Gencom: Difference between revisions

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== Example ==

Consider [[baldy]], the temperament tempering out 225/224, 325/324, and 640/637 in the 2.9.5.7.13 subgroup. This is every other step of [[garibaldi|garibaldi/cassandra]] in the 13-limit, without prime 11. With normalized generators ~2 and ~9, the gencom is [2 9; 225/224 325/324 640/637]. Converting it to a matrix of monzos, we get [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 2 0 0 0 0 }}; {{monzo| -5 2 2 -1 0 0 }}, {{monzo| -2 -4 2 0 0 1 }}, {{monzo| 7 0 1 -2 0 -1 }}]. Taking the pseudoinverse and canonicalizing it, the extended gencom mapping is found to be [{{val| 1 0 15 25 0 28 }}, {{val| 0 1/2 -4 -7 0 10 }}, {{val| 0 0 2 3 0 4 }}, {{val| 0 0 -1 -2 0 3 }}, {{val| 0 0 -1 -2 0 2 }}]. Since this is a rank-2 temperament, the gencom mapping is the first two row thereof, {{mapping| 1 0 15 25 0 28 | 0 1/2 -4 -7 0 10 }}.

Consider [[baldy]], the temperament tempering out 225/224, 325/324, and 640/637 in the 2.9.5.7.13 subgroup. This is every other step of [[garibaldi|garibaldi/cassandra]] in the 13-limit, without prime 11. With normalized generators ~2 and ~9, the gencom is [2 9; 225/224 325/324 640/637]. Converting it to a matrix of monzos, we get [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 2 0 0 0 0 }}; {{monzo| -5 2 2 -1 0 0 }}, {{monzo| -2 -4 2 0 0 1 }}, {{monzo| 7 0 1 -2 0 -1 }}]. Taking the pseudoinverse and canonicalizing it, the extended gencom mapping is found to be [{{val| 1 0 15 25 0 28 }}, {{val| 0 1/2 -4 -7 0 10 }}; {{val| 0 0 2 3 0 4 }}, {{val| 0 0 -1 -2 0 3 }}, {{val| 0 0 -1 -2 0 2 }}]. Since this is a rank-2 temperament, the gencom mapping is the first two row thereof, {{mapping| 1 0 15 25 0 28 | 0 1/2 -4 -7 0 10 }}.

With this mapping we can insert the monzo of 9, {{monzo| 0 2 }}, to the mapping and see it is represented by +1 generator step. Further, we can see prime 3, not in the subgroup, must be "1/2" generator step. Through the same process we find prime 5 is -4 steps and prime 7 is -7 steps, which correspond to -8 and -14 steps of garibaldi. Prime 11 is not in the temperament so it is signified by "0" steps. Finally, prime 13 is +10 steps, corresponding to +20 steps of garibaldi/cassandra.

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 == Example ==
-Consider [[baldy]], the temperament tempering out 225/224, 325/324, and 640/637 in the 2.9.5.7.13 subgroup. This is every other step of [[garibaldi|garibaldi/cassandra]] in the 13-limit, without prime 11. With normalized generators ~2 and ~9, the gencom is [2 9; 225/224 325/324 640/637]. Converting it to a matrix of monzos, we get [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 2 0 0 0 0 }}; {{monzo| -5 2 2 -1 0 0 }}, {{monzo| -2 -4 2 0 0 1 }}, {{monzo| 7 0 1 -2 0 -1 }}]. Taking the pseudoinverse and canonicalizing it, the extended gencom mapping is found to be [{{val| 1 0 15 25 0 28 }}, {{val| 0 1/2 -4 -7 0 10 }}, {{val| 0 0 2 3 0 4 }}, {{val| 0 0 -1 -2 0 3 }}, {{val| 0 0 -1 -2 0 2 }}]. Since this is a rank-2 temperament, the gencom mapping is the first two row thereof, {{mapping| 1 0 15 25 0 28 | 0 1/2 -4 -7 0 10 }}.
+Consider [[baldy]], the temperament tempering out 225/224, 325/324, and 640/637 in the 2.9.5.7.13 subgroup. This is every other step of [[garibaldi|garibaldi/cassandra]] in the 13-limit, without prime 11. With normalized generators ~2 and ~9, the gencom is [2 9; 225/224 325/324 640/637]. Converting it to a matrix of monzos, we get [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 2 0 0 0 0 }}; {{monzo| -5 2 2 -1 0 0 }}, {{monzo| -2 -4 2 0 0 1 }}, {{monzo| 7 0 1 -2 0 -1 }}]. Taking the pseudoinverse and canonicalizing it, the extended gencom mapping is found to be [{{val| 1 0 15 25 0 28 }}, {{val| 0 1/2 -4 -7 0 10 }}; {{val| 0 0 2 3 0 4 }}, {{val| 0 0 -1 -2 0 3 }}, {{val| 0 0 -1 -2 0 2 }}]. Since this is a rank-2 temperament, the gencom mapping is the first two row thereof, {{mapping| 1 0 15 25 0 28 | 0 1/2 -4 -7 0 10 }}.
 With this mapping we can insert the monzo of 9, {{monzo| 0 2 }}, to the mapping and see it is represented by +1 generator step. Further, we can see prime 3, not in the subgroup, must be "1/2" generator step. Through the same process we find prime 5 is -4 steps and prime 7 is -7 steps, which correspond to -8 and -14 steps of garibaldi. Prime 11 is not in the temperament so it is signified by "0" steps. Finally, prime 13 is +10 steps, corresponding to +20 steps of garibaldi/cassandra.