Date: Sat, 28 Jun 1997 23:49:13 -0700 From: Yraina Chantres 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? ============================================================================== Date: Mon, 21 Jul 1997 13:29:28 -0500 (CDT) From: Dave 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 ==============================================================================