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The keenanisma is the 11-limit comma 385/384 = |-7 -1 1 1 1 | The '''keenanisma''' is the [[11-limit]] comma '''385/384''' = {{Monzo| -7 -1 1 1 1 }} of 4.503 [[cent]]s. Tempering it out leads to a temperament of the 11-limit rank four [[Keenanismic family]]. | ||
The keenanisma equates 48/35 with 11/8 and 35/24 with 16/11; these are 7-limit intervals of low complexity, lying across from 1/1 in the hexanies 8/7-6/5-48/35-8/5-12/7-2 and 7/6-5/4-35/24-5/3-7/4-2. Hence keenanismic tempering allows the hexany to be viewed as containing some 11-limit harmony. The hexany is a fundamental construct in the 3D lattice of [[The_Seven_Limit_Symmetrical_Lattices|7-limit pitch classes]], the "deep holes" of the lattice as opposed to the "holes" represented by major and minor tetrads, and in terms of the [[The_Seven_Limit_Symmetrical_Lattices|cubic lattice of 7-limit tetrads]], the otonal tetrad with root 11 (or 11/8) is represented by [-2 0 0]: 1-6/5-48/35-12/7-2. In terms of 7-limit chord relationships, this complexity is as low as possible for an 11-limit projection comma, equaling the [0 1 -1] of 56/55 and less than the other alternatives. Since keenanismic temperament is also quite accurate, this singles it out as being of special interest. | The keenanisma equates [[48/35]] with [[11/8]] and [[35/24]] with [[16/11]]; these are [[7-limit]] intervals of low complexity, lying across from 1/1 in the hexanies 8/7-6/5-48/35-8/5-12/7-2 and 7/6-5/4-35/24-5/3-7/4-2. Hence keenanismic tempering allows the hexany to be viewed as containing some 11-limit harmony. The hexany is a fundamental construct in the 3D lattice of [[The_Seven_Limit_Symmetrical_Lattices|7-limit pitch classes]], the "deep holes" of the lattice as opposed to the "holes" represented by major and minor tetrads, and in terms of the [[The_Seven_Limit_Symmetrical_Lattices|cubic lattice of 7-limit tetrads]], the otonal tetrad with root 11 (or 11/8) is represented by [-2 0 0]: 1-6/5-48/35-12/7-2. In terms of 7-limit chord relationships, this complexity is as low as possible for an 11-limit projection comma, equaling the [0 1 -1] of 56/55 and less than the other alternatives. Since keenanismic temperament is also quite accurate, this singles it out as being of special interest. | ||
EDOs with [[ | EDOs with [[patent val]]s tempering out the keenansima include {{EDOs| 15, 19, 22, 31, 34, 37, 41, 53, 68, 72, 118, 159, 190, 212 and 284}}. | ||
Characteristic of keenanismic tempering are the [[ | Characteristic of keenanismic tempering are the [[keenanismic tetrads]], 385/384-tempered versions of 1-5/4-3/2-12/7, 1-5/4-10/7-12/7, 1-6/5-3/2-7/4, 1-5/4-16/11-7/4, and 1-14/11-16/11-7/4. These are essentially tempered [[Dyadic_chord|dyadic chord]]s, where every dyad of the chord is a keenanismic tempered version of an interval of the 11-limit [[tonality diamond]], and hence regarded as an 11-limit consonance. | ||
[[File: | [[File:keenanismic tetrads in 31edo sym.png]] | ||
[[Category:11-limit]] | |||
[[Category:31edo]] | [[Category:31edo]] | ||
[[Category: | [[Category:Comma]] | ||
[[Category: | [[Category:Keenanismic]] | ||
Revision as of 20:27, 11 September 2020
The keenanisma is the 11-limit comma 385/384 = [-7 -1 1 1 1⟩ of 4.503 cents. Tempering it out leads to a temperament of the 11-limit rank four Keenanismic family.
The keenanisma equates 48/35 with 11/8 and 35/24 with 16/11; these are 7-limit intervals of low complexity, lying across from 1/1 in the hexanies 8/7-6/5-48/35-8/5-12/7-2 and 7/6-5/4-35/24-5/3-7/4-2. Hence keenanismic tempering allows the hexany to be viewed as containing some 11-limit harmony. The hexany is a fundamental construct in the 3D lattice of 7-limit pitch classes, the "deep holes" of the lattice as opposed to the "holes" represented by major and minor tetrads, and in terms of the cubic lattice of 7-limit tetrads, the otonal tetrad with root 11 (or 11/8) is represented by [-2 0 0]: 1-6/5-48/35-12/7-2. In terms of 7-limit chord relationships, this complexity is as low as possible for an 11-limit projection comma, equaling the [0 1 -1] of 56/55 and less than the other alternatives. Since keenanismic temperament is also quite accurate, this singles it out as being of special interest.
EDOs with patent vals tempering out the keenansima include 15, 19, 22, 31, 34, 37, 41, 53, 68, 72, 118, 159, 190, 212 and 284.
Characteristic of keenanismic tempering are the keenanismic tetrads, 385/384-tempered versions of 1-5/4-3/2-12/7, 1-5/4-10/7-12/7, 1-6/5-3/2-7/4, 1-5/4-16/11-7/4, and 1-14/11-16/11-7/4. These are essentially tempered dyadic chords, where every dyad of the chord is a keenanismic tempered version of an interval of the 11-limit tonality diamond, and hence regarded as an 11-limit consonance.
