Tetracot family: Difference between revisions

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 <<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.

= Tetracot =

Comma: 20000/19683

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

EDOs: {{EDOs| 14c, 27, 34, 75, 109, 470b, 579b }}

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.

Monkey, the monkey puzzle tree temperament, tempers out the keema and has a wedgie <<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&41 temperament, if the vals in question are taken to be [[patent val]]s, 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, the bunya-bunya tree temperament, adds 225/224 to the list of commas and may be described as the 41&75 temperament. It has <<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.

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 =

Commas: 875/864, 5120/5103

[[POTE_tuning|POTE generator]]: 175.659

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

EDOs: {{EDOs| 7, 34, 41, 321cd }}

Badness: 0.0734

==11-limit==

Commas: 100/99, 243/242, 385/384

[[POTE_tuning|POTE generator]]: 175.570

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

EDOs: {{EDOs| 7, 34, 41, 123c }}

Badness: 0.0388

== 13-limit ==

[[POTE_tuning|POTE generator]]: 175.622

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

EDOs: {{EDOs| 7, 34, 41 }}

Badness: 0.0284

Commas: 225/224, 15625/15309

[[POTE_tuning|POTE generator]]: 175.741

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

EDOs: {{EDOs| 34d, 41, 116, 157c, 198c }}

Badness: 0.0629

==11-limit==

Commas: 100/99, 225/224, 1344/1331

[[POTE_tuning|POTE generator]]: 175.777

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

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

Badness: 0.0313

Commas: 100/99, 144/143, 225/224, 243/242

[[POTE_tuning|POTE generator]]: 175.886

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

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

Badness: 0.0249

= Modus =

POTE generator: ~10/9 = 177.203

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

EDOs: {{EDOs| 7, 27, 61d, 88bcd }}

Badness: 0.0682

Commas: 64/63, 100/99, 243/242

POTE generator: ~10/9 = 177.053

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

EDOs: {{EDOs| 7, 20ce, 27e, 34d, 61de }}

Badness: 0.0351

Commas: 64/63, 78/77, 100/99, 144/143

POTE generator: ~10/9 = 176.953

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

EDOs: {{EDOs| 7, 27e, 34d, 61de }}

Badness: 0.0238

* [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]

Commas: 55/54, 64/63, 363/350

POTE generator: ~10/9 = 177.200

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

EDOs: {{EDOs| 7, 20c, 27, 61de, 88bcde }}

Badness: 0.0631

Commas: 55/54, 64/63, 66/65, 143/140

POTE generator: ~10/9 = 177.197

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

EDOs: {{EDOs| 7, 20c, 27, 61de, 88bcde }}

Badness: 0.039

= Wollemia =

Commas: 126/125, 2240/2187

POTE generator: ~10/9 = 177.357

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

Wedgie: <<4 9 19 5 19 19||

EDOs: {{EDOs| 27, 61, 88bc, 115bc }}

Badness: 0.0705

== 11-limit ==

Commas: 56/55, 100/99, 243/242

POTE generator: ~10/9 = 177.413

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

EDOs: {{EDOs| 27e, 34, 61e }}

Badness: 0.0376

== 13-limit ==

Commas: 56/55, 91/90, 100/99, 352/351

POTE generator: ~10/9 = 177.231

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

EDOs: {{EDOs| 27e, 34, 61e }}

Badness: 0.0312

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 <<8 18 11 10 -5 -25|| and 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.

Commas: 245/243, 2401/2400

[[POTE_tuning|POTE generator]]: 88.076

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

EDOs: {{EDOs| 14c, 27, 41, 68, 109 }}

Badness: 0.0338

== 11-limit ==

Commas: 100/99, 243/242, 245/242

[[POTE_tuning|POTE generator]]: 87.975

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

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

Badness: 0.0241

{{see also| Chords of octacot }}

Commas: 100/99, 144/143, 196/195, 243/242

[[POTE_tuning|POTE generator]]: ~22/21 = 88.106

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

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

Badness: 0.0233

Commas: 78/77, 91/90, 100/99, 245/242

POTE generator: ~22/21 = 88.179

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

EDOs: {{EDOs| 27e, 41f, 68ef }}

Badness: 0.0276

=== Octopod ===

Commas: 100/99, 105/104, 243/242, 245/242

POTE generator: ~22/21 = 87.697

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

EDOs: {{EDOs| 41, 137cd, 178cd }}

Badness: 0.0283

=== Dificot ===

Commas: 100/99, 243/242, 245/242, 343/338

POTE generator: ~13/9 = 643.989

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

EDOs: {{EDOs| 41 }}

Badness: 0.0519

Commas: 3125/3087, 10976/10935

POTE generator: ~28/27 = 58.675

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

Wedgie: <<12 27 37 15 25 10||

[[Category:Rank 2]]

[[Category:Listen]]