94edo: Difference between revisions

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== 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.

=== 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.

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! 13-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]]

! Long-form WOFED

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=== 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.

{| class="wikitable" style="text-align: center;"

! colspan="2" |Steps

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| rowspan="2" |<big>{{sagittal|(|(}}</big>

| rowspan="2" |<big>{{sagittal|/|\}}</big>

|<~~small~~>{{sagittal|~~#}}{{sagittal~~|~~\!/~~}}</~~small~~>

| rowspan="2" |<big>{{sagittal|(|)}}</big>

|~~~~{{sagittal|#}}{{sagittal|~~(!(~~}}~~~~

|{{sagittal|(!(}}{{sagittal|#}}

|~~~~{{sagittal|#}}{{sagittal|\!}}~~~~

|{{sagittal|\!}}{{sagittal|#}}

|~~~~{{sagittal|#}}{{sagittal|~~~!(~~}}~~~~

|{{sagittal|~!(}}{{sagittal|#}}

|~~~~{{sagittal|#}}~~~~

|{{sagittal|#}}

|-

!Revo

~~|<big>{{sagittal|(|)}}</big>~~

|<big>{{sagittal|~||(}}</big>

|<big>{{sagittal|||\}}</big>

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| [[Kleischismic]]

|}

<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal ~~froms~~|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

Temperaments to which 94et can be detempered:

* [[Satin]] ({{nowrap|94 & 217}})

* [[Satin]] ({{nowrap| 94 & 217 }})

* [[Gariwizmic]] (94 & 176)

* [[Gariwizmic]] ({{nowrap| 94 & 176 }})

* {{nowrap|94 & 422}}

* {{nowrap| 94 & 422 }}

=== Commas ===

94et [[~~Tempering~~ out|tempers out]] the following [[~~Comma|commas~~]] using its patent [[val]], ~~⟨94~~ 149 218 264 325 348 384 399 425].

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 }}.

{| class="commatable wikitable center-1 center-2 right-3 center-6"

! [[Harmonic limit|Prime ~~mit~~]]

! [[Harmonic limit|Prime limit]]

! [[Ratio]]<ref>Ratios with more than 8 digits are presented by placeholders with informative hints</ref>

! [[Cents]]

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| <abbr title="36893488147419103232/36472996377170786403">(90 digits)</abbr>

| 16.22

| {{~~monzo~~| 149 -94 }}

| {{Monzo| 149 -94 }}

| Wa-94

| 94-edo

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| 1urry

| Swetisma

|-

~~| 11~~

~~| [[4225/4224]]~~

~~| 0.41~~

~~| {{Monzo| -7 -1 2 0 -1 2 }}~~

~~| Thotholuyoyo~~

~~| 3oo1uyy~~

~~| Leprechaun comma~~

|-

| 11

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| s3urg

| Minisma

|-

| 13

| [[4225/4224]]

| 0.41

| {{Monzo| -7 -1 2 0 -1 2 }}

| Thotholuyoyo

| 3oo1uyy

| Leprechaun comma

|-

| 17

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| s19og

| Eratosthenes' comma

|-

| 19

| [[11859211/11859210|(16 digits)]]

| 0.00

| {{Monzo| -1 -4 -1 1 -4 1 0 4 }}

| Quadno-athoquadlu-azogu

| 9o43o1u4zg

| 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

|}

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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.
+edo 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.
 === As a tuning of other temperaments ===
-edo 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]].
+edo 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]].
+edo'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]].
 edo 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
 ! 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]]
 ! Long-form WOFED
@@ Line 908: / Line 910: @@
 === Sagittal ===
-edo 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.
+edo 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.
 {| class="wikitable" style="text-align: center;"
 ! colspan="2" |Steps
@@ Line 929: / Line 931: @@
 | rowspan="2" |<big>{{sagittal|(|(}}</big>
 | rowspan="2" |<big>{{sagittal|/|\}}</big>
-|<small>{{sagittal|#}}{{sagittal|\!/}}</small>
+| rowspan="2" |<big>{{sagittal|(|)}}</big>
-|<small>{{sagittal|#}}{{sagittal|(!(}}</small>
+|{{sagittal|(!(}}{{sagittal|#}}
-|<small>{{sagittal|#}}{{sagittal|\!}}</small>
+|{{sagittal|\!}}{{sagittal|#}}
-|<small>{{sagittal|#}}{{sagittal|~!(}}</small>
+|{{sagittal|~!(}}{{sagittal|#}}
-|<small>{{sagittal|#}}</small>
+|{{sagittal|#}}
 |-
 !Revo
-|<big>{{sagittal|(|)}}</big>
 |<big>{{sagittal|~||(}}</big>
 |<big>{{sagittal|||\}}</big>
@@ Line 1,112: / Line 1,113: @@
 | [[Kleischismic]]
 |}
-<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal froms|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
 Temperaments to which 94et can be detempered:
-* [[Satin]] ({{nowrap|94 & 217}})
+* [[Satin]] ({{nowrap| 94 & 217 }})
-* [[Gariwizmic]] (94 & 176)
+* [[Gariwizmic]] ({{nowrap| 94 & 176 }})
-* {{nowrap|94 & 422}}
+* {{nowrap| 94 & 422 }}
 === Commas ===
-et [[Tempering out|tempers out]] the following [[Comma|commas]] using its patent [[val]], ⟨94 149 218 264 325 348 384 399 425].
+et [[tempering out|tempers out]] the following [[comma]]s using its 23-limit patent [[val]], {{val| 94 149 218 264 325 348 384 399 425 }}.
 {| class="commatable wikitable center-1 center-2 right-3 center-6"
-! [[Harmonic limit|Prime<br>mit]]
+! [[Harmonic limit|Prime<br>limit]]
 ! [[Ratio]]<ref>Ratios with more than 8 digits are presented by placeholders with informative hints</ref>
 ! [[Cents]]
@@ Line 1,132: / Line 1,134: @@
 | <abbr title="36893488147419103232/36472996377170786403">(90 digits)</abbr>
 | 16.22
-| {{monzo| 149 -94 }}
+| {{Monzo| 149 -94 }}
 | Wa-94
 | 94-edo
@@ Line 1,208: / Line 1,210: @@
 | 1urry
 | Swetisma
-|-
-| 11
-| [[4225/4224]]
-| 0.41
-| {{Monzo| -7 -1 2 0 -1 2 }}
-| Thotholuyoyo
-| 3oo1uyy
-| Leprechaun comma
 |-
 | 11
@@ Line 1,296: / Line 1,290: @@
 | s3urg
 | Minisma
+|-
+| 13
+| [[4225/4224]]
+| 0.41
+| {{Monzo| -7 -1 2 0 -1 2 }}
+| Thotholuyoyo
+| 3oo1uyy
+| Leprechaun comma
 |-
 | 17
@@ Line 1,336: / Line 1,338: @@
 | 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
 |}