Talk:Normal forms: Difference between revisions

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Sensi is a great example to show how all these will differ from each other:

# Canonical form: [{{~~val~~| 1 6 8 11 ~~}}, {{val~~| 0 7 9 13 }}] with generators ~2, ~9/14

# Canonical form: {{mapping| 1 6 8 11 | 0 7 9 13 }} with generators ~2, ~9/14

# Positive generator form: [{{~~val~~| 1 6 8 11 ~~}}, {{val~~| 0 -7 -9 -13 }}] with generators ~2, ~14/9

# Positive generator form: {{mapping| 1 6 8 11 | 0 -7 -9 -13 }} with generators ~2, ~14/9

# Equave-reduced generator form: [{{~~val~~| 1 -1 -1 -2 ~~}}, {{val~~| 0 7 9 13 }}] with generators ~2, ~9/7

# Equave-reduced generator form: {{mapping| 1 -1 -1 -2 | 0 7 9 13 }} with generators ~2, ~9/7

# Positive equave-reduced generator form: [{{~~val~~| 1 6 8 11 ~~}}, {{val~~| 0 -7 -9 -13 }}] with generators ~2, ~14/9

# Positive equave-reduced generator form: {{mapping| 1 6 8 11 | 0 -7 -9 -13 }} with generators ~2, ~14/9

# Minimal generator form: [{{~~val~~| 1 -1 -1 -2 ~~}}, {{val~~| 0 7 9 13 }}] with generators ~2, ~9/7

# Minimal generator form: {{mapping| 1 -1 -1 -2 | 0 7 9 13 }} with generators ~2, ~9/7

My expectation is that the positive equave-reduced generator form should be left largely obsolete since musicians will work with the new equave-reduced generator form. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 08:00, 16 January 2023 (UTC)

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: Makes sense. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 13:09, 15 June 2025 (UTC)

== I propose an even better form/rule ==

A combo of the positive generator form and equave-reduced generator form (there should technically be a hyphen between ''positive''/''equave-reduced'' and ''generator'', right?): it makes sure the generators are positive not by flipping but by octave-reducing. Iow it's the canonical form if the canonical gens are positive, but octave-reduced form if the canonical gens are negative. This also makes sure the first nonzero entry of each row in the mapping is always positive; for rank-2 temps with primes 2 and 3 this means prime 3 is always a positive number of gensteps.

* For meantone this gives {{mapping| 1 0 -4 -13 | 0 1 4 10 }} with gens ~2, ~3, same as canonical and positive generator forms.

* For sensi this gives {{mapping| 1 -1 -1 -2 | 0 7 9 13 }} with gens ~2, ~9/7, same as equave-reduced and minimal generator forms.

* For würschmidt this gives {{mapping| 1 -1 2 | 0 8 1 }} with gens ~2, ~5/4(!), same as equave-reduced and minimal generator forms.

But that is still not good enough, for it gives {{mapping| 1 -1 -2 | 0 3 5 }} for porcupine, with gens ~2, ~9/5, yet we're generally more familiar with ~10/9 as the gen. So here's an additional rule based on the omega extension of ploidacot: if the temperament is omega-''n''-cot i.e. splits ~4/3 into three or more steps, we should stick to the positive generator form, so that porcupine remains {{mapping| 1 2 3 | 0 -3 -5 }}. I'm still contemplating how this should be extended to no-2 and/or no-3 temps.

I think it's very humanized and fits well for the wiki, tho perhaps it's more like a rule for choosing forms than an individual form.

[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 13:12, 15 June 2025 (UTC)

@@ Line 181: / Line 181: @@
 Sensi is a great example to show how all these will differ from each other:
-# Canonical form: [{{val| 1 6 8 11 }}, {{val| 0 7 9 13 }}] with generators ~2, ~9/14
+# Canonical form: {{mapping| 1 6 8 11 | 0 7 9 13 }} with generators ~2, ~9/14
-# Positive generator form: [{{val| 1 6 8 11 }}, {{val| 0 -7 -9 -13 }}] with generators ~2, ~14/9
+# Positive generator form: {{mapping| 1 6 8 11 | 0 -7 -9 -13 }} with generators ~2, ~14/9
-# Equave-reduced generator form: [{{val| 1 -1 -1 -2 }}, {{val| 0 7 9 13 }}] with generators ~2, ~9/7
+# Equave-reduced generator form: {{mapping| 1 -1 -1 -2 | 0 7 9 13 }} with generators ~2, ~9/7
-# Positive equave-reduced generator form: [{{val| 1 6 8 11 }}, {{val| 0 -7 -9 -13 }}] with generators ~2, ~14/9
+# Positive equave-reduced generator form: {{mapping| 1 6 8 11 | 0 -7 -9 -13 }} with generators ~2, ~14/9
-# Minimal generator form: [{{val| 1 -1 -1 -2 }}, {{val| 0 7 9 13 }}] with generators ~2, ~9/7
+# Minimal generator form: {{mapping| 1 -1 -1 -2 | 0 7 9 13 }} with generators ~2, ~9/7
 My expectation is that the positive equave-reduced generator form should be left largely obsolete since musicians will work with the new equave-reduced generator form. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 08:00, 16 January 2023 (UTC)
@@ Line 200: / Line 200: @@
 : Makes sense. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 13:09, 15 June 2025 (UTC)
+== I propose an even better form/rule ==
+A combo of the positive generator form and equave-reduced generator form (there should technically be a hyphen between ''positive''/''equave-reduced'' and ''generator'', right?): it makes sure the generators are positive not by flipping but by octave-reducing. Iow it's the canonical form if the canonical gens are positive, but octave-reduced form if the canonical gens are negative. This also makes sure the first nonzero entry of each row in the mapping is always positive; for rank-2 temps with primes 2 and 3 this means prime 3 is always a positive number of gensteps.
+* For meantone this gives {{mapping| 1 0 -4 -13 | 0 1 4 10 }} with gens ~2, ~3, same as canonical and positive generator forms.
+* For sensi this gives {{mapping| 1 -1 -1 -2 | 0 7 9 13 }} with gens ~2, ~9/7, same as equave-reduced and minimal generator forms.
+* For würschmidt this gives {{mapping| 1 -1 2 | 0 8 1 }} with gens ~2, ~5/4(!), same as equave-reduced and minimal generator forms.
+But that is still not good enough, for it gives {{mapping| 1 -1 -2 | 0 3 5 }} for porcupine, with gens ~2, ~9/5, yet we're generally more familiar with ~10/9 as the gen. So here's an additional rule based on the omega extension of ploidacot: if the temperament is omega-''n''-cot i.e. splits ~4/3 into three or more steps, we should stick to the positive generator form, so that porcupine remains {{mapping| 1 2 3 | 0 -3 -5 }}. I'm still contemplating how this should be extended to no-2 and/or no-3 temps.
+I think it's very humanized and fits well for the wiki, tho perhaps it's more like a rule for choosing forms than an individual form.
+[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 13:12, 15 June 2025 (UTC)