Tetracot family: Difference between revisions

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Octacot ~~cuts~~ the ~~Gordian knot of deciding~~ between the [[#Monkey|monkey]] and [[#Bunya|bunya]] mappings for 7 by cutting the generator in half ~~and splitting the difference~~. It adds [[245/243]] to the normal comma list, and also tempers out [[2401/2400]]. It may also be described as {{nowrap| 41 & 68 }}. [[68edo]] or [[109edo]] can be used as tunings, as can (5/2)<sup>1/18</sup>, which gives just major thirds. Another tuning is [[150edo]], which has a generator, 11\150, of exactly 88 cents. This relates octacot to the [[88cET]] non-octave temperament, which like [[Carlos Alpha]] arguably makes more sense viewed as part of a rank-2 temperament with octaves rather than rank-1 without them.

Octacot splits the difference between the [[#Monkey|monkey]] and [[#Bunya|bunya]] mappings for 7 by cutting the generator in half. It adds [[245/243]] to the normal comma list, and also tempers out [[2401/2400]]. It may also be described as {{nowrap| 41 & 68 }}. [[68edo]] or [[109edo]] can be used as tunings, as can (5/2)<sup>1/18</sup>, which gives just major thirds. Another tuning is [[150edo]], which has a generator, 11\150, of exactly 88 cents. This relates octacot to the [[88cET]] non-octave temperament, which like [[Carlos Alpha]] arguably makes more sense viewed as part of a rank-2 temperament with octaves rather than rank-1 without them.

Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas. Generators of 3\41, 8\109 and 11\150 (88 cents) are all good choices for the 7, 11 and 13 limits.

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=== 13-limit ===

The canonical mapping finds 13/8 at +15 generators rather than using the regular tetracot mapping, in order to find [[15/13]] as being half of [[4/3]].

Subgroup: 2.3.5.7.11.13

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=== Weasly ===

The alternative extension uses the same mapping of 13 as in tetracot, though many other intervals of 13 take more generators to reach as a result.

Subgroup: 2.3.5.7.11.13

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==== 17-limit ====

Subgroup: 2.3.5.7.11.13

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 85/84, 99/98, 144/143, 243/242

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==== 19-limit ====

Subgroup: 2.3.5.7.11.13

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 50/49, 85/84, 99/98, 144/143, 190/189, 243/242

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* CWE: ~7/5 = 600.000{{c}}, ~10/9 = 175.593{{c}}

Badness (Sintel): 1.48

@@ Line 459: / Line 459: @@
 {{See also| Chords of octacot }}
-Octacot cuts the Gordian knot of deciding between the [[#Monkey|monkey]] and [[#Bunya|bunya]] mappings for 7 by cutting the generator in half and splitting the difference. It adds [[245/243]] to the normal comma list, and also tempers out [[2401/2400]]. It may also be described as {{nowrap| 41 & 68 }}. [[68edo]] or [[109edo]] can be used as tunings, as can (5/2)<sup>1/18</sup>, which gives just major thirds. Another tuning is [[150edo]], which has a generator, 11\150, of exactly 88 cents. This relates octacot to the [[88cET]] non-octave temperament, which like [[Carlos Alpha]] arguably makes more sense viewed as part of a rank-2 temperament with octaves rather than rank-1 without them.
+Octacot splits the difference between the [[#Monkey|monkey]] and [[#Bunya|bunya]] mappings for 7 by cutting the generator in half. It adds [[245/243]] to the normal comma list, and also tempers out [[2401/2400]]. It may also be described as {{nowrap| 41 & 68 }}. [[68edo]] or [[109edo]] can be used as tunings, as can (5/2)<sup>1/18</sup>, which gives just major thirds. Another tuning is [[150edo]], which has a generator, 11\150, of exactly 88 cents. This relates octacot to the [[88cET]] non-octave temperament, which like [[Carlos Alpha]] arguably makes more sense viewed as part of a rank-2 temperament with octaves rather than rank-1 without them.
 Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas. Generators of 3\41, 8\109 and 11\150 (88 cents) are all good choices for the 7, 11 and 13 limits.
@@ Line 791: / Line 791: @@
 === 13-limit ===
+The canonical mapping finds 13/8 at +15 generators rather than using the regular tetracot mapping, in order to find [[15/13]] as being half of [[4/3]].
 Subgroup: 2.3.5.7.11.13
@@ Line 837: / Line 839: @@
 === Weasly ===
 {{Todo|review|unify precision}}
+The alternative extension uses the same mapping of 13 as in tetracot, though many other intervals of 13 take more generators to reach as a result.
 Subgroup: 2.3.5.7.11.13
@@ Line 852: / Line 856: @@
 ==== 17-limit ====
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17
 Comma list: 50/49, 85/84, 99/98, 144/143, 243/242
@@ Line 867: / Line 871: @@
 ==== 19-limit ====
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17.19
 Comma list: 50/49, 85/84, 99/98, 144/143, 190/189, 243/242
@@ Line 877: / Line 881: @@
 * CWE: ~7/5 = 600.000{{c}}, ~10/9 = 175.593{{c}}
-{{Optimal ET sequence|legend=0| 14c, 20cdehh, 34dh, 48 }}
+{{Optimal ET sequence|legend=0| 14c, 34dh, 48 }}
 Badness (Sintel): 1.48