94edo: Difference between revisions

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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 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 it dramatically improves on prime 3, as well as primes 11, 19 and 23~~ to ~~a lesser degree. It can also be thought of as the "sum"~~ of [[~~41edo]~~] ~~and [[53edo~~]~~] (41 + 53 = 94)~~, ~~both of~~ which ~~are known for their approximation of [[Pythagorean~~ tuning~~]]. Therefore 94edo's fifth is the [[mediant]] of these two tunings' fifths; it is slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's~~.

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

94edo~~'s step size is close to that~~ of [[~~144/143~~]], ~~which is consistently represented in this tuning system~~.

94edo 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 it dramatically improves on prime 3, as well as primes 11, 19 and 23 to a lesser degree.

~~The list~~ of ~~23-limit commas it tempers out is huge~~, but ~~it is worth noting that it tempers out~~ [[~~32805/32768~~]] ~~and~~ is ~~thus~~ a [[~~schismatic~~]] ~~system~~, ~~that it tempers~~ out [[225/224]] and [[385/384]] and ~~so is~~ a [[~~marvel~~]] ~~system~~, and ~~that~~ it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for 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]].

===Significance of cassandra===

94edo can also be thought of as the "sum" of [[41edo]] and [[53edo]] (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 temperament]] known as [[Schismatic family#Cassandra|cassandra]] (which is itself a variety of [[Schismatic family#Garibaldi|garibaldi]]), tempering out [[32805/32768]], [[225/224]], and [[385/384]]. Therefore, 94edo's fifth is the [[mediant]] of these two edos' fifths; it is 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 temperament properties===

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 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.

=== Prime harmonics ===

===Prime harmonics===

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 {{EDO intro|94}}
-== Theory ==
+==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 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 it dramatically improves on prime 3, as well as primes 11, 19 and 23 to a lesser degree. It can also be thought of as the "sum" of [[41edo]] and [[53edo]] (41 + 53 = 94), both of which are known for their approximation of [[Pythagorean tuning]]. Therefore 94edo's fifth is the [[mediant]] of these two tunings' fifths; it is slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's.
+Its step size is close to that of [[144/143]], which is consistently represented in this tuning system.
-edo's step size is close to that of [[144/143]], which is consistently represented in this tuning system.
+edo 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 it dramatically improves on prime 3, as well as primes 11, 19 and 23 to a lesser degree.
-The list of 23-limit commas it tempers out is huge, but it is worth noting that it tempers out [[32805/32768]] and is thus a [[schismatic]] system, that it tempers out [[225/224]] and [[385/384]] and so is a [[marvel]] system, and that it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for 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]].
+===Significance of cassandra===
+edo can also be thought of as the "sum" of [[41edo]] and [[53edo]] (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 temperament]] known as [[Schismatic family#Cassandra|cassandra]] (which is itself a variety of [[Schismatic family#Garibaldi|garibaldi]]), tempering out [[32805/32768]], [[225/224]], and [[385/384]]. Therefore, 94edo's fifth is the [[mediant]] of these two edos' fifths; it is 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 temperament properties===
+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 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.
-=== Prime harmonics ===
+===Prime harmonics===
 {{Harmonics in equal|94|columns=11}}