otonal to utonal

Paul Erlich:
> Unfortunately, Partch's glossary definitions (reproduced in Monz's
> dictionary) are pretty useless. It's pretty simple, though. Here are some
> frequency ratios for otonal chords and their utonal counterparts:
>
> otonal utonal
> 4:5:6 1/6:1/5:1/4=10:12:15
> 6:7:9 1/9:1/7:1/6=14:18:21
> 16:19:24 1/24:1/19:1/16=38:48:57
>
> Otonal chords are best though of as overtones over a (possibly absent but
> often still audible) fundamental, while utonal chords are best thought of as
> a set of fundamentals with a single common overtone (called the guide tone).
> For example, 4:5:6 is the 4th, 5th, and 6th harmonics of 1, while
> 1/6:1/5:1/4 has a guide tone of 1 which is the 6th harmonic of the first
> note, the 5th harmonic of the second note, and the 4th harmonic of the third
> note. Clearly, your tones need to have integer harmonics for utonal chords
> to "work", while otonal chords will "work" even if you have no harmonics at
> all or even a mild set of inharmonic partials.

How to translate an otonal triad to its utonal equivalent ?
(This is not new, it's just to help those who do not know.)

Formula Example
given triad x:y:z given triad 18:21:27
x' = x / GCD(x,y,z) x' = 18/GCD(18,21,27) = 18/3 = 6
y' = y / GCD(x,y,z) y' = 21/GCD(18,21,27) = 18/3 = 7
z' = z / GCD(x,y,z) z' = 27/GCD(18,21,27) = 18/3 = 9

x'' = y' * z' x'' = 7*9 = 63
y'' = x' * z' y'' = 6*9 = 54
z'' = x' * y' z'' = 6*7 = 42

x''' = x'' / GCD(x'',y'',z'') x''' = 63/GCD(63,54,42) = 63/3 = 21
y''' = y'' / GCD(x'',y'',z'') y''' = 54/GCD(63,54,42) = 63/3 = 18
z''' = z'' / GCD(x'',y'',z'') z''' = 42/GCD(63,54,42) = 63/3 = 14

x'''' = 1 / x''' x'''' = 1/21
y'''' = 1 / y''' y'''' = 1/18
z'''' = 1 / z''' z'''' = 1/14

result
18:21:27 = 6:7:9 = 1/21:1/18:1/14

Peter