243/242: Difference between revisions
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'''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: | '''243/242''', the '''rastma''' or the '''neutral third comma''', is a [[small comma|small]] [[11-limit]] (specifically [[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-subgroup intervals, namely: between [[11/9]] and its [[fifth complement]], [[27/22]], between [[3/2]] and a stack of two [[11/9]]'s (as a [[semiparticular]] comma), or equivalently between a [[2187/2048|Pythagorean apotome (2187/2048)]] and a stack of two [[33/32]]'s. | ||
In the full 11-limit, it is also the difference between [[45/44]] and [[55/54]], which shows it as the [[subchromatic|subchroma]] that separates greater and lesser 11-limit [[chroma]]s – 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)]]. | 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)]]. | ||
Latest revision as of 10:17, 12 November 2026
| Interval information |
neutral third comma
1uuM, luluma
reduced
243/242, the rastma or the neutral third comma, is a small 11-limit (specifically 2.3.11-subgroup) comma of roughly 7.1 cents. It is significant in just intonation as the difference between many 2.3.11-subgroup intervals, namely: between 11/9 and its fifth complement, 27/22, between 3/2 and a stack of two 11/9's (as a semiparticular comma), or equivalently between a Pythagorean apotome (2187/2048) and a stack of two 33/32's.
In the full 11-limit, it is also the difference between 45/44 and 55/54, which shows it as the 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).
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, enabling rastmic chords in either case.
Tempering out the components of the S-expression alone (81/80 and 121/120) leads to mohaha, 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.