Vulture family: Difference between revisions

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== Buzzard ==

Buzzard is the main extension to vulture of practical interest, finding 7 at only 3 generators down so that the generator is interpreted as a sharp ~21/16, but is more of a full 13-limit system in its own right ~~as will be explained~~. It is most naturally described as 53 & 58 (though [[48edo]] is an interesting higher-damage tuning of it for some purposes). As one might expect, 111edo is a great tuning for it.

Buzzard is the main extension to vulture of practical interest, finding prime 7 at only 3 generators down so that the generator is interpreted as a sharp ~21/16, but is more of a full 13-limit system in its own right. It is most naturally described as 53 & 58 (though [[48edo]] is an interesting higher-damage tuning of it for some purposes). As one might expect, 111edo is a great tuning for it.

Its S-expression-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]} ~~which because of the equality [[36/35|S6]]] = [[64/63|S8]] * [[81/80|S9]] tells us that~~ in any ~~nontrivial~~ tuning ~~(any tuning that observes~~ [[64/~~63|S8~~]] ~~and [[81/80|S9]]) it~~ naturally finds an interval between 6/5 and 7/6 which in the 7-limit is 32/27 and in the 13-limit is 13/11~~, then because~~ [[~~1728/1715|S6~~/S7]] is ~~tempered out~~, ~~that~~ is also split into two so that it also finds an interval between 7/6 and 8/7 which in the 7-limit is ~~[[~]][[8/~~7~~]] times [[~]][[81~~/~~80]][[~]][64~~/~~63]] or [[~]][[~~7~~/6]] divided~~ by ~~[[~]][[81/80]][[~]][64/63]]~~, and in the 13-limit is [~~[~]]~~[15/13]], so that it's clear this system naturally wants to be extended to and interpreted in the full 13-limit.

Its S-expression-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]}. Like in any hemifamity tuning, [[36/35]], the septimal quartertone, is split into two syntonic~septimal commas, so buzzard naturally finds an interval between [[6/5]] and [[7/6]] which in the 7-limit is [[32/27]] and in the 13-limit is [[13/11]]. Then the vanish of the orwellisma implies [[49/48]], the large septimal diesis, is tuned to the same step as 36/35, so 49/48 is also split into two so that the system also finds an interval between 7/6 and 8/7 which in the 7-limit is 7/6 inflected down by a comma or 8/7 inflected up by a comma, and in the 13-limit is ~[[15/13]], so that it is clear this system naturally wants to be extended to and interpreted in the full 13-limit.

[[Subgroup]]: 2.3.5.7

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 == Buzzard ==
 {{Main| Buzzard }}
-{{See also|No-fives subgroup temperaments#Buzzard}}
+{{See also| No-fives subgroup temperaments #Buzzard }}
-Buzzard is the main extension to vulture of practical interest, finding 7 at only 3 generators down so that the generator is interpreted as a sharp ~21/16, but is more of a full 13-limit system in its own right as will be explained. It is most naturally described as 53 & 58 (though [[48edo]] is an interesting higher-damage tuning of it for some purposes). As one might expect, 111edo is a great tuning for it.
+Buzzard is the main extension to vulture of practical interest, finding prime 7 at only 3 generators down so that the generator is interpreted as a sharp ~21/16, but is more of a full 13-limit system in its own right. It is most naturally described as 53 & 58 (though [[48edo]] is an interesting higher-damage tuning of it for some purposes). As one might expect, 111edo is a great tuning for it.
-Its S-expression-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]} which because of the equality [[36/35|S6]]] = [[64/63|S8]] * [[81/80|S9]] tells us that in any nontrivial tuning (any tuning that observes [[64/63|S8]] and [[81/80|S9]]) it naturally finds an interval between 6/5 and 7/6 which in the 7-limit is 32/27 and in the 13-limit is 13/11, then because [[1728/1715|S6/S7]] is tempered out, that is also split into two so that it also finds an interval between 7/6 and 8/7 which in the 7-limit is [[~]][[8/7]] times [[~]][[81/80]][[~]][64/63]] or [[~]][[7/6]] divided by [[~]][[81/80]][[~]][64/63]], and in the 13-limit is [[~]][15/13]], so that it's clear this system naturally wants to be extended to and interpreted in the full 13-limit.
+Its S-expression-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]}. Like in any hemifamity tuning, [[36/35]], the septimal quartertone, is split into two syntonic~septimal commas, so buzzard naturally finds an interval between [[6/5]] and [[7/6]] which in the 7-limit is [[32/27]] and in the 13-limit is [[13/11]]. Then the vanish of the orwellisma implies [[49/48]], the large septimal diesis, is tuned to the same step as 36/35, so 49/48 is also split into two so that the system also finds an interval between 7/6 and 8/7 which in the 7-limit is 7/6 inflected down by a comma or 8/7 inflected up by a comma, and in the 13-limit is ~[[15/13]], so that it is clear this system naturally wants to be extended to and interpreted in the full 13-limit.
 [[Subgroup]]: 2.3.5.7