Keemic temperaments: Difference between revisions

This is a collection of [[rank-2 temperament|linear]] [[regular temperament|temperaments]] that [[tempering out|temper out]] the [[keema]] ({{monzo|legend=1| -5 -3 3 1 }}, [[ratio]]: 875/864), with [[S-expression]] S5/S6. Its 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 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.

* ''[[Sycamore]]'' (+686/675) → [[Sycamore family #Septimal sycamore|Sycamore family]]

Discussed below are quasitemp, chromo, barbad, pentadecal, undeka, hyperkleismic, and sevond, in the order of increasing [[TE logflat badness]].

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

Quasitemp tempers out [[2401/2400]] in addition to 875/864 and may be described as the {{nowrap| 37 & 41 }} temperament. It is characterized by equating the interval between the pental and septimal thirds ([[36/35]]) with the classical chromatic semitone ([[25/24]]), and by tempering together the septimal dieses of [[49/48]] and [[50/49]]. In that sense, it is opposed to [[orwellismic temperaments]], in particular [[myna]], where the distance between the pental and septimal thirds is the same as the septimal dieses and different from the classical chromatic semitone.

Quasitemp can also be thought of as a [[strong extension]] of the 2.5/3.7/3-subgroup temperament called [[gariberttet]], which is 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]]. This implies that 3:5:7 and 5:6:7 chords are reached rather quickly. Quasitemp tempering out 875/864 entails that [[8/7]] is found after 9 generators, from which the mappings of 3 and 5 follow.  

Note that the generator is given as 25/21's octave complement, 42/25, in the data that follow, since a stack of 14 such generators octave-reduced is the perfect fifth, whence the temperament's [[ploidacot]] is iota-14-cot. This generator is equated to [[22/13]] for the 13-limit extension, tempering out [[275/273]].

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 1 -9 -6 -4 | 0 14 11 9 }}

: mapping generators: ~2, ~42/25

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.9237{{c}}, ~42/25 = 907.9887{{c}}

: [[error map]]: {{val| +0.924 +1.573 -3.981 -0.623 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 907.3471{{c}}

: error map: {{val| 0.000 +0.905 -5.495 -2.702 }}

[[Badness]] (Sintel): 1.53

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 1375/1372

Mapping: {{mapping| 1 -9 -6 -4 8 | 0 14 11 9 -6 }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Badness (Sintel): 1.36

 

 

 

Mapping: {{mapping| 1 -9 -6 -4 -23 -22 | 0 14 11 9 35 34 }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Badness (Sintel): 1.66

Subgroup: 2.3.5.7.11.13

 

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

 

 

* WE: ~2 = 1199.7960{{c}}, ~20/13 = 731.6053{{c}}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Badness]] (Sintel): 3.59

Subgroup: 2.3.5.7.11

Comma list: 100/99, 352/343, 385/384

Mapping: {{mapping| 11 0 8 31 38 | 0 1 1 0 0 }}

* WE: ~21/20 = 109.0321{{c}}, ~3/2 = 706.3870{{c}}

* CWE: ~21/20 = 109.0909{{c}}, ~3/2 = 706.4785{{c}}

{{Optimal ET sequence|legend=0| 11c, 22 }}

Badness (Sintel): 2.27

Subgroup: 2.3.5.7.11.17

 

 

 

 

Badness (Sintel): 1.82

 

[[Subgroup]]: 2.3.5.7

 

 

 

: [[error map]]: {{val| +0.029 +2.358 -5.759 +2.597 }}

 

 

 

 

 

 

 

 

Subgroup: 2.3.5.7.11.13

 

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

 

 

* WE: ~2 = 1200.0524{{c}}, ~6/5 = 323.8039{{c}}

 

Badness (Sintel): 1.48

: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sevond (5-limit)]].''

[[Subgroup]]: 2.3.5.7

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

: mapping generators: ~10/9, ~3

[[Optimal tuning]]s:  

* [[WE]]: ~10/9 = 171.4007{{c}}, ~3/2 = 705.4982{{c}}

: [[error map]]: {{val| -0.195 +3.348 -4.112 -0.499 }}

: error map: {{val| 0.000 +3.651 -3.674 +0.071 }}

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

[[Badness]] (Sintel): 5.23

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 tunings:  

* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.4973{{c}}

{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}

Badness (Sintel): 2.33

 

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 tunings:  

* WE: ~11/10 = 171.4163{{c}}, ~3/2 = 705.2930{{c}}

* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.3402{{c}}

{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}

Badness (Sintel): 1.70

[[Category:Temperament collections]]

[[Category:Keemic temperaments| ]] <!-- main article -->