Normal forms: Difference between revisions

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Consider the case of septimal meantone. As we know, its defactored Hermite normal form is {{rket| {{map| 1 0 -4 -13 }} {{map| 0 1 4 10 }} }} which corresponds to generators of ~2/1 and ~3/1. In this case, as is typical, the formal prime represented by the first column of the matrix is 2, and so the equave is the octave. Therefore, all generators must be octave-reduced. But our second generator is ~3/1, which is not octave-reduced. We must alter the mapping in such a way that this row represents a generator of ~3/2 instead. We can do that here by adding the second row of the mapping to the first: {{rket| {{map| 1 1 0 -3 }} {{map| 0 1 4 10 }} }}. So that is septimal meantone's equave-reduced generator form, corresponding to generators of ~2/1 and ~3/2.

Probably the most reliable way to achieve equave-reduced generator form in general, however, is not to work with a [[generator preimage ~~transversal~~]] such as ~3/1 and ~3/2. Instead the Frobenius generators may be used, as described in the positive generator form section just above, and reduction can be accomplished by calling modulo on their cents.

Probably the most reliable way to achieve equave-reduced generator form in general, however, is not to work with a [[generator preimage]] such as ~3/1 and ~3/2. Instead the Frobenius generators may be used, as described in the positive generator form section just above, and reduction can be accomplished by calling modulo on their cents.

For a general discussion of how to manipulate the sizes of generators in this way, see [[Generator size manipulation]].

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 Consider the case of septimal meantone. As we know, its defactored Hermite normal form is {{rket| {{map| 1 0 -4 -13 }} {{map| 0 1 4 10 }} }} which corresponds to generators of ~2/1 and ~3/1. In this case, as is typical, the formal prime represented by the first column of the matrix is 2, and so the equave is the octave. Therefore, all generators must be octave-reduced. But our second generator is ~3/1, which is not octave-reduced. We must alter the mapping in such a way that this row represents a generator of ~3/2 instead. We can do that here by adding the second row of the mapping to the first: {{rket| {{map| 1 1 0 -3 }} {{map| 0 1 4 10 }} }}. So that is septimal meantone's equave-reduced generator form, corresponding to generators of ~2/1 and ~3/2.
-Probably the most reliable way to achieve equave-reduced generator form in general, however, is not to work with a [[generator preimage transversal]] such as ~3/1 and ~3/2. Instead the Frobenius generators may be used, as described in the positive generator form section just above, and reduction can be accomplished by calling modulo on their cents.
+Probably the most reliable way to achieve equave-reduced generator form in general, however, is not to work with a [[generator preimage]] such as ~3/1 and ~3/2. Instead the Frobenius generators may be used, as described in the positive generator form section just above, and reduction can be accomplished by calling modulo on their cents.
 For a general discussion of how to manipulate the sizes of generators in this way, see [[Generator size manipulation]].