2.5.7 subgroup: Difference between revisions
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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"). | 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"). | ||
== Chords and harmony == | |||
The fundamental consonances of this subgroup may be taken to be [[4:7:10]] and [[14:20:35]]. The otonal chord is [[DR]] and the utonal chord is the inverse of the otonal chord, similar to [[4:5:6]] and [[10:12:15]] in the [[5-limit]]. They are bounded by [[5/2]] and the middle voices contrast by [[49/40]]. | |||
4:7:10 may be extended to [[4:7:10:13]], preserving the DR property and implying adding prime [[13/1|13]] to this subgroup. | |||
== Properties == | == Properties == | ||
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=== Scales === | === Scales === | ||
{{Todo| | {{Todo|complete section|inline=1}} | ||
== Regular temperaments == | == Regular temperaments == | ||
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=== Rank-1 temperaments (edos) === | === Rank-1 temperaments (edos) === | ||
A list of edos with progressively better | A list of edos with progressively better tunings for the 2.5.7 subgroup (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs| '''6''', 15, 16, 21, 25, '''31''', 99, 109, 140, 171, 208, 239, 348, '''379''', 410, '''789''', 6691, 7480, 8269, 9058, 9847, … }}. | ||
As [[31edo]] is very strong in the 2.5.7 subgroup so that it is a [[weakly consistent circle]] of [[5/4]]'s and [[7/4]]'s (and thus [[8/5]]'s and [[8/7]]'s) and a [[strongly consistent circle]] of [[35/32]]'s (and thus [[64/35]]'s), it makes sense for those interested in high-complexity [[fractional-octave temperaments]] to consider [[31st-octave temperaments]] (temperaments with a 1\31 [[period]]) that preserve this representation of 2.5.7, which can be seen as combining the simplificatory logics of [[didacus]], [[rainy]] and [[mercy]], which is the 2.5.7-subgroup restriction of [[miracle]]. See [[31st-octave temperaments #Birds]] for details on the canonical extension of it to the full [[19-limit]] that utilises 31edo's good approximation of the interval [[11/9]] and of the 13:17:19 chord. | |||
As [[31edo]] is very strong in the 2.5.7 subgroup so that it is a [[weakly consistent circle]] of [[5/4]]'s and [[7/4]]'s (and thus [[8/5]]'s and [[8/7]]'s) and a [[strongly consistent circle]] of [[35/32]]'s (and thus [[64/35]]'s), it makes sense for those interested in high-complexity [[fractional-octave temperaments]] to consider [[31st-octave temperaments]] (temperaments with a 1\31 [[period]]) that preserve this representation of 2.5.7, which can be seen as combining the simplificatory logics of [[didacus]], [[rainy]] and | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{{Main|Tour of regular temperaments#Clans defined by a 2.5.7 (yaza nowa) comma}} | {{Main|Tour of regular temperaments#Clans defined by a 2.5.7 (yaza nowa) comma}} | ||
Some important [[diesis|dieses]] in the 2.5.7 subgroup are: | |||
* [[128/125|2/(5/4)<sup>3</sup>]], the [[enharmonic diesis]] (or just "diesis"), leading to [[augmented]] if tempered out. | |||
* [[16807/16384|2/(8/7)<sup>5</sup>]], the [[cloudy comma]], leading to [[cloudy]] if tempered, which can be likened in structural role to the enharmonic diesis but for prime 7 instead of prime 5. Equating it with the enharmonic diesis to form a general-purpose diesis results in [[rainy]] temperament, as their difference is the [[rainy comma]]. | |||
* [[50/49|(10/7)/(7/5)]], the [[septimal tritonic diesis]], leading to [[jubilic]] if tempered out. | |||
* [[256/245|(8/7)<sup>2</sup>/(5/4)]], which can be likened in structural role to the septimal tritonic diesis, leading to [[bapbo clan|bapbo]] if tempered out, considered below: | |||
==== Didacus (3136/3125) ==== | ==== Didacus (3136/3125) ==== | ||
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<nowiki />* In 2.5.7-targeted DKW tuning | <nowiki />* In 2.5.7-targeted DKW tuning | ||
==== | ==== Exotemperaments ==== | ||
The [[bapbo clan|bapbo]] [[exotemperament]] tempers out [[256/245]] = S6⋅S8 = S4/S7, equating [[35/32]] with [[8/7]], and making [[5/4]] the square of 8/7, which hence serves as the generator. This comma is over 70 cents, and pathologically equates the outer edges of the clusters around 1\6 and 5\6 to each other without also equating the middle interval with them (if this is done, the tuning reduces to 6edo). Therefore, bapbo can safely be considered an exotemperament – perhaps an analogue to [[mavila]]. | The [[bapbo clan|bapbo]] [[exotemperament]] tempers out [[256/245]] = S6⋅S8 = S4/S7, equating [[35/32]] with [[8/7]], and making [[5/4]] the square of 8/7, which hence serves as the generator. This comma is over 70 cents, and pathologically equates the outer edges of the clusters around 1\6 and 5\6 to each other without also equating the middle interval with them (if this is done, the tuning reduces to 6edo). Therefore, bapbo can safely be considered an exotemperament – perhaps an analogue to [[mavila]]. | ||