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'''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 [[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 property that becomes especially apparent when one works extensively with the 11-limit. | '''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 [[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 property that becomes especially apparent when one works extensively with the 11-limit. | ||
== | == Temperaments == | ||
Tempering it out leads to | 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]]. | ||
=== Rastmic temperament === | |||
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 generator]]s: ~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> | |||
== See also == | == See also == | ||
* [[Rastmic temperaments]], a collection of rank-3 temperaments where it is tempered out | |||
* [[Rastmic chords]] | * [[Rastmic chords]] | ||
* [[Small comma]] | * [[Small comma]] | ||
Revision as of 09:10, 19 April 2021
| Interval information |
neutral third comma
reduced
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 cents. It is significant in Just Intonation as the amount by which a stack of two 33/32 quartertones falls short of an 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 subchroma that separates greater and lesser 11-limit chromas – a property that becomes especially apparent when one works extensively with 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 neutral temperament, enabling neutral chords.
Rastmic temperament
Subgroup: 2.3.5.7.11
Comma list: 243/242
Mapping: [⟨1 1 0 0 2], ⟨0 2 0 0 5], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]
POTE generators: ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464
Badness: 0.05088 × 10-6
See also
- Rastmic temperaments, a collection of rank-3 temperaments where it is tempered out
- Rastmic chords
- Small comma
- List of superparticular intervals