352/351: Difference between revisions

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'''352/351''', the '''major minthma''', '''major gentle comma''' or '''11/13-kleisma''' is a [[small comma|small]] [[13-limit]] (also 2.3.11.13 [[subgroup]]) [[comma]] measuring about 4.9{{cent}}. This comma can be described in a number of ways. First, it is the difference between the tridecimal minor third of [[13/11]] and the Pythagorean minor third of [[32/27]], hence the name ''11/13''-kleisma. This means that when this comma is [[tempering out|tempered out]], 11/8 plus three perfect fourths octave reduces to 13/8; as such, this comma can be understood as relating harmonics 11 and 13 by a chain of fourths. Second, it is the difference between various tridecimal intervals and their adjacent undecimal intervals such as:

* ~~Between the~~ tridecimal quartertone of [[1053/1024]] and the undecimal quartertone of [[33/32]];

* The tridecimal quartertone of [[1053/1024]] and the undecimal quartertone of [[33/32]];

* ~~Between~~ [[16/13]] and [[27/22]]; and

* [[88/81]] and [[13/12]];

* ~~Between~~ [[39/32]] and [[11/9]].

* [[128/117]] and [[12/11]];

* [[11/9]] and [[39/32]];

* [[16/13]] and [[27/22]];

* [[11/8]] and [[351/256]];

* [[243/176]] and [[18/13]];

And their [[octave complement]]<nowiki/>s.

352/351 and [[351/350]], the ratwolfsma, are extremely close in size and make up a consecutive pair of 13-limit [[superparticular]] commas. Their difference is [[chalmersia|123201/123200]], the chalmersia, the smallest 13-limit superparticular comma; their sum is [[176/175]], the valinorsma, an 11-limit superparticular comma.

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== Temperaments ==

Tempering out this comma leads to a form of '''major minthmic temperament''' (full 13-limit rank-5) or '''majorminth temperament''' (2.3.11.13 subgroup rank-3), which enables [[major minthmic chords]]. Mintha was a nymph turned into a mint plant by a goddess whom she got in the way of<ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0104%3Aalphabetic+letter%3DM%3Aentry+group%3D26%3Aentry%3Dmintha-bio-1 ''Dictionary of Greek and Roman Biography and Mythology''] – William Smith (1849)</ref><ref>[https://web.archive.org/web/20221230130135/http://mythandreligion.upatras.gr/english/__trashed/ ''Dictionary of Classical Mythology''] – M. R. Wright (2012)</ref><ref>[https://www.greekboston.com/culture/mythology/minthe ''All About Minthe – Myth of Greek Mythology''] – Greek Boston (2024)</ref>, and major minthmic tempering has the slightly sharp, minty-fresh fifths some people appreciate. [[Margo Schulter]] has expressed great fondness for this comma, for example.

It is tempered out in a large number of notable temperaments such as [[meanpop]], [[srutal]], [[countercata]], [[cassandra]], [[diaschismic]], [[hemififths]], [[amity]], [[rodan]], [[buzzard]] and higher rank temperaments like [[parapyth]], [[Aberschismic family#Overview to extensions|aberschismic extensions]], [[gentle]], [[Catalog of rank-4 temperaments#Akua (352/351, 847/845)|akua]] or [[valinorsmic]].

== Notation ==

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 }}
 '''352/351''', the '''major minthma''', '''major gentle comma''' or '''11/13-kleisma''' is a [[small comma|small]] [[13-limit]] (also 2.3.11.13 [[subgroup]]) [[comma]] measuring about 4.9{{cent}}. This comma can be described in a number of ways. First, it is the difference between the tridecimal minor third of [[13/11]] and the Pythagorean minor third of [[32/27]], hence the name ''11/13''-kleisma. This means that when this comma is [[tempering out|tempered out]], 11/8 plus three perfect fourths octave reduces to 13/8; as such, this comma can be understood as relating harmonics 11 and 13 by a chain of fourths. Second, it is the difference between various tridecimal intervals and their adjacent undecimal intervals such as:
-* Between the tridecimal quartertone of [[1053/1024]] and the undecimal quartertone of [[33/32]];
+* The tridecimal quartertone of [[1053/1024]] and the undecimal quartertone of [[33/32]];
-* Between [[16/13]] and [[27/22]]; and
+* [[88/81]] and [[13/12]];
-* Between [[39/32]] and [[11/9]].
+* [[128/117]] and [[12/11]];
+* [[11/9]] and [[39/32]];
+* [[16/13]] and [[27/22]];
+* [[11/8]] and [[351/256]];
+* [[243/176]] and [[18/13]];
+And their [[octave complement]]<nowiki/>s.
 /351 and [[351/350]], the ratwolfsma, are extremely close in size and make up a consecutive pair of 13-limit [[superparticular]] commas. Their difference is [[chalmersia|123201/123200]], the chalmersia, the smallest 13-limit superparticular comma; their sum is [[176/175]], the valinorsma, an 11-limit superparticular comma.
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 == Temperaments ==
 Tempering out this comma leads to a form of '''major minthmic temperament''' (full 13-limit rank-5) or '''majorminth temperament''' (2.3.11.13 subgroup rank-3), which enables [[major minthmic chords]]. Mintha was a nymph turned into a mint plant by a goddess whom she got in the way of<ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0104%3Aalphabetic+letter%3DM%3Aentry+group%3D26%3Aentry%3Dmintha-bio-1 ''Dictionary of Greek and Roman Biography and Mythology''] – William Smith (1849)</ref><ref>[https://web.archive.org/web/20221230130135/http://mythandreligion.upatras.gr/english/__trashed/ ''Dictionary of Classical Mythology''] – M. R. Wright (2012)</ref><ref>[https://www.greekboston.com/culture/mythology/minthe ''All About Minthe – Myth of Greek Mythology''] – Greek Boston (2024)</ref>, and major minthmic tempering has the slightly sharp, minty-fresh fifths some people appreciate. [[Margo Schulter]] has expressed great fondness for this comma, for example.
+It is tempered out in a large number of notable temperaments such as [[meanpop]], [[srutal]], [[countercata]], [[cassandra]], [[diaschismic]], [[hemififths]], [[amity]], [[rodan]], [[buzzard]] and higher rank temperaments like [[parapyth]], [[Aberschismic family#Overview to extensions|aberschismic extensions]], [[gentle]], [[Catalog of rank-4 temperaments#Akua (352/351, 847/845)|akua]] or [[valinorsmic]].
 == Notation ==