User:Dummy index/Semitritave

Interval information
Expression	[math]\displaystyle{ (3/1)^(1/2) }[/math]
Monzo	[0 1/2⟩
Size in cents	950.9775¢
Name	semitritave
Special properties	reduced

Interval

Semitritave, square root of 3:1, is an interseptimal interval. It divide tritave into two equal parts. Every even-numbered EDT has this interval.　It is strongly related to island comma, 676/675, via 13-limit approximant 26/15 and 45/26.

Merciful intonation

Semitritave is an candidate for "practically merciful intonation", because it is [math]\displaystyle{ [1; 1, 2, 1, 2, ...] }[/math] in continued fraction, have many gradually proximal ratios, 7/4, 19/11, 26/15, 71/41, ..., makes rich dissonance.

Approximating it by noble number:

[math]\displaystyle{ [1; 1, 2, 1, 1, 1, ...] }[/math] - 942.5 cents, between 12/7 and 19/11.
[math]\displaystyle{ [1; 1, 2, 1, 2, 1, 1, 1, ...] }[/math] - 950.4 cents, between 45/26 and 71/41.
[math]\displaystyle{ [1; 1, 2, 1, 3, 1, 1, 1, ...] }[/math] - 954.6 cents, between 26/15 and 33/19.

False octave

Semitritave is available for false octave. Differ from acoustic phi or ed7/4, two equave makes 3:1, well-known equave.

2*N-edt

Every even-numbered EDT has semitritave interval. Treating it as equave. Another preferable intervals...

5edt - 380 cents major third
6edt - 317 cents minor third
so 30edt?

To do mechanical translation from diatonic scores, "fifth" sound is preferred to be consonance. 7/5 is better, but it makes 3L 2s. 11/8 corresponds to meantone region. (for this purpose, 7/5 ≈ 3\5 of ed7/4 and 7/5 ≈ 4\7 of ed9/5 are both extreme...)

24edt - simple. "Fifth" is "7\12" ≈ 11/8, off by 3 cents.
36edt - approximately stretched-23edo. This have two "fifth," "11\18" ≈ 7/5 and "10\18" ≈ 19/14. 6/5 and 7/6 are good.
38edt - approximately 24edo. "Fifth" is "11\19" ≈ 11/8. Can convert easily from 19edo.
46edt - approximately 29edo. Two "fifth," "14\23" ≈ 7/5, "13\23" ≈ 15/11. 13/11 and 15/13 are precise.
54edt - approximately 34edo. Two "fifth," "16\27" ≈ 18/13 and "15\27" ≈ 19/14 are precise. Together with "9\27" ≈ 6/5 and "11\27" ≈ 5/4, seems good for micro- augene[12].
62edt - approximately 39edo. "Fifth" is "18\31" ≈ 11/8, and "wolf fifth" is "19\31" ≈ 7/5. By the way, "upmajor 3rd" and "downminor 3rd" approximate 17/14 and 17/15, where (17/14)*(17/15) = (11/8)*(1156/1155).

Memo

3.5/2.11/8 => 24edt, 3.5.7.13 => 30edt, 3.5/2.7/2 => 36edt, 3.2.11.17 => 38edt, 3.2.11/5.13/5 => 46edt, 3.10.14.13/8.34 => 52edt, 3.2.5.13.17 => 54edt, 3.10.14.17.11/8 => 62edt