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