Wedgie/Archived version: Difference between revisions

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== How the period and generator falls out of a rank-2 wedgie ==

The following is a procedure for finding a period and a generator for a rank-2 regular temperament on the 2.''q''1.(…).q''n'' [[JI subgroup]], with basis '''2''', '''q'''1, ..., '''q'''''n'' (We're writing bold letters and numbers to represent elements of the JI lattice, viewed as vectors; so, for example, 3/2 = '''3''' − '''2''' in the 2.3 lattice). We also give a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the [[equave]] is the octave, but non-octave JI equaves can be substituted for the octave if needed, by substituting the appropriate JI ratio for 2/1.

The following is a procedure for finding a period and a generator for a rank-2 regular temperament on the 2.''q''1.(…).q''n'' [[JI subgroup]]. We also give a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the [[equave]] is the octave, but non-octave JI equaves can be substituted for the octave if needed, by substituting the appropriate JI ratio for 2/1.

The following assumes that:

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=== Proof (a bit technical) ===

The following additionally assumes that you know what the words "basis", "linear map", and "determinant" mean.

Consider the 2.''q''1.(…).q''n'' [[JI subgroup]], with basis '''2''', '''q'''1, ..., '''q'''''n'' (Here we use bold letters and numbers to represent elements of the JI lattice, viewed as vectors; so, for example, 3/2 = '''3''' − '''2''' in the 2.3 lattice. Italicized letters represent scalars and sets, and roman letters represent maps.)

The period '''p''' (fraction of octave) and generator '''g''' form a basis for all the intervals of a rank-2 temperament. For example, '''p''' = 2/1 and '''g''' = 3/2 form a basis for meantone. But from a purely linear-algebra perspective, there's nothing special about the basis {'''p''', '''g'''}; I could have chosen another basis, for example '''p'''' = 3/1 for my "period" and '''g'''' = 2/1 for my "generator". What makes the wedgie a unique identifier for a temperament is that rather than specify a basis directly, the wedgie specifies a ''constraint'' that any basis for the temperament must satisfy: namely, that a basis '''e'''1, '''e'''2 must satisfy W('''e'''1, '''e'''2) = ±1.

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 == How the period and generator falls out of a rank-2 wedgie ==
-The following is a procedure for finding a period and a generator for a rank-2 regular temperament on the 2.''q''<sub>1</sub>.(…).q<sub>''n''</sub> [[JI subgroup]], with basis '''2''', '''q'''<sub>1</sub>, ..., '''q'''<sub>''n''</sub> (We're writing bold letters and numbers to represent elements of the JI lattice, viewed as vectors; so, for example, 3/2 = '''3''' &minus; '''2''' in the 2.3 lattice). We also give a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the [[equave]] is the octave, but non-octave JI equaves can be substituted for the octave if needed, by substituting the appropriate JI ratio for 2/1.
+The following is a procedure for finding a period and a generator for a rank-2 regular temperament on the 2.''q''<sub>1</sub>.(…).q<sub>''n''</sub> [[JI subgroup]]. We also give a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the [[equave]] is the octave, but non-octave JI equaves can be substituted for the octave if needed, by substituting the appropriate JI ratio for 2/1.
 The following assumes that:
@@ Line 55: / Line 55: @@
 === Proof (a bit technical) ===
 The following additionally assumes that you know what the words "basis", "linear map", and "determinant" mean.
+Consider the 2.''q''<sub>1</sub>.(…).q<sub>''n''</sub> [[JI subgroup]], with basis '''2''', '''q'''<sub>1</sub>, ..., '''q'''<sub>''n''</sub> (Here we use bold letters and numbers to represent elements of the JI lattice, viewed as vectors; so, for example, 3/2 = '''3''' &minus; '''2''' in the 2.3 lattice. Italicized letters represent scalars and sets, and roman letters represent maps.)
 The period '''p''' (fraction of octave) and generator '''g''' form a basis for all the intervals of a rank-2 temperament. For example, '''p''' = 2/1 and '''g''' = 3/2 form a basis for meantone. But from a purely linear-algebra perspective, there's nothing special about the basis {'''p''', '''g'''}; I could have chosen another basis, for example '''p'''' = 3/1 for my "period" and '''g'''' = 2/1 for my "generator". What makes the wedgie a unique identifier for a temperament is that rather than specify a basis directly, the wedgie specifies a ''constraint'' that any basis for the temperament must satisfy: namely, that a basis '''e'''<sub>1</sub>, '''e'''<sub>2</sub> must satisfy W('''e'''<sub>1</sub>, '''e'''<sub>2</sub>) = ±1.