On Thu, Mar 19, 2009 at 12:02 PM, Mathieu Bouchard matju@artengine.ca wrote:
On Wed, 18 Mar 2009, Matt Barber wrote:
If you're doing mod 12 operations, there is one more pitch operator -- multiplication by 5 or 7 -- which maps the chromatic scale to the circle of fifths and vice-versa.
The "vice-versa" part is quite cool. Actually, apart from 1,5,7,11, all modulo-multiplications are not undoable, because they forget part of what was the original note, so, the undo would be ambiguous. 5 undoes itself because 5*5=1 in mod 12, 7 undoes itself because 7*7=1 in mod 12, but then 5=-7 and 7=-5 as well, just like 1=-11 and 11=-1.
The undoability depends on whether the greatest common divisor of the modulo and of the multiplicator is 1 or not. If you use the 22 equal temperament, for example, there are 10 invertibles, and with 43 equal temperament, there are 42 of them; the proportion of undoables vs non-undoables varies greatly from modulo to modulo.
I'm not into microtonal stuff, but I studied the modulo theory and I think that people who can care about microtonal music are lucky to have a nice application of that theory in their hands :)
Right, in mod-12, the other multiplications are not strictly operations (there is no inverse). I used to like to joke with friends that I was "really into the multiplication by 0 mapping."
Recently I've been writing music in various 19-tone equal temperaments, which, since it's prime, has a complete multiplicative group. 19 per octave is nice because you get really "pure" thirds. I've also been experimenting with 19 per perfect 12th (octave and a fifth), the smallest intervals of which work out almost exactly to standard 12-tone half-steps (check the 12th root of 2 and the 19th root of 3).
In addition each modulus has strikingly different voice-leading possibilities.