2.5.7 subgroup: Difference between revisions
note birds under rank 1 temperaments, split temperaments into comma-based (normal) & diesis-based (exo), add missing comma-based temperaments of note |
m →Commas & rank-2 temperaments: add a notable temp of 2.5.7 *not* supported by 31edo / birds |
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=== [[Rainy]] ([[2100875/2097152]]) === | === [[Rainy]] ([[2100875/2097152]]) === | ||
[[Rainy]] is related to a number of high-limit rank 3 temperaments such as [[valentine]], [[dwynwyn]] and [[tertiaseptal]] in rank 2, and [[eros]] and [[sophia]] in rank 3, though is good as a (rank 2) pure 2.5.7 temperament also. Its generator is [[~]][[256/245]] sharpened by approximately 1{{cent}} which acts as the square root of [[~]][[35/32]], the cube root of [[~]][[8/7]] and the fifth root of [[~]][[5/4]]. Four generators reaches an ambiguous interval whose 2.5.7 interpretation is rather complex, being 2048/1715[[~]]1225/1024, which is the starting point for extensions. The most simple extension, [[valentine]], interprets this interval as a flat [[~]][[6/5]] by tempering (6/5)/(1225/1024) = [[6144/6125]], and the generator as a sharp [[~]][[25/24]], so that [[3/2]] is found at 9 generators, that is, at ([[~]][[8/7]])<sup>3</sup>, so also tempering [[1029/1024|1029/1024 = S7/S8]] = (6/5)/(2048/1715). Meanwhile, a much more accurate and complex mapping is to find [[4/3]] at 22 generators octave reduced, which is the strategy taken by [[tertiaseptal]], notable as the very high-limit [[140edo|140]] & [[311edo|311]] temperament. These two mappings of 3 merge in [[31edo]] (which serves as a trivial tuning of tertiaseptal, as another tuning of tertiaseptal is 311edo - 140edo = [[171edo]], and 171 - 140 = 31). | [[Rainy]] is related to a number of high-limit rank 3 temperaments such as [[valentine]], [[dwynwyn]] and [[tertiaseptal]] in rank 2, and [[eros]] and [[sophia]] in rank 3, though is good as a (rank 2) pure 2.5.7 temperament also. Its generator is [[~]][[256/245]] sharpened by approximately 1{{cent}} which acts as the square root of [[~]][[35/32]], the cube root of [[~]][[8/7]] and the fifth root of [[~]][[5/4]]. Four generators reaches an ambiguous interval whose 2.5.7 interpretation is rather complex, being 2048/1715[[~]]1225/1024, which is the starting point for extensions. The most simple extension, [[valentine]], interprets this interval as a flat [[~]][[6/5]] by tempering (6/5)/(1225/1024) = [[6144/6125]], and the generator as a sharp [[~]][[25/24]], so that [[3/2]] is found at 9 generators, that is, at ([[~]][[8/7]])<sup>3</sup>, so also tempering [[1029/1024|1029/1024 = S7/S8]] = (6/5)/(2048/1715). Meanwhile, a much more accurate and complex mapping is to find [[4/3]] at 22 generators octave reduced, which is the strategy taken by [[tertiaseptal]], notable as the very high-limit [[140edo|140]] & [[311edo|311]] temperament. These two mappings of 3 merge in [[31edo]] (which serves as a trivial tuning of tertiaseptal, as another tuning of tertiaseptal is 311edo - 140edo = [[171edo]], and 171 - 140 = 31). | ||
=== 2.5.7[6 & 60] = 2.5.7-subgroup restriction of [[Waage]] ([[244140625/240945152]]) === | |||
This temperament sharpens [[~]][[28/25]] by 3.8{{cent}} to make it equal to 1\6 so that [[6edo]] is made a [[strongly consistent circle]] of 28/25's, so it is one of the [[6th-octave temperaments]]. It contrasts the close relation of the 2.5.7 subgroup to hexatonic structure in an intriguing way by contrasting it with ''equalized'' hexatonic structure, chosen to represent ~28/25. | |||
=== [[Cloudy]] ([[16807/16384]]) === | === [[Cloudy]] ([[16807/16384]]) === | ||