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 pure tritave into two equal parts. Every even-numbered EDT has this interval.

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, 5/3, 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 preferrable 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 meantone region. (for this purpose, ed9/5 is good answer.)

• 24edt - simple. "Fifth" is "7\12" ≈ 11/8, off by 3 cents.
• 46edt - approximately 29edo. This have two "fifth," "14\23" ≈ 7/5, "13\23" ≈ 15/11.
• 38edt - approximately 24edo. "Fifth" is "11\19" ≈ 11/8. Can convert easily from 19edo.
• 36edt - approximately stretched 23edo. Another candidate with "fifth" 7/5...
• 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 to good for micro- augene[12].