292edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Fredg999 category edits (talk | contribs)
autopatrolled, Bots, emailconfirmed
4,405 edits
m Removing from Category:Theory using Cat-a-lot
Expansion (+infobox; +prime error table; +RTT table and rank-2 temperaments)
Line 1: Line 1:
The '''292 equal divisions of the octave''' ('''292edo''') is the [[EDO|equal division of the octave]] into 292 parts of 4.1095 [[cent]]s each.
{{Infobox ET
| Prime factorization = 2<sup>2</sup> × 73
| Step size = 4.10959¢
| Fifth = 171\292 (702.74¢)
| Semitones = 29:21 (119.18¢ : 86.30¢)
| Consistency = 9
}}
{{EDO intro|292}}


== Theory ==
292edo is closely related to [[146edo]], but the patent vals differ on the mapping for 3. It tempers out {{monzo| 3 -18 11 }} (quartonic comma) and {{monzo| 38 -2 -15 }} (luna/hemithirds comma) in the [[5-limit]]; 5120/5103 ([[5120/5103|hemifamity]]), 390625/388962 ([[dimcomp comma|dimcomp]]), 420175/419904 (wizma), and 4802000/4782969 ([[canousma]]) in the [[7-limit]]; 1375/1372, 5632/5625, [[6250/6237]], [[9801/9800]] and [[14641/14580]] in the [[11-limit]]; [[352/351]], [[625/624]], [[847/845]], [[1716/1715]], and [[2080/2079]] in the [[13-limit]].  
292edo is closely related to [[146edo]], but the patent vals differ on the mapping for 3. It tempers out {{monzo| 3 -18 11 }} (quartonic comma) and {{monzo| 38 -2 -15 }} (luna/hemithirds comma) in the [[5-limit]]; 5120/5103 ([[5120/5103|hemifamity]]), 390625/388962 ([[dimcomp comma|dimcomp]]), 420175/419904 (wizma), and 4802000/4782969 ([[canousma]]) in the [[7-limit]]; 1375/1372, 5632/5625, [[6250/6237]], [[9801/9800]] and [[14641/14580]] in the [[11-limit]]; [[352/351]], [[625/624]], [[847/845]], [[1716/1715]], and [[2080/2079]] in the [[13-limit]].  


It provides the [[optimal patent val]] for the [[undim]] temperament in the 7-, 11-, and 13-limit, and notably [[support]]s [[Hemifamity temperaments #Semiseptiquarter|semiseptiquarter]] and [[Luna family #Semiluna|semiluna]].  
It provides the [[optimal patent val]] for the [[undim]] temperament in the 7-, 11-, and 13-limit, and notably [[support]]s [[Hemifamity temperaments #Semiseptiquarter|semiseptiquarter]] and [[semiluna]].
 
{{Harmonics in equal|292}}
 
== Regular temperament properties ==
{| 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| 463 -292 }}
| [{{val| 292 463 }}]
| -0.2476
| 0.2475
| 6.02
|-
| 2.3.5
| {{monzo| 3 -18 11 }}, {{monzo| 38 -2 -15 }}
| [{{val| 292 463 678 }}]
| -0.1633
| 0.2346
| 5.71
|-
| 2.3.5.7
| 5120/5103, 390625/388962, 420175/419904
| [{{val| 292 463 678 820 }}]
| -0.2148
| 0.2219
| 5.40
|-
| 2.3.5.7.11
| 1375/1372, 5120/5103, 5632/5625, 14641/14580
| [{{val| 292 463 678 820 1010 }}]
| -0.1353
| 0.2544
| 6.19
|-
| 2.3.5.7.11.13
| 352/351, 625/624, 847/845, 1716/1715, 14641/14580
| [{{val| 292 463 678 820 1010 1081 }}]
| -0.3480
| 0.2736
| 6.66
|-
| 2.3.5.7.11.13.17
| 352/351, 625/624, 715/714, 847/845, 1225/1224, 2025/2023
| [{{val| 292 463 678 820 1010 1081 1194 }}]
| -0.2376
| 0.2696
| 6.56
|}
 
=== 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
| 11\292
| 45.21
| 250/243
| [[Quartonic]] (5-limit)
|-
| 1
| 47\292
| 193.15
| 262144/234375
| [[Luna]]
|-
| 1
| 59\292
| 242.47
| 147/128
| [[Septiquarter]]
|-
| 1
| 111\292
| 456.16
| 125/96
| [[Qak]]
|-
| 2
| 47\292
| 193.15
| 121/108
| [[Semiluna]]
|-
| 2
| 59\292
| 242.47
| 121/105
| [[Semiseptiquarter]]
|-
| 4
| 121\292
| 497.26
| 4/3
| [[Undim]]
|}


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

Revision as of 14:25, 28 April 2022

← 291edo 292edo 293edo →
Prime factorization 22 × 73
Step size 4.10959 ¢ 
Fifth 171\292 (702.74 ¢)
Semitones (A1:m2) 29:21 (119.2 ¢ : 86.3 ¢)
Consistency limit 9
Distinct consistency limit 9

Template:EDO intro

Theory

292edo is closely related to 146edo, but the patent vals differ on the mapping for 3. It tempers out [3 -18 11⟩ (quartonic comma) and [38 -2 -15⟩ (luna/hemithirds comma) in the 5-limit; 5120/5103 (hemifamity), 390625/388962 (dimcomp), 420175/419904 (wizma), and 4802000/4782969 (canousma) in the 7-limit; 1375/1372, 5632/5625, 6250/6237, 9801/9800 and 14641/14580 in the 11-limit; 352/351, 625/624, 847/845, 1716/1715, and 2080/2079 in the 13-limit.

It provides the optimal patent val for the undim temperament in the 7-, 11-, and 13-limit, and notably supports semiseptiquarter and semiluna.


Approximation of prime harmonics in 292edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.78 -0.01 +1.04 -0.63 +1.94 +1.89 -1.62 +0.49 +1.93 +1.54
Relative (%) +0.0 +19.1 -0.3 +25.2 -15.4 +47.2 +46.1 -39.5 +12.0 +47.0 +37.5
Steps
(reduced) 		292
(0) 		463
(171) 		678
(94) 		820
(236) 		1010
(134) 		1081
(205) 		1194
(26) 		1240
(72) 		1321
(153) 		1419
(251) 		1447
(279)

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [463 -292⟩ [⟨292 463]] -0.2476 0.2475 6.02
2.3.5 [3 -18 11⟩, [38 -2 -15⟩ [⟨292 463 678]] -0.1633 0.2346 5.71
2.3.5.7 5120/5103, 390625/388962, 420175/419904 [⟨292 463 678 820]] -0.2148 0.2219 5.40
2.3.5.7.11 1375/1372, 5120/5103, 5632/5625, 14641/14580 [⟨292 463 678 820 1010]] -0.1353 0.2544 6.19
2.3.5.7.11.13 352/351, 625/624, 847/845, 1716/1715, 14641/14580 [⟨292 463 678 820 1010 1081]] -0.3480 0.2736 6.66
2.3.5.7.11.13.17 352/351, 625/624, 715/714, 847/845, 1225/1224, 2025/2023 [⟨292 463 678 820 1010 1081 1194]] -0.2376 0.2696 6.56

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 11\292 45.21 250/243 Quartonic (5-limit)
1 47\292 193.15 262144/234375 Luna
1 59\292 242.47 147/128 Septiquarter
1 111\292 456.16 125/96 Qak
2 47\292 193.15 121/108 Semiluna
2 59\292 242.47 121/105 Semiseptiquarter
4 121\292 497.26 4/3 Undim
Retrieved from "https://en.xen.wiki/index.php?title=292edo&oldid=91604"