Vulture family: Difference between revisions

Line 33:

=== 2.3.5.19 ===

It can be observed that the generator of vulture is very close to [[25/19]]; this corresponds to tempering out [[1216/1215]] = ([[19/15]])/([[9/8|18/16]])<sup>2</sup> = S16/S18. It results in a surprising decrease in Dirichlet badness, and (up to [[octave equivalence]]) finds [[19/16]] at 41 generators so that [[19/10]] is found at 20 generators, [[38/27]] is found at 18, [[19/15]] is found at 16 (as 3 is found at 4) and 76/45 is found at 12 so that it's equated with [[27/16]], which is tuned slightly sharp, as 76/45 is 1216/1215 above it. As a result of the interpretation of 1 gen as ~25/19, the 3 gen interval of ~226.6{{cent}} is interpreted as ([[3/2]])/([[25/19]]) = [[~]][[57/50]] which is tuned ~0.2{{cent}} flat. (Interpreting this interval as a damaged [[~]][[8/7]] leads to [[#Buzzard]].) Note that unless you are fine with the low accuracy* tuning offered by [[53edo]], you cannot temper out the [[schisma]], nor can you equate 32/27 with 19/16 or 24/19 with 19/15, meaning both the schisma and [[513/512]][[~]][[361/360]] (resp.) are observed. * Compared to what this microtemperament is capable of. This means that the step size of [[270edo]] is especially ideal, being between 361/360 and 513/512, with [[217edo]] exaggerating the comma to be slightly sharp of 361/360. Also note that 164 - 53 = 53 + 58 = [[111edo]] is a possible tuning which doesn't appear in the optimal ET sequence because it's less accurate than 53edo on the 2.3.5.19 subgroup.

It can be observed that the generator of vulture is very close to [[25/19]]; this corresponds to tempering out [[1216/1215]] = ([[19/15]])/([[9/8|18/16]])<sup>2</sup> = S16/S18. It results in a surprising decrease in Dirichlet badness, and (up to [[octave equivalence]]) finds [[19/16]] at 41 generators so that [[19/10]] is found at 20 generators, [[38/27]] is found at 18, [[19/15]] is found at 16 (as 3 is found at 4) and 76/45 is found at 12 so that it's equated with [[27/16]], which is tuned slightly sharp, as 76/45 is 1216/1215 above it. As a result of the interpretation of 1 gen as ~25/19, the 3 gen interval of ~226.6{{cent}} is interpreted as ([[3/2]])/([[25/19]]) = [[~]][[57/50]] which is tuned ~0.2{{cent}} flat. (Interpreting this interval as a damaged [[~]][[8/7]] leads to [[#Buzzard]].) Note that unless you are fine with the low accuracy* tuning offered by [[53edo]], you cannot temper out the [[schisma]], nor can you equate 32/27 with 19/16 or 24/19 with 19/15, meaning both the schisma and [[513/512]][[~]][[361/360]] (resp.) are observed. {{nowrap|* Compared}} to what this microtemperament is capable of. This means that the step size of [[270edo]] is especially ideal, being between 361/360 and 513/512, with [[217edo]] exaggerating the comma to be slightly sharp of 361/360. Also note that 164 - 53 = 53 + 58 = [[111edo]] is a possible tuning which doesn't appear in the optimal ET sequence because it's less accurate than 53edo on the 2.3.5.19 subgroup.

Subgroup: 2.3.5.19

@@ Line 33: / Line 33: @@
 === 2.3.5.19 ===
-It can be observed that the generator of vulture is very close to [[25/19]]; this corresponds to tempering out [[1216/1215]] = ([[19/15]])/([[9/8|18/16]])<sup>2</sup> = S16/S18. It results in a surprising decrease in Dirichlet badness, and (up to [[octave equivalence]]) finds [[19/16]] at 41 generators so that [[19/10]] is found at 20 generators, [[38/27]] is found at 18, [[19/15]] is found at 16 (as 3 is found at 4) and 76/45 is found at 12 so that it's equated with [[27/16]], which is tuned slightly sharp, as 76/45 is 1216/1215 above it. As a result of the interpretation of 1 gen as ~25/19, the 3 gen interval of ~226.6{{cent}} is interpreted as ([[3/2]])/([[25/19]]) = [[~]][[57/50]] which is tuned ~0.2{{cent}} flat. (Interpreting this interval as a damaged [[~]][[8/7]] leads to [[#Buzzard]].) Note that unless you are fine with the low accuracy* tuning offered by [[53edo]], you cannot temper out the [[schisma]], nor can you equate 32/27 with 19/16 or 24/19 with 19/15, meaning both the schisma and [[513/512]][[~]][[361/360]] (resp.) are observed. * Compared to what this microtemperament is capable of. This means that the step size of [[270edo]] is especially ideal, being between 361/360 and 513/512, with [[217edo]] exaggerating the comma to be slightly sharp of 361/360. Also note that 164 - 53 = 53 + 58 = [[111edo]] is a possible tuning which doesn't appear in the optimal ET sequence because it's less accurate than 53edo on the 2.3.5.19 subgroup.
+It can be observed that the generator of vulture is very close to [[25/19]]; this corresponds to tempering out [[1216/1215]] = ([[19/15]])/([[9/8|18/16]])<sup>2</sup> = S16/S18. It results in a surprising decrease in Dirichlet badness, and (up to [[octave equivalence]]) finds [[19/16]] at 41 generators so that [[19/10]] is found at 20 generators, [[38/27]] is found at 18, [[19/15]] is found at 16 (as 3 is found at 4) and 76/45 is found at 12 so that it's equated with [[27/16]], which is tuned slightly sharp, as 76/45 is 1216/1215 above it. As a result of the interpretation of 1 gen as ~25/19, the 3 gen interval of ~226.6{{cent}} is interpreted as ([[3/2]])/([[25/19]]) = [[~]][[57/50]] which is tuned ~0.2{{cent}} flat. (Interpreting this interval as a damaged [[~]][[8/7]] leads to [[#Buzzard]].) Note that unless you are fine with the low accuracy* tuning offered by [[53edo]], you cannot temper out the [[schisma]], nor can you equate 32/27 with 19/16 or 24/19 with 19/15, meaning both the schisma and [[513/512]][[~]][[361/360]] (resp.) are observed. {{nowrap|* Compared}} to what this microtemperament is capable of. This means that the step size of [[270edo]] is especially ideal, being between 361/360 and 513/512, with [[217edo]] exaggerating the comma to be slightly sharp of 361/360. Also note that 164 - 53 = 53 + 58 = [[111edo]] is a possible tuning which doesn't appear in the optimal ET sequence because it's less accurate than 53edo on the 2.3.5.19 subgroup.
 Subgroup: 2.3.5.19