(no subject)

I have posted a new paper on my web site, http://members.aol.com/mckyyy. The
title of the paper is "Chord Recognition". I needed illustrations to present
the subject clearly, so I used a .pdf file. I would appreciate and comments.

I'd like to contribute the following JI approximation for 144 et.

1/1 208/207 104/103 70/69 52/51 42/41 35/34 30/29 26/25 23/22 21/20 19/18
18/17 16/15 15/14 14/13 27/25 13/12 12/11 23/21 11/10 21/19 10/9 19/17 9/8
44/39 17/15 41/36 8/7 23/20 15/13 36/31 7/6 34/29 20/17 13/11 19/16 49/41 6/5
35/29 17/14 28/23 11/9 16/13 21/17 36/29 5/4 79/63 29/23 19/15 14/11 23/18
9/7 40/31 35/27 13/10 17/13 25/19 37/28 89/67 4/3 55/41 31/23 23/17 15/11
26/19 11/8 29/21 18/13 46/33 7/5 38/27 17/12 27/19 10/7 33/23 13/9 42/29
16/11 19/13 25/17 31/21 46/31 85/57 3/2 137/91 59/39 38/25 26/17 23/15 17/11
31/20 14/9 36/23 11/7 30/19 27/17 67/42 8/5 29/18 34/21 13/8 18/11 23/14
33/20 63/38 5/3 77/46 37/22 22/13 17/10 29/17 12/7 31/18 19/11 40/23 7/4
72/41 30/17 39/22 16/9 34/19 9/5 47/26 20/11 31/17 11/6 35/19 50/27 13/7
43/23 15/8 17/9 19/10 21/11 23/12 25/13 29/15 33/17 41/21 51/26 69/35 103/52
207/104

A few LCM basics:

A simple way to find the least common multiple: I'll use the 10:12:15 ratio
for this example.

Factor the numbers: 10=2*5,12=2^2*3,15=3*5.

Go through and find the largest exponent for each prime factor: 2^2*3*5=60.

This works well for small numbers. Unfortunately, factoring large numbers is
not trivial. Fortunately there is an easy way to get around that, Euclid's
Algorithm. Euclid's algorithm really only finds the greatest common divisor,
but if you know the greatest common divisor, then you can find the least
common multiple. I'll use Euclid's Algorithm to find the least common
multiple between 144 and 80.

Euclid's algorithm consists of finding the remainder between two numbers over
and over again until you get to a remainder of 0.

144 divided by 80 = 1 with a remainder of 64.

We could also say:

144 mod 80 = 64

The whole algorithm:

144 mod 80=64
80 mod 64=16
16 mod 64=0

Now we know that the greatest common divisor between 144 and 80 is 16,
therefore the least common multiple is 144*80/16, or 720. We can check with
the factoring method.

144=2^4*3^2
80=2^4*5

LCM=2^4*3^2*5=720

Also to clarify how the "lattice" function in the Aliq program works. I'll
explain using an example.

6 9
5 8
4 6

The fundamental of the scale is obtained by multiplying the bottom elements
of the chords: 4*6=24. All other numbers are octave adjusted to be in the
octave defined by this number. The next step is to multiply all the elements
of each chord with every element of every other chord. In this case those
products would be 4*8=32, 4*9=36, 5*6=30, 5*8=40, 5*9=45, 6*6=36, 6*8=48,
6*9=54.

Since the 48 is merely the octave, it is discarded. The duplicate 36 is
removed. Since the 54 is greater than 48, the octave limit, it is divide by
two and becomes 27.

The scale:

LCM: 4,320

7 45 15/8
6 40 5/3
5 36 3/2
4 32 4/3
3 30 5/4
2 27 9/8
1 24 1/1

I started using this method back when I was trying to design frequency
divider networks using a minimum of parts back in the late 60's. I came up
with a complete 7-tone musical instrument that did major and minor and had
automatic harmony and used only 32 flip-flops. Sometime in the early 80's we
had samples of a custom integrated circuit made but were never able to get it
into production. It was these instruments, called Holophones, which led to
my discussions with Wurlitzer. The chip had a single channel frequency
divider that did chord modulation, and a three-channel divider that could be
switched between major and minor.

Marion.