243/242: Difference between revisions

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{{Infobox Interval

| Name = rastma, neutral third comma

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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 an [[11-limit]] (also 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 fall short of [[3/2]] just perfect fifth, 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 [[~~Diatonic, Chromatic, Enharmonic, 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.

[[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.

== Temperaments ==

Tempering it out in the full 11-limit leads to the ~~rastmic temperament, enabling~~ [[rastmic ~~chords~~]], or, in the 2.3.11 subgroup, the [[~~Subgroup temperaments #Neutral~~|neutral]] temperament, enabling [[~~Neutral tetrad|neutral~~ chords]].

[[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.

~~=== Rastmic ===~~

~~Subgroup: 2.3.5.7.11~~

~~[[Comma list]]: 243/242~~

~~[[Mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 0 0 5 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 0}}]~~

~~[[POTE tuning]]: ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464~~

~~{{Val list|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }}~~

~~[[Badness]]: 0.05088 × 10<sup>-6</sup>~~

~~=== Neutral ===~~

~~Subgroup: 2.3.11~~

~~[[Comma list]]: 243/242~~

~~[[Sval]] [[mapping]]: [{{val| 1 1 2 }}, {{val| 0 2 5 }}]~~

~~[[Gencom]] [[mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 0 0 5 }}]~~

~~Gencom: [2 11/9; 243/242]~~

~~[[POTE tuning]]: ~11/9 = 350.525~~

~~{{Val list|legend=1| 7, 10, 17, 24, 65, 89, 202, 291, 380 }}~~

[[~~Tp tuning #T2 tuning~~|~~RMS error~~]]~~: 0~~.~~3021 cents~~

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

== See also ==

* [[Rank-4 temperament #Rastmic (243/242)]]

* [[Rastmic rank three clan]], the clan of rank-3 temperaments where it is tempered out

* [[Rastmic chords]]

* [[Small comma]]

* [[List of superparticular intervals]]

[[Category:Rastmic]]

[[Category:Neutral]]

[[Category:Commas named after musical traditions]]

[[Category:Commas named for the generator of their temperament]]

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+{{interwiki
+| de = 243/242
+| en = 243/242
+| es =
+| ja =
+}}
 {{Infobox Interval
 | Name = rastma, neutral third comma
@@ Line 4: / 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 an [[11-limit]] (also 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 fall short of [[3/2]] just perfect fifth, 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 [[Diatonic, Chromatic, Enharmonic, 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.
+[[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.
 == Temperaments ==
-Tempering it out in the full 11-limit leads to the rastmic temperament, enabling [[rastmic chords]], or, in the 2.3.11 subgroup, the [[Subgroup temperaments #Neutral|neutral]] temperament, enabling [[Neutral tetrad|neutral chords]].
+[[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.
-=== Rastmic ===
-Subgroup: 2.3.5.7.11
-[[Comma list]]: 243/242
-[[Mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 0 0 5 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 0}}]
-[[POTE tuning]]: ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464
-{{Val list|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }}
-[[Badness]]: 0.05088 × 10<sup>-6</sup>
-=== Neutral ===
-Subgroup: 2.3.11
-[[Comma list]]: 243/242
-[[Sval]] [[mapping]]: [{{val| 1 1 2 }}, {{val| 0 2 5 }}]
-[[Gencom]] [[mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 0 0 5 }}]
-Gencom: [2 11/9; 243/242]
-[[POTE tuning]]: ~11/9 = 350.525
-{{Val list|legend=1| 7, 10, 17, 24, 65, 89, 202, 291, 380 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.3021 cents
+See [[Rastmic clan]] for the clan of rank-2 temperaments where it is tempered out. See [[Rastmic rank three clan|Rastmic rank-3 clan]] for the clan of rank-3 temperaments where it is tempered out.
 == See also ==
-* [[Rank-4 temperament #Rastmic (243/242)]]
-* [[Rastmic rank three clan]], the clan of rank-3 temperaments where it is tempered out
-* [[Rastmic chords]]
-* [[Small comma]]
 * [[List of superparticular intervals]]
 [[Category:Rastmic]]
 [[Category:Neutral]]
+[[Category:Commas named after musical traditions]]
+[[Category:Commas named for the generator of their temperament]]