Tetracot family: Difference between revisions

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__FORCETOC__
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 dual of this comma is the wedgie {{multival| 4 9 5 }}, which tells us [[10/9]] is a generator, and that four of them 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 temperament [[tempering out]] [[20000/19683]] = {{monzo| 5 -9 4 }}, the minimal diesis or tetracot comma. The dual of this comma is the wedgie &lt;&lt;4 9 5||, which tells us 10/9 is a generator, and that four of them 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).
The name comes from members of the Araucaria family of conifers, which have four cotyledons (though sometimes these are fused).


= Tetracot =
= Tetracot =
Comma: 20000/19683
Comma list: 20000/19683


[[POTE generator]]: 176.160
[[POTE generator]]: ~10/9 = 176.160


Map: [&lt;1 1 1|, &lt;0 4 9|]
[[Mapping]]: [&lt;1 1 1|, &lt;0 4 9|]


EDOs: {{EDOs| 14c, 27, 34, 75, 109, 470b, 579b }}
{{Val list|legend=1| 14c, 27, 34, 75, 109, 470b, 579b }}


== Seven limit children ==
== Extensions ==
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding [[875/864]], the keema, gives monkey, and 179200/177147 (or equivalently [[225/224]]) gives bunya. Adding [[245/243]] gives octacot, which splits the generator in half.
The second comma of the [[Normal lists|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;
* [[245/243]] gives octacot, which splits the generator in half.


=== Monkey and Bunya ===
=== Monkey and Bunya ===
Monkey tempers out the keema and has a wedgie &lt;&lt;4 9 -15 5 -35 -60||. 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, if the vals in question are taken to be [[patent val]]s, meaning that ''n''×log<sub>2</sub>(prime) rounded to the nearest integer gives the mapping. [[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 34&amp;41 temperament, if the vals in question are taken to be [[patent val]]s, meaning that ''n''×log<sub>2</sub>(prime) rounded to the nearest integer gives the mapping. [[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 41&amp;75 temperament. It has &lt;&lt;4 9 26 5 30 35|| as a wedgie, and [[41edo]] can again be used as a tuning, in which case it is the same as monkey. However an excellent alternative is (14)^(1/26) as a generator, giving just 7s 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 41&amp;75 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 7s 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, &lt;&lt;4 9 -15 10 …|| and 11-limit banya, &lt;&lt;4 9 26 10...||. 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} temperament, dispensing with 7. However 11-limit bunya, like 7-limit bunya, profits a little from a slightly sharper fifth, such as the (14)^(1/26) generator supplies, or even sharper yet, as for instance by the val &lt;355 563 823 997 1230|, with a 52/355 generator.
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, {{multival| 4 9 -15 10 … }} and 11-limit banya, {{multival| 4 9 26 10 … }}. 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 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.


Since 16/13 is shy of (10/9)^2 by just 325/324, it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us &lt;&lt;4 9 -15 10 -2 …|| for 13-limit monkey and &lt;&lt;4 9 26 10 -2 …|| for 13-limit bunya. Once again, 41 is recommended as a tuning for monkey, while banyan can with advantage tune the fifth sharper: 17/116 as a generator with a fifth a cent and a half sharp or 11/75 with a fifth two cents sharp.
Since [[16/13]] is shy of (10/9)<sup>2</sup> by just [[325/324]], it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us {{multival| 4 9 -15 10 -2 … }} for 13-limit monkey and {{multival| 4 9 26 10 -2 … }} for 13-limit bunya. Once again, 41 is recommended as a tuning for monkey, while banyan can with advantage tune the fifth sharper: 17/116 as a generator with a fifth a cent and a half sharp or 11/75 with a fifth two cents sharp.


= Monkey =
= Monkey =
Commas: 875/864, 5120/5103
[[Comma list]]: 875/864, 5120/5103


[[POTE_tuning|POTE generator]]: 175.659
[[Mapping]]: [&lt;1 1 1 5|, &lt;0 4 9 -15|]


Map: [&lt;1 1 1 5|, &lt;0 4 9 -15|]
{{Multival|legend=1| 4 9 -15 5 -35 -60 }}


EDOs: {{EDOs| 7, 34, 41, 321cd }}
[[POTE generator]]: ~10/9 = 175.659


Badness: 0.0734
{{Val list|legend=1| 7, 34, 41, 321cd }}


==11-limit==
[[Badness]]: 0.0734
Commas: 100/99, 243/242, 385/384


[[POTE_tuning|POTE generator]]: 175.570
== 11-limit ==
Comma list: 100/99, 243/242, 385/384
 
Mapping: [&lt;1 1 1 5 2|, &lt;0 4 9 -15 10|]


Map: [&lt;1 1 1 5 2|, &lt;0 4 9 -15 10|]
POTE generator: ~10/9 = 175.570


EDOs: {{EDOs| 7, 34, 41, 123c }}
Vals: {{Val list| 7, 34, 41, 123c }}


Badness: 0.0388
Badness: 0.0388


== 13-limit ==
== 13-limit ==
Commas: 100/99, 105/104, 144/143, 243/242
Comma list: 100/99, 105/104, 144/143, 243/242


[[POTE_tuning|POTE generator]]: 175.622
Mapping: [&lt;1 1 1 5 2 4|, &lt;0 4 9 -15 10 -2|]


Map: [&lt;1 1 1 5 2 4|, &lt;0 4 9 -15 10 -2|]
POTE generator: ~10/9 = 175.622


EDOs: {{EDOs| 7, 34, 41 }}
Vals: {{Val list| 7, 34, 41 }}


Badness: 0.0284
Badness: 0.0284


= Bunya =
= Bunya =
Commas: 225/224, 15625/15309
[[Comma list]]: 225/224, 15625/15309
 
[[Mapping]]: [&lt;1 1 1 -1|, &lt;0 4 9 26|]


[[POTE_tuning|POTE generator]]: 175.741
{{Multival|legend=1| 4 9 26 5 30 35 }}


Map: [&lt;1 1 1 -1|, &lt;0 4 9 26|]
[[POTE generator]]: ~10/9 = 175.741


EDOs: {{EDOs| 34d, 41, 116, 157c, 198c }}
{{Val list|legend=1| 34d, 41, 116, 157c, 198c }}


Badness: 0.0629
[[Badness]]: 0.0629


==11-limit==
== 11-limit ==
Commas: 100/99, 225/224, 1344/1331
Comma list: 100/99, 225/224, 1344/1331


[[POTE_tuning|POTE generator]]: 175.777
Mapping: [&lt;1 1 1 -1 2|, &lt;0 4 9 26 10|]


Map: [&lt;1 1 1 -1 2|, &lt;0 4 9 26 10|]
POTE generator: ~10/9 = 175.777


EDOs: {{EDOs| 34d, 41, 116e, 157ce }}
Vals: {{Val list| 34d, 41, 116e, 157ce }}


Badness: 0.0313
Badness: 0.0313


== 13-limit ==
== 13-limit ==
Commas: 100/99, 144/143, 225/224, 243/242
Comma list: 100/99, 144/143, 225/224, 243/242


[[POTE_tuning|POTE generator]]: 175.886
Mapping: [&lt;1 1 1 -1 2 4|, &lt;0 4 9 26 10 -2|]


Map: [&lt;1 1 1 -1 2 4|, &lt;0 4 9 26 10 -2|]
POTE generator: ~10/9 = 175.886


EDOs: {{EDOs| 34d, 41, 75e, 116ef }}
Vals: {{Val list| 34d, 41, 75e, 116ef }}


Badness: 0.0249
Badness: 0.0249


= Modus =
= Modus =
Commas: 64/63, 4375/4374
Comma list: 64/63, 4375/4374
 
Mapping: [&lt;1 1 1 4|, &lt;0 4 9 -8|]


POTE generator: ~10/9 = 177.203
POTE generator: ~10/9 = 177.203


Map: [&lt;1 1 1 4|, &lt;0 4 9 -8|]
{{Val list|legend=1| 7, 27, 61d, 88bcd }}
 
EDOs: {{EDOs| 7, 27, 61d, 88bcd }}


Badness: 0.0682
Badness: 0.0682


== 11-limit ==
== 11-limit ==
Commas: 64/63, 100/99, 243/242
Comma list: 64/63, 100/99, 243/242
 
Mapping: [&lt;1 1 1 4 2|, &lt;0 4 9 -8 10|]


POTE generator: ~10/9 = 177.053
POTE generator: ~10/9 = 177.053


Map: [&lt;1 1 1 4 2|, &lt;0 4 9 -8 10|]
Vals: {{Val list| 7, 20ce, 27e, 34d, 61de }}
 
EDOs: {{EDOs| 7, 20ce, 27e, 34d, 61de }}


Badness: 0.0351
Badness: 0.0351


=== 13-limit ===
=== 13-limit ===
Commas: 64/63, 78/77, 100/99, 144/143
Comma list: 64/63, 78/77, 100/99, 144/143
 
Mapping: [&lt;1 1 1 4 2 4|, &lt;0 4 9 -8 10 -2|]


POTE generator: ~10/9 = 176.953
POTE generator: ~10/9 = 176.953


Map: [&lt;1 1 1 4 2 4|, &lt;0 4 9 -8 10 -2|]
Vals: {{Val list| 7, 27e, 34d, 61de }}
 
EDOs: {{EDOs| 7, 27e, 34d, 61de }}


Badness: 0.0238
Badness: 0.0238
Line 132: Line 138:
The error of 11 is about the same as that of Modus, but flat instead of sharp, and much more abundant. Since the other primes are all sharp, however, this leads to a much larger error for other intervals involving 11.
The error of 11 is about the same as that of Modus, but flat instead of sharp, and much more abundant. Since the other primes are all sharp, however, this leads to a much larger error for other intervals involving 11.


Commas: 55/54, 64/63, 363/350
Comma list: 55/54, 64/63, 363/350
 
Mapping: [&lt;1 1 1 4 3|, &lt;0 4 9 -8 3|]


POTE generator: ~10/9 = 177.200
POTE generator: ~10/9 = 177.200


Map: [&lt;1 1 1 4 3|, &lt;0 4 9 -8 3|]
Vals: {{Val list| 7, 20c, 27, 61de, 88bcde }}
 
EDOs: {{EDOs| 7, 20c, 27, 61de, 88bcde }}


Badness: 0.0631
Badness: 0.0631


=== 13-limit ===
=== 13-limit ===
Commas: 55/54, 64/63, 66/65, 143/140
Comma list: 55/54, 64/63, 66/65, 143/140
 
Mapping: [&lt;1 1 1 4 3 4|, &lt;0 4 9 -8 3 -2|]


POTE generator: ~10/9 = 177.197
POTE generator: ~10/9 = 177.197


Map: [&lt;1 1 1 4 3 4|, &lt;0 4 9 -8 3 -2|]
Vals: {{Val list| 7, 20c, 27, 61de, 88bcde }}
 
EDOs: {{EDOs| 7, 20c, 27, 61de, 88bcde }}


Badness: 0.039
Badness: 0.039


= Wollemia =
= Wollemia =
Commas: 126/125, 2240/2187
[[Comma list]]: 126/125, 2240/2187


POTE generator: ~10/9 = 177.357
[[Mapping]]: [&lt;1 1 1 0|, &lt;0 4 9 19|]


Map: [&lt;1 1 1 0|, &lt;0 4 9 19|]
{{Multival|legend=1| 4 9 19 5 19 19 }}


Wedgie: &lt;&lt;4 9 19 5 19 19||
[[POTE generator]]: ~10/9 = 177.357


EDOs: {{EDOs| 27, 61, 88bc, 115bc }}
{{Val list|legend=1| 27, 61, 88bc, 115bc }}


Badness: 0.0705
[[Badness]]: 0.0705


== 11-limit ==
== 11-limit ==
Commas: 56/55, 100/99, 243/242
Comma list: 56/55, 100/99, 243/242
 
Mapping: [&lt;1 1 1 0 2|, &lt;0 4 9 19 10|]


POTE generator: ~10/9 = 177.413
POTE generator: ~10/9 = 177.413


Map: [&lt;1 1 1 0 2|, &lt;0 4 9 19 10|]
Vals: {{Val list| 27e, 34, 61e }}
 
EDOs: {{EDOs| 27e, 34, 61e }}


Badness: 0.0376
Badness: 0.0376


== 13-limit ==
== 13-limit ==
Commas: 56/55, 91/90, 100/99, 352/351
Comma list: 56/55, 91/90, 100/99, 352/351
 
Mapping: [&lt;1 1 1 0 2 4|, &lt;0 4 9 19 10 -2|]


POTE generator: ~10/9 = 177.231
POTE generator: ~10/9 = 177.231


Map: [&lt;1 1 1 0 2 4|, &lt;0 4 9 19 10 -2|]
Vals: {{Val list| 27e, 34, 61e }}
 
EDOs: {{EDOs| 27e, 34, 61e }}


Badness: 0.0312
Badness: 0.0312


= Octacot =
= 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 has wedgie &lt;&lt;8 18 11 10 -5 -25|| and 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 two temperament with octaves rather than rank one without them.
{{see also| Chords of octacot }}


Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas, giving &lt;&lt;8 18 11 20 -4 …|| as the octave part of the wedgie. Generators of 3/41, 8/109 and 11/150 (88 cents) are all good choices for the 7, 11 and 13 limits.
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 two temperament with octaves rather than rank one without them.


Commas: 245/243, 2401/2400
Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas, giving {{multival| 8 18 11 20 -4 … }} as the octave part of the wedgie. Generators of 3\41, 8\109 and 11\150 (88 cents) are all good choices for the 7, 11 and 13 limits.


[[POTE_tuning|POTE generator]]: 88.076
[[Comma list]]: 245/243, 2401/2400


Map: [&lt;1 1 1 2|, &lt;0 8 18 11|]
[[Mapping]]: [&lt;1 1 1 2|, &lt;0 8 18 11|]


EDOs: {{EDOs| 14c, 27, 41, 68, 109 }}
{{Multival|legend=1| 8 18 11 10 -5 -25 }}


Badness: 0.0338
[[POTE generator]]: ~21/20 = 88.076
 
{{Val list|legend=1| 14c, 27, 41, 68, 109 }}
 
[[Badness]]: 0.0338


== 11-limit ==
== 11-limit ==
Commas: 100/99, 243/242, 245/242
Comma list: 100/99, 243/242, 245/242


[[POTE_tuning|POTE generator]]: 87.975
Mapping: [&lt;1 1 1 2 2|, &lt;0 8 18 11 20|]


Map: [&lt;1 1 1 2 2|, &lt;0 8 18 11 20|]
POTE generator: ~21/20 = 87.975


EDOs: {{EDOs| 27e, 41, 109e, 150e, 191e }}
Vals: {{Val list| 27e, 41, 109e, 150e, 191e }}


Badness: 0.0241
Badness: 0.0241
{{see also| Chords of octacot }}


=== 13-limit ===
=== 13-limit ===
Commas: 100/99, 144/143, 196/195, 243/242
Comma list: 100/99, 144/143, 196/195, 243/242


[[POTE_tuning|POTE generator]]: ~22/21 = 88.106
Mapping: [&lt;1 1 1 2 2 4|, &lt;0 8 18 11 20 -4|]


Map: [&lt;1 1 1 2 2 4|, &lt;0 8 18 11 20 -4|]
POTE generator: ~21/20 = 88.106


EDOs: {{EDOs| 27e, 41, 68e, 109ef }}
Vals: {{Val list| 27e, 41, 68e, 109ef }}


Badness: 0.0233
Badness: 0.0233


=== Octocat ===
=== Octocat ===
Commas: 78/77, 91/90, 100/99, 245/242
Comma list: 78/77, 91/90, 100/99, 245/242
 
Mapping: [&lt;1 1 1 2 2 2|, &lt;0 8 18 11 20 23|]


POTE generator: ~22/21 = 88.179
POTE generator: ~22/21 = 88.179


Map: [&lt;1 1 1 2 2 2|, &lt;0 8 18 11 20 23|]
Vals: {{Val list| 27e, 41f, 68ef }}
 
EDOs: {{EDOs| 27e, 41f, 68ef }}


Badness: 0.0276
Badness: 0.0276


=== Octopod ===
=== Octopod ===
Commas: 100/99, 105/104, 243/242, 245/242
Comma list: 100/99, 105/104, 243/242, 245/242
 
Mapping: [&lt;1 1 1 2 2 1|, &lt;0 8 18 11 20 37|]


POTE generator: ~22/21 = 87.697
POTE generator: ~22/21 = 87.697


Map: [&lt;1 1 1 2 2 1|, &lt;0 8 18 11 20 37|]
Vals: {{Val list| 41, 137cd, 178cd }}
 
EDOs: {{EDOs| 41, 137cd, 178cd }}


Badness: 0.0283
Badness: 0.0283


=== Dificot ===
=== Dificot ===
Commas: 100/99, 243/242, 245/242, 343/338
Comma list: 100/99, 243/242, 245/242, 343/338
 
Mapping: [&lt;1 9 19 13 22 19|, &lt;0 -16 -36 -22 -40 -33|]


POTE generator: ~13/9 = 643.989
POTE generator: ~13/9 = 643.989


Map: [&lt;1 9 19 13 22 19|, &lt;0 -16 -36 -22 -40 -33|]
Vals: {{Val list| 41 }}
 
EDOs: {{EDOs| 41 }}


Badness: 0.0519
Badness: 0.0519


= Dodecacot =
= Dodecacot =
Commas: 3125/3087, 10976/10935
[[Comma list]]: 3125/3087, 10976/10935


POTE generator: ~28/27 = 58.675
[[Mapping]]: [&lt;1 1 1 1|, &lt;0 12 27 37|]


Map: [&lt;1 1 1 1|, &lt;0 12 27 37|]
{{Multival|legend=1| 12 27 37 15 25 10 }}


Wedgie: &lt;&lt;12 27 37 15 25 10||
[[POTE generator]]: ~28/27 = 58.675


EDOs: {{EDOs| 41, 184, 225, 409bcd }}
{{Val list|legend=1| 41, 184, 225, 409bcd }}


Badness: 0.1198
[[Badness]]: 0.1198


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Tetracot]]
[[Category:Tetracot]]

Revision as of 15:56, 28 April 2021

The parent of the tetracot family is tetracot, the 5-limit temperament tempering out 20000/19683 = [5 -9 4⟩, the minimal diesis or tetracot comma. The dual of this comma is the wedgie ⟨⟨ 4 9 5 ]], which tells us 10/9 is a generator, and that four of them 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).

Tetracot

Comma list: 20000/19683

POTE generator: ~10/9 = 176.160

Mapping: [<1 1 1|, <0 4 9|]

Template:Val list

Extensions

The second comma of the 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;
  • 245/243 gives octacot, which splits the generator in half.

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&41 temperament, if the vals in question are taken to be patent vals, meaning that n×log2(prime) rounded to the nearest integer gives the mapping. 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 41&75 temperament. 41edo can again be used as a tuning, in which case it is the same as monkey. However an excellent alternative is 14(1/26) as a generator, giving just 7s 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, ⟨⟨ 4 9 -15 10 … ]] and 11-limit banya, ⟨⟨ 4 9 26 10 … ]]. 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 temperament, dispensing with 7. However 11-limit bunya, like 7-limit bunya, profits a little from a slightly sharper fifth, such as the 14(1/26) generator supplies, or even sharper yet, as for instance by the val ⟨355 563 823 997 1230], with a 52/355 generator.

Since 16/13 is shy of (10/9)2 by just 325/324, it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us ⟨⟨ 4 9 -15 10 -2 … ]] for 13-limit monkey and ⟨⟨ 4 9 26 10 -2 … ]] for 13-limit bunya. Once again, 41 is recommended as a tuning for monkey, while banyan can with advantage tune the fifth sharper: 17/116 as a generator with a fifth a cent and a half sharp or 11/75 with a fifth two cents sharp.

Monkey

Comma list: 875/864, 5120/5103

Mapping: [<1 1 1 5|, <0 4 9 -15|]

Wedgie: ⟨⟨ 4 9 -15 5 -35 -60 ]]

POTE generator: ~10/9 = 175.659

Template:Val list

Badness: 0.0734

11-limit

Comma list: 100/99, 243/242, 385/384

Mapping: [<1 1 1 5 2|, <0 4 9 -15 10|]

POTE generator: ~10/9 = 175.570

Vals: Template:Val list

Badness: 0.0388

13-limit

Comma list: 100/99, 105/104, 144/143, 243/242

Mapping: [<1 1 1 5 2 4|, <0 4 9 -15 10 -2|]

POTE generator: ~10/9 = 175.622

Vals: Template:Val list

Badness: 0.0284

Bunya

Comma list: 225/224, 15625/15309

Mapping: [<1 1 1 -1|, <0 4 9 26|]

Wedgie: ⟨⟨ 4 9 26 5 30 35 ]]

POTE generator: ~10/9 = 175.741

Template:Val list

Badness: 0.0629

11-limit

Comma list: 100/99, 225/224, 1344/1331

Mapping: [<1 1 1 -1 2|, <0 4 9 26 10|]

POTE generator: ~10/9 = 175.777

Vals: Template:Val list

Badness: 0.0313

13-limit

Comma list: 100/99, 144/143, 225/224, 243/242

Mapping: [<1 1 1 -1 2 4|, <0 4 9 26 10 -2|]

POTE generator: ~10/9 = 175.886

Vals: Template:Val list

Badness: 0.0249

Modus

Comma list: 64/63, 4375/4374

Mapping: [<1 1 1 4|, <0 4 9 -8|]

POTE generator: ~10/9 = 177.203

Template:Val list

Badness: 0.0682

11-limit

Comma list: 64/63, 100/99, 243/242

Mapping: [<1 1 1 4 2|, <0 4 9 -8 10|]

POTE generator: ~10/9 = 177.053

Vals: Template:Val list

Badness: 0.0351

13-limit

Comma list: 64/63, 78/77, 100/99, 144/143

Mapping: [<1 1 1 4 2 4|, <0 4 9 -8 10 -2|]

POTE generator: ~10/9 = 176.953

Vals: Template:Val list

Badness: 0.0238

Musical Examples

Ponens

The error of 11 is about the same as that of Modus, but flat instead of sharp, and much more abundant. Since the other primes are all sharp, however, this leads to a much larger error for other intervals involving 11.

Comma list: 55/54, 64/63, 363/350

Mapping: [<1 1 1 4 3|, <0 4 9 -8 3|]

POTE generator: ~10/9 = 177.200

Vals: Template:Val list

Badness: 0.0631

13-limit

Comma list: 55/54, 64/63, 66/65, 143/140

Mapping: [<1 1 1 4 3 4|, <0 4 9 -8 3 -2|]

POTE generator: ~10/9 = 177.197

Vals: Template:Val list

Badness: 0.039

Wollemia

Comma list: 126/125, 2240/2187

Mapping: [<1 1 1 0|, <0 4 9 19|]

Wedgie: ⟨⟨ 4 9 19 5 19 19 ]]

POTE generator: ~10/9 = 177.357

Template:Val list

Badness: 0.0705

11-limit

Comma list: 56/55, 100/99, 243/242

Mapping: [<1 1 1 0 2|, <0 4 9 19 10|]

POTE generator: ~10/9 = 177.413

Vals: Template:Val list

Badness: 0.0376

13-limit

Comma list: 56/55, 91/90, 100/99, 352/351

Mapping: [<1 1 1 0 2 4|, <0 4 9 19 10 -2|]

POTE generator: ~10/9 = 177.231

Vals: Template:Val list

Badness: 0.0312

Octacot

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&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 two temperament with octaves rather than rank one without them.

Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas, giving ⟨⟨ 8 18 11 20 -4 … ]] as the octave part of the wedgie. Generators of 3\41, 8\109 and 11\150 (88 cents) are all good choices for the 7, 11 and 13 limits.

Comma list: 245/243, 2401/2400

Mapping: [<1 1 1 2|, <0 8 18 11|]

Wedgie: ⟨⟨ 8 18 11 10 -5 -25 ]]

POTE generator: ~21/20 = 88.076

Template:Val list

Badness: 0.0338

11-limit

Comma list: 100/99, 243/242, 245/242

Mapping: [<1 1 1 2 2|, <0 8 18 11 20|]

POTE generator: ~21/20 = 87.975

Vals: Template:Val list

Badness: 0.0241

13-limit

Comma list: 100/99, 144/143, 196/195, 243/242

Mapping: [<1 1 1 2 2 4|, <0 8 18 11 20 -4|]

POTE generator: ~21/20 = 88.106

Vals: Template:Val list

Badness: 0.0233

Octocat

Comma list: 78/77, 91/90, 100/99, 245/242

Mapping: [<1 1 1 2 2 2|, <0 8 18 11 20 23|]

POTE generator: ~22/21 = 88.179

Vals: Template:Val list

Badness: 0.0276

Octopod

Comma list: 100/99, 105/104, 243/242, 245/242

Mapping: [<1 1 1 2 2 1|, <0 8 18 11 20 37|]

POTE generator: ~22/21 = 87.697

Vals: Template:Val list

Badness: 0.0283

Dificot

Comma list: 100/99, 243/242, 245/242, 343/338

Mapping: [<1 9 19 13 22 19|, <0 -16 -36 -22 -40 -33|]

POTE generator: ~13/9 = 643.989

Vals: Template:Val list

Badness: 0.0519

Dodecacot

Comma list: 3125/3087, 10976/10935

Mapping: [<1 1 1 1|, <0 12 27 37|]

Wedgie: ⟨⟨ 12 27 37 15 25 10 ]]

POTE generator: ~28/27 = 58.675

Template:Val list

Badness: 0.1198

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