385/384: Difference between revisions

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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]]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]], 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.

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]]s, where every dyad of the chord is a keenanismic tempered version of an interval of the [[11-odd-limit]] [[tonality diamond]], and hence regarded as an 11-odd-limit consonance.

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

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

== See also ==

* [[Keenanismic chords]]

* [[Keenanismic family]]

* [[Keenanismic family]], the rank-4 temperament family where it is tempered out

* [[~~Comma~~]]

* [[Keenanismic temperaments]], a collection of rank-3 temperaments where it is tempered out

* [[Small comma]]

* [[List of superparticular intervals]]

[[Category:11-limit]]

~~[[Category:31edo]]~~

[[Category:Small comma]]

[[Category:Keenanismic]]

[[Category:Superparticular]]

@@ Line 12: / Line 12: @@
 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]]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]], 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.
+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]]s, where every dyad of the chord is a keenanismic tempered version of an interval of the [[11-odd-limit]] [[tonality diamond]], and hence regarded as an 11-odd-limit consonance.
-[[File:keenanismic tetrads in 31edo sym.png]]
+[[File:keenanismic tetrads in 31edo sym.png|thumb]]
 == See also ==
 * [[Keenanismic chords]]
-* [[Keenanismic family]]
+* [[Keenanismic family]], the rank-4 temperament family where it is tempered out
-* [[Comma]]
+* [[Keenanismic temperaments]], a collection of rank-3 temperaments where it is tempered out
+* [[Small comma]]
 * [[List of superparticular intervals]]
 [[Category:11-limit]]
-[[Category:31edo]]
 [[Category:Small comma]]
 [[Category:Keenanismic]]
 [[Category:Superparticular]]