94edo: Difference between revisions

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 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]].

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]].

![[Ups and downs notation|Ups and downs]]

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.  

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

|<small>{{sagittal|#}}{{sagittal|(!(}}</small>

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

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

|<small>{{sagittal|#}}</small>

| {{monzo| 149 -94 }}

| {{mapping| 94 149 }}

| {{mapping| 94 149 218 }}

| {{mapping| 94 149 218 264 }}

| {{mapping| 94 149 218 264 325 }}

| {{mapping| 94 149 218 264 325 348 }}

| {{mapping| 94 149 218 264 325 348 384 }}

| {{mapping| 94 149 218 264 325 348 384 399 }}

| {{mapping| 94 149 218 264 325 348 384 399 425 }}

| [[Tsaharuk]] / [[quanic]] / [[glacier]]

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

 

 

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

* [[Gariwizmic]] (94 & 176)

* {{nowrap|94 & 422}}  

![[Harmonic limit|Prime]]

[[Harmonic limit|limit]]

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

![[Cents]]

![[Monzo]]

! colspan="2" |[[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

|r³y⁵

|Gariboh comma

|7

|[[4000/3969]]

|13.47

|{{monzo|5 -4 3 -2}}

|Rurutriyo

|rry³

|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

|[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

|[[4225/4224]]

|0.41

|[-7 -1 2 0 -1 2⟩

|Thotholuyoyo

|3oo1uyy

|Leprechaun comma

|11

|[[9801/9800]]

|0.17

|[-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

|17

|[[289/288]]

|6.00

|[5 -2 0 0 0 0 2&#x27E9;

|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