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[Mathieu Bouchard]

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 :)

See also http://en.wikipedia.org/wiki/Euler_phi_function

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| Mathieu Bouchard - tél:+1.514.383.3801, Montréal, Québec