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