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{{Infobox Interval | {{Infobox Interval | ||
| Name = keenanisma, undecimal kleisma | |||
| Color name = 1ozy1, lozoyo 1sn,<br>Lozoyo comma | |||
| Comma = yes | |||
| Name = undecimal kleisma | |||
| Color name = | |||
| | |||
}} | }} | ||
'''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]]. | |||
== 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 hexanies 8/ | 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 }}. | |||
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>. | |||
The history of the name "keenanisma" is less clear. It is possible that "Keenan's kleisma" remained in use, and eventually was altered to "keenanisma" following a pattern used for many commas named for people. Another possibility is that when a temperament based on this comma was being named, "undecimal kleisma" was seen as unfit to base the name upon, and so Keenan's name was referenced instead, leading to "keenanismic", and then later "keenanisma" was formed from that. | |||
== See also == | == See also == | ||
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== References == | == References == | ||
<references/> | <references /> | ||
[[Category:Keenanismic]] | [[Category:Keenanismic]] | ||
[[Category: | [[Category:Commas named after their interval size]] | ||
[[Category:Commas named after music theorists]] | |||