Normal forms: Difference between revisions

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=== Equave-reduced generator form ===

The '''equave-reduced generator form''' has the matrix modified from the defactored Hermite normal form so that each generator is equave-reduced, where the [[equave]] can be found as the formal prime represented by the first ''column'' of the matrix (which is usually the octave). This is closely related to [[ploidacot]]s, so it may be casually called the ''ploidacot form''~~. For more information, see [[Octave reduction #Generalization]]~~.

The '''equave-reduced generator form''' has the matrix modified from the defactored Hermite normal form so that each generator is equave-reduced, where the [[equave]] can be found as the formal prime represented by the first ''column'' of the matrix, which is usually the octave. For more information, see [[Octave reduction #Generalization]]. This form is closely related to [[ploidacot]]s, so it may be casually called the ''ploidacot form''.

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.

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 === Equave-reduced generator form ===
-The '''equave-reduced generator form''' has the matrix modified from the defactored Hermite normal form so that each generator is equave-reduced, where the [[equave]] can be found as the formal prime represented by the first ''column'' of the matrix (which is usually the octave). This is closely related to [[ploidacot]]s, so it may be casually called the ''ploidacot form''. For more information, see [[Octave reduction #Generalization]].
+The '''equave-reduced generator form''' has the matrix modified from the defactored Hermite normal form so that each generator is equave-reduced, where the [[equave]] can be found as the formal prime represented by the first ''column'' of the matrix, which is usually the octave. For more information, see [[Octave reduction #Generalization]]. This form is closely related to [[ploidacot]]s, so it may be casually called the ''ploidacot form''.
 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.