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User:Overthink/Asymptotic consistency score: Difference between revisions

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reworked lists after debugging code so second-best mappings work properly
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→The consistency metric: Fixed lists again, after more bug fixes
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We want to give each edo a single score for how well it does in terms of consistency. Given n-edo, we start with the trivial mapping ⟨n| 0]. We then add each odd one by one, and look at how many additional intervals are consistent and inconsistent. When we add odd q, each consistent interval increases the score by 1/q<sup>2</sup>, and each inconsistent interval decreases the score by 3/q<sup>2</sup>. However, it is impossible to calculate this score precisely, as there would be infinitely many terms. Using a program I made on scratch, considering EDOs up to 311 and odds up to 511, here is a sequence of EDOs that have better consistency scores:
We want to give each edo a single score for how well it does in terms of consistency. Given n-edo, we start with the trivial mapping ⟨n| 0]. We then add each odd one by one, and look at how many additional intervals are consistent and inconsistent. When we add odd q, each consistent interval increases the score by 1/q<sup>2</sup>, and each inconsistent interval decreases the score by 3/q<sup>2</sup>. However, it is impossible to calculate this score precisely, as there would be infinitely many terms. Using a program I made on scratch, considering EDOs up to 311 and odds up to 511, here is a sequence of EDOs that have better consistency scores:


{{edos|(1, 2, 3,) 7, 10, 24, 31, 38, 39, 45.}}
{{edos|(1, 2, 3, 4, 5,) 10, 15, 22, 34, 37, 41, 46, 53, 58, 80, 183, 217, 270, 311.}}


Here's the same list with odds up to 255:
Here's the same list with odds up to 255:


{{edos|(1, 2, 3,) 5, 7, 10, 24, 31, 41, 45, 270.}}
{{edos|(1, 2, 3, 4, 5,) 10, 15, 22, 31, 34, 37, 41, 46, 53, 58, 80, 121, 183, 217, 270, 311.}}


In the 127-odd-limit:
In the 127-odd-limit:


{{Edos|(1, 3,) 5, 7, 10, 15, 24, 29, 31, 37, 41, 45, 46, 53, 87, 183, 270, 311.}}
{{Edos|(1, 2, 3, 4, 5,) 10, 15, 22, 34, 37, 41, 46, 53, 58, 80, 183, 217, 270, 311.}}


63-odd-limit:
63-odd-limit:


{{Edos|(1, 3,) 5, 10, 15, 22, 24, 29, 41, 87, 159, 183, 217, 270, 311.}}
{{Edos|(1, 2, 3, 4, 5,) 10, 15, 22, 31, 34, 37, 41, 58, 80, 183, 217, 270, 311.}}


31-odd-limit:
31-odd-limit:


{{Edos|(1, 3, 4,) 5, 10, 15, 22, 29, 41, 80, 87, 159, 217, 282, 311.}}
{{Edos|(1, 3, 4, 5,) 10, 15, 22, 41, 58, 80, 159, 217, 282, 311.}}


63-odd-limit, up to 20567edo (outdated):
63-odd-limit, up to 20567edo:


{{Edos|(1, 3, 4,) 5, 10, 15, 22, 29, 31, 41, 87, 159, 183, 270, 311, 388, 525, 653, 718, 1600, 2554, 3889, 4380, 10257, 14348.}}
{{Edos|(1, 2, 3, 4, 5,) 10, 15, 22, 31, 34, 37, 41, 58, 80, 183, 217, 270, 311, 388, 422, 718, 1600, 2554, 3395, 3889, 4380, 10257, 14348.}}
 
For some reason, 45edo does extremely well in high limits?
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