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Sun Jan 2 13:05:26 EST 2011

Music: scales and intervals: almost equal composite numbers

The fact that there are different tuning systems for the chromatic
scale indicates that there is something quite wrong here.  The
chromatic scale is a happy accident because our brain glosses over the
differences between intervals that are close but quite different.

There are many examples.  Take the minor 7th[1] for example.  It
corresponds to 

   16 / 9  = 1.777...  
    9 / 5  = 1.800...

both harmonic intervals have nothing to do with each other, but
melodically they are very close, about 1%.  (A minor step is about
5%).

It seems that music relies on this kind of gloss-over substitution
when tones take on different relations in chords and melodies.  I've
always found this to be one of the weirdest things about the theory
behind music.  I've never seen it being mentioned so explicitly
either, only in the context of tuning systems where it is mentioned as
a nuisance.  I think Bach even called it a Devine Joke or something
(TODO: find quote).

In short, it seems as if we can "teleport" between ratios that are
related by near-1 fractions.

I.e. for the minor 7th above we have a difference of

  16 * 5 / 9^2 =  80 / 81

In general this phenomenon is recognized as a comma[3].  The ratio
above is called the syntonic comma[2].

So one, how to enumerate the commas? Since the intervals in music all
use limited prime ratios (i.e. up to 7), and limited amount of
octaves, there is a limit to the amount of mistune one can get.

And second, I wonder if this can be turned around.  Given a chord
written in "human" chromatic notation glossing over commas, is it
possible to "distill" its different harmonic meanings by finding
intervals that match in certain ways?

I.e.:
    16 / 9 = (4 / 3)^2   : two perfect fourths
    9 / 5  = ...


[1] http://en.wikipedia.org/wiki/Minor_seventh
[2] http://en.wikipedia.org/wiki/Syntonic_comma
[3] http://en.wikipedia.org/wiki/Comma_%28music%29



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