mm. When I was studying music theory we used to reserve "operation" for a function that was 1 to 1 and onto; I think that usage has been pretty standard in music theory since 1987, through the work of David Lewin. Music theorists often screw up standard math terms though, so I never know what to call anything in what company, and always suspect it will be meaningless/wrong... =o)
On Thu, Mar 19, 2009 at 1:09 PM, Mathieu Bouchard matju@artengine.ca wrote:
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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