243/242
Ratio | 243/242 |
Factorization | 2^{-1} × 3^{5} × 11^{-2} |
Monzo | [-1 5 0 0 -2⟩ |
Size in cents | 7.1391196¢ |
Names | rastma, neutral third comma |
Color name | 1uu1, lulu 1sn, Lulu comma |
FJS name | [math]\text{A1}_{11,11}[/math] |
Special properties | superparticular, reduced |
Tenney height (log_{2} n⋅d) | 15.8437 |
Weil height (max(n, d)) | 243 |
Benedetti height (n⋅d) | 58806 |
Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~2.61322 bits |
Comma size | small |
S-expression | S9 / S11 |
open this interval in xen-calc |
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 GPV 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 GPV 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