Intervals list (beating frequency)

Lorenzo Frizzera (Tue, 29 Oct) said:

> Ciao. I'm working with a friend on a software for guitar and I need a
> list of the intervals in 12-ET from the more consonant to more
> dissonant. With "consonant" I mean beatless, empty. ...
> Anyone can help me? Maybe with a mathematic
> law on beating and interval spacing (spacing: 10M is more consonant
> than 3M). Ciao.

Beating is dependent upon frequency as well as interval size, so you'll
have to set this according to your own purposes. The formula will vary
according to the just (beatless) interval which a tempered interval most
nearly approximates. Let's represent the just interval as "x/y", and
indicate the size of the tempered interval close to this as "t"; the
frequency of the lower pitch in the interval t we shall call "f". In
this case the upper pitch of t will be ft. Take harmonic number y of the
upper pitch, which will have the frequency fty, and subtract from this
the harmonic number x of the lower pitch, which will have the frequency
fx, i.e. f(ty - x). A positive result means that the tempered interval
is sharper than the neighbouring just interval, and a negative result
will signify flatness.

To give an example: say you wish to find the rate at which the 12-TET
fifth above A = 220Hz beats. In this case x = 3, y = 2, t = 2^(7/12) and
f = 220. You will then have to calculate 220(2[2^(7/12)] - 3). The
result is nearly -0.75, so you should listen for 3 beats every 4
seconds.

I hope this enables you to complete your task, Lorenzo. Ciao!

--
Jonathan Walker
Queen's University Belfast
mailto:kollos@cavehill.dnet.co.uk
http://www.music.qub.ac.uk/~walker/

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