8019/8000

Interval information

Ratio	8019/8000
Factorization	2-6 × 36 × 5-3 × 11
Monzo	[-6 6 -3 0 1⟩
Size in cents	4.106806¢
Name	trimitone comma
Color name	L1og31, lalotrigu 1sn, Lalotrigu comma
FJS name	[math]\displaystyle{ \text{d1}^{11}_{5,5,5} }[/math]
Special properties	reduced
Tenney height (log2 nd)	25.935
Weil height (log2 max(n, d))	25.9384
Wilson height (sopfr(nd))	56
Comma size	small
S-expression	S9 / S10
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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.

See also

• Small comma