6656/6655

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Interval information
Ratio 6656/6655
Factorization 29 × 5-1 × 11-3 × 13
Monzo [9 0 -1 0 -3 1
Size in cents 0.26012081¢
Name jacobin comma
Color name thotrilu-agu comma
FJS name [math]\text{m2}^{13}_{5,11,11,11}[/math]
Special properties superparticular,
reduced
Tenney height (log2 n⋅d) 25.4007
Weil height (max(n, d)) 6656
Benedetti height (n⋅d) 44295680
Harmonic entropy
(Shannon, [math]\sqrt{n\cdot d}[/math])
~2.39778 bits
Comma size unnoticeable
open this interval in xen-calc

6656/6655, the jacobin comma, apparently named by Gene Ward Smith in 2014, is a 13-limit (also 2.5.11.13 subgroup) superparticular interval of about 0.26 ¢. It is the difference between a stack of three 11/8 superfourths and one 13/10 naiadic plus an octave. In terms of commas, it is the difference between 364/363 and 385/384, between 2080/2079 and 3025/3024 as well as between 4096/4095 and 10648/10647. In the 17-limit, it factors neatly into 12376/12375 × 14400/14399.

Temperaments

By tempering it out, the jacobin temperament is defined. Perhaps most remarkably, 1789edo is an edo that supports the jacobin temperament. You may find a list of good JI-approximating edos that support this temperament below. Although it is more rational to use such edos for this temperament, 1789edo has a unique position due to its number of steps being a hallmark year of the French Revolution.

Subgroup: 2.3.5.7.11.13

Mapping:
[1 0 0 0 0 -9],
0 1 0 0 0 0],
0 0 1 0 0 1],
0 0 0 1 0 0],
0 0 0 0 1 3]]

Mapping generators: ~2, ~3, ~5, ~7, ~11

Optimal GPV sequence9, 15, 22, 26, 31f, 37, 39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee

See also