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### Boethius and Harmonics

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Date: 09/28/97 at 03:34:23
From: Jules MacDonell
Subject: Boethius and harmonics

Can you explain Boethius and his theory of harmonics of music? A lot
of what I've found on the Net is in Latin!

Thank you very much!

Jules
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Date: 09/28/97 at 07:02:25
From: Doctor Mitteldorf
Subject: Re: Boethius and harmonics

Dear Jules,

First, I suggest that you get your hands on this book:

_Emblems of Mind: The Inner Life of Music and Mathematics_,
by Edward Rothstein.

It is full of interesting information, is fairly easy to understand
without assuming too much math knowledge, and has a nice bibliography.

We also did a search of the Math Forum site and came up with some
interesting facts to start you off:

If you move along the piano, each note gets higher than the next.
Instead of equal steps as you move from one note to the next, the
notes are actually in EQUAL RATIOS from one to the next. In fact, from
a C to a C#, the ratio of the pitches of the two notes is 1.05946.
The ratio of a D to a C# is exactly the same, and the ratio from a
D to a D# is again identical.

This number, 1.05946, is calculated in such a way that twelve such
steps (which carries you a full octave from one C to the next) is
exactly a factor of 2. Tuning the piano in exactly this fashion is an
innovation that J.S. Bach came up with. It's called "well-tempered
tuning."

A series of numbers in which each one is bigger (or smaller) than the
last by a constant factor is called a "harmonic series," precisely
because such series were first studied in the context of music.

So if you take a string and pluck it, you get a pitch. If you divide
the string in half and pluck each half, you get a pitch that's exactly
twice as high, and the sound is an octave higher.

It just so happens that 7 of the 12 steps, each of which is a factor
of 1.05946, gets you to a number that is very close to 1.5, or 3/2.
This is the sound called a "perfect fifth," the sound of a C and a G
on the piano.

You can get the sound of a fifth from a violin string by dividing the
string in thirds with your finger - the sound that is produced is a
fifth + an octave above the fundamental.

Furthermore, the ratio produced by four piano steps is close to 1.25,
or 5/4. This means that the notes CEG, which are a basic "chord" on
the piano, are in the ratio 6:5:4. All such chords, whether they start
on C or any other note, have the same ratio 6:5:4.

-Doctors Melissa and Mitteldorf,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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