94edo: Difference between revisions

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=== 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. Therefore, 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, 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]].

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 === 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. Therefore, 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, 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]].