Gencom: Difference between revisions

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== Invariance ==

Converting a gencom to a [[Normal lists #Normal interval list|normal interval list]] gives a canonical form for the subgroup, which is an {{w|Invariant (mathematics)|invariant}} of the temperament. Doing the same to just the commas produces another invariant, and together these determine the temperament: it is the unique temperament on the given group tempering out the given commas. The normal list defined by the generators alone is not an invariant of the temperament, since the generators give only a transversal for the tempered intervals of the temperament, not the full set of intervals being tempered. Hence, for instance, [2 40/27; 81/80] and [2 3/2; 81/80] both define 5-limit meantone, but the normal list for [2 40/27] is 2.27/5 and for [2 3/2] is 2.3. However, the extended gencom mapping can be used to determine if an interval ''q'' is in the group of the temperament. Suppose [''c''1 ''c''2 … ''c''n] is a gencom and [''v''1 ''v''2 … ''v''''n''] is the corresponding extended mapping. Then each of ''v''1 (''q''), ''v''2 (''q'') … ''v''''n'' (''q'') must be an integer, and moreover we must have ''q'' = ''c''1^''v''1 (''q'') · ''c''2^''v''2 (''q'') · … · ''c''''n''^''v''''n'' (''q''). This provides sufficient conditions as well as necessary ones.

== 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 }}.

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.

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.

[[Category:Generator]]

[[Category:Regular temperament theory]]

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 == Invariance ==
 Converting a gencom to a [[Normal lists #Normal interval list|normal interval list]] gives a canonical form for the subgroup, which is an {{w|Invariant (mathematics)|invariant}} of the temperament. Doing the same to just the commas produces another invariant, and together these determine the temperament: it is the unique temperament on the given group tempering out the given commas. The normal list defined by the generators alone is not an invariant of the temperament, since the generators give only a transversal for the tempered intervals of the temperament, not the full set of intervals being tempered. Hence, for instance, [2 40/27; 81/80] and [2 3/2; 81/80] both define 5-limit meantone, but the normal list for [2 40/27] is 2.27/5 and for [2 3/2] is 2.3. However, the extended gencom mapping can be used to determine if an interval ''q'' is in the group of the temperament. Suppose [''c''<sub>1</sub> ''c''<sub>2</sub> … ''c''<sub>n</sub>] is a gencom and [''v''<sub>1</sub> ''v''<sub>2</sub> … ''v''<sub>''n''</sub>] is the corresponding extended mapping. Then each of ''v''<sub>1</sub> (''q''), ''v''<sub>2</sub> (''q'') … ''v''<sub>''n''</sub> (''q'') must be an integer, and moreover we must have ''q'' = ''c''<sub>1</sub>^''v''<sub>1</sub> (''q'') · ''c''<sub>2</sub>^''v''<sub>2</sub> (''q'') · … · ''c''<sub>''n''</sub>^''v''<sub>''n''</sub> (''q''). This provides sufficient conditions as well as necessary ones.
+== 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 }}.
+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.
+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.
 [[Category:Generator]]
 [[Category:Regular temperament theory]]