Gencom: Difference between revisions
m Moving from Category:Temperament to Category:Regular temperament theory using Cat-a-lot |
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Converting a gencom to a [[Normal lists #Normal interval list|normal interval list]] gives a canonical form for the subgroup, which is an 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. | Converting a gencom to a [[Normal lists #Normal interval list|normal interval list]] gives a canonical form for the subgroup, which is an 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. | ||
[[Category:Generator]] | [[Category:Generator]] | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||