source file: mills3.txt Subject: RE: Basic dimensionality, and related issues From: gbreed@cix.compulink.co.uk I wrote: >If the dissonance of a composite number is independent of its >factors, why use lattices at all? and Paul replied: >What an odd question! What could you possibly mean by it? Okay, the first part of my question is a condition. If you're going to give a new direction to 9, you may also give one to 15, 21 and so on until every number has its own diection. Nobody has suggested this, but then nobody has said why we should stop at 9 either. Using the harmonic distance on a rectangular lattice as a measure of dissonance is an economical theory. All octave invariant, 5-prime-limit intervals can be expressed in 2 dimensions on this lattice. The dissonance of these intervals can be calculated using 2 free parameters: the step lengths in the 3- and 5- directions. The method can be generalised into any number of prime dimensions with a new free parameter for every new dimension. Switch to a triangular lattice, and you need n-1 new free parameters for n-dimensional harmony, giving n(n-1) in total. These parameters can all be produced from the same formula, like log(max(m,n)). Give each integer its own direction, though, and you need n free parameters to assess n+1 intervals (including 1/1). You could just as easily write down a list of the intervals you plan to use, and say how dissonant each is. So, why use a lattice? Or, if 9 is special, why? Actually, since I asked that original question something did occur to me about these composite lattices. They are geting very close to being a Hilbert space. I think this may be important, but I will not explain myself here because I don't expect most list subscribers will know what a Hilbert space is, let alone what its significance might be. Approximate scales where 9/1 approximates to be different to twice 3/1 are an entirely different idea. Don't use arguments for one to justify the other. There is nothing wrong with the basic dimension of a scale changing with an approximation. Incidentally, Paul, why do you think your previous comment on this contradicts what I said? All equal temperaments have a basic dimension of 1. If 9 approximates to an odd number, how would you tune 1:3:9? If 3/1 and 9/1 are optimised, 9/3=3/1 must be wrong. Or do you plan to avoid all chords where 9 and 3 both occur? Otherwise, you *could* equally say that you have two different 3- directions. This problem doesn't arise with harmonic dimensionality. You could curve your triangular, 3-5 lattice and add extra links between the 9's. This will be quantitatively the same as giving 9 a new dimension. $AdditionalHeaders: Received: from ns.ezh.nl by notesrv2.ezh.nl (Lotus SMTP MTA v1.1 (385.6 5-6-1997)) with SMTP id C12564DB.00281032; Mon, 21 Jul 1997 09:17:35 +0200