2.3.7 subgroup: Difference between revisions

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== Regular temperaments ==

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

A list of edos with progressively better tunings for the 2.3.7 subgroup: {{EDOs| 5, 12, 14, 17, 22, 31, 36, 77, 94, 130, 135, 171, 265, 306, 400, 571, 706, 1277 }} ~~and so on.~~

A list of edos with progressively better tunings for the 2.3.7 subgroup (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs| '''5''', 12, 14, 17, 22, 31, '''36''', 77, 94, 130, '''135''', 171, 265, 306, 400, '''571''', 706, '''1277''', … }}

Another list of edos which provides relatively good tunings for the 2.3.7 subgroup (relative error < 2.5%): {{EDOs| 36, 41, 77, 94, 99, 130, 135, 171, 207, 229, 265, 301, 306, 364, 400, 436, 441, 477, 494, 535, 571, 576, 607, 648, 665, 670, 701, 706, 742, 747, 783, 836, 841, 877, 913, 935, 971, 976, 1007, 1012, 1048, 1106, 1147, 1178, 1183, 1236, 1241, 1277 }} and so on.

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Semaphore tempers out the comma [[49/48]] {{S|7}} in the 2.3.7 subgroup, which equates [[8/7]] with [[7/6]], creating a single neutral semifourth which serves as the generator. Similarly to [[dicot]], semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup. From the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does, though the comma involved is half the size of dicot's [[25/24]].

Semaphore tempers out the comma [[49/48]] ({{S|7}}) in the 2.3.7 subgroup, which equates [[8/7]] with [[7/6]], creating a single neutral semifourth which serves as the generator. Similarly to [[dicot]], semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup. From the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does, though the comma involved is half the size of dicot's [[25/24]].

The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 696.230{{c}}; a chart of mistunings of simple intervals is below.

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Archy tempers out the comma [[64/63]] {{S|8}} in the 2.3.7 subgroup, which equates [[9/8]] with [[8/7]], and [[4/3]] with [[21/16]]. It serves as a septimal analogue of [[meantone]], favoring fifths sharp of just rather than flat.

Archy tempers out the comma [[64/63]] ({{S|8}}) in the 2.3.7 subgroup, which equates [[9/8]] with [[8/7]], and [[4/3]] with [[21/16]]. It serves as a septimal analogue of [[meantone]], favoring fifths sharp of just rather than flat.

The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 712.585{{c}} though most other optimizations tune this a few cents flatter; a chart of mistunings of simple intervals is below.

@@ Line 33: / Line 33: @@
 == Regular temperaments ==
 === Rank-1 temperaments (edos) ===
-A list of edos with progressively better tunings for the 2.3.7 subgroup: {{EDOs| 5, 12, 14, 17, 22, 31, 36, 77, 94, 130, 135, 171, 265, 306, 400, 571, 706, 1277 }} and so on.
+A list of edos with progressively better tunings for the 2.3.7 subgroup (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs| '''5''', 12, 14, 17, 22, 31, '''36''', 77, 94, 130, '''135''', 171, 265, 306, 400, '''571''', 706, '''1277''', … }}
 Another list of edos which provides relatively good tunings for the 2.3.7 subgroup (relative error < 2.5%): {{EDOs| 36, 41, 77, 94, 99, 130, 135, 171, 207, 229, 265, 301, 306, 364, 400, 436, 441, 477, 494, 535, 571, 576, 607, 648, 665, 670, 701, 706, 742, 747, 783, 836, 841, 877, 913, 935, 971, 976, 1007, 1012, 1048, 1106, 1147, 1178, 1183, 1236, 1241, 1277 }} and so on.
@@ Line 45: / Line 45: @@
 {{Main| Semaphore and godzilla }}
-Semaphore tempers out the comma [[49/48]] {{S|7}} in the 2.3.7 subgroup, which equates [[8/7]] with [[7/6]], creating a single neutral semifourth which serves as the generator. Similarly to [[dicot]], semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup. From the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does, though the comma involved is half the size of dicot's [[25/24]].
+Semaphore tempers out the comma [[49/48]] ({{S|7}}) in the 2.3.7 subgroup, which equates [[8/7]] with [[7/6]], creating a single neutral semifourth which serves as the generator. Similarly to [[dicot]], semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup. From the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does, though the comma involved is half the size of dicot's [[25/24]].
 The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 696.230{{c}}; a chart of mistunings of simple intervals is below.
@@ Line 85: / Line 85: @@
 {{Main| Superpyth }}
-Archy tempers out the comma [[64/63]] {{S|8}} in the 2.3.7 subgroup, which equates [[9/8]] with [[8/7]], and [[4/3]] with [[21/16]]. It serves as a septimal analogue of [[meantone]], favoring fifths sharp of just rather than flat.
+Archy tempers out the comma [[64/63]] ({{S|8}}) in the 2.3.7 subgroup, which equates [[9/8]] with [[8/7]], and [[4/3]] with [[21/16]]. It serves as a septimal analogue of [[meantone]], favoring fifths sharp of just rather than flat.
 The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 712.585{{c}} though most other optimizations tune this a few cents flatter; a chart of mistunings of simple intervals is below.