243/242

Interval information

Ratio	243/242
Factorization	2-1 × 35 × 11-2
Monzo	[-1 5 0 0 -2⟩
Size in cents	7.13912¢
Names	rastma, neutral third comma
FJS name	[math]\displaystyle{ \text{A1}_{121} }[/math]
Special properties	superparticular, reduced
Tenney height (log2 nd)	15.8437
Weil height (log2 max(n, d))	15.8496
Wilson height (sopfr(nd))	39
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 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

Neutral temperament

Subgroup: 2.3.11

Comma list: 243/242

POL2 generator: ~11/9 = 350.525

Gencom: [2 11/9; 243/242]

Gencom mapping: [<1 1 0 0 2|, <0 2 0 0 5|]

Map: [<1 1 2|, <0 2 5|]

EDOs: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380

RMS error: 0.3021 cents

See also

• Rastmic temperaments, a collection of rank-3 temperaments where it is tempered out
• Rastmic chords
• Small comma
• List of superparticular intervals