385/384: Difference between revisions

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'''385/384''', the '''keenanisma''' or '''undecimal kleisma''', is an [[11-limit]] [[comma]] of 4.503 [[cent]]s. It is both the interval that separates [[77/64]] and [[6/5]], and, the sum of the [[schisma]] and the [[symbiotic comma]]. Tempering it out leads to a temperament of the 11-limit rank-4 [[keenanismic family]].

~~The~~ '''keenanisma''' or '''undecimal kleisma''' is ~~the~~ [[11-limit]] [[comma]] ~~'''385/384''' = {{Monzo| -7 -1 1 1 1 }}~~ of 4.503~~{{nbhsp}}~~[[cent]]s. It is both the interval that separates [[77/64]] and [[6/5]], and, the sum of the [[schisma]] and the [[symbiotic comma]]. Tempering it out leads to a temperament of the 11-limit rank-4 [[keenanismic family]].

In addition to equating [[77/64]] and [[6/5]], tempering out the keenanisma equates [[48/35]] with [[11/8]], [[35/24]] with [[16/11]], and [[12/11]] with [[35/32]], which 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 {{nowrap|[-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 {{nowrap|[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.

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+'''385/384''', the '''keenanisma''' or '''undecimal kleisma''', is an [[11-limit]] [[comma]] of 4.503 [[cent]]s. It is both the interval that separates [[77/64]] and [[6/5]], and, the sum of the [[schisma]] and the [[symbiotic comma]]. Tempering it out leads to a temperament of the 11-limit rank-4 [[keenanismic family]].
-The '''keenanisma''' or '''undecimal kleisma''' is the [[11-limit]] [[comma]] '''385/384'''&nbsp;=&nbsp;{{Monzo| -7 -1 1 1 1 }} of 4.503{{nbhsp}}[[cent]]s. It is both the interval that separates [[77/64]] and [[6/5]], and, the sum of the [[schisma]] and the [[symbiotic comma]]. Tempering it out leads to a temperament of the 11-limit rank-4 [[keenanismic family]].
 In addition to equating [[77/64]] and [[6/5]], tempering out the keenanisma equates [[48/35]] with [[11/8]], [[35/24]] with [[16/11]], and [[12/11]] with [[35/32]], which 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 {{nowrap|[-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 {{nowrap|[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.