8019/8000
Ratio | 8019/8000 |
Factorization | 2^{-6} × 3^{6} × 5^{-3} × 11 |
Monzo | [-6 6 -3 0 1⟩ |
Size in cents | 4.106806¢ |
Name | trimitone comma |
Color name | L1og^{3}1, lalotrigu 1sn, Lalotrigu comma |
FJS name | [math]\text{d1}^{11}_{5,5,5}[/math] |
Special properties | reduced |
Tenney height (log_{2} nd) | 25.935 |
Weil height (log_{2} max(n, d)) | 25.9384 |
Wilson height (sopfr (nd)) | 56 |
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~1.35522 bits |
Comma size | small |
S-expression | S9 / S10 |
open this interval in xen-calc |
8019/8000, the trimitone comma (for "triple minor (whole) tone"), is the comma in the 11-limit (also 2.3.5.11 subgroup) by which a stack of three instances of 10/9 fall short of 11/8, thus leading to the formulation of (11/8)/(10/9)^{3}. It is also the interval separating the syntonic comma and the ptolemisma because of being an ultraparticular.
In the 13-limit, it factors neatly into (729/728)(1001/1000).
Temperaments
In the full 11-limit, tempering it out leads to the rank-4 trimitone temperament (→ Rank-4 temperament #Trimitone). Due to the factorization above, it extends neatly to the 13-limit.
In terms of microtempering the 2.3.5.11 subgroup, it may combine well with the schisma as doing so gives lower-complexity interpretations to the 5-limit "tritones" of (10/9)^{3} and (9/8)^{3} and their octave-complements, which results in the 53&65 (or equivalently 12&53) temperament in the 2.3.5.11 subgroup. (The term "tritones" is being used here in the sense of stacking 3 tones, as calling (10/9)^{3} a "tritone" is questionable.) For optimising this temperament, 183edo is recommendable, although 65edo provides a less accurate tuning at the benefit of a more manageable number of tones (and at the benefit of being a superset of 5edo and 13edo, thus potentially making it easier to conceptualise). This temperament is therefore great for 8:9:10:11:12 chords. If extended to the full 11- or 13-limit, it is closely related to bischismic, which also tempers 3136/3125.