1156/1155: Difference between revisions

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== Temperaments ==

Tempering out this comma in the 17-limit results in the rank-6 '''quadrantonismic temperament''', or in the 2.3.5.7.11.17 subgroup, the rank-5 '''quadrantonic temperament'''. In either case [[35/33]] is split into two equal parts, each representing 34/33~35/34, and enables [[quadrantonismic chords]]. If [[9801/9800]] is also added to the comma list, the quartertone above becomes literally a quarter of [[9/8]] and is tuned exactly middle of [[33/32]], the undecimal quartertone, and [[36/35]], the septimal quartertone.

[[Tempering out]] this comma in the 17-limit results in the rank-6 '''quadrantonismic temperament''', or in the 2.3.5.7.11.17 subgroup, the rank-5 '''quadrantonic temperament'''. In either case [[35/33]] is split into two equal parts, each representing 34/33~35/34, and enables [[quadrantonismic chords]]. If [[9801/9800]] is also added to the comma list, the quartertone above becomes literally a quarter of [[9/8]] and is tuned exactly middle of [[33/32]], the undecimal quartertone, and [[36/35]], the septimal quartertone.

=== Quadrantonic ===

[[Subgroup]]: 2.3.5.7.11.17

[[~~Sval~~]] [[mapping]]: <br>

[[Subgroup val|Subgroup-val]] [[mapping]]: <br>

{| class="right-all"

|-

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|}

: ~~sval~~ mapping generators: ~2, ~3, ~5, ~7, ~34/11

: mapping generators: ~2, ~3, ~5, ~7, ~34/11

[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 701.9948, ~5/4 = 386.3991, ~7/4 = 968.9508, ~17/11 = 752.9186

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9948{{c}}, ~5/4 = 386.3991{{c}}, ~7/4 = 968.9508{{c}}, ~17/11 = 752.9186{{c}}

{{Optimal ET sequence|legend=1| 17cg, 19eg, 22, 27eg, 39dg, 43, 46, 65d, 68, 72, 118, 171, 183, 239, 282, 301, 311, 400, 422, 472, 494, 894, 1012g, 1205, 1388 }}

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: mapping generators: ~2, ~3, ~5, ~7, ~34/11, ~13

[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 701.9948, ~5/4 = 386.3991, ~7/4 = 968.9508, ~17/11 = 752.9186, ~13/8

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9948{{c}}, ~5/4 = 386.3991{{c}}, ~7/4 = 968.9508{{c}}, ~17/11 = 752.9186{{c}}, ~13/8

{{Optimal ET sequence|legend=1| 17cg, 19eg, 22, 26, 27eg, 29g, 39dfg, 43, 46, 65d, 68, 72, 111, 121, 140, 171, 183, 217, 282, 301, 311, 354, 400, 422, 494, 894, 1012g, 1133, 1205, 1506g, 1627e }}

@@ Line 19: / Line 19: @@
 == Temperaments ==
-Tempering out this comma in the 17-limit results in the rank-6 '''quadrantonismic temperament''', or in the 2.3.5.7.11.17 subgroup, the rank-5 '''quadrantonic temperament'''. In either case [[35/33]] is split into two equal parts, each representing 34/33~35/34, and enables [[quadrantonismic chords]]. If [[9801/9800]] is also added to the comma list, the quartertone above becomes literally a quarter of [[9/8]] and is tuned exactly middle of [[33/32]], the undecimal quartertone, and [[36/35]], the septimal quartertone.
+[[Tempering out]] this comma in the 17-limit results in the rank-6 '''quadrantonismic temperament''', or in the 2.3.5.7.11.17 subgroup, the rank-5 '''quadrantonic temperament'''. In either case [[35/33]] is split into two equal parts, each representing 34/33~35/34, and enables [[quadrantonismic chords]]. If [[9801/9800]] is also added to the comma list, the quartertone above becomes literally a quarter of [[9/8]] and is tuned exactly middle of [[33/32]], the undecimal quartertone, and [[36/35]], the septimal quartertone.
 === Quadrantonic ===
 [[Subgroup]]: 2.3.5.7.11.17
-[[Sval]] [[mapping]]: <br>
+[[Subgroup val|Subgroup-val]] [[mapping]]: <br>
 {| class="right-all"
 |-
@@ Line 38: / Line 38: @@
 |}
-: sval mapping generators: ~2, ~3, ~5, ~7, ~34/11
+: mapping generators: ~2, ~3, ~5, ~7, ~34/11
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9948, ~5/4 = 386.3991, ~7/4 = 968.9508, ~17/11 = 752.9186
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9948{{c}}, ~5/4 = 386.3991{{c}}, ~7/4 = 968.9508{{c}}, ~17/11 = 752.9186{{c}}
 {{Optimal ET sequence|legend=1| 17cg, 19eg, 22, 27eg, 39dg, 43, 46, 65d, 68, 72, 118, 171, 183, 239, 282, 301, 311, 400, 422, 472, 494, 894, 1012g, 1205, 1388 }}
@@ Line 65: / Line 65: @@
 : mapping generators: ~2, ~3, ~5, ~7, ~34/11, ~13
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9948, ~5/4 = 386.3991, ~7/4 = 968.9508, ~17/11 = 752.9186, ~13/8
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9948{{c}}, ~5/4 = 386.3991{{c}}, ~7/4 = 968.9508{{c}}, ~17/11 = 752.9186{{c}}, ~13/8
 {{Optimal ET sequence|legend=1| 17cg, 19eg, 22, 26, 27eg, 29g, 39dfg, 43, 46, 65d, 68, 72, 111, 121, 140, 171, 183, 217, 282, 301, 311, 354, 400, 422, 494, 894, 1012g, 1133, 1205, 1506g, 1627e }}