en.xen.wiki/index.php?title=935edo&oldid=231547"