Parimo
Ratio | 1771561/1771470 |
Factorization | 2^{-1} × 3^{-11} × 5^{-1} × 11^{6} |
Monzo | [-1 -11 -1 0 6⟩ |
Size in cents | 0.088930804¢ |
Name | parimo |
Color name | satribilo-agu comma |
FJS name | [math]\text{ddd1}^{11,11,11,11,11,11}_{5}[/math] |
Special properties | reduced |
Tenney height (log_{2} nd) | 41.5131 |
Weil height (log_{2} max(n, d)) | 41.5132 |
Wilson height (sopfr (nd)) | 106 |
Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~2.39752 bits |
Comma size | unnoticeable |
open this interval in xen-calc |
The parimo (ratio: 1771561/1771470, monzo: [-1 -11 -1 0 6⟩) is an unnoticeable 11-limit comma with a value of approximately 0.09 ¢. It is the amount by which an octave-reduced stack of six 11/9 neutral thirds exceeds 5/3, as well as the amount by which a stack of three rastmas falls short of a syntonic comma.
Temperaments
Tempering it out leads to a form of parimic temperament, which can be said to be the undecimal counterpart of the metric temperament in that the syntonic comma is split into three equal parts in both systems. However, in parimic temperament, the parts represent different intervals – one part represents the rastma, and two represent the biyatisma. The corresponding 2.3.5.11 subgroup temperament is tritomere.
Parimic
Subgroup: 2.3.5.7.11
Comma list: 1771561/1771470
Mapping: [⟨1 0 5 0 1], ⟨0 1 1 0 2], ⟨0 0 -6 0 -1], ⟨0 0 0 1 0]]
- mapping generators: ~2, ~3, ~18/11, ~7
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9581, ~18/11 = 852.6073
Optimal ET sequence: 14c, 17c, 24, 31, 90e, 107cde, 114e, 121, 145, 152, 311, 342, 836, 1178, 1354, 1506, 1817, 1848, 2684, 3665, 4007, 4501, 4843, 6349, 6691, 21921, 28612, 35303, 48343, 63573, 70264e, 76955e, 98876e, 105567e
Badness: 0.0279 × 10^{-6}
Tritomere
Subgroup: 2.3.5.11
Comma list: 1771561/1771470
Sval mapping: [⟨1 0 5 1], ⟨0 1 1 2], ⟨0 0 -6 -1]]
- sval mapping generators: ~2, ~3, ~18/11
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9581, ~18/11 = 852.6073
Optimal ET sequence: 7, 17c, 24, 31, 69e, 90e, 97, 107ce, 114e, 121, 145, 152, 311, 335, 342, 494, 677, 1019, 1171, 3855, 4349, 5026, 5520, 6691, 7862, 36474, 37645, 44336, 45507, 53369, 61231
Badness: 0.00712 × 10^{-3}
Etymology
It is yet to be found out how this comma was named. However, as Petr Pařízek was the first to take note of it^{[1]} and as it was tempered out in the mohaha temperament, it could be conjectured that the name was a contraction of Pařízek and mohaha.