8019/8000: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Icon =
| Name = trimitone comma
| Ratio = 8019/8000
| Color name = L1og<sup>3</sup>1, lalotrigu 1sn,<br>Lalotrigu comma
| Monzo = -6 6 -3 0 1
| Comma = yes
| Cents = 4.1068
| Name =  
| Color name =  
| FJS name =
| Sound =  
}}
}}
'''8019/8000''' is a comma in the 2.3.5.11 subgroup, equal to ([[11/8]])/([[10/9]])<sup>3</sup>.
'''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)<sup>3</sup>.  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 ==
== Temperaments ==
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 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. 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 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]].


== See also ==
==See also==
*[[Small comma]]


* [[Unnoticeable comma]]
[[Category:Trimitone]]
* [[Category:11-limit]]
[[Category:Commas named for the intervals they stack]]
* [[Category:Ratio]]

Latest revision as of 11:08, 24 January 2025

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
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. 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