source file: m1389.txt
Date: Sat, 18 Apr 1998 17:11:28 -0400

Subject: microtonal math

From: monz@juno.com (Joseph L Monzo)

> ...What the world desperately needs is a web site
> dealing with math for music,sort of a 'Math for Absolute
> Dummies ' !!! I'm telling you, a lot of people could be
> put off by the math involved in microtonality.Good 
> thing I'm the persistant type !

1)
I'm working on a webpage to introduce my theory
and it will have a good deal of microtonal math.
(a couple more months, I hope...)

2)
Thought you might find this usefu:

The formula to calculate cents from any ratio.
The format I use here can be pasted directly into
an Excel spreadsheet; just replace RATIO with
the cell reference to the ratio:

=MOD((LOG(RATIO))*(1200/LOG(2)),1200)

For example, to find the cents of the "perfect 5th",
mod((log(3/2)*(1200/log(2)),1200).  This returns
702 cents.

The use of logarithms is necessary in converting
from just-intonation to equal-temperament because
equal-temperament is based on a just ratio
(normally the "octave" 2/1) which is divided equally.
When measured rationally, the frequencies  keep
increasing geometrically, so roots or logarithms
must be used to compute the tempered intervals.

The ratio of a tempered interval can be calculated
easily as follows:

=DIVISOR^(DEGREE/DIVISIONS)

For example:

The just "perfect 5th" is 3/2, which equals 1.5.

The 12-eq "perfect 5th" is the 7th degree
of a scale which is measured by dividing the "octave"
(with the ratio 2/1) into 12 equal parts, so its formula
is (2/1)^(7/12).  This returns 1.498307...

The 22-eq "perfect 5th" is the 13th degree of a scale
which also divides the "octave", this time in 22 equal
parts, so its formula is (2/1)^(13/22).  This returns
1.506196...

The 13th degee in the  "13th root of 3/2" scale
provides an exact "perfect 5th" because (3/2)^( 13/13)
returns 3/2 itself, or 1.5.

   Joseph L. Monzo
    monz@juno.com

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