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 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 {{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 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 ==
{{main| Tetracot }}
{{Main| Tetracot }}


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 20000/19683
[[Comma list]]: 20000/19683
Line 12: Line 12:
[[Mapping]]: [{{val| 1 1 1 }}, {{val| 0 4 9 }}]
[[Mapping]]: [{{val| 1 1 1 }}, {{val| 0 4 9 }}]


[[POTE generator]]: ~10/9 = 176.160
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 176.160


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 5-odd-limit: ~10/9 = {{monzo| -1/9 0 1/9 }}
* [[5-odd-limit]]: ~10/9 = {{monzo| -1/9 0 1/9 }}
: Eigenmonzos (unchanged intervals): 2, 5
: [[Eigenmonzo basis]]: 2.5


{{Val list|legend=1| 7, 20c, 27, 34, 75, 109, 470b, 579b }}
{{Val list|legend=1| 7, 20c, 27, 34, 75, 109, 470b, 579b }}
Line 22: Line 22:
[[Badness]]: 0.048518
[[Badness]]: 0.048518


Scales: [[Tetracot7]], [[Tetracot13]], [[Tetracot20]]
=== Overview to extensions ===
 
=== 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 lists|normal comma list]] defines which 7-limit family member we are looking at.  
* [[875/864]], the keema, gives monkey;  
* [[875/864]], the keema, gives monkey;  
Line 31: Line 29:


==== Monkey and bunya ====
==== 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, 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. [[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. 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.
''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.


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 bunya, {{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 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 bunya, {{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 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.


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, 41edo is recommended as a tuning for monkey, while bunya 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, 41edo is recommended as a tuning for monkey, while bunya 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.


=== Subgroup temperament ===
=== 2.3.5.11 subgroup ===
{{see also| No-sevens subgroup temperaments #Tetracot }}
{{Main| Tetracot }}
{{See also| No-sevens subgroup temperaments #Tetracot }}


The tetracot temperament works well for the 2.3.5.11.13 subgroup, in which tempering out 100/99, 144/143 and 243/242. In this temperament, 3/2 is divided into four equal parts, one of which represents both 10/9 and 11/10.
As discussed above, tetracot works well for the 2.3.5.11.13 subgroup, in which tempering out 100/99, 144/143 and 243/242.  


Subgroup: 2.3.5.11
[[Subgroup]]: 2.3.5.11


[[Comma list]]: 100/99, 243/242
[[Comma list]]: 100/99, 243/242
[[Sval]] [[mapping]]: [{{val| 1 1 1 2 }}, {{val| 0 4 9 10 }}]


[[Gencom]]: [2 10/9; 100/99 243/242]
[[Gencom]]: [2 10/9; 100/99 243/242]


[[Mapping|Sval mapping]]: [{{val|1 1 1 2}}, {{val|0 4 9 10}}]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 175.985
 
[[Tp tuning|POL2 generator]]: ~10/9 = 175.985


{{Val list|legend=1| 7, 27e, 34, 41, 75e }}
{{Val list|legend=1| 7, 27e, 34, 41, 75e }}


Scales: [[Tetracot7]], [[Tetracot13]], [[Tetracot20]]
==== 2.3.5.11.13 subgroup ====
 
==== 2.3.5.11.13 ====
Subgroup: 2.3.5.11.13
Subgroup: 2.3.5.11.13


[[Comma list]]: 100/99, 144/143, 243/242
Comma list: 100/99, 144/143, 243/242


[[Gencom]]: [2 10/9; 100/99 144/143 243/242]
Sval mapping: [{{val| 1 1 1 2 4 }}, {{val| 0 4 9 10 -2 }}]


[[Mapping|Sval mapping]]: [{{val|1 1 1 2 4}}, {{val|0 4 9 10 -2}}]
Gencom: [2 10/9; 100/99 144/143 243/242]


[[Tp tuning|POL2 generator]]: ~10/9 = 176.196
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 176.196


{{Val list|legend=1| 7, 27e, 34, 41, 75e }}
{{Val list|legend=1| 7, 27e, 34, 41, 75e }}
Scales: [[Tetracot7]], [[Tetracot13]], [[Tetracot20]]


== Monkey ==
== Monkey ==
{{see also| Tetracot }}
{{See also| Tetracot }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 875/864, 5120/5103
[[Comma list]]: 875/864, 5120/5103
Line 84: Line 79:
{{Multival|legend=1| 4 9 -15 5 -35 -60 }}
{{Multival|legend=1| 4 9 -15 5 -35 -60 }}


[[POTE generator]]: ~10/9 = 175.659
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 175.659


{{Val list|legend=1| 7, 27d, 34, 41, 321ccdd }}
{{Val list|legend=1| 7, 27d, 34, 41, 321ccdd }}
Line 97: Line 92:
Mapping: [{{val| 1 1 1 5 2 }}, {{val| 0 4 9 -15 10 }}]
Mapping: [{{val| 1 1 1 5 2 }}, {{val| 0 4 9 -15 10 }}]


POTE generator: ~10/9 = 175.570
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 175.570


Optimal GPV sequence: {{Val list| 7, 27de, 34, 41 }}
Optimal GPV sequence: {{Val list| 7, 27de, 34, 41 }}
Line 110: Line 105:
Mapping: [{{val| 1 1 1 5 2 4 }}, {{val| 0 4 9 -15 10 -2 }}]
Mapping: [{{val| 1 1 1 5 2 4 }}, {{val| 0 4 9 -15 10 -2 }}]


POTE generator: ~10/9 = 175.622
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 175.622


Optimal GPV sequence: {{Val list| 7, 27de, 34, 41 }}
Optimal GPV sequence: {{Val list| 7, 27de, 34, 41 }}
Line 117: Line 112:


== Bunya ==
== Bunya ==
{{see also| Tetracot }}
{{See also| Tetracot }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 15625/15309
[[Comma list]]: 225/224, 15625/15309
Line 127: Line 122:
{{Multival|legend=1| 4 9 26 5 30 35 }}
{{Multival|legend=1| 4 9 26 5 30 35 }}


[[POTE generator]]: ~10/9 = 175.741
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 175.741


{{Val list|legend=1| 34d, 41, 116, 157c, 198c }}
{{Val list|legend=1| 34d, 41, 116, 157c, 198c }}
Line 140: Line 135:
Mapping: [{{val| 1 1 1 -1 2 }}, {{val| 0 4 9 26 10 }}]
Mapping: [{{val| 1 1 1 -1 2 }}, {{val| 0 4 9 26 10 }}]


POTE generator: ~10/9 = 175.777
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 175.777


Optimal GPV sequence: {{Val list| 34d, 41, 116e, 157ce }}
Optimal GPV sequence: {{Val list| 34d, 41, 116e, 157ce }}
Line 153: Line 148:
Mapping: [{{val| 1 1 1 -1 2 4 }}, {{val| 0 4 9 26 10 -2 }}]
Mapping: [{{val| 1 1 1 -1 2 4 }}, {{val| 0 4 9 26 10 -2 }}]


POTE generator: ~10/9 = 175.886
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 175.886


Optimal GPV sequence: {{Val list| 34d, 41, 75e, 116ef }}
Optimal GPV sequence: {{Val list| 34d, 41, 75e, 116ef }}
Line 160: Line 155:


== Modus ==
== Modus ==
{{see also| Tetracot }}
{{See also| Tetracot }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 4375/4374
[[Comma list]]: 64/63, 4375/4374
Line 168: Line 163:
[[Mapping]]: [{{val| 1 1 1 4 }}, {{val| 0 4 9 -8 }}]
[[Mapping]]: [{{val| 1 1 1 4 }}, {{val| 0 4 9 -8 }}]


[[POTE generator]]: ~10/9 = 177.203
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 177.203


{{Val list|legend=1| 7, 20c, 27, 61d, 88bcd }}
{{Val list|legend=1| 7, 20c, 27, 61d, 88bcd }}
Line 181: Line 176:
Mapping: [{{val| 1 1 1 4 2 }}, {{val| 0 4 9 -8 10 }}]
Mapping: [{{val| 1 1 1 4 2 }}, {{val| 0 4 9 -8 10 }}]


POTE generator: ~10/9 = 177.053
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 177.053


Optimal GPV sequence: {{Val list| 7, 20ce, 27e, 34d, 61de }}
Optimal GPV sequence: {{Val list| 7, 20ce, 27e, 34d, 61de }}
Line 194: Line 189:
Mapping: [{{val| 1 1 1 4 2 4 }}, {{val| 0 4 9 -8 10 -2 }}]
Mapping: [{{val| 1 1 1 4 2 4 }}, {{val| 0 4 9 -8 10 -2 }}]


POTE generator: ~10/9 = 176.953
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 176.953


Optimal GPV sequence: {{Val list| 7, 20ce, 27e, 34d, 61de }}
Optimal GPV sequence: {{Val list| 7, 20ce, 27e, 34d, 61de }}
Line 200: Line 195:
Badness: 0.023806
Badness: 0.023806


; Musical examples
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Perc-Sitar.mp3 Tetracot Perc-Sitar] by [http://soundcloud.com/dustin-schallert/tetracot-perc-sitar Dustin Schallert]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Perc-Sitar.mp3 ''Tetracot Perc-Sitar''] by [http://soundcloud.com/dustin-schallert/tetracot-perc-sitar Dustin Schallert]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Jam.mp3 Tetracot Jam] by [http://soundcloud.com/dustin-schallert/tetracot-jam Dustin Schallert]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Jam.mp3 ''Tetracot Jam''] by [http://soundcloud.com/dustin-schallert/tetracot-jam Dustin Schallert]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Pump.mp3 Tetracot Pump] by [http://soundcloud.com/dustin-schallert/tetracot-pump Dustin Schallert] all in [[27edo]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Pump.mp3 ''Tetracot Pump''] by [http://soundcloud.com/dustin-schallert/tetracot-pump Dustin Schallert] all in [[27edo]]


=== Ponens ===
=== Ponens ===
Line 214: Line 209:
Mapping: [{{val| 1 1 1 4 3 }}, {{val| 0 4 9 -8 3 }}]
Mapping: [{{val| 1 1 1 4 3 }}, {{val| 0 4 9 -8 3 }}]


POTE generator: ~10/9 = 177.200
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 177.200


Optimal GPV sequence: {{Val list| 7, 20c, 27, 61dee, 88bcdee }}
Optimal GPV sequence: {{Val list| 7, 20c, 27, 61dee, 88bcdee }}
Line 227: Line 222:
Mapping: [{{val| 1 1 1 4 3 4 }}, {{val| 0 4 9 -8 3 -2 }}]
Mapping: [{{val| 1 1 1 4 3 4 }}, {{val| 0 4 9 -8 3 -2 }}]


POTE generator: ~10/9 = 177.197
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 177.197


Optimal GPV sequence: {{Val list| 7, 20c, 27, 61dee, 88bcdee }}
Optimal GPV sequence: {{Val list| 7, 20c, 27, 61dee, 88bcdee }}
Line 234: Line 229:


== Wollemia ==
== Wollemia ==
{{see also| Tetracot }}
{{See also| Tetracot }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 2240/2187
[[Comma list]]: 126/125, 2240/2187
Line 244: Line 239:
{{Multival|legend=1| 4 9 19 5 19 19 }}
{{Multival|legend=1| 4 9 19 5 19 19 }}


[[POTE generator]]: ~10/9 = 177.357
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 177.357


{{Val list|legend=1| 27, 61, 88bc, 115bc }}
{{Val list|legend=1| 27, 61, 88bc, 115bc }}
Line 257: Line 252:
Mapping: [{{val| 1 1 1 0 2 }}, {{val| 0 4 9 19 10 }}]
Mapping: [{{val| 1 1 1 0 2 }}, {{val| 0 4 9 19 10 }}]


POTE generator: ~10/9 = 177.413
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 177.413


Optimal GPV sequence: {{Val list| 27e, 34, 61e }}
Optimal GPV sequence: {{Val list| 27e, 34, 61e }}
Line 270: Line 265:
Mapping: [{{val| 1 1 1 0 2 4 }}, {{val| 0 4 9 19 10 -2 }}]
Mapping: [{{val| 1 1 1 0 2 4 }}, {{val| 0 4 9 19 10 -2 }}]


POTE generator: ~10/9 = 177.231
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 177.231


Optimal GPV sequence: {{Val list| 27e, 34, 61e }}
Optimal GPV sequence: {{Val list| 27e, 34, 61e }}
Line 277: Line 272:


== Octacot ==
== Octacot ==
{{see also| Chords of 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&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.
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.


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.
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.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 2401/2400
[[Comma list]]: 245/243, 2401/2400
Line 291: Line 286:
{{Multival|legend=1| 8 18 11 10 -5 -25 }}
{{Multival|legend=1| 8 18 11 10 -5 -25 }}


[[POTE generator]]: ~21/20 = 88.076
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 88.076


{{Val list|legend=1| 14c, 27, 41, 68, 109 }}
{{Val list|legend=1| 14c, 27, 41, 68, 109 }}
Line 304: Line 299:
Mapping: [{{val| 1 1 1 2 2 }}, {{val| 0 8 18 11 20 }}]
Mapping: [{{val| 1 1 1 2 2 }}, {{val| 0 8 18 11 20 }}]


POTE generator: ~21/20 = 87.975
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 87.975


Optimal GPV sequence: {{Val list| 27e, 41, 109e, 150e, 191e }}
Optimal GPV sequence: {{Val list| 27e, 41, 109e, 150e, 191e }}
Line 317: Line 312:
Mapping: [{{val| 1 1 1 2 2 4 }}, {{val| 0 8 18 11 20 -4 }}]
Mapping: [{{val| 1 1 1 2 2 4 }}, {{val| 0 8 18 11 20 -4 }}]


POTE generator: ~21/20 = 88.106
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.106


Optimal GPV sequence: {{Val list| 27e, 41, 68e, 109ef }}
Optimal GPV sequence: {{Val list| 27e, 41, 68e, 109ef }}
Line 330: Line 325:
Mapping: [{{val| 1 1 1 2 2 4 3 }}, {{val| 0 8 18 11 20 -4 15 }}]
Mapping: [{{val| 1 1 1 2 2 4 3 }}, {{val| 0 8 18 11 20 -4 15 }}]


POTE generator: ~18/17 = 88.102
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 88.102


Optimal GPV sequence: {{Val list| 14c, 27eg, 41, 68egg, 109efgg }}
Optimal GPV sequence: {{Val list| 14c, 27eg, 41, 68egg, 109efgg }}
Line 343: Line 338:
Mapping: [{{val| 1 1 1 2 2 4 3 3 }}, {{val| 0 8 18 11 20 -4 15 17 }}]
Mapping: [{{val| 1 1 1 2 2 4 3 3 }}, {{val| 0 8 18 11 20 -4 15 17 }}]


POTE generator: ~18/17 = 88.111
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 88.111


Optimal GPV sequence: {{Val list| 14c, 27eg, 41, 68egg, 109efgg }}
Optimal GPV sequence: {{Val list| 14c, 27eg, 41, 68egg, 109efgg }}
Line 356: Line 351:
Mapping: [{{val| 1 1 1 2 2 2 }}, {{val| 0 8 18 11 20 23 }}]
Mapping: [{{val| 1 1 1 2 2 2 }}, {{val| 0 8 18 11 20 23 }}]


POTE generator: ~21/20 = 88.179
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.179


Optimal GPV sequence: {{Val list| 27e, 41f, 68ef }}
Optimal GPV sequence: {{Val list| 27e, 41f, 68ef }}
Line 369: Line 364:
Mapping: [{{val| 1 1 1 2 2 1 }}, {{val| 0 8 18 11 20 37 }}]
Mapping: [{{val| 1 1 1 2 2 1 }}, {{val| 0 8 18 11 20 37 }}]


POTE generator: ~21/20 = 87.697
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 87.697


Optimal GPV sequence: {{Val list| 41, 137cd, 178cd }}
Optimal GPV sequence: {{Val list| 41, 137cd, 178cd }}
Line 382: Line 377:
Mapping: [{{val| 1 9 19 13 22 19 }}, {{val| 0 -16 -36 -22 -40 -33 }}]
Mapping: [{{val| 1 9 19 13 22 19 }}, {{val| 0 -16 -36 -22 -40 -33 }}]


POTE generator: ~13/9 = 643.989
Optimal tuning (POTE): ~2 = 1\1, ~13/9 = 643.989


Optimal GPV sequence: {{Val list| 13cdeef, 28ccdef, 41 }}
Optimal GPV sequence: {{Val list| 13cdeef, 28ccdef, 41 }}
Line 395: Line 390:
Mapping: [{{val|1 1 1 2 5}}, {{val|0 8 18 11 -21}}]
Mapping: [{{val|1 1 1 2 5}}, {{val|0 8 18 11 -21}}]


POTE generator: ~21/20 = 88.035
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.035


Optimal GPV sequence: {{Val list| 27, 41, 68, 109, 150, 259 }}
Optimal GPV sequence: {{Val list| 27, 41, 68, 109, 150, 259 }}
Line 408: Line 403:
Mapping: [{{val|1 1 1 2 5 4}}, {{val|0 8 18 11 -21 -4}}]
Mapping: [{{val|1 1 1 2 5 4}}, {{val|0 8 18 11 -21 -4}}]


POTE generator: ~21/20 = 88.075
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.075


Optimal GPV sequence: {{Val list| 27, 41, 68, 109f }}
Optimal GPV sequence: {{Val list| 27, 41, 68, 109f }}
Line 421: Line 416:
Mapping: [{{val|1 1 1 2 5 4 6}}, {{val|0 8 18 11 -21 -4 -26}}]
Mapping: [{{val|1 1 1 2 5 4 6}}, {{val|0 8 18 11 -21 -4 -26}}]


POTE generator: ~21/20 = 88.104
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.104


Optimal GPV sequence: {{Val list| 27, 41, 68, 109f }}
Optimal GPV sequence: {{Val list| 27, 41, 68, 109f }}
Line 434: Line 429:
Mapping: [{{val|1 1 1 2 5 4 6 3}}, {{val|0 8 18 11 -21 -4 -26 17}}]
Mapping: [{{val|1 1 1 2 5 4 6 3}}, {{val|0 8 18 11 -21 -4 -26 17}}]


POTE generator: ~19/18 = 88.113
Optimal tuning (POTE): ~2 = 1\1, ~19/18 = 88.113


Optimal GPV sequence: {{Val list| 27, 41, 68, 109f, 177ffg }}
Optimal GPV sequence: {{Val list| 27, 41, 68, 109f, 177ffg }}
Line 441: Line 436:


== Dodecacot ==
== Dodecacot ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3125/3087, 10976/10935
[[Comma list]]: 3125/3087, 10976/10935
Line 462: Line 457:
Mapping: [{{val| 1 1 1 1 2 }}, {{val| 0 12 27 37 30 }}]
Mapping: [{{val| 1 1 1 1 2 }}, {{val| 0 12 27 37 30 }}]


POTE generator: ~28/27 = 58.665
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 58.665


Optimal GPV sequence: {{Val list| 20cde, 41 }}
Optimal GPV sequence: {{Val list| 20cde, 41 }}
Line 475: Line 470:
Mapping: [{{val| 1 1 1 1 2 2 }}, {{val| 0 12 27 37 30 35 }}]
Mapping: [{{val| 1 1 1 1 2 2 }}, {{val| 0 12 27 37 30 35 }}]


POTE generator: ~27/26 = 58.639
Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 58.639


Optimal GPV sequence: {{Val list| 20cdef, 41 }}
Optimal GPV sequence: {{Val list| 20cdef, 41 }}
Line 482: Line 477:


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Tetracot]]
[[Category:Tetracot family| ]] <!-- main article -->
[[Category:Tetracot family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]

Revision as of 13:34, 1 May 2023

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

Main article: Tetracot

Subgroup: 2.3.5

Comma list: 20000/19683

Mapping: [⟨1 1 1], ⟨0 4 9]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 176.160

Minimax tuning:

Eigenmonzo basis: 2.5

Template:Val list

Badness: 0.048518

Overview to 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. 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 & 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, ⟨⟨ 4 9 -15 10 … ]] and 11-limit bunya, ⟨⟨ 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 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 ⟨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, 41edo is recommended as a tuning for monkey, while bunya 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.

2.3.5.11 subgroup

Main article: Tetracot
See also: No-sevens subgroup temperaments § Tetracot

As discussed above, tetracot works well for the 2.3.5.11.13 subgroup, in which tempering out 100/99, 144/143 and 243/242.

Subgroup: 2.3.5.11

Comma list: 100/99, 243/242

Sval mapping: [⟨1 1 1 2], ⟨0 4 9 10]]

Gencom: [2 10/9; 100/99 243/242]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 175.985

Template:Val list

2.3.5.11.13 subgroup

Subgroup: 2.3.5.11.13

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

Sval mapping: [⟨1 1 1 2 4], ⟨0 4 9 10 -2]]

Gencom: [2 10/9; 100/99 144/143 243/242]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 176.196

Template:Val list

Monkey

See also: Tetracot

Subgroup: 2.3.5.7

Comma list: 875/864, 5120/5103

Mapping: [⟨1 1 1 5], ⟨0 4 9 -15]]

Wedgie: ⟨⟨ 4 9 -15 5 -35 -60 ]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 175.659

Template:Val list

Badness: 0.073437

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 175.570

Optimal GPV sequence: Template:Val list

Badness: 0.038836

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 175.622

Optimal GPV sequence: Template:Val list

Badness: 0.028410

Bunya

See also: Tetracot

Subgroup: 2.3.5.7

Comma list: 225/224, 15625/15309

Mapping: [⟨1 1 1 -1], ⟨0 4 9 26]]

Wedgie: ⟨⟨ 4 9 26 5 30 35 ]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 175.741

Template:Val list

Badness: 0.062897

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 175.777

Optimal GPV sequence: Template:Val list

Badness: 0.031332

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 175.886

Optimal GPV sequence: Template:Val list

Badness: 0.024886

Modus

See also: Tetracot

Subgroup: 2.3.5.7

Comma list: 64/63, 4375/4374

Mapping: [⟨1 1 1 4], ⟨0 4 9 -8]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 177.203

Template:Val list

Badness: 0.068184

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 1 4 2], ⟨0 4 9 -8 10]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 177.053

Optimal GPV sequence: Template:Val list

Badness: 0.035149

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 176.953

Optimal GPV sequence: Template:Val list

Badness: 0.023806

Music

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.

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 1 4 3], ⟨0 4 9 -8 3]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 177.200

Optimal GPV sequence: Template:Val list

Badness: 0.063077

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 177.197

Optimal GPV sequence: Template:Val list

Badness: 0.039043

Wollemia

See also: Tetracot

Subgroup: 2.3.5.7

Comma list: 126/125, 2240/2187

Mapping: [⟨1 1 1 0], ⟨0 4 9 19]]

Wedgie: ⟨⟨ 4 9 19 5 19 19 ]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 177.357

Template:Val list

Badness: 0.070522

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 1 0 2], ⟨0 4 9 19 10]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 177.413

Optimal GPV sequence: Template:Val list

Badness: 0.037551

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 91/90, 100/99, 243/242

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

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 177.231

Optimal GPV sequence: Template:Val list

Badness: 0.031219

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-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, 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.

Subgroup: 2.3.5.7

Comma list: 245/243, 2401/2400

Mapping: [⟨1 1 1 2], ⟨0 8 18 11]]

Wedgie: ⟨⟨ 8 18 11 10 -5 -25 ]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.076

Template:Val list

Badness: 0.033845

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 87.975

Optimal GPV sequence: Template:Val list

Badness: 0.024078

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.106

Optimal GPV sequence: Template:Val list

Badness: 0.023276

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 120/119, 144/143, 154/153, 189/187

Mapping: [⟨1 1 1 2 2 4 3], ⟨0 8 18 11 20 -4 15]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 88.102

Optimal GPV sequence: Template:Val list

Badness: 0.021088

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 120/119, 133/132, 144/143, 154/153, 189/187

Mapping: [⟨1 1 1 2 2 4 3 3], ⟨0 8 18 11 20 -4 15 17]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 88.111

Optimal GPV sequence: Template:Val list

Badness: 0.016652

Octocat

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.179

Optimal GPV sequence: Template:Val list

Badness: 0.027601

Octopod

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 87.697

Optimal GPV sequence: Template:Val list

Badness: 0.028326

Dificot

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~13/9 = 643.989

Optimal GPV sequence: Template:Val list

Badness: 0.051876

October

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 1375/1372

Mapping: [⟨1 1 1 2 5], ⟨0 8 18 11 -21]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.035

Optimal GPV sequence: Template:Val list

Badness: 0.039643

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 275/273, 385/384

Mapping: [⟨1 1 1 2 5 4], ⟨0 8 18 11 -21 -4]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.075

Optimal GPV sequence: Template:Val list

Badness: 0.031136

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 154/153, 170/169, 196/195, 245/243, 256/255

Mapping: [⟨1 1 1 2 5 4 6], ⟨0 8 18 11 -21 -4 -26]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.104

Optimal GPV sequence: Template:Val list

Badness: 0.026833

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 154/153, 170/169, 190/189, 196/195, 209/208, 245/243

Mapping: [⟨1 1 1 2 5 4 6 3], ⟨0 8 18 11 -21 -4 -26 17]]

Optimal tuning (POTE): ~2 = 1\1, ~19/18 = 88.113

Optimal GPV sequence: Template:Val list

Badness: 0.020511

Dodecacot

Subgroup: 2.3.5.7

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: 0.119761

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 243/242, 1375/1372

Mapping: [⟨1 1 1 1 2], ⟨0 12 27 37 30]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 58.665

Optimal GPV sequence: Template:Val list

Badness: 0.059528

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 243/242, 275/273

Mapping: [⟨1 1 1 1 2 2], ⟨0 12 27 37 30 35]]

Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 58.639

Optimal GPV sequence: Template:Val list

Badness: 0.043645

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