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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 | '''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]]. | ||
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/ | == Temperaments == | ||
[[Tempering out]] this comma 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 [[hexany|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. | |||
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 }}. | 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 | 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 interval 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|thumb]] | [[File:keenanismic tetrads in 31edo sym.png|thumb]] | ||
== | == Etymology == | ||
Originally this comma was recommended by [[Paul Erlich]] to be named "Keenan's kleisma", after [[Dave Keenan]], due to "it figur[ing] particularly heavily in his many postings about microtemperament"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_1161.html#1284 Yahoo! Tuning Group | ''72 owns the 11-limit'']</ref>. Dave himself initially resisted this eponymous naming, recommending a more descriptive name<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7286.html#7286 Yahoo! Tuning Group | ''Eponyms'']</ref>. And so undecimal kleisma was adopted<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7286.html#7296 Yahoo! Tuning Group | ''Eponyms'']</ref>, and to this day, undecimal kleisma is a name in ''Stichting Huygens–Fokker records''<ref>[https://www.huygens-fokker.org/docs/intervals.html Stichting Huygens–Fokker | ''List of Intervals'']</ref>. | Originally this comma was recommended by [[Paul Erlich]] to be named "Keenan's kleisma", after [[Dave Keenan]], due to "it figur[ing] particularly heavily in his many postings about microtemperament"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_1161.html#1284 Yahoo! Tuning Group | ''72 owns the 11-limit'']</ref>. Dave himself initially resisted this eponymous naming, recommending a more descriptive name<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7286.html#7286 Yahoo! Tuning Group | ''Eponyms'']</ref>. And so undecimal kleisma was adopted<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7286.html#7296 Yahoo! Tuning Group | ''Eponyms'']</ref>, and to this day, undecimal kleisma is a name in ''Stichting Huygens–Fokker records''<ref>[https://www.huygens-fokker.org/docs/intervals.html Stichting Huygens–Fokker | ''List of Intervals'']</ref>. | ||