Tetracot family: Difference between revisions

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The parent of the '''tetracot family''' is [[tetracot]], the 5-limit ~~temperament [[tempering out~~]] [[~~20000/19683]] = {{monzo~~| ~~5 -9 4 }}, the minimal diesis or tetracot comma. The generator is [[~~~]][[~~10/9~~]]~~, and that four of these give~~ [[~]][[~~3/2~~]]~~. In fact, (10/9)4 =~~ 20000/19683 ~~× 3/2. We also have (10/9)9~~ = ~~(20000/19683)2 ×~~ 5~~/2. From this it is evident we should flatten the generator a bit, and [[34edo]] does this and makes for a recommendable tuning. Another possibility is to use (5/2)1/~~9~~ for a generator. The 13-note [[mos]] gives enough space for eight triads, with the 20-note mos supplying many more.~~

The parent of the '''tetracot family''' is [[tetracot]], the [[5-limit]] [[regular temperament|temperament]] [[tempering out]] the [[tetracot comma]] ([[ratio]]: 20000/19683, {{monzo|legend=1| 5 -9 4 }}).

~~The name comes from members of the Araucaria family of conifers, which have four cotyledons (though sometimes these are fused~~).

== Tetracot ==

The [[generator]] of tetracot is [[~]][[10/9]], and that four of these give [[~]][[3/2]]. In fact, (10/9)4 = (20000/19683)⋅(3/2). We also have (10/9)9 = (20000/19683)2⋅(5/2). From this it is evident we should flatten the generator a bit, and [[34edo]] does this and makes for a recommendable tuning. Another possibility is to use (5/2)1/9 for a generator. The 13-note [[mos]] gives enough space for eight triads, with the 20-note mos supplying many more.

The name comes from members of the Araucaria family of conifers, which have four cotyledons (though sometimes these are fused).

[[Subgroup]]: 2.3.5

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=== Overview to extensions ===

The second comma of the [[~~Normal lists~~|normal comma list]] defines which 7-limit family member we are looking at.

The second comma of the [[normal forms|normal comma list]] defines which 7-limit family member we are looking at.

* [[875/864]], the keema, gives monkey;

* 179200/177147 (or equivalently [[225/224]]) gives bunya;

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==== Monkey and bunya ====

''Monkey'' tempers out the keema. The keema, 875/864, is the amount by which three just minor thirds fall short of 7/4, and tells us the ~7/4 of monkey is reached by three minor thirds in succession. It can be described as the 34 &~~amp;~~ 41 temperament. [[41edo]] is an excellent tuning for monkey, and has the effect of making monkey identical to bunya with the same tuning.

Monkey tempers out the keema. The keema, 875/864, is the amount by which three just minor thirds fall short of 7/4, and tells us the ~7/4 of monkey is reached by three minor thirds in succession. It can be described as the {{nowrap| 34 & 41 }} temperament. [[41edo]] is an excellent tuning for monkey, and has the effect of making monkey identical to bunya with the same tuning.

''Bunya'' adds 225/224 to the list of commas and may be described as the 34d &~~amp;~~ 41 temperament. 41edo can again be used as a tuning, in which case it is the same as monkey. However an excellent alternative is 141/26 as a generator, giving just ~7's and an improved value for ~5, at the cost of a slightly sharper, but still less than a cent sharp, fifth. Octave stretching, if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.

Bunya adds 225/224 to the list of commas and may be described as the {{nowrap| 34d & 41 }} temperament. 41edo can again be used as a tuning, in which case it is the same as monkey. However an excellent alternative is 141/26 as a generator, giving just ~7's and an improved value for ~5, at the cost of a slightly sharper, but still less than a cent sharp, fifth. Octave stretching, if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.

Since the generator in all cases is between 10/9 and 11/10, it is natural to extend these temperaments to the 11-limit by tempering out (10/9)/(11/10) = [[100/99]]. This gives 11-limit monkey and 11-limit bunya. Again, 41edo can be used as a tuning, making the two identical, which is also the case if we turn to the 2.3.5.11 subgroup temperament, dispensing with 7. However, 11-limit bunya, like 7-limit bunya, profits a little from a slightly sharper fifth, such as the 141/26 generator supplies, or even sharper yet, as for instance by the val {{val| 355 563 823 997 1230 }}, with a 52/355 generator.

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Optimal tunings:

* CTE: ~2 = 1\1, ~10/9 = 176.818

* POTE:~2 = 1\1, ~10/9 = 177.203

* POTE: ~2 = 1\1, ~10/9 = 177.203

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Octacot cuts the Gordian knot of deciding between the monkey and 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 41 &~~amp;~~ 68. [[68edo]] or [[109edo]] can be used as tunings, as can (5/2)1/18, 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 cuts the Gordian knot of deciding between the monkey and 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)1/18, 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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Optimal tunings:

* CTE: ~2 = 1\1, ~21/20 = 88.023

* POTE:~2 = 1\1, ~21/20 = 88.076

* POTE: ~2 = 1\1, ~21/20 = 88.076

{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109, 150, 259, 409bc, 668bbcc }}

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Optimal tunings:

* CTE: ~2 = 1\1, ~21/20 = 87.910

* POTE ~2 = 1\1, ~21/20 = 87.975

* POTE: ~2 = 1\1, ~21/20 = 87.975

Optimal ET sequence: {{Optimal ET sequence| 14c, 27e, 41, 150ee, 191ee, 232cee }}

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Optimal tunings:

* CTE: ~2 = 1\1, ~21/20 = 87.926

* POTE ~2 = 1\1, ~21/20 = 88.106

* POTE: ~2 = 1\1, ~21/20 = 88.106

Optimal ET sequence: {{Optimal ET sequence| 14c, 27e, 41, 150eef, 191eeff, 232ceeff }}

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Optimal tunings:

* CTE: ~2 = 1\1, ~18/17 = 87.842

* POTE ~2 = 1\1, ~18/17 = 88.102

* POTE: ~2 = 1\1, ~18/17 = 88.102

Optimal ET sequence: {{Optimal ET sequence| 14c, 27eg, 41 }}

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Optimal tunings:

* CTE: ~2 = 1\1, ~18/17 = 87.866

* POTE ~2 = 1\1, ~18/17 = 88.111

* POTE: ~2 = 1\1, ~18/17 = 88.111

Optimal ET sequence: {{Optimal ET sequence| 14c, 27eg, 41 }}

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Optimal tunings:

* CTE: ~2 = 1\1, ~21/20 = 88.090

* POTE ~2 = 1\1, ~21/20 = 88.179

* POTE: ~2 = 1\1, ~21/20 = 88.179

Optimal ET sequence: {{Optimal ET sequence| 14cf, 27e, 41f, 68ef, 109eff }}

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=== Devisemi (2.3.5.7.19) ===

==== 2.3.5.19 subgroup ====

Subgroup: 2.3.5.19

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Optimal tunings:

* CTE: ~2 = 1\1, ~28/27 = 58.648

* POTE:~2 = 1\1, ~28/27 = 58.675

* POTE: ~2 = 1\1, ~28/27 = 58.675

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Optimal tunings:

* CTE: ~2 = 1\1, ~28/27 = 58.602

* POTE ~2 = 1\1, ~28/27 = 58.665

* POTE: ~2 = 1\1, ~28/27 = 58.665

Optimal ET sequence: {{Optimal ET sequence| 20cde, 41 }}

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Optimal tunings:

* CTE: ~2 = 1\1, ~27/26 = 58.551

* POTE ~2 = 1\1, ~27/26 = 58.639

* POTE: ~2 = 1\1, ~27/26 = 58.639

Optimal ET sequence: {{Optimal ET sequence| 20cdef, 21cdef, 41 }}

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== Byhearted ==

~~: ''~~See also~~: [[Jubilismic clan]] and [[No-fives subgroup temperaments #Skwares~~|No-fives subgroup temperaments #Byhearted~~]]''~~

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~7/5, ~10/9

: ~~Mapping~~ generators: ~7/5, ~10/9

Optimal tuning (CTE): ~7/5 = 1\2, ~10/9 = 175.472

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 {{Technical data page}}
-The parent of the '''tetracot family''' is [[tetracot]], the 5-limit temperament [[tempering out]] [[20000/19683]] = {{monzo| 5 -9 4 }}, the minimal diesis or tetracot comma. The generator is [[~]][[10/9]], and that four of these give [[~]][[3/2]]. In fact, (10/9)<sup>4</sup> = 20000/19683 × 3/2. We also have (10/9)<sup>9</sup> = (20000/19683)<sup>2</sup> × 5/2. From this it is evident we should flatten the generator a bit, and [[34edo]] does this and makes for a recommendable tuning. Another possibility is to use (5/2)<sup>1/9</sup> for a generator. The 13-note [[mos]] gives enough space for eight triads, with the 20-note mos supplying many more.
+The parent of the '''tetracot family''' is [[tetracot]], the [[5-limit]] [[regular temperament|temperament]] [[tempering out]] the [[tetracot comma]] ([[ratio]]: 20000/19683, {{monzo|legend=1| 5 -9 4 }}).
-The name comes from members of the Araucaria family of conifers, which have four cotyledons (though sometimes these are fused).
 == Tetracot ==
 {{Main| Tetracot }}
+The [[generator]] of tetracot is [[~]][[10/9]], and that four of these give [[~]][[3/2]]. In fact, (10/9)<sup>4</sup> = (20000/19683)⋅(3/2). We also have (10/9)<sup>9</sup> = (20000/19683)<sup>2</sup>⋅(5/2). From this it is evident we should flatten the generator a bit, and [[34edo]] does this and makes for a recommendable tuning. Another possibility is to use (5/2)<sup>1/9</sup> for a generator. The 13-note [[mos]] gives enough space for eight triads, with the 20-note mos supplying many more.
+The name comes from members of the Araucaria family of conifers, which have four cotyledons (though sometimes these are fused).
 [[Subgroup]]: 2.3.5
@@ Line 26: / Line 28: @@
 === Overview to extensions ===
-The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at.
+The second comma of the [[normal forms|normal comma list]] defines which 7-limit family member we are looking at.
 * [[875/864]], the keema, gives monkey;
 * 179200/177147 (or equivalently [[225/224]]) gives bunya;
@@ Line 32: / Line 34: @@
 ==== Monkey and bunya ====
-''Monkey'' tempers out the keema. The keema, 875/864, is the amount by which three just minor thirds fall short of 7/4, and tells us the ~7/4 of monkey is reached by three minor thirds in succession. It can be described as the 34 &amp; 41 temperament. [[41edo]] is an excellent tuning for monkey, and has the effect of making monkey identical to bunya with the same tuning.
+Monkey tempers out the keema. The keema, 875/864, is the amount by which three just minor thirds fall short of 7/4, and tells us the ~7/4 of monkey is reached by three minor thirds in succession. It can be described as the {{nowrap| 34 & 41 }} temperament. [[41edo]] is an excellent tuning for monkey, and has the effect of making monkey identical to bunya with the same tuning.
-''Bunya'' adds 225/224 to the list of commas and may be described as the 34d &amp; 41 temperament. 41edo can again be used as a tuning, in which case it is the same as monkey. However an excellent alternative is 14<sup>1/26</sup> as a generator, giving just ~7's and an improved value for ~5, at the cost of a slightly sharper, but still less than a cent sharp, fifth. Octave stretching, if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.
+Bunya adds 225/224 to the list of commas and may be described as the {{nowrap| 34d & 41 }} temperament. 41edo can again be used as a tuning, in which case it is the same as monkey. However an excellent alternative is 14<sup>1/26</sup> as a generator, giving just ~7's and an improved value for ~5, at the cost of a slightly sharper, but still less than a cent sharp, fifth. Octave stretching, if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.
 Since the generator in all cases is between 10/9 and 11/10, it is natural to extend these temperaments to the 11-limit by tempering out (10/9)/(11/10) = [[100/99]]. This gives 11-limit monkey and 11-limit bunya. Again, 41edo can be used as a tuning, making the two identical, which is also the case if we turn to the 2.3.5.11 subgroup temperament, dispensing with 7. However, 11-limit bunya, like 7-limit bunya, profits a little from a slightly sharper fifth, such as the 14<sup>1/26</sup> generator supplies, or even sharper yet, as for instance by the val {{val| 355 563 823 997 1230 }}, with a 52/355 generator.
@@ Line 248: / Line 250: @@
 Optimal tunings:
 * CTE: ~2 = 1\1, ~10/9 = 176.818
-* POTE:~2 = 1\1, ~10/9 = 177.203
+* POTE: ~2 = 1\1, ~10/9 = 177.203
 {{Optimal ET sequence|legend=1| 7, 20c, 27, 34d, 61d, 95dd }}
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 {{See also| Chords of octacot }}
-Octacot cuts the Gordian knot of deciding between the monkey and 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 41 &amp; 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 cuts the Gordian knot of deciding between the monkey and 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.
 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 461: / Line 463: @@
 Optimal tunings:
 * CTE: ~2 = 1\1, ~21/20 = 88.023
-* POTE:~2 = 1\1, ~21/20 = 88.076
+* POTE: ~2 = 1\1, ~21/20 = 88.076
 {{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109, 150, 259, 409bc, 668bbcc }}
@@ Line 476: / Line 478: @@
 Optimal tunings:
 * CTE: ~2 = 1\1, ~21/20 = 87.910
-* POTE ~2 = 1\1, ~21/20 = 87.975
+* POTE: ~2 = 1\1, ~21/20 = 87.975
 Optimal ET sequence: {{Optimal ET sequence| 14c, 27e, 41, 150ee, 191ee, 232cee }}
@@ Line 491: / Line 493: @@
 Optimal tunings:
 * CTE: ~2 = 1\1, ~21/20 = 87.926
-* POTE ~2 = 1\1, ~21/20 = 88.106
+* POTE: ~2 = 1\1, ~21/20 = 88.106
 Optimal ET sequence: {{Optimal ET sequence| 14c, 27e, 41, 150eef, 191eeff, 232ceeff }}
@@ Line 506: / Line 508: @@
 Optimal tunings:
 * CTE: ~2 = 1\1, ~18/17 = 87.842
-* POTE ~2 = 1\1, ~18/17 = 88.102
+* POTE: ~2 = 1\1, ~18/17 = 88.102
 Optimal ET sequence: {{Optimal ET sequence| 14c, 27eg, 41 }}
@@ Line 521: / Line 523: @@
 Optimal tunings:
 * CTE: ~2 = 1\1, ~18/17 = 87.866
-* POTE ~2 = 1\1, ~18/17 = 88.111
+* POTE: ~2 = 1\1, ~18/17 = 88.111
 Optimal ET sequence: {{Optimal ET sequence| 14c, 27eg, 41 }}
@@ Line 536: / Line 538: @@
 Optimal tunings:
 * CTE: ~2 = 1\1, ~21/20 = 88.090
-* POTE ~2 = 1\1, ~21/20 = 88.179
+* POTE: ~2 = 1\1, ~21/20 = 88.179
 Optimal ET sequence: {{Optimal ET sequence| 14cf, 27e, 41f, 68ef, 109eff }}
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 === Devisemi (2.3.5.7.19) ===
 ==== 2.3.5.19 subgroup ====
 Subgroup: 2.3.5.19
@@ Line 729: / Line 730: @@
 Optimal tunings:
 * CTE: ~2 = 1\1, ~28/27 = 58.648
-* POTE:~2 = 1\1, ~28/27 = 58.675
+* POTE: ~2 = 1\1, ~28/27 = 58.675
 {{Optimal ET sequence|legend=1| 41, 143d, 184, 225, 266c }}
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 Optimal tunings:
 * CTE: ~2 = 1\1, ~28/27 = 58.602
-* POTE ~2 = 1\1, ~28/27 = 58.665
+* POTE: ~2 = 1\1, ~28/27 = 58.665
 Optimal ET sequence: {{Optimal ET sequence| 20cde, 41 }}
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 Optimal tunings:
 * CTE: ~2 = 1\1, ~27/26 = 58.551
-* POTE ~2 = 1\1, ~27/26 = 58.639
+* POTE: ~2 = 1\1, ~27/26 = 58.639
 Optimal ET sequence: {{Optimal ET sequence| 20cdef, 21cdef, 41 }}
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 == Byhearted ==
-: ''See also: [[Jubilismic clan]] and [[No-fives subgroup temperaments #Skwares|No-fives subgroup temperaments #Byhearted]]''
+{{See also| No-fives subgroup temperaments #Byhearted }}
 [[Subgroup]]: 2.3.5.7
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 {{Mapping|legend=1| 2 2 2 3 | 0 4 9 9 }}
+: mapping generators: ~7/5, ~10/9
-: Mapping generators: ~7/5, ~10/9
 Optimal tuning (CTE): ~7/5 = 1\2, ~10/9 = 175.472