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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 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 | === Rastmic === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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[[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}}] | [[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 | [[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 }} | {{Val list|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }} | ||
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[[Badness]]: 0.05088 × 10<sup>-6</sup> | [[Badness]]: 0.05088 × 10<sup>-6</sup> | ||
=== Neutral | === Neutral === | ||
Subgroup: 2.3.11 | Subgroup: 2.3.11 | ||
[[Comma list]]: 243/242 | [[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] | 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 also == | == See also == | ||
* [[Rank-4 temperament]] | |||
* [[Rastmic temperaments]], a collection of rank-3 temperaments where it is tempered out | * [[Rastmic temperaments]], a collection of rank-3 temperaments where it is tempered out | ||
* [[Rastmic chords]] | * [[Rastmic chords]] | ||
Revision as of 13:45, 6 October 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 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 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 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 neutral temperament, enabling neutral chords.
Rastmic
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 tuning: ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464
Badness: 0.05088 × 10-6
Neutral
Subgroup: 2.3.11
Comma list: 243/242
Sval mapping: [⟨1 1 2], ⟨0 2 5]]
Gencom mapping: [⟨1 1 0 0 2], ⟨0 2 0 0 5]]
Gencom: [2 11/9; 243/242]
POTE tuning: ~11/9 = 350.525
RMS error: 0.3021 cents
See also
- Rank-4 temperament
- Rastmic temperaments, a collection of rank-3 temperaments where it is tempered out
- Rastmic chords
- Small comma
- List of superparticular intervals