microtonal math

> ...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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Drew Skyfyre writes:

> I was on Amazon's web site looking up Helmholtz' On the Sensations
> of Tone' ($ 12.76) and found they didn't have much info there about
> it.Their page says "Publication date: December 1954".Is there more
> than one edition of this book around or is this the same one,@570
> pages,translated,with additions,by Alexander Ellis ?

The only one I've seen (and the one I have) is published by Dover.
Based on the price you mentioned, I'd guess this is "it", and it
does have the additions by Ellis.

> Also would this be a "if you can only buy one book,this is it "
> category book ? Any other suggestions ?

Helmholtz and Partch's "Genesis of a Music" were my main [and only]
references until very recently.

I'm sure that there's tons of material out there, and those more
familiar with it will provide a bibliography, but based on what I
have, I'd suggest spending for a stack of Xenharmonicons [my personal
collection runs from #6 to the present].

As for "must haves", I'd definately recommend Doty's "The Just
Intonation Primer", available from the Just Intonation Network.

Steve