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[Mathieu Bouchard]

On Thu, 19 Mar 2009, Matt Barber wrote:

> Right, in mod-12, the other multiplications are not strictly operations 
> (there is no inverse).

They are called operations anyway. I don't know your definition of 
operation.

They're usually called "non-invertible operations", but in a Group 
(of Group Theory), all elements are invertible.

Group Theory also has an operator (written as a small straight "x" in 
exponent) that makes a multiplication-wise group from an addition-wise 
group. For Z/12Z (the mod 12 integers), this gives you a group make of 
1,5,7,11, which behaves like (Z/2Z)^2, which is are the 2-D vectors made 
of Z/2Z (mod 2 integers):

1  -> (0,0)
5  -> (0,1)
7  -> (1,0)
11 -> (1,1)

> Recently I've been writing music in various 19-tone equal temperaments, 
> which, since it's prime, has a complete multiplicative group.

yes... and as a bonus, this multiplicative group acts just like Z/18Z !!!

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