243/242: Difference between revisions

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| Comma = yes

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'''243/242''', the '''rastma''' or the '''neutral third comma''', is a [[small comma|small]] [[11-limit]] (~~also~~ [[2.3.11 subgroup|2.3.11-subgroup]]) [[comma]] ~~with a value~~ of roughly 7.1 [[cent]]s. It is significant in [[just intonation]] as the ~~amount by which a stack of two [[~~11~~/9]] neutral thirds (that is an Alpharabian narrow fifth~~

'''243/242''', the '''rastma''' or the '''neutral third comma''', is a [[small comma|small]] [[11-limit]] ( [[2.3.11 subgroup|2.3.11-subgroup]]) [[comma]] of roughly 7.1 [[cent]]s. It is significant in [[just intonation]] as the difference between many 2.3.11 intervals, namely:

[[~~121~~/81]]~~) fall short~~ of [[3/2]] ~~just perfect fifth as well~~ as ~~the amount by which~~ a ~~stack of two~~ [[~~27/22~~]] ~~neutral thirds (that is an Alpharabian wide fifth~~ [[~~729~~/~~484~~]]~~) exceeds a 3/2 just perfect fifth, the amount by which a stack of two~~ [[33/32]] ~~quartertones falls short of~~ an [[2187/2048|apotome (2187/2048)]] ~~as well as the difference between 11/9~~ and 27/~~22, and the difference between~~ [[45/44]] and [[55/54]]. The last ~~makes the theory where~~ the rastma ~~serves~~ as the [[subchromatic|subchroma]] that separates greater and lesser 11-limit chromas – a ~~theory~~ that becomes especially useful when one works extensively with high-fidelity renditions of the 11-limit.

* [[3/2]] and a stack of two [[11/9]] – as a [[semiparticular]] comma

* [[27/22]] and [[11/9]]

* an [[2187/2048|apotome (2187/2048)]] and a stack of two [[33/32]]

* [[45/44]] and [[55/54]]

The last difference shows the rastma as the [[subchromatic|subchroma]] that separates greater and lesser 11-limit chromas – a demonstration that becomes especially useful when one works extensively with high-fidelity renditions of the 11-limit.

Its square rooted value, about 3.57 cents, is also a tiny comma that is used in [[Neutral FJS]] as the formal comma of prime 11; the difference between [[11/8]] and the half-augmented fourth sqrt(243/128), or between 11/9 or 27/22 and the hemipythagorean neutral third [[sqrt(3/2)]].

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== Temperaments ==

[[Tempering out]] this comma in the full 11-limit leads to the [[rastmic]] rank-4 temperament, or in the 2.3.11 subgroup, the [[neutral (temperament)|neutral]] temperament, enabling [[rastmic chords]] in either case.

Tempering out the components of the S-expression alone ([[81/80]] and [[121/120]]) leads to [[mohajira]], a structurally very clean rendition of the [[2.3.5.11 subgroup]] combining [[meantone]] and rastmic.

See [[Rastmic clan]] for the clan of rank-2 temperaments where it is tempered out. See [[Rastmic rank-3 clan]] for the clan of rank-3 temperaments where it is tempered out.

@@ Line 10: / Line 10: @@
 | Comma = yes
 }}
-'''243/242''', the '''rastma''' or the '''neutral third comma''', is a [[small comma|small]] [[11-limit]] (also [[2.3.11 subgroup|2.3.11-subgroup]]) [[comma]] with a value of roughly 7.1 [[cent]]s. It is significant in [[just intonation]] as the amount by which a stack of two [[11/9]] neutral thirds (that is an Alpharabian narrow fifth
+'''243/242''', the '''rastma''' or the '''neutral third comma''', is a [[small comma|small]] [[11-limit]] ( [[2.3.11 subgroup|2.3.11-subgroup]]) [[comma]] of roughly 7.1 [[cent]]s. It is significant in [[just intonation]] as the difference between many 2.3.11 intervals, namely:
-[[121/81]]) fall short of [[3/2]] just perfect fifth as well as the amount by which a stack of two [[27/22]] neutral thirds (that is an Alpharabian wide fifth [[729/484]]) exceeds a 3/2 just perfect fifth, the amount by which a stack of two [[33/32]] quartertones falls short of an [[2187/2048|apotome (2187/2048)]] as well as the difference between 11/9 and 27/22, and the difference between [[45/44]] and [[55/54]]. The last makes the theory where the rastma serves as the [[subchromatic|subchroma]] that separates greater and lesser 11-limit chromas – a theory that becomes especially useful when one works extensively with high-fidelity renditions of the 11-limit.
+* [[3/2]] and a stack of two [[11/9]] – as a [[semiparticular]] comma
+* [[27/22]] and [[11/9]]
+* an [[2187/2048|apotome (2187/2048)]] and a stack of two [[33/32]]
+* [[45/44]] and [[55/54]]
+The last difference shows the rastma as the [[subchromatic|subchroma]] that separates greater and lesser 11-limit chromas – a demonstration that becomes especially useful when one works extensively with high-fidelity renditions of the 11-limit.
 Its square rooted value, about 3.57 cents, is also a tiny comma that is used in [[Neutral FJS]] as the formal comma of prime 11; the difference between [[11/8]] and the half-augmented fourth sqrt(243/128), or between 11/9 or 27/22 and the hemipythagorean neutral third [[sqrt(3/2)]].
@@ Line 17: / Line 23: @@
 == Temperaments ==
 [[Tempering out]] this comma in the full 11-limit leads to the [[rastmic]] rank-4 temperament, or in the 2.3.11 subgroup, the [[neutral (temperament)|neutral]] temperament, enabling [[rastmic chords]] in either case.
+Tempering out the components of the S-expression alone ([[81/80]] and [[121/120]]) leads to [[mohajira]], a structurally very clean rendition of the [[2.3.5.11 subgroup]] combining [[meantone]] and rastmic.
 See [[Rastmic clan]] for the clan of rank-2 temperaments where it is tempered out. See [[Rastmic rank-3 clan]] for the clan of rank-3 temperaments where it is tempered out.

243/242: Difference between revisions

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