385/384: Difference between revisions

Line 4:

| Monzo = -7 -1 1 1 1

| Cents = 4.50256

| Name = ~~undecimal kleisma~~, <br>~~keenanisma~~

| Name = keenanisma,<br>undecimal kleisma

| Color name =

| FJS name = P1<sup>5, 7, 11</sup>

Line 10:

}}

The '''~~undecimal kleisma~~'''~~<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7286.html#7296</ref><ref>https://www.huygens-fokker.org/docs/intervals.html</ref>~~ or '''~~keenanisma~~''' is the [[11-limit]] [[comma]] '''385/384''' = {{Monzo| -7 -1 1 1 1 }} 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 four [[Keenanismic family]].

The '''keenanisma''' or '''undecimal kleisma''' is the [[11-limit]] [[comma]] '''385/384''' = {{Monzo| -7 -1 1 1 1 }} 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 four [[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 [-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.

Line 19:

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

== Name ==

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</ref>. Dave himself initially resisted this eponymous naming, recommending a more descriptive name<ref> https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7286.html#7286</ref>. And so undecimal kleisma was adopted<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7286.html#7296</ref>, and to this day, undecimal kleisma in the Stichting Huygens-Fokker records<ref>https://www.huygens-fokker.org/docs/intervals.html</ref>.

The history of the name "keenanisma" is less clear. It's 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 ==

@@ Line 4: / Line 4: @@
 | Monzo = -7 -1 1 1 1
 | Cents = 4.50256
-| Name = undecimal kleisma, <br>keenanisma
+| Name = keenanisma,<br>undecimal kleisma
 | Color name =
 | FJS name = P1<sup>5, 7, 11</sup>
@@ Line 10: / Line 10: @@
 }}
-The '''undecimal kleisma'''<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7286.html#7296</ref><ref>https://www.huygens-fokker.org/docs/intervals.html</ref> or '''keenanisma''' is the [[11-limit]] [[comma]] '''385/384''' = {{Monzo| -7 -1 1 1 1 }} 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 four [[Keenanismic family]].
+The '''keenanisma''' or '''undecimal kleisma''' is the [[11-limit]] [[comma]] '''385/384''' = {{Monzo| -7 -1 1 1 1 }} 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 four [[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 [-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.
@@ Line 19: / Line 19: @@
 [[File:keenanismic tetrads in 31edo sym.png|thumb]]
+== Name ==
+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</ref>. Dave himself initially resisted this eponymous naming, recommending a more descriptive name<ref> https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7286.html#7286</ref>. And so undecimal kleisma was adopted<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7286.html#7296</ref>, and to this day, undecimal kleisma in the Stichting Huygens-Fokker records<ref>https://www.huygens-fokker.org/docs/intervals.html</ref>.
+The history of the name "keenanisma" is less clear. It's 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 ==