243/242: Difference between revisions
Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
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{{Infobox Interval | {{Infobox Interval | ||
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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 [[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. | '''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. | ||
== Temperaments == | == Temperaments == | ||
Revision as of 19:56, 3 August 2023
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| Interval information |
neutral third comma
Lulu 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
Optimal ET sequence: 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202
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
Optimal ET sequence: 7, 10, 17, 24, 65, 89, 202, 291, 380
RMS error: 0.3021 cents
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