source file: mills2.txt Date: Tue, 1 Jul 1997 14:27:59 +0200 Subject: Lattice, LCM, and Aliquot Parts From: Mckyyy@aol.com Hi Graham, On octave invariance, it has just occurred to me that, in LCM terms, the octave has a property that no other ratio shares. That is two notes that differ only by octaves can be mixed without making the length of the resulting "interference" pattern longer than the waveform of the lowest note. If you mix two notes of wavelength 1 and 8, the wavelength of the resulting pattern is still 8, but if you mix two notes of wavelength 2 and 3, the resulting pattern has a wavelength of 6. This property makes it much easier to sing in octaves than any other interval, because the basic perception of the fundamental frequency of a melody is not disturbed, and is a justification for the principle of octave invariance. Another form of invariance, ratio invariance, is important in LCM analysis. This became clear to me during a discussion I had with Paul E. some time ago. Thinking back over that, it seems to me that our differences of opinion on the subject of the length of LCM patterns was mostly due to different assumptions about ratio invariance. I was assuming ratio invariance, he was not. I'm still not absolutely clear on all this and would appreciate any comments. Aliquot parts are simply a list of all the possible numbers that can be derived by multiplying subsets of the prime factors of a given number. For example, here is a list of the aliquot parts of 2880: 1 1 2 2 3 3 4 2^2 5 5 6 2*3 8 2^3 9 3^2 10 2*5 12 2^2*3 15 3*5 16 2^4 18 2*3^2 20 2^2*5 24 2^3*3 30 2*3*5 32 2^5 36 2^2*3^2 40 2^3*5 45 3^2*5 48 2^4*3 60 2^2*3*5 64 2^6 72 2^3*3^2 80 2^4*5 90 2*3^2*5 96 2^5*3 120 2^3*3*5 144 2^4*3^2 160 2^5*5 180 2^2*3^2*5 192 2^6*3 240 2^4*3*5 288 2^5*3^2 320 2^6*5 360 2^3*3^2*5 480 2^5*3*5 576 2^6*3^2 720 2^4*3^2*5 960 2^6*3*5 1440 2^5*3^2*5 2880 2^6*3^2*5 This list contains the seven-tone Zarlino scale, and 2880 is the LCM of that scale. That is not a coincidence. A list of aliquot parts contains all the scales that can be made from a given LCM. Most of the utilities distributed with my FasTrak Sequencer use this principle. To classify the chords in a given scale, I just sort them by their LCM's, or by their pattern length, if I do not want to think in terms of ratio invariance. I find this to be much simpler and easier to work with than multidimensional ratio analysis. But the big advantage, from my exact JI point of view, is that it gives me an easy way to analyze the musical possibilities of a given frequency divider network, and all modern electronic musical instruments use some form of frequency division in the sense that all the musical output is phase locked to some high frequency signal in the system, therefore these systems have the capability of producing exact musical intervals if programmed properly. Note that octave inversion of a chord can double its LCM. Marion Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 1 Jul 1997 15:10 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA15776; Tue, 1 Jul 1997 15:10:43 +0200 Date: Tue, 1 Jul 1997 15:10:43 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA15771 Received: (qmail 10512 invoked from network); 1 Jul 1997 13:10:35 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 1 Jul 1997 13:10:35 -0000 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu