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2.3.7 subgroup: Difference between revisions

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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.  
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.  


=== Rank-2 temperaments ===
=== Commas and rank-2 temperaments ===
{{Main|Tour of regular temperaments#Clans defined by a 2.3.7 (za) comma}}
{{Main|Tour of regular temperaments#Clans defined by a 2.3.7 (za) comma}}
=== Semaphore ===
'''Semaphore''' temperament tempers out the comma [[49/48]] = S7 in the 2.3.7 subgroup, which equates [[8/7]] with [[7/6]], creating a single neutral semifourth. 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 696.230c (though most other optimizations tune this a few cents flatter); a chart of mistunings of simple intervals is below.
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Semaphore (49/48)
|-
! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings
|-
! Optimal tuning !! Deviation
|-
| 9/8 || 203.910 || 192.460 || -11.450
|-
| 8/7 || 231.174 || 251.885 || +20.711
|-
| 7/6 || 266.871 || 251.885 || -14.986
|-
| 9/7 || 435.084 || 444.345 || +9.261
|-
| 21/16 || 470.781 || 444.345 || -26.436
|-
| 4/3 || 498.045 || 503.770 || +5.725
|-
| 3/2 || 701.955 || 696.230 || -5.725
|-
| 32/21 || 729.219 || 755.655 || +26.436
|-
| 14/9 || 764.916 || 755.655 || -9.261
|-
| 12/7 || 933.129 || 948.115 || +14.986
|-
| 7/4 || 968.826 || 948.115 || -20.711
|-
| 16/9 || 996.090 || 1007.540 || +11.450
{{table notes|cols=4
| In 2.3.7-targeted DKW tuning
}}
|}
=== Archy ===
'''Archy''' temperament tempers out the comma [[64/63]] = S8 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 712.585c (though most other optimizations tune this a few cents flatter); a chart of mistunings of simple intervals is below.
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Archy (64/63)
|-
! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings
|-
! Optimal tuning !! Deviation
|-
| 9/8 || 203.910 || 225.171 || +21.261
|-
| 8/7 || 231.174 || 225.171 || -6.003
|-
| 7/6 || 266.871 || 262.244 || -4.627
|-
| 9/7 || 435.084 || 450.341 || +15.257
|-
| 21/16 || 470.781 || 487.415 || +16.634
|-
| 4/3 || 498.045 || 487.415 || -10.630
|-
| 3/2 || 701.955 || 712.585 || +10.630
|-
| 32/21 || 729.219 || 712.585 || -16.634
|-
| 14/9 || 764.916 || 749.659 || -15.257
|-
| 12/7 || 933.129 || 937.756 || +4.627
|-
| 7/4 || 968.826 || 974.829 || +6.003
|-
| 16/9 || 996.090 || 974.829 || -21.261
{{table notes|cols=4
| In 2.3.7-targeted DKW tuning
}}
|}
=== Gamelic ===
'''Gamelic''' temperament, better known as [[slendric]], tempers out the comma [[1029/1024]] = S7/S8 in the 2.3.7 subgroup, which splits the perfect fifth into three intervals of [[8/7]]. It is one of the most accurate temperaments of its simplicity.
The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 699.126c, and therefore ~8/7 to 233.042c; a chart of mistunings of simple intervals is below.
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Gamelic (1029/1024)
|-
! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings
|-
! Optimal tuning !! Deviation
|-
| 9/8 || 203.910 || 198.253 || -5.657
|-
| 8/7 || 231.174 || 233.042 || +1.868
|-
| 7/6 || 266.871 || 267.831 || +0.960
|-
| 9/7 || 435.084 || 431.295 || -3.789
|-
| 21/16 || 470.781 || 466.084 || -4.697
|-
| 4/3 || 498.045 || 500.874 || +2.829
|-
| 3/2 || 701.955 || 699.126 || -2.829
|-
| 32/21 || 729.219 || 733.916 || +4.697
|-
| 14/9 || 764.916 || 768.705 || +3.789
|-
| 12/7 || 933.129 || 932.169 || -0.960
|-
| 7/4 || 968.826 || 966.958 || -1.868
|-
| 16/9 || 996.090 || 1001.747 || +5.657
{{table notes|cols=4
| In 2.3.7-targeted DKW tuning
}}
|}


== Music ==
== Music ==
Retrieved from "https://en.xen.wiki/w/2.3.7_subgroup"