LCMs of periods

I didn't receive this tuning digest from Paul E until he recently
forwarded it to me. In it he said.

>I thought we had decided that our "differences of opinion" were
>simply an error on your part. You were claiming that the LCM of
>a chord was proportional to the pattern length of the chord. I
>pointed out that this was only true under certain wild
>assumptions about the relative register of the chords. For
>example, the major triad (4:5:6) and the minor triad (10:12:15)
>both have an LCM of 60. For both chords, the pattern length of
>the chord as a whole is that of the frequency represented by the
>number 1. So in order to give both chords the same wavelength,
>you have to play the minor triad an octave plus a major third
>higher than the major triad. If you play them in the same
>register, the wavelength of the minor triad is 10/4 = 2.5 times
>longer than that of the major triad.

It's true that I was confused about this for awhile, actually,
quite a long while, and I would again like to thank you, Paul,
for bringing my error to my attention. The pattern length of
chords is proportional to the LCM of their PERIODS, not the LCM
of their frequencies. I discovered this when I was designing
frequency divider JI instruments in the mid 60's, but when I
wrote my sequencer for FM synthesis, which is frequency rather
than period oriented, I lost track of the distinction. The
oversight was masked by the fact that chords with small lcms of
frequencies also tend to have small lcms of periods, since they
are all made up of the same set of prime factors.

Below is an ascii illustration of the periods of the triads
mentioned by Paul. Those who are put off by long lists of
numbers would be well advised to skip the rest of this post.

1 1 1 1 1 1 1
2 2 2 2 2 2 2
3 3 3 3 3 3 3
4 1 4 4 4 4 4
5 2 1 5 5 5 5
6 3 2 1 6 6 6
7 1 3 2 7 7 7
8 2 4 3 8 8 8
9 3 1 4 9 9 9
10 1 2 5 10 10 10
11 2 3 1 1 11 11
12 3 4 2 2 12 12
13 1 1 3 3 1 13
14 2 2 4 4 2 14
15 3 3 5 5 3 15
16 1 4 1 6 4 1
17 2 1 2 7 5 2
18 3 2 3 8 6 3
19 1 3 4 9 7 4
20 2 4 5 10 8 5
21 3 1 1 1 9 6
22 1 2 2 2 10 7
23 2 3 3 3 11 8
24 3 4 4 4 12 9
25 1 1 5 5 1 10
26 2 2 1 6 2 11
27 3 3 2 7 3 12
28 1 4 3 8 4 13
29 2 1 4 9 5 14
30 3 2 5 10 6 15
31 1 3 1 1 7 1
32 2 4 2 2 8 2
33 3 1 3 3 9 3
34 1 2 4 4 10 4
35 2 3 5 5 11 5
36 3 4 1 6 12 6
37 1 1 2 7 1 7
38 2 2 3 8 2 8
39 3 3 4 9 3 9
40 1 4 5 10 4 10
41 2 1 1 1 5 11
42 3 2 2 2 6 12
43 1 3 3 3 7 13
44 2 4 4 4 8 14
45 3 1 5 5 9 15
46 1 2 1 6 10 1
47 2 3 2 7 11 2
48 3 4 3 8 12 3
49 1 1 4 9 1 4
50 2 2 5 10 2 5
51 3 3 1 1 3 6
52 1 4 2 2 4 7
53 2 1 3 3 5 8
54 3 2 4 4 6 9
55 1 3 5 5 7 10
56 2 4 1 6 8 11
57 3 1 2 7 9 12
58 1 2 3 8 10 13
59 2 3 4 9 11 14
60 3 4 5 10 12 15

I think of the above diagram as a representation of a bunch of
frequency dividers hooked to a common frequency source. As the
high-frequency source steps through its 60 states, the counters
step through 3, 4, 5 ... .etc states.

I thought we had this one straightened out. As I remember it you
agreed with one of my tables of numbers, but maybe I'm not
remembering correctly.

The other issue, at least in my mind, was ratio invariance. I
suppose it might have been better to use the term frequency
invariance. To cover that one you need to go to the scale level.

Let's take an example 12-tone, 7-limit scale, which I believe is
the 12-tone, 7-limit scale with the smallest frequencies.

70 72 80 84 90 96 105 108 112 120 126 135

Here is that scales inversion, or the expression of the scale in
period form rather than frequency form.

224 240 252 270 280 288 315 336 360 378 420 432

This is the output of a program I wrote after my conversations
with Paul that takes the LCMs of the periods of a scale and
constructs a minimum set of chords without using the principle of
ratio/frequency invariance.

In other words, the same chord at a higher frequency has a
shorter period than one at a lower frequency, and thus comes up
first in the sort.

12 224
11 240
10 252
9 270
8 280
7 288
6 315
5 336
4 360
3 378
2 420
1 432

1- 5- 9 ; 1260 3 4 5
3- 6-10 ; 1440 4 5 6
1- 4-10 ; 1680 4 5 7
1- 4- 7 ; 1680 4 5 6
2- 5- 8 ; 1890 5 6 7
4- 9-11 ; 2016 6 8 9
0- 3-10 ; 2160 5 6 9

The numbers after the semicolons represent the periods of the
chords, not the LCMS adjusted for frequency. Thus the two 4:5:7
chords are rated differently. This is an interesting concept
even though it reversed the usual practice of rating similar
chords lower in the scale as more "fundamental" or "harmonious".

For example consider the many distinctions that are made in the
literature of the three major chords in the 7-tone Zarlino Scale.

I plan to do more work on this subject as soon as I can find the
time. I want to do a program that does period analysis on
frequency representations so that I can use it to generate chord
patterns for use with the my FM sequencer. Also, I am still
working on my DSP music project. That one uses period
representation throughout, but who knows when I'll get it done?
Too many fun things to do, and not enough time to do them in is
not really the worst problem to have.

Thanks again, Paul, for taking the time to analyze my work. As
you find other problems, I would be very happy to hear about
them.

Marion



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