Keemic temperaments: Difference between revisions

These temper out the keema, {{monzo| -5 -3 3 1 }} = [[875/864]] = {{S|5/S6}}, whose fundamental equivalence entails that [[6/5]] is sharpened so that it stacks three times to reach [[7/4]], and the interval between 6/5 and [[5/4]] is compressed so that [[7/6]] - 6/5 - 5/4 - [[9/7]] are set equidistant from each other. As the [[Keemic family#Undecimal supermagic|canonical extension]] of rank-3 keemic to the [[11-limit]] tempers out the commas [[100/99]] and [[385/384]] (whereby ([[6/5]])² is identified with [[16/11]]), this provides a clean way to extend the various keemic temperaments to the 11-limit as well.

* ''[[Undeka]]'' (+3200/3087) → [[11th-octave temperaments #Undeka|11th-octave temperaments]]

: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Quasitemp]].''

Quasitemp is a full 7-limit strong extension of [[gariberttet]], the 2.5/3.7/3 subgroup temperament defined by tempering out [[3125/3087]]. In gariberttet, three generators reach [[5/3]] and five reach [[7/3]], so that the generator itself has the interpretation of [[25/21]] (which is equated to [[13/11]] in the 13-limit extension). This implies that 3:5:7 and 5:6:7 chords are reached rather quickly. In quasitemp, tempering out 875/864 entails that [[8/7]] is found after 9 generators, from which the mappings of 3 and 5 follow.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 2401/2400

{{Mapping|legend=1| 1 5 5 5 | 0 -14 -11 -9 }}

: Mapping generators: ~2, ~25/21

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 292.710

{{Optimal ET sequence|legend=1| 4, 37, 41 }}

[[Badness]]: 0.060269

Mapping: {{mapping| 1 5 5 5 2 | 0 -14 -11 -9 6 }}

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.547

{{Optimal ET sequence|legend=1| 4, 37, 41, 119 }}

Badness: 0.043209

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 275/273, 385/384

Mapping: {{mapping| 1 5 5 5 2 2 | 0 -14 -11 -9 6 7 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.457

{{Optimal ET sequence|legend=1| 4, 37, 41, 78, 119f }}

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 625/616

Mapping: {{mapping| 1 5 5 5 12 | 0 -14 -11 -9 -35 }}

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.851

{{Optimal ET sequence|legend=1| 41, 127cd, 168cd }}

Badness: 0.041170

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 243/242, 275/273, 325/324

Mapping: {{mapping| 1 5 5 5 12 12 | 0 -14 -11 -9 -35 -34 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.928

{{Optimal ET sequence|legend=1| 41, 86ce, 127cd }}

Badness: 0.030081

Chromo represents the [[13edf]] chain as a rank-2 temperament, with [[6/5]] and [[5/4]] mapped to 6 and 7 steps, respectively. Since the difference of those two intervals is abbreviated considerably from just, keemic provides the most meaningful 7-limit extension (setting [[7/6]], 6/5, 5/4, [[9/7]] equidistant) so that the temperament then approximates the [[4:5:6:7]] tetrad with 0:7:13:18 generator steps.

Note that if one allows a more complex mapping for prime 7 and wants a larger prime limit, one may prefer [[escapade]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 2430/2401

{{Mapping|legend=1| 1 1 2 2 | 0 13 7 18 }}

: Mapping generators: ~2, ~25/24

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/24 = 53.816

{{Optimal ET sequence|legend=1| 22, 45, 67c }}

[[Badness]]: 0.090769

{{Mapping|legend=1| 1 9 7 11 | 0 -19 -12 -21 }}

: Mapping generators: ~2, ~98/75

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98/75 = 468.331

[[Badness]]: 0.110448

Comma list: 245/242, 540/539, 625/616

Mapping: {{mapping| 1 9 7 11 14 | 0 -19 -12 -21 -27 }}

Optimal tuning (POTE): ~2 = 1\1, ~98/75 = 468.367

Comma list: 144/143, 196/195, 245/242, 275/273

Mapping: {{mapping| 1 9 7 11 14 8 | 0 -19 -12 -21 -27 -11 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 468.270

Badness: 0.039183

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 51200/50421

{{Mapping|legend=1| 1 -3 -2 2 | 0 17 16 3 }}

: Mapping generators: ~2, ~6/5

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 323.780

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 2420/2401

Mapping: {{mapping| 1 -3 -2 2 4 | 0 17 16 3 -2}}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.796

{{Optimal ET sequence|legend=1| 26, 37, 63 }}

Badness: 0.065356

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 169/168, 275/273, 385/384

Mapping: {{mapping| 1 -3 -2 2 4 1 | 0 17 16 3 -2 10 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.790

== Sevond ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 327680/321489

{{Mapping|legend=1| 7 0 -6 53 | 0 1 2 -3 }}

: Mapping generators: ~10/9, ~3

[[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~3/2 = 705.613

{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}

[[Badness]]: 0.206592

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 6655/6561

Mapping: {{mapping| 7 0 -6 53 2 | 0 1 2 -3 2 }}

Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.518

{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}

Badness: 0.070437

Comma list: 100/99, 169/168, 352/351, 385/384

Mapping: {{mapping| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}

Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.344

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