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 |
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The '''2.5.7 subgroup''', or the '''no-threes 7-limit''' | The '''2.5.7 subgroup''', or the '''no-threes 7-limit''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 5, and 7 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 5, and 7. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[7/4]], [[8/7]], [[7/5]], [[28/25]], [[35/32]], and so on. | ||
The 2.5.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 3. Its simplest expansion is the [[2.5.7.11 subgroup]], which adds prime 11. | The 2.5.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 3. Its simplest expansion is the [[2.5.7.11 subgroup]], which adds prime 11. | ||
A notable subset of the 2.5.7 subgroup is the 1.5.7 [[tonality diamond]], comprised of all intervals in which 1, 5 and 7 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. The complete list of intervals in the 1.5.7 tonality diamond within the octave is [[1/1]], [[8/7]], [[5/4]], [[7/5]], [[10/7]], [[8/5]], [[7/4]], and [[2/1]]. | A notable subset of the 2.5.7 subgroup is the 1.5.7 [[tonality diamond]], comprised of all intervals in which 1, 5 and 7 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. The complete list of intervals in the 1.5.7 tonality diamond (which is the 7-odd-limit (1.3.5.7) with intervals of 3 removed) within the octave is [[1/1]], [[8/7]], [[5/4]], [[7/5]], [[10/7]], [[8/5]], [[7/4]], and [[2/1]]. | ||
Another such subset is the 1.5.7.25.35 tonality diamond, which adds the following intervals to the previous list: [[25/16]], [[25/14]], [[35/32]], [[64/35]], [[28/25]], and [[32/25]]. | Another such subset is the 1.5.7.25.35 tonality diamond, which adds the following intervals to the previous list: [[25/16]], [[25/14]], [[35/32]], [[64/35]], [[28/25]], and [[32/25]]. | ||
When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 5 and 7, which can be represented in a 2-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]]. | When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 5 and 7, which can be represented in a 2-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]]. | ||
In [[color notation]], this subgroup may be called '''yaza nowa''', which means that it is the intersection of 2.3.5 and 2.3.7 ("yaza"), but without 3 ("nowa"). | |||
== Properties == | == Properties == | ||
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| 64/35 || 1044.860 || 1041.573 || -3.288 | | 64/35 || 1044.860 || 1041.573 || -3.288 | ||
|} | |} | ||
<nowiki />* In 2.5.7-targeted DKW tuning | |||
=== [[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 relates 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]]) === | ||
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| 64/35 || 1044.860 || 1018.297 || -26.563 | | 64/35 || 1044.860 || 1018.297 || -26.563 | ||
|} | |} | ||
<nowiki />* In 2.5.7-targeted DKW tuning | |||
=== [[Jubilic]] ([[50/49]]) === | === [[Jubilic]] ([[50/49]]) === | ||
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| 64/35 || 1044.860 || 1029.995 || -14.865 | | 64/35 || 1044.860 || 1029.995 || -14.865 | ||
|} | |} | ||
<nowiki />* In 2.5.7-targeted DKW tuning | |||
== Dieses & rank-2 [[exotemperament]]s == | == Dieses & rank-2 [[exotemperament]]s == | ||
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| 64/35 || 1044.860 || 994.614 || -50.247 | | 64/35 || 1044.860 || 994.614 || -50.247 | ||
|} | |} | ||
<nowiki />* In 2.5.7-targeted DKW tuning | |||
== Music == | == Music == | ||