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== Theory ==
== Theory ==
94edo is a remarkable all-around utility tuning system, good from low [[prime limit]] to very high prime limit situations. It is the first edo to be [[consistent]] through the [[23-odd-limit]], and no other edo is so consistent until [[282edo|282]] and [[311edo|311]] make their appearance.
94edo is a remarkable well-rounded tuning system, good from low [[prime limit]] to very high prime limit situations. It is the first edo to be [[consistent]] through the [[23-odd-limit]], and no other edo is so consistent until [[282edo|282]] and [[311edo|311]] make their appearance.


Its step size is close to that of [[144/143]], which is consistently represented in this tuning system.
Its step size is close to that of [[144/143]], which is consistently represented in this tuning system.


=== As a tuning of other temperaments ===
=== As a tuning of other temperaments ===
94edo can also be thought of as the "sum" of [[41edo]] and [[53edo]] {{nowrap|(41 + 53 {{=}} 94)}}, both of which are not only known for their approximation of [[Pythagorean tuning]], but also support a variety of [[Schismatic family|schismatic temperaments]], like [[Schismatic family#Cassandra|cassandra]] (which is itself a variety of [[Schismatic family#Garibaldi|garibaldi]]), tempering out [[32805/32768]], [[225/224]], and [[385/384]], and tempering together the [[81/80|syntonic]], [[Septimal comma|septimal]], and [[pythagorean comma]] into the same interval. Therefore, 94edo's fifth is the [[mediant]] of these two edos' fifths; it is ever so slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's, and acts as a generator for a highly optimized and high-prime-limit form of cassandra. Few, if any, edos that support schismatic by [[Val|patent val]] have at least as high of a consistency limit as 94edo while also having a fifth that can stack to reach any interval in it. Other non-garibaldi schismatic notable edos in the patent val are [[118edo|118]], [[159edo|159]], [[171edo|171]], [[224edo|224]], and [[460edo|460]].
94edo is the sum of [[41edo]] and [[53edo]], both of which are not only known for their approximation of [[Pythagorean tuning]], but also support a variety of [[Schismatic family|schismatic temperaments]], like [[Schismatic family#Cassandra|cassandra]] (which is itself a variety of [[Schismatic family#Garibaldi|garibaldi]]), tempering out [[32805/32768]], [[225/224]], and [[385/384]], and tempering together the [[81/80|syntonic]], [[Septimal comma|septimal]], and [[pythagorean comma]] into the same interval.


The list of 23-limit commas it tempers out is huge, and in lower prime limits, it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for gassormic, the rank-5 temperament tempering out [[275/273]] (despite one edostep being very close in size to this comma), and for a number of other temperaments, such as [[isis]].
94edo's fifth is the [[mediant]] of these two edos' fifths; it is ever so slightly sharp of just and only a hair less accurate than 53edo's fifth, but more accurate than 41edo's, and acts as a generator for a highly optimized and high-prime-limit form of cassandra. Few, if any, edos that support schismatic by [[Val|patent val]] have at least as high of a consistency limit as 94edo while also having a fifth that can stack to reach any interval in it. Other non-garibaldi schismatic notable edos in the patent val are [[118edo|118]], [[159edo|159]], [[171edo|171]], [[224edo|224]], and [[460edo|460]].
 
The list of 23-limit commas it tempers out is huge (see below), and in lower prime limits, it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for gassormic, the rank-5 temperament tempering out [[275/273]] (despite one edostep being very close in size to this comma), and for a number of other temperaments, such as [[isis]].


94edo is an excellent edo for [[Carlos Beta]] scale, since the difference between 1 step of Carlos Beta and 5 steps of 94edo is only 0.00314534 cents.
94edo is an excellent edo for [[Carlos Beta]] scale, since the difference between 1 step of Carlos Beta and 5 steps of 94edo is only 0.00314534 cents.
Line 31: Line 33:
! 13-limit
! 13-limit
! 23-limit
! 23-limit
![[Ups and downs notation|Ups and downs]]
! [[Ups and downs notation|Ups and downs]]
! Short-form [[SKULO interval names#WOFED interval names|WOFED]]
! Short-form [[SKULO interval names#WOFED interval names|WOFED]]
! Long-form WOFED
! Long-form WOFED
Line 908: Line 910:


=== Sagittal ===
=== Sagittal ===
94edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal_notation#Athenian_extension_single-shaft|Athenian extension]], with the apotome equal to 9 edosteps and the limma to 7 edosteps.
94edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal_notation#Athenian|Athenian set]], with the apotome equal to 9 edosteps and the limma to 7 edosteps.  
 
Thus, the JI chord 16:17:18:19:20:21:22:23:24:25:26:27:28:30 from D would be written D{{sagittal||//|}}:E{{sagittal|(!!(}}:E{{sagittal||//|}}:F{{sagittal||//|}}:F{{sagittal|||\}}:G{{sagittal|\!}}:G{{sagittal|/|\}}:G{{sagittal|~|||(}}:A{{sagittal||//|}}:A{{sagittal|(|)}}:B{{sagittal|(!)}}:B{{sagittal||//|}}:C{{sagittal|\!}}:C{{sagittal|||\}}. Music that doesn't modulate much in the 2.3.5.7.11.13.19 subgroup can be notated by only using {{sagittal|/|}} {{sagittal|/|\}} / {{sagittal|\!}} {{sagittal|\!/}} and their apotome complements; where naturals are used for 3 and 19, {{sagittal|/|}} / {{sagittal|\!}} for 5 and 7, and {{sagittal|/|\}} / {{sagittal|\!/}} for 11 and 13.
{| class="wikitable" style="text-align: center;"
{| class="wikitable" style="text-align: center;"
!Degree
! colspan="2" |Steps
 
!'''0'''
| '''−9'''
! 1
| −8
! 2
| −7
! 3
| −6
! 4
| −5
! 5
| −4
! 6
| −3
! 7
| −2
! 8
| −1
! '''9'''
|'''0'''
| +1
| +2
| +3
| +4
| +5
| +6
| +7
| +8
| '''+9'''
|-
|-
! rowspan="2" |Symbol
!Evo
!Evo
|{{sagittal|b}}
| rowspan="2" |<big>{{sagittal||//|}}</big>
|{{sagittal|b}}{{sagittal|~|(}}
| rowspan="2" |<big>{{sagittal|~|(}}</big>
|{{sagittal|b}}{{sagittal|/|}}
| rowspan="2" |<big>{{sagittal|/|}}</big>
|{{sagittal|b}}{{sagittal|(|(}}
| rowspan="2" |<big>{{sagittal|(|(}}</big>
|{{sagittal|b}}{{sagittal|/|\}}
| rowspan="2" |<big>{{sagittal|/|\}}</big>
| rowspan="2" |{{sagittal|\!/}}
| rowspan="2" |<big>{{sagittal|(|)}}</big>
| rowspan="2" |{{sagittal|(!(}}
|{{sagittal|(!(}}{{sagittal|#}}
| rowspan="2" |{{sagittal|\!}}
|{{sagittal|\!}}{{sagittal|#}}
| rowspan="2" |{{sagittal|~!(}}
|{{sagittal|~!(}}{{sagittal|#}}
| rowspan="2" |{{sagittal||//|}}
| rowspan="2" |{{sagittal|~|(}}
| rowspan="2" |{{sagittal|/|}}
| rowspan="2" |{{sagittal|(|(}}
| rowspan="2" |{{sagittal|/|\}}
|{{sagittal|#}}{{sagittal|\!/}}
|{{sagittal|#}}{{sagittal|(!(}}
|{{sagittal|#}}{{sagittal|\!}}
|{{sagittal|#}}{{sagittal|~!(}}
|{{sagittal|#}}
|{{sagittal|#}}
|-
|-
!Revo
!Revo
|{{sagittal|\!!/}}
|<big>{{sagittal|~||(}}</big>
|{{sagittal|(!!(}}
|<big>{{sagittal|||\}}</big>
|{{sagittal|!!/}}
|<big>{{sagittal|(||(}}</big>
|{{sagittal|~!!(}}
|<big>{{sagittal|/||\}}</big>
|{{sagittal|(!)}}
|{{sagittal|(|)}}
|{{sagittal|~||(}}
|{{sagittal|||\}}
|{{sagittal|(||(}}
|{{sagittal|/||\}}
|}
|}
The following enharmonics from the Athenian set are present (comma tempered out):
* {{sagittal|//|}} = {{Sagittal|/|)}} = {{Sagittal|/|\}} ([[325/324]], [[352/351]])
* {{sagittal|/|}} = {{sagittal||)}} = {{sagittal||\}} ([[225/224]], [[2200/2187]])
* {{sagittal|)|(}} = {{sagittal|~|(}} ([[3680721/3670016]])
* {{sagittal|(|}} = {{sagittal|(|(}} ([[5120/5103]])
* {{sagittal||(}} = {{sagittal||//|}} ([[5120/5103]])
See [[Sagittal notation#Revo|apotome complements]] for equivalent accidental pairs.
The JI chord 16:17:18:19:20:21:22:23:24:25:26:27:28:30 from D would be written D{{sagittal||//|}}:E{{sagittal|(!!(}}:E{{sagittal||//|}}:F{{sagittal||//|}}:F{{sagittal|||\}}:G{{sagittal|\!}}:G{{sagittal|/|\}}:G{{sagittal|~|||(}}:A{{sagittal||//|}}:A{{sagittal|(|)}}:B{{sagittal|(!)}}:B{{sagittal||//|}}:C{{sagittal|\!}}:C{{sagittal|||\}}. Music that doesn't modulate much in the 2.3.5.7.11.13.19 subgroup can be notated by only using {{sagittal|/|}} {{sagittal|/|\}} / {{sagittal|\!}} {{sagittal|\!/}} and their apotome complements; where naturals are used for 3 and 19, {{sagittal|/|}} / {{sagittal|\!}} for 5 and 7, and {{sagittal|/|\}} / {{sagittal|\!/}} for 11 and 13.


== Regular temperament properties ==
== Regular temperament properties ==
Line 981: Line 968:
|-
|-
| 2.3
| 2.3
| {{monzo| 149 -94 }}
| {{Monzo| 149 -94 }}
| {{mapping| 94 149 }}
| {{Mapping| 94 149 }}
| −0.054
| −0.054
| 0.054
| 0.054
Line 989: Line 976:
| 2.3.5
| 2.3.5
| 32805/32768, 9765625/9565938
| 32805/32768, 9765625/9565938
| {{mapping| 94 149 218 }}
| {{Mapping| 94 149 218 }}
| +0.442
| +0.442
| 0.704
| 0.704
Line 996: Line 983:
| 2.3.5.7
| 2.3.5.7
| 225/224, 3125/3087, 118098/117649
| 225/224, 3125/3087, 118098/117649
| {{mapping| 94 149 218 264 }}
| {{Mapping| 94 149 218 264 }}
| +0.208
| +0.208
| 0.732
| 0.732
Line 1,003: Line 990:
| 2.3.5.7.11
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 2200/2187
| 225/224, 385/384, 1331/1323, 2200/2187
| {{mapping| 94 149 218 264 325 }}
| {{Mapping| 94 149 218 264 325 }}
| +0.304
| +0.304
| 0.683
| 0.683
Line 1,010: Line 997:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 225/224, 275/273, 325/324, 385/384, 1331/1323
| 225/224, 275/273, 325/324, 385/384, 1331/1323
| {{mapping| 94 149 218 264 325 348 }}
| {{Mapping| 94 149 218 264 325 348 }}
| +0.162
| +0.162
| 0.699
| 0.699
Line 1,017: Line 1,004:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 170/169, 225/224, 275/273, 289/288, 325/324, 385/384
| 170/169, 225/224, 275/273, 289/288, 325/324, 385/384
| {{mapping| 94 149 218 264 325 348 384 }}
| {{Mapping| 94 149 218 264 325 348 384 }}
| +0.238
| +0.238
| 0.674
| 0.674
Line 1,024: Line 1,011:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 170/169, 190/189, 225/224, 275/273, 289/288, 325/324, 385/384
| 170/169, 190/189, 225/224, 275/273, 289/288, 325/324, 385/384
| {{mapping| 94 149 218 264 325 348 384 399 }}
| {{Mapping| 94 149 218 264 325 348 384 399 }}
| +0.323
| +0.323
| 0.669
| 0.669
Line 1,031: Line 1,018:
| 2.3.5.7.11.13.17.19.23
| 2.3.5.7.11.13.17.19.23
| 170/169, 190/189, 209/208, 225/224, 275/273, 289/288, 300/299, 323/322
| 170/169, 190/189, 209/208, 225/224, 275/273, 289/288, 300/299, 323/322
| {{mapping| 94 149 218 264 325 348 384 399 425 }}
| {{Mapping| 94 149 218 264 325 348 384 399 425 }}
| +0.354
| +0.354
| 0.637
| 0.637
Line 1,059: Line 1,046:
| 25/24
| 25/24
| [[Betic]]
| [[Betic]]
|-
| 1
| 7\94
| 89.36
| 21/20
| [[Slithy]]
|-
|-
| 1
| 1
Line 1,064: Line 1,057:
| 140.43
| 140.43
| 13/12
| 13/12
| [[Tsaharuk]] / [[quanic]] / [[glacier]]
| [[Tsaharuk]] / [[quanic]]
|-
|-
| 1
| 1
Line 1,077: Line 1,070:
| 147/128
| 147/128
| [[Septiquarter]]
| [[Septiquarter]]
|-
| 1
| 25\94
| 319.15
| 6/5
| [[Dhaivatic]]
|-
|-
| 1
| 1
Line 1,114: Line 1,113:
| [[Kleischismic]]
| [[Kleischismic]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for [[garibaldi temperament]], shows us how to extend it to the 23-limit.
Temperaments to which 94et can be detempered:
* [[Satin]] ({{nowrap| 94 & 217 }})
* [[Gariwizmic]] ({{nowrap| 94 & 176 }})
* {{nowrap| 94 & 422 }}


* {{nowrap|46 &amp; 94}}
=== Commas ===
* {{nowrap|68 &amp; 94}}
94et [[tempering out|tempers out]] the following [[comma]]s using its 23-limit patent [[val]], {{val| 94 149 218 264 325 348 384 399 425 }}.
* {{nowrap|53 &amp; 94}}  (one garibaldi)
* {{nowrap|41 &amp; 94}}  (another garibaldi, only differing in the mappings of 17 and 23)
* {{nowrap|135 &amp; 94}} (another garibaldi)
* {{nowrap|130 &amp; 94}}  (a pogo extension)
* {{nowrap|58 &amp; 94}}  (a supers extension)
* {{nowrap|50 &amp; 94}}
* {{nowrap|72 &amp; 94}}  (a gizzard extension)
* {{nowrap|80 &amp; 94}}
* 94 solo  (a rank one temperament!)


Temperaments to which 94et can be detempered:
{| class="commatable wikitable center-1 center-2 right-3 center-6"
 
! [[Harmonic limit|Prime<br>limit]]
* [[Satin]] ({{nowrap|94 & 311}})  
! [[Ratio]]<ref>Ratios with more than 8 digits are presented by placeholders with informative hints</ref>
* {{nowrap|94 & 422}}  
! [[Cents]]
! [[Monzo]]
! colspan="2" | [[Kite's color notation|Color name]]
! Name(s)
|-
| 3
| <abbr title="36893488147419103232/36472996377170786403">(90 digits)</abbr>
| 16.22
| {{Monzo| 149 -94 }}
| Wa-94
| 94-edo
| [[94-comma]]
|-
| 5
| [[32805/32768|(10 digits)]]
| 1.95
| {{Monzo| -15 8 1 }}
| Layo
| Ly
| [[Schisma]]
|-
| 7
| [[3125/3087]]
| 21.18
| {{Monzo| 0 -2 5 -3 }}
| Triru-aquinyo
| r<sup>3</sup>y<sup>5</sup>
| Gariboh comma
|-
| 7
| [[4000/3969]]
| 13.47
| {{Monzo| 5 -4 3 -2 }}
| Rurutriyo
| rry<sup>3</sup>
| Octagar comma
|-
| 7
| [[225/224]]
| 7.71
| {{monzo| -5 2 2 -1 }}
| Ruyoyo
| ryy
| Marvel comma
|-
| 7
| <abbr title="36893488147419103232/36472996377170786403">(12 digits)</abbr>
| 6.59
| {{monzo| 1 10 0 -6 }}
| Latribiru
| L6r
| Stearnsma
|-
| 7
| [[5120/5103]]
| 5.76
| {{monzo| 10 -6 1 -1 }}
| Saruyo
| sry
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| 3.80
| {{Monzo| 25 -14 0 -1 }}
| Sasaru
| ssr
| [[Garischisma]]
|-
| 11
| [[385/384]]
| 4.50
| {{Monzo| -7 -1 1 1 1 }}
| Lozoyo
| 1ozg
| Keenanisma
|-
| 11
| [[540/539]]
| 3.21
| {{Monzo| 2 3 1 -2 -1 }}
| Lururuyo
| 1urry
| Swetisma
|-
| 11
| [[9801/9800]]
| 0.17
| {{Monzo| -3 4 -2 -2 2 }}
| Bilorugu
| 1oorrgg-2
| Kalisma
|-
| 13
| [[275/273]]
| 12.64
| {{Monzo| 0 -1 2 -1 1 -1 }}
| Thuloruyoyo
| 3u1oryy
| Gassorma
|-
| 13
| [[640/637]]
| 8.13
| {{Monzo| 7 0 1 -2 0 -1 }}
| Thururuyo
| 3urry
| Huntma
|-
| 13
| [[1188/1183]]
| 7.30
| {{Monzo| 2 3 0 -1 1 -2 }}
| Thuthuloru
| 3uu1or
| Kestrel comma
|-
| 13
| [[325/324]]
| 5.34
| {{Monzo| -2 -4 2 0 0 1 }}
| Thoyoyo
| 3oyy
| Marveltwin comma
|-
| 13
| [[352/351]]
| 4.93
| {{Monzo| 5 -3 0 0 1 -1 }}
| Thulo
| 3u1o
| Major minthma
|-
| 13
| [[847/845]]
| 4.09
| {{Monzo| 0 0 -1 1 2 -2 }}
| Thuthulolozogu
| 3uu1oozg
| Cuthbert comma
|-
| 13
| [[729/728]]
| 2.38
| {{Monzo| -3 6 0 -1 0 -1 }}
| Lathuru
| L3ur
| Squbema
|-
| 13
| [[2080/2079]]
| 0.83
| {{Monzo| 5 -3 1 -1 -1 1 }}
| Tholuruyo
| 3o1ury
| Ibnsinma, sinaisma
|-
| 13
| [[4096/4095]]
| 0.42
| {{Monzo| 12 -2 -1 -1 0 -1 }}
| Sathurugu
| s3urg
| Minisma
|-
| 13
| [[4225/4224]]
| 0.41
| {{Monzo| -7 -1 2 0 -1 2 }}
| Thotholuyoyo
| 3oo1uyy
| Leprechaun comma
|-
| 17
| [[289/288]]
| 6.00
| {{Monzo| 5 -2 0 0 0 0 2 }}
| Soso
| 17oo
| Semitonisma
|-
| 17
| [[715/714]]
| 2.42
| {{Monzo| -1 -1 1 -1 1 1 -1 }}
| Sutholoruyo
| 17u3o1ory
| Septendecimal bridge comma
|-
| 19
| [[361/360]]
| 4.80
| {{Monzo| -3 -2 -1 0 0 0 0 2 }}
| Nonogu
| 19oog2
| Go comma
|-
| 19
| [[513/512]]
| 3.38
| {{Monzo| -9 3 0 0 0 0 0 1 }}
| Lano
| L19o
| Boethius' comma
|-
| 19
| [[1216/1215]]
| 1.42
| {{Monzo| 6 -5 -1 0 0 0 0 1 }}
| Sanogu
| s19og
| Eratosthenes' comma
|-
| 19
| [[11859211/11859210|(16 digits)]]
| 0.00
| {{Monzo| -1 -4 -1 1 -4 1 0 4 }}
| <small>Quadno-athoquadlu-azogu</small>
| <small>9o<sup>4</sup>3o1u<sup>4</sup>zg</small>
| Tredekisma
|-
| 23
| [[300/299]]
| 5.78
| {{Monzo| 2 1 2 0 0 -1 0 0 -1 }}
| Twethuthuyoyo
| 23u3uyy
| Major naiadvicema
|-
| 23
| [[323/322]]
| 5.37
| {{Monzo| -1 0 0 -1 0 0 1 1 -1 }}
| Twethunosoru
| 23u19o17or
| Major semivicema
|-
| 23
| [[391/390]]
| 4.43
| {{Monzo| -1 -1 -1 0 0 -1 1 0 1 }}
| Twethosothugu
| 23o17o3ug
| Minor naiadvicema
|-
| 23
| [[460/459]]
| 3.77
| {{Monzo| 2 -3 1 0 0 0 -1 0 1 }}
| Twethosuyo
| 23o17uy
| Scanisma, vicewolf comma
|-
| 23
| [[484/483]]
| 3.58
| {{Monzo| 2 -1 0 -1 2 0 0 0 -1 }}
| Twethuloloru
| 23u1oor
| Pittsburghisma
|}


== Scales ==
== Scales ==
Line 1,155: Line 1,408:
* [https://www.youtube.com/shorts/HMD8QFpB2-U ''94edo improv''] (2025)
* [https://www.youtube.com/shorts/HMD8QFpB2-U ''94edo improv''] (2025)
* [https://www.youtube.com/watch?v=KrPQ_tPsvzQ ''Waltz in 94edo''] (2025)
* [https://www.youtube.com/watch?v=KrPQ_tPsvzQ ''Waltz in 94edo''] (2025)
* [https://www.youtube.com/shorts/KJ5dbF4aH2A ''Twinleaf Town - Pokémon Diamond and Pearl (microtonal cover in 94edo)''] (2026)


; [[Eufalesio]]
; [[Eufalesio]]
Retrieved from "https://en.xen.wiki/w/94edo"