User talk:FloraC/Archive 2022

Normalized mapping vs minimum generator

Why is the generator wider than a half octave in some temperaments? Why did you edit mappings to normalize?

• [⟨1 0 -4 -13], ⟨0 1 4 10]] (generator: ~3 = 1896.5 cents) vs [⟨1 2 4 7], ⟨0 -1 -4 -10]] (generator: ~4/3 = 503.5 cents) in the meantone temperament (12&19)
• [⟨1 0 -4 2], ⟨0 2 8 1]] (generator: ~7/4 = 947.4 cents) vs [⟨1 2 4 3], ⟨0 -2 -8 -1]] (generator: ~7/6 = 252.6 cents) in the godzilla temperament (5&14c)
• [⟨1 0 -13 -3], ⟨0 3 29 11]] (generator: ~81/56 = 634.0 cents) vs [⟨1 3 16 8], ⟨0 -3 -29 -11]] (generator: ~112/81 = 566.0 cents) in the tricot temperament (53&70)
• [⟨1 7 3 15], ⟨0 -8 -1 -18]] (generator: ~8/5 = 812.6 cents) vs [⟨1 -1 2 -3], ⟨0 8 1 18]] (generator: ~5/4 = 387.4 cents) in the würschmidt temperament (31&96)
• [⟨1 12 56 -2], ⟨0 -13 -67 6]] (generator: ~256/147 = 961.4 cents) vs [⟨1 -1 -11 4], ⟨0 13 67 -6]] (generator: ~147/128 = 238.6 cents) in the tokko temperament (5&166)
• [⟨1 16 32 -15], ⟨0 -17 -35 21]] (generator: ~9/5 = 1017.5 cents) vs [⟨1 -1 -3 6], ⟨0 17 35 -21]] (generator: ~10/9 = 182.5 cents) in the mitonic temperament (46&125)
• [⟨1 25 -31 -8], ⟨0 -26 37 12]] (generator: ~28/15 = 1080.7 cents) vs [⟨1 -1 6 4], ⟨0 26 -37 -12]] (generator: ~15/14 = 119.3 cents) in the septidiasemi temperament (10&161)
• [⟨1 17 9 10], ⟨0 -30 -13 -14]] (generator: ~10/7 = 616.6 cents) vs [⟨1 -13 -4 -4], ⟨0 30 13 14]] (generator: ~7/5 = 583.4 cents) in the cotritone temperament (37&72)
• [⟨2 0 11 31], ⟨0 1 -2 -8]] (generator: ~3 = 1903.7 cents) vs [⟨2 3 5 7], ⟨0 1 -2 -8]] (generator: ~16/15 = 103.7 cents) in the diaschismic temperament (46&58)
• [⟨2 1 9 -2], ⟨0 2 -4 7]] (generator: ~35/24 = 652.8 cents) vs [⟨2 3 5 5], ⟨0 2 -4 7]] (generator: ~36/35 = 52.8 cents) in the shrutar temperament (22&46)
• [⟨3 0 7 18], ⟨0 1 0 -2]] (generator: ~3 = 1909.3 cents) vs [⟨3 5 7 8], ⟨0 -1 0 2]] (generator: ~16/15 = 90.7 cents) in augene temperament (12&15)
• [⟨9 1 1 12], ⟨0 2 3 2]] (generator: ~5/3 = 884.3 cents) vs [⟨9 15 22 26], ⟨0 -2 -3 -2]] (generator: ~36/35 = 49.0 cents) in the ennealimmic temperament (27&45)

There are an infinite of mappings of each temperaments including normalized form (left) and minimum generator form (right). In the normalized form, a₂ in the mapping [⟨a₁ a₂ a₃ …], ⟨0 b₂ b₃ …]] takes 0 ≤ a₂ < abs(b₂) if b₂ ≠ 0. The minimum generator form ("Reduced Mapping" in the Temperament finding scripts by Graham Breed, taking 0 ≤ g ≤ p/2 where p is the period and g is the generator) can be yielded by Euclidean algorithm. Which form are you favor? --Xenllium (talk) 13:48, 29 January 2022 (UTC)

I'm aware of all the normal forms. I participated in the rework on the Normal lists page, after all (see also the corresponding talk page). The positive generator form is what I prefer, and with mapping generators showing the corresponding ratios. Reasons? First, Gene has always chosen that form. Second, it makes sense in higher ranks, whereas the minimum generator form doesn't. That said, I'm less sure about the POTE generator line. This line is more practical and sometimes really used to tune things. I hope octave-reduced form for this line isn't a bad choice. We're used to meantone being generated by fifths, not fourths. We may also add minimum generator form in parentheses when appropriate. FloraC (talk) 14:08, 29 January 2022 (UTC)

Reasonable commas extension

Hi there,

I recently stumbled upon your "reasonable commas" page, and I wanted to know a few things:

- What are/were your motivations for this page?
- What is the difference between the two definitions on that page?
- What is the algorithm you used? (as to extend to higher limits)

Thank you --Royalmilktea (talk) 07:27, 28 September 2022 (UTC)

> What are/were your motivations for this page?

It seems like a good criterion for whether a comma is an efficient one.

> What is the difference between the two definitions on that page?

My redefinition is more strict. For example, 135/128 would be a reasonable comma in the original definition cuz none of 129, 130, 131, 132, 133, 134 is 5-limit. In my redefinition 135/128 isn't one since 135/128 = (25/24)(81/80), factored into two simpler commas.

> What is the algorithm you used? (as to extend to higher limits)

Dead Shaman somehow generated the lists of commas according to his original definition. I simply checked each comma manually. So unfortunately I don't have an algorithm to share.

FloraC (talk) 05:50, 29 September 2022 (UTC)

Optimal GPV sequence template/module

Is there a way to actually implement your temperament evaluator python files to find a temperament's optimal GPV sequence into a template on this site for better ease of use? Or for all of your temperament evaluator files? --Royalmilktea (talk) 04:28, 12 October 2022 (UTC)

I have no idea how to implement it in lua. That said, I might make a separate python script for this particular functionality. FloraC (talk) 09:37, 12 October 2022 (UTC)