Re: Why are there 7 discrete notes? A possibly stupid question about sound...

From: dilvie (dilvie_at_dilvie.remove-this-to-reply.com)
Date: 06/27/04 

Date: Sun, 27 Jun 2004 20:08:42 GMT

You're both right, and you're both wrong.

Yes, it does have to do with math, and yes, different cultures came up
with different scales, but the development of the seven-note diatonic
scale began with pythagoras.

According to legend, Pythagoras discovered the foundations of music by
listening to the sounds of four blacksmith's hammers, which produced
consonance and dissonance when they were struck simultaneously.
Specifically, he noticed that hammer A produced consonance with hammer B
when they were struck together, and hammer C produced consonance with
hammer A, but hammers B and C produced dissonance with each other.
Hammer D produced such perfect consonance with hammer A that they seemed
to be "singing" the same note! Pythagoras rushed into the blacksmith to
discover why, and he found that the explanation was in the weight
ratios. The hammers weighed 12, 9, 8, and 6 pounds respectively. Hammers
A and D were in a ratio of 2:1, which is the ratio of the octave.
Hammers B and C weighed 9 and 8 pounds. Their ratios with hammer A were
(12:9 = 4:3 = musical fourth) and (12:8 = 3:2 = musical fifth).
Interestingly, if you invert the ratio of hammer B (making it 3:4 =
12:8) it becomes the ratio of hammer C, and vise-verse, thus, the
musical fourth can be described as an inverted fifth, and vise verse.
The space between B and C is a ratio of 9:8, which is equal to musical
whole tone, or whole step interval.

The ratio 2:1 produces perfect consonance, that is, for each cycle of
the lower frequency, there are exactly two cycles of the frequency one
octave higher. The ratio 3:2 produces a very similar kind of
consonance. It's all about symetry. The development of western music
is very much like biological evolution. A variety of musicians,
scholars, and methematicians over the course of many centuries stumbled
accross the natural laws that make music sound good. It isn't purely
cultural -- that is, there are mathematical reasons that those specific
notes sound good, but the seven-tone diatonic scale didn't simply pop
into being on the discovery of any one person.

The modern scales you hear today in popular music came from the
development of the equal-tempered scale. Before equal and well-tempered
scales came about, different keys sounded more in tune -- as long as you
played only in that key. That's because they were tuned for perfect
consonance in the 5ths and 3rds for that specific scale. The problem
was, the different scales sounded out of tune with each other, meaning
that when you modulate keys, the key you modulate to sounds out-of-tune.

The well-tempered clavier was an attempt to confront this problem. J.S.
Bach championed it with "the well tempered clavier" -- A selection of
fugues and preludes that is still very popular today.

Equal temperement didn't become a standard until the late 1800's. The
idea was to make every scale equally out of tune. They did this by
dividing the octave into an equal number of cents (cents being a measure
of pitch variation). This division was optimized to make 5ths and 3rds
sound particularly good with the octaves for every scale. It wasn't an
arbitrary choice. It had a very specific purpose.

This is just a rough overview. If you want all the details (including
how you can mathematically discover the whole 7-tone diatonic scale), I
refer you to these excellent sources:

James, Jamie _ The Music of the Spheres: Music, Science, and the Natural
Order of the Universe _. New York, Grove Press, 1993

"Pythagorean hammers" _ Harvard Dictionary of Music _. Second Edition,
Massachusets: Harvard University Press 

-- 
~
<http://www.dilvie.com/>

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