RE: LCMs and minimum chords

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


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