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]].

Since 94 factors into primes as {{nowrap| 2 × 47 }}, 94edo contains [[2edo]] and [[47edo]] as subset edos. It can be thought of as two sets of 47edo offset by one step of 94edo. It inherits from 47edo's good approximations of primes 5, 7, 13, and 17, while dramatically improving on prime 3, as well as primes 11, 19, and 23 to a lesser degree.

![[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.  

!Degree

|'''0'''

| +1

| +2

| +3

| +4

| +5

| +6

| +7

| +8

| '''+9'''

|<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

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.

* {{nowrap|68 &amp; 94}}

* {{nowrap|41 &amp; 94}}  (another garibaldi, only differing in the mappings of 17 and 23)

* {{nowrap|135 &amp; 94}} (another garibaldi)

 

* [[Satin]] ({{nowrap|94 & 311}})  

* {{nowrap|94 & 422}}