Date: Sat, 28 Jun 1997 23:49:13 -0700
From: Yraina Chantres <yraina@fog.net>
To: rusin@math.niu.edu
Subject: Music/Math: Sharps and flats

Hi, I found your web site via Yahoo looking up music theory.  I am
trying to teach myself a little of the mathematical theory behind
musical scales so that I can try to figure out if there is some harmonic
reason behind the various Indian and Middle-Eastern scales which are
frequntly 8-tone but include notes constructed "ostensibly" from notes
in an equal-tempered 24 tone scale.

First, I need to make sure I really understand what the "meaning" of
sharps and flats are.  I have heard that, in just intonation, the flat
of one note is NOT exactly the same pitch as the sharp of the one below
it or, in the case of C or F, the not below it and that they are merely
close enough to each other to have a single good approximation in the
equal-tempered scale.

Is this true?  If it is true, what is the specific harmonic "meaning" of
a sharp or a flat?  Also, do you know or have any insight into the
possible harmonic basis for scales which contain some 3/4 tone
intervals?

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Date: Mon, 21 Jul 1997 13:29:28 -0500 (CDT)
From: Dave Rusin <rusin>
To: yraina@fog.net
Subject: Re:  Music/Math: Sharps and flats

I'm not sure if I ever responded to your mail (I've been travelling
a lot lately). Some composers intend that e.g. D-sharp and E-flat
be a little different in tone, but as far as I know most melodic
systems in the world divide an octave into a fixed set of intervals
(i.e. a fixed set of ratios of frequencies); ancient Chinese
music used five roughly equal ratios, ancient Greek music used
7 unequal ones, eventually expanded to 12 roughly equal ones. This
was cemented in the Renaissance to 12 exactly equal ratios, although
the ratios are then not rational numbers (fractions) and that
sounds a little out of tune to a well-trained ear. 

As I understand it, those musicians whose instruments allow it
(e.g. violins) will play a chord which sounds more pure even if
it is not really consistent with the exactly-divided 12-tone octave.
Thus, if they must play an E-flat above an A-flat, they can get a
pure fifth (meaning, two notes whose frequencies are exactly in a
3:2 ratio) if they play that E-flat a little sharper than would
come from a pretuned instrument like a piano; conversely, if they
must play a D-sharp under an A-sharp, they want that sharp to be
a little flatter than the pretuned instrument would play. So
while D-sharp and E-flat are the same to the pianist, the
violinist can take advantage of the infinite tunability of
his or her instrument, and play them a little different on the
two occasions, so that the two chords sound better. The pianist must
accept a compromise note.

Of course if the violin and piano play together, they must play
the same note, so the violinist cannot try this trick except in solo.

dave
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