Gencom: Difference between revisions

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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.

With this mapping we can insert the monzo of an arbitrary interval to see its number of generator steps. For instance 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. Collecting the numbers of generator steps of all the monzos in the subgroup basis we can convert the gencom mapping to the sval mapping: {{mapping| 1 0 15 25 -28 | 0 1 -4 -7 10 }}.

For a more complicated case let us consider [[edson]], the temperament tempering out 196/195, 352/351, and 364/363 in the 2.3.7/5.11/5.13/5 subgroup. Using normalized generators ~2 and ~3, its gencom mapping is {{mapping| 1 0 -49/4 -9/4 19/4 39/4 | 0 1 29/4 5/4 -11/4 -23/4 }}. Like before, we can see 7/5 is mapped to -6 generator steps by inserting its monzo {{monzo| 0 0 -1 1 }} to the mapping. However, if we insert the monzo of 625, {{monzo| 0 0 4 }}, it will also return a seemingly meaningful +29 steps ~~as a result~~, ~~but 625~~ is not in the subgroup to begin with.

For a more complicated case let us consider [[edson]], the temperament tempering out 196/195, 352/351, and 364/363 in the 2.3.7/5.11/5.13/5 subgroup. Using normalized generators ~2 and ~3, its gencom mapping is {{mapping| 1 0 -49/4 -9/4 19/4 39/4 | 0 1 29/4 5/4 -11/4 -23/4 }}. Like before, we can see 7/5 is mapped to -6 generator steps by inserting its monzo {{monzo| 0 0 -1 1 }} to the mapping. However, if we insert the monzo of 625, {{monzo| 0 0 4 }}, it will also return a seemingly meaningful result of +29 steps, which should ''not'' be so interpreted because the interval is not in the subgroup to begin with.

[[Category:Generator]]

[[Category:Regular temperament theory]]

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 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.
+With this mapping we can insert the monzo of an arbitrary interval to see its number of generator steps. For instance 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. Collecting the numbers of generator steps of all the monzos in the subgroup basis we can convert the gencom mapping to the sval mapping: {{mapping| 1 0 15 25 -28 | 0 1 -4 -7 10 }}.
-For a more complicated case let us consider [[edson]], the temperament tempering out 196/195, 352/351, and 364/363 in the 2.3.7/5.11/5.13/5 subgroup. Using normalized generators ~2 and ~3, its gencom mapping is {{mapping| 1 0 -49/4 -9/4 19/4 39/4 | 0 1 29/4 5/4 -11/4 -23/4 }}. Like before, we can see 7/5 is mapped to -6 generator steps by inserting its monzo {{monzo| 0 0 -1 1 }} to the mapping. However, if we insert the monzo of 625, {{monzo| 0 0 4 }}, it will also return a seemingly meaningful +29 steps as a result, but 625 is not in the subgroup to begin with.
+For a more complicated case let us consider [[edson]], the temperament tempering out 196/195, 352/351, and 364/363 in the 2.3.7/5.11/5.13/5 subgroup. Using normalized generators ~2 and ~3, its gencom mapping is {{mapping| 1 0 -49/4 -9/4 19/4 39/4 | 0 1 29/4 5/4 -11/4 -23/4 }}. Like before, we can see 7/5 is mapped to -6 generator steps by inserting its monzo {{monzo| 0 0 -1 1 }} to the mapping. However, if we insert the monzo of 625, {{monzo| 0 0 4 }}, it will also return a seemingly meaningful result of +29 steps, which should ''not'' be so interpreted because the interval is not in the subgroup to begin with.
 [[Category:Generator]]
 [[Category:Regular temperament theory]]