User:Hkm/Sandbox: Difference between revisions

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Let min_cents ~ 35, error_power ~ 1.5, complexity_fondness ~ 0.93 (if we say that step a/b has complexity a+b), and magic_number ~ 2. Let goodness_measurer be a variable.

Let a "step" be any JI interval. We say that is the score of a "step" is equal to 1/(min_cents + (the step's error in cents)**error_power) * complexity_fondness**(the complexity of the step) / goodness_measurer. We then say that the score of a "path" is equal to the product of the scores of the steps. The score of a temperament with a list of generator tunings is equal to the sum of the scores of all paths that reach the original interval times those path lengths. The goodness of a temperament with a list of generator tunings is the goodness_measurer necessary to get a score of magic_number. (This also works for scales without JI interpretations; we assign ~~a JI interpretation~~ to each ~~pair of notes~~ and compute the goodness of the best assignment.) The goodness of a temperament on its own is the highest goodness that that temperament achieves; we can find optimal tunings for any temperament through this algorithm.

Let a "step" be any JI interval. We say that is the score of a "step" is equal to 1/(min_cents + (the step's error in cents)**error_power) * complexity_fondness**(the complexity of the step) / goodness_measurer. We then say that the score of a "path" is equal to the product of the scores of the steps. The score of a temperament with a list of generator tunings is equal to the sum of the scores of all paths that reach the original interval times those path lengths. The goodness of a temperament with a list of generator tunings is the goodness_measurer necessary to get a score of magic_number. (This also works for scales without JI interpretations; we assign to each note and just interval a tempered interval rooted on that note, and compute the goodness of the best assignment.) The goodness of a temperament on its own is the highest goodness that that temperament achieves; we can find optimal tunings for any temperament through this algorithm.

Revision as of 23:32, 25 June 2025 view source Hkm (talk \| contribs) 934 edits →Badness Tag: Visual edit ← Older edit		Revision as of 23:37, 25 June 2025 view source Hkm (talk \| contribs) 934 edits →Badness Tag: Visual edit Newer edit →
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	Let min_cents ~ 35, error_power ~ 1.5, complexity_fondness ~ 0.93 (if we say that step a/b has complexity a+b), and magic_number ~ 2. Let goodness_measurer be a variable.		Let min_cents ~ 35, error_power ~ 1.5, complexity_fondness ~ 0.93 (if we say that step a/b has complexity a+b), and magic_number ~ 2. Let goodness_measurer be a variable.

	Let a "step" be any JI interval. We say that is the score of a "step" is equal to 1/(min_cents + (the step's error in cents)*error_power) complexity_fondness**(the complexity of the step) / goodness_measurer. We then say that the score of a "path" is equal to the product of the scores of the steps. The score of a temperament with a list of generator tunings is equal to the sum of the scores of all paths that reach the original interval times those path lengths. The goodness of a temperament with a list of generator tunings is the goodness_measurer necessary to get a score of magic_number. (This also works for scales without JI interpretations; we assign ~~a JI interpretation~~ to each ~~pair of notes~~ and compute the goodness of the best assignment.) The goodness of a temperament on its own is the highest goodness that that temperament achieves; we can find optimal tunings for any temperament through this algorithm.		Let a "step" be any JI interval. We say that is the score of a "step" is equal to 1/(min_cents + (the step's error in cents)*error_power) complexity_fondness**(the complexity of the step) / goodness_measurer. We then say that the score of a "path" is equal to the product of the scores of the steps. The score of a temperament with a list of generator tunings is equal to the sum of the scores of all paths that reach the original interval times those path lengths. The goodness of a temperament with a list of generator tunings is the goodness_measurer necessary to get a score of magic_number. (This also works for scales without JI interpretations; we assign to each note and just interval a tempered interval rooted on that note, and compute the goodness of the best assignment.) The goodness of a temperament on its own is the highest goodness that that temperament achieves; we can find optimal tunings for any temperament through this algorithm.