Re: difference tones - utonal and otonal

Hi Paul,

I too wouldn't mind some kind of clarification.

I understand about diff. tones - that if two notes belong to the same
harmonic series then one hears the fundamental of that series as a
difference tone.

For instance, if one plays a 5/4 and a 3/2, one will hear a
1/4 diff tone beneath them.

Kind of quite a quiet note, but easily noticeable with the right voices.
As you add the second note in, you hear the low difference tone
join in with it.

Also understand that major chord harmonics 4 5 6 is otonal,
(= 1/1 5/4 3/2 on transposing down 1/4)
and minor chord can be thought of as 1/4 1/5 1/6
(= 3/2 6/5 1/1 on transposing up *6)

But now go to 1,3,5,7 hexany:
15/8 21/16 35/32
= 60/32 42/32 35/32
- you can find them all in the same harmonic series,
but the numbers are rather large, and there are no
other factors to divide out by.

Is this what it means to be an otonal triad?

Then a utonal triad would be one where the powers of two
are all on the top of the ratios, after suitable multiplying out?

I know this is a beginner's question, but prob. there may be some others new
to the topic who have joined the tuning list recently as well,

Robert

Robert Walker wrote,

>Is this what it means to be an otonal triad?

Yes.

>Then a utonal triad would be one where the powers of two
>are all on the top of the ratios, after suitable multiplying out?

Right -- with smaller numbers in the _bottom_ of the ratios than if you
tried to express the triad as an otonal one. For example, the major triad is
4:5:6 otonal and 1/15:1/12:1/10 utonal, so it's considered otonal, while the
reverse is true for the minor triad.

See Partch's _Genesis of a Music_ for more.