8019/8000: Difference between revisions
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== Temperaments == | == Temperaments == | ||
In the full 11-limit, tempering it out leads to the rank-4 | In the full 11-limit, tempering it out leads to the rank-4 [[Rank-4 temperament #Trimitone (8019/8000)|trimitone temperament]]. 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)<sup>3</sup> and [[729/512|(9/8)<sup>3</sup>]] 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)<sup>3</sup> 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-limit|11-]] or [[13-limit]], it is closely related to [[Schismatic family #Bischismic|bischismic]], which also tempers [[3136/3125]]. | 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)<sup>3</sup> and [[729/512|(9/8)<sup>3</sup>]] 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)<sup>3</sup> 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-limit|11-]] or [[13-limit]], it is closely related to [[Schismatic family #Bischismic|bischismic]], which also tempers [[3136/3125]]. | ||
== See also == | ==See also== | ||
* [[Small comma]] | *[[Small comma]] | ||
[[Category:Trimitone]] | [[Category:Trimitone]] | ||
[[Category:Commas named for the intervals they stack]] | [[Category:Commas named for the intervals they stack]] | ||
Latest revision as of 11:08, 24 January 2025
| Interval information |
Lalotrigu comma
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. 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.