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").

== 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 ==

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=== Scales ===

== Regular temperaments ==

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=== Rank-1 temperaments (edos) ===

A list of edos with progressively better~~<sup>*</sup>~~ tunings for the 2.5.7 subgroup: {{EDOs| 6, 15, 16, 21, 25, ~~27,~~ 31, 68, ~~103, 134~~, 140, 171, 239, 379, 410, ~~550, 618,~~ 789~~, 5902~~, 6691, 7480, 8269, 9058, 9847 ~~}}, and so on.~~

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, … }}.

~~Another list of edos is those with progressively smaller relative error for the 2.5.7 subgroup: {{EDOs| 1~~, ~~2, 4, 6, 31, 379, 789, 103169~~ }}~~, and so on~~.

~~<sup>*</sup> in absolute DKW distance~~

~~=== Birds ===~~

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 ~~[[Quince clan|quince]]/~~[[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.

=== Rank-2 temperaments ===

@@ Line 10: / Line 10: @@
 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]].
+: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 ==
@@ Line 15: / Line 20: @@
 === Scales ===
-{{Todo|expand|inline=1}}
+{{Todo|complete section|inline=1}}
 == Regular temperaments ==
@@ Line 21: / Line 26: @@
 === Rank-1 temperaments (edos) ===
-A list of edos with progressively better<sup>*</sup> tunings for the 2.5.7 subgroup: {{EDOs| 6, 15, 16, 21, 25, 27, 31, 68, 103, 134, 140, 171, 239, 379, 410, 550, 618, 789, 5902, 6691, 7480, 8269, 9058, 9847 }}, and so on.
+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, … }}.
-Another list of edos is those with progressively smaller relative error for the 2.5.7 subgroup: {{EDOs| 1, 2, 4, 6, 31, 379, 789, 103169 }}, and so on.
-<sup>*</sup> in absolute DKW distance
-=== Birds ===
+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 [[Quince clan|quince]]/[[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.
 === Rank-2 temperaments ===