2.5.7 subgroup: Difference between revisions
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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. | ||
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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]]) === | ||
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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 == | ||