Date: Tue, 25 Nov 1997 22:10:43 -0500
From: John Savage
Subject: What to do with log3/log2?
To: Dave Rusin
Dave,
Ok... I more or less understand where log3/log2 comes from, but
what exactly do you *do* with it?
-John Savage
==============================================================================
Date: Tue, 25 Nov 1997 21:54:26 -0600 (CST)
From: Dave Rusin
To: JSavage3@compuserve.com
Subject: Re: What to do with log3/log2?
You mean regarding musical scales? You try to put in a good number of notes per
octave ("q" say) so that if you count up some number "p" of tones from the
bottom of the octave you get a good approximation of the fifth. Well, the
interval you'll create has end frequencies in a ratio of (2^(1/q))^p -to-1.
Is that close to a fifth? A fifth has frequencies in a 3-to-2 ratio, so is
2^(p/q) close to 3/2 ? Take logs: the log of the first is (p/q)*log(2); the
log of the second is log(3) - log(2). So are _these_ two things close? Now
you're down to asking if
[(p/q) + 1]*log(2) is close to log(3)
which requires that [(p/q)+1] be a good approximation to... log(3)/log(2).
So! the issue at handis what good p's and q's there are which allow decent
approximations to 1.58496250...
Well, it's pretty close to 1.5 = 3/2, so things work pretty well if
(p/q)+1 = 1.5, that is, p=1 and q=2. So divide the octave into two equal
parts; a fifth is obtained with the middle note. Got some circuitry to
produce that tone? It's not all so good. You'll create frequencies in a
(2^(1/2))-to-1 ratio -- that's 1.414..., and not really so close to a
3-to-2 ratio.
Let's try again. 1.585.. is close to 1.6 = 8/5, which is (p/q)+1 if p=3 and
q=5. So we try breaking an octave into 5 equal parts; three together make
an interval which should be about a "fifth". How good is it? The actual
ratio will be (2^(3/5))-to-1, about 1.516: now pretty close to a perfect
3-to-2 ratio. This is classical Chinese music (5 intervals per octave;
3 intervals makes a fair-sounding "fifth").
Still not satisfied? How about noting how close 1.5849... is to 1.58333=19/12.
This leads to p=7 and q=12. So divide the scale into (duh!) 12 equal parts;
count off seven intervals (C->C#->D->D#->E->F->F#->G) and you should have a
pretty fine "fifth". How'd we do? Our attained ratio is (2^(7/12))-to-1, about
1.4983... to 1. This is really quite close to 3-to-2 ! Most people can't
hear the difference between a 1.4983...-to-1 ratio and a 1.5-to-1 ratio.
You can do better with larger p's and q's of course. There's a big chunk of
mathematics to help you find the best p's and q's. You'll never get it
perfect, however, as log(3)/log(2) isn't rational (i.e. not a ratio of
two integers).
dave