Normal forms: Difference between revisions

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The first positive-generator form, called <code>"flip"</code> in the Temperament Evaluator, changes the signs of every entry in a row if the corresponding generator is negative.

Another form, called <code>"shift"</code>, also makes sure the generators are positive, but it maniplates the mapping to period-shift the negative generators instead, except when the temperament's generator is (-''p'')-sheared where ''p'' is the number of periods per equave, in which case it uses the <code>"flip"</code> routine. This form may be more musically useful as the prime harmonics are more often in positive numbers of generator steps.

Another form, called <code>"shift"</code>, also makes sure the generators are positive, but it maniplates the mapping to period-shift the negative generators instead, except when the temperament's generator is (''c'' - ''p'')-sheared where ''c'' is the cot (number of generators to reach the prime in question) and ''p'' is the ploid (number of periods per equave), in which case it uses the <code>"flip"</code> routine. This form may be more musically useful as the prime harmonics are more often in positive numbers of generator steps.

~~'''~~Note~~:'''~~ Most mappings (though not the "[[mapping to lattice]]") listed on temperament data pages of this wiki are in the <code>"shift"</code> form, though some are still in <code>"flip"</code> form.

{{Note| Most mappings (though not the "[[mapping to lattice]]") listed on temperament data pages of this wiki are in the <code>"shift"</code> form, though some are still in <code>"flip"</code> form. }}

The generators in defactored Hermite form of septimal meantone is positive already, so its positive-generator forms are the same as its defactored Hermite form, {{mapping| 1 0 -4 -13 | 0 1 4 10 }}, corresponding to generators of ~2/1 and ~3/1. An example of positive-generator form that is different from the defactored Hermite form is the porcupine temperament, elaborated below.

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=== Minimal-generator form ===

The ''minimal-generator form'' (or '''mingen form''') is a form specific to rank-2 ~~temperaments~~, where the generator is positive and no greater than half the period.<ref group="note">This is somewhat like octave reduction combined with octave inversion, because you can't just add or subtract half octaves until it's between 0 and 600 cents; you have to add or subtract octaves until it's between −600 and +600 cents, then multiply by −1 if it's negative.</ref><ref group="note">You could always find a smaller and smaller generator by going negative, so this assumes positive generators.</ref>

The ''minimal-generator form'' (or '''mingen form''') is a form specific to [[rank-2 temperament]]s, where the generator is positive and no greater than half the period.<ref group="note">This is somewhat like octave reduction combined with octave inversion, because you can't just add or subtract half octaves until it's between 0 and 600 cents; you have to add or subtract octaves until it's between −600 and +600 cents, then multiply by −1 if it's negative.</ref><ref group="note">You could always find a smaller and smaller generator by going negative, so this assumes positive generators.</ref>

[[Graham Breed]]'s [http://x31eq.com/temper/ ~~temperament finder~~] uses this form for all rank-2 temperaments. Septimal meantone in minimal-generator form is {{mapping| 1 2 4 7 | 0 -1 -4 -10 }}, corresponding to generators of ~2/1 and ~4/3. [[Kite Giedraitis]]'s [[pergen]] also uses this form, except for when the generator is a perfect fifth.

[[Graham Breed]]'s [http://x31eq.com/temper/ Temperament Finder] uses this form for all rank-2 temperaments. Septimal meantone in minimal-generator form is {{mapping| 1 2 4 7 | 0 -1 -4 -10 }}, corresponding to generators of ~2/1 and ~4/3. [[Kite Giedraitis]]'s [[pergen]] also uses this form, except for when the generator is a perfect fifth.

Beyond rank-2, the mingen form of a temperament is no longer unique. You can always get smaller and smaller generators. This is why on Graham Breed's temperament finding tool, beyond rank-2 he simply uses the Hermite Normal Form.

@@ Line 47: / Line 47: @@
 The first positive-generator form, called <code>"flip"</code> in the Temperament Evaluator, changes the signs of every entry in a row if the corresponding generator is negative.
-Another form, called <code>"shift"</code>, also makes sure the generators are positive, but it maniplates the mapping to period-shift the negative generators instead, except when the temperament's generator is (-''p'')-sheared where ''p'' is the number of periods per equave, in which case it uses the <code>"flip"</code> routine. This form may be more musically useful as the prime harmonics are more often in positive numbers of generator steps.
+Another form, called <code>"shift"</code>, also makes sure the generators are positive, but it maniplates the mapping to period-shift the negative generators instead, except when the temperament's generator is (''c'' - ''p'')-sheared where ''c'' is the cot (number of generators to reach the prime in question) and ''p'' is the ploid (number of periods per equave), in which case it uses the <code>"flip"</code> routine. This form may be more musically useful as the prime harmonics are more often in positive numbers of generator steps.
-'''Note:''' Most mappings (though not the "[[mapping to lattice]]") listed on temperament data pages of this wiki are in the <code>"shift"</code> form, though some are still in <code>"flip"</code> form.
+{{Note| Most mappings (though not the "[[mapping to lattice]]") listed on temperament data pages of this wiki are in the <code>"shift"</code> form, though some are still in <code>"flip"</code> form. }}
 The generators in defactored Hermite form of septimal meantone is positive already, so its positive-generator forms are the same as its defactored Hermite form, {{mapping| 1 0 -4 -13 | 0 1 4 10 }}, corresponding to generators of ~2/1 and ~3/1. An example of positive-generator form that is different from the defactored Hermite form is the porcupine temperament, elaborated below.
@@ Line 161: / Line 161: @@
 === Minimal-generator form ===
-The ''minimal-generator form'' (or '''mingen form''') is a form specific to rank-2 temperaments, where the generator is positive and no greater than half the period.<ref group="note">This is somewhat like octave reduction combined with octave inversion, because you can't just add or subtract half octaves until it's between 0 and 600 cents; you have to add or subtract octaves until it's between −600 and +600 cents, then multiply by −1 if it's negative.</ref><ref group="note">You could always find a smaller and smaller generator by going negative, so this assumes positive generators.</ref>
+The ''minimal-generator form'' (or '''mingen form''') is a form specific to [[rank-2 temperament]]s, where the generator is positive and no greater than half the period.<ref group="note">This is somewhat like octave reduction combined with octave inversion, because you can't just add or subtract half octaves until it's between 0 and 600 cents; you have to add or subtract octaves until it's between −600 and +600 cents, then multiply by −1 if it's negative.</ref><ref group="note">You could always find a smaller and smaller generator by going negative, so this assumes positive generators.</ref>
-[[Graham Breed]]'s [http://x31eq.com/temper/ temperament finder] uses this form for all rank-2 temperaments. Septimal meantone in minimal-generator form is {{mapping| 1 2 4 7 | 0 -1 -4 -10 }}, corresponding to generators of ~2/1 and ~4/3. [[Kite Giedraitis]]'s [[pergen]] also uses this form, except for when the generator is a perfect fifth.
+[[Graham Breed]]'s [http://x31eq.com/temper/ Temperament Finder] uses this form for all rank-2 temperaments. Septimal meantone in minimal-generator form is {{mapping| 1 2 4 7 | 0 -1 -4 -10 }}, corresponding to generators of ~2/1 and ~4/3. [[Kite Giedraitis]]'s [[pergen]] also uses this form, except for when the generator is a perfect fifth.
 Beyond rank-2, the mingen form of a temperament is no longer unique. You can always get smaller and smaller generators. This is why on Graham Breed's temperament finding tool, beyond rank-2 he simply uses the Hermite Normal Form.