source file: mills2.txt Date: Fri, 17 Nov 1995 22:55:46 -0800 Subject: RE: LCMs and minimum chords From: Mmcky@aol.com Many on this list are teachers, and therefore much better qualified to explain Least Common Multiple than I, but, since I am the one who constantly uses the concept, I will make an attempt. Any help would be appreciated. The reason I think LCM is important to music, is that it describes a physical reality. If A single tone is a repeating waveform, the period of the signal is simply the inverse of its frequency. When you mix two or more tones, you get a signal whose period is the least common multiple of the periods of the individual tones. There are at least two fairly simple ways of computing the least common multiple of a set of integers. One is to factor the integers, and take the highest power of any single prime. For example to find the LCM of 12, 15, and 20, we factor the three numbers into 2^2*3=12, 3*5=15, and 2^2*5=20. The result is 2^2*3*5, or 60. This may be the simplest method when computing LCM's by hand for small numbers with small prime factors. The other method is better for dealing with large numbers with large prime factors, and is much faster for these problems, but it is a little more difficult to program. One applies Euclid's algorithm to two of the numbers, and finds the greatest common divisor. Then apply the same method to the LCM just calculated and the next number. Iteration of this method gives the LMC of a large set of number fairly rapidly. Here's a brief description of Euclid's Algorithm. Divide the smallest number into the larger, throw away the answer and keep the remainder. Divide the remainder you just calculated into the smaller of the original numbers. Keep doing this until you reach a zero remainder. The last number you generated before you reached the zero divisor is the greatest common divisor. For 12 and 15: 12/15 = 1 remainder 3 12/3 = 4 remainder 0 The gcd is 3. Then take (12*15)/3=60. The LCM of 12 and 15 is 60. Now we need to find the LCM of 60 and 20. For 20 and 60: 60/20 = 3 remainder 0 The gcd of 60 and 20 is 20. (20*60)/20=60. The LMC of 60 and 20 is 60. A minimum chord set is one that has the lowest LCM's and spans the octave. It requires computing the LCM's for all the possible chords of interest in a scale, (I use triads.), and then sorting the chords according to their LCM's. I also sort according to the smallest numbers in the triad, and then according to the notes in the scales on which the triad is built. This gives me an unambiguous definition of the minimum chord set, but it is not really clear to me that this actually produces the best minimum chord set. It might be interesting to sort all possible minimum chord sets according to a composite, or sum, of their LCMs. I hope this helps those of you who want to learn more about this approach to scale analysis. Marion Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 20 Nov 1995 14:54 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id EAA01093; Mon, 20 Nov 1995 04:54:29 -0800 Date: Mon, 20 Nov 1995 04:54:29 -0800 Message-Id: <9511200453.aa11876@cyber.cyber.net> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu