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.

Full [[7-limit]] keemic temperaments discussed elsewhere are:  

* [[Superkleismic]] (+1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]

* ''[[Fifives]]'' (+83349/81920) → [[Fifive family #Fifives|Fifive family]]

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

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.  

{{Mapping|legend=0| 1 -9 -6 -4 8 | 0 14 11 9 -6 }}

{{Mapping|legend=0| 1 -9 -6 -4 8 9 | 0 14 11 9 -6 -7 }}

{{Mapping|legend=0| 1 -9 -6 -4 -23 | 0 14 11 9 35 }}

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

{{Mapping|legend=0| 1 -10 -5 -10 -13 | 0 19 12 21 27 }}

{{Mapping|legend=0| 1 -10 -5 -10 -13 -3 | 0 19 12 21 27 11 }}

{{Mapping|legend=0| 1 -3 -2 2 4 | 0 17 16 3 -2}}

{{Mapping|legend=0| 1 -3 -2 2 4 1 | 0 17 16 3 -2 10 }}

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

{{Mapping|legend=0| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}

[[Category:Catalogs of rank-2 temperaments]]