385/384: Difference between revisions

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The keenanisma is the 11-limit comma 385/384 = |-7 -1 1 1 1~~>~~ of 4.503 ~~cents~~. Tempering it out leads to the 11-limit rank four [[~~Keenanismic_family|keenanismic temperament~~]].

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 [[~~Patent_val|~~patent val]]s tempering out the keenansima include ~~[[15edo~~|15]], ~~[[19edo|~~19]], ~~[[22edo|~~22]], ~~[[31edo|~~31]], ~~[[34edo|~~34]], ~~[[37edo|~~37]], ~~[[41edo|~~41]], ~~[[53edo|~~53]], ~~[[68edo|~~68]], ~~[[72edo|~~72]], ~~[[118edo|~~118]], ~~[[159edo|~~159]], ~~[[190edo|~~190]], ~~[[212edo|~~212]] and ~~[[284edo|~~284]].

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 [[~~keenanismic_tetrads|~~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|~~tonality diamond]], and hence regarded as an 11-limit consonance.

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:~~keenanismic_tetrads_in_31edo_sym.png|alt=keenanismic_tetrads_in_31edo_sym.png|keenanismic_tetrads_in_31edo_sym~~.png]] [[Category:11-limit]]

[[File:keenanismic tetrads in 31edo sym.png]]

[[Category:11-limit]]

[[Category:31edo]]

[[Category:~~comma~~]]

[[Category:Comma]]

[[Category:~~keenanismic]]~~

[[Category:Keenanismic]]

~~[[Category:tetrad~~]]

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-The keenanisma is the 11-limit comma 385/384 = |-7 -1 1 1 1&gt; of 4.503 cents. Tempering it out leads to the 11-limit rank four [[Keenanismic_family|keenanismic temperament]].
+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 [[Patent_val|patent val]]s tempering out the keenansima include [[15edo|15]], [[19edo|19]], [[22edo|22]], [[31edo|31]], [[34edo|34]], [[37edo|37]], [[41edo|41]], [[53edo|53]], [[68edo|68]], [[72edo|72]], [[118edo|118]], [[159edo|159]], [[190edo|190]], [[212edo|212]] and [[284edo|284]].
+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 [[keenanismic_tetrads|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|tonality diamond]], and hence regarded as an 11-limit consonance.
+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:keenanismic_tetrads_in_31edo_sym.png|alt=keenanismic_tetrads_in_31edo_sym.png|keenanismic_tetrads_in_31edo_sym.png]]      [[Category:11-limit]]
+[[File:keenanismic tetrads in 31edo sym.png]]
+[[Category:11-limit]]
 [[Category:31edo]]
-[[Category:comma]]
+[[Category:Comma]]
-[[Category:keenanismic]]
+[[Category:Keenanismic]]
-[[Category:tetrad]]