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[Matt Barber]

On Thu, Mar 19, 2009 at 12:02 PM, Mathieu Bouchard <matju at 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 :)
>
> See also http://en.wikipedia.org/wiki/Euler_phi_function
>

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.