User:Overthink/Asymptotic consistency score: Difference between revisions

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== The consistency metric ==

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/q2, and each inconsistent interval decreases the score by 3/q2. 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 ~~255~~, 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/q2, and each inconsistent interval decreases the score by 3/q2. 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, 3~~, 4~~,) 5, 10, 15, 22, 29, 31, 41, 46, 53, 87, ~~130~~, ~~140~~, ~~183~~, 270~~, 311~~.}}

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

In the 127-odd-limit:

{{Edos|(1, 3~~, 4~~,) 5, 10, 15, 22, 29, 31, 41, 46, 53, 87, 183, 270, 311.}}

{{Edos|(1, 3,) 5, 7, 10, 15, 24, 29, 31, 37, 41, 45, 46, 53, 87, 183, 270, 311.}}

63-odd-limit:

{{Edos|(1, 3~~, 4~~,) 5, 10, 15, 22, 24, 29~~, 31~~, 41, 87, 159, 183, 270, 311.}}

{{Edos|(1, 3,) 5, 10, 15, 22, 24, 29, 41, 87, 159, 183, 217, 270, 311.}}

31-odd-limit:

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

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

63-odd-limit, up to 20567edo (~~took a few minutes to generate~~):

63-odd-limit, up to 20567edo (outdated):

{{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.}}

~~Surprisingly~~, ~~20567edo itself isn't on the last list.~~

For some reason, 45edo does extremely well in high limits?

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 == The consistency metric ==
-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 255, 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, 3, 4,) 5, 10, 15, 22, 29, 31, 41, 46, 53, 87, 130, 140, 183, 270, 311.}}
+{{edos|(1, 2, 3,) 7, 10, 24, 31, 38, 39, 45.}}
+Here's the same list with odds up to 255:
+{{edos|(1, 2, 3,) 5, 7, 10, 24, 31, 41, 45, 270.}}
 In the 127-odd-limit:
-{{Edos|(1, 3, 4,) 5, 10, 15, 22, 29, 31, 41, 46, 53, 87, 183, 270, 311.}}
+{{Edos|(1, 3,) 5, 7, 10, 15, 24, 29, 31, 37, 41, 45, 46, 53, 87, 183, 270, 311.}}
 -odd-limit:
-{{Edos|(1, 3, 4,) 5, 10, 15, 22, 24, 29, 31, 41, 87, 159, 183, 270, 311.}}
+{{Edos|(1, 3,) 5, 10, 15, 22, 24, 29, 41, 87, 159, 183, 217, 270, 311.}}
 -odd-limit:
-{{Edos|(1, 3, 4,) 5, 10, 15, 22, 29, 41, 87, 159, 217, 282, 311.}}
+{{Edos|(1, 3, 4,) 5, 10, 15, 22, 29, 41, 80, 87, 159, 217, 282, 311.}}
--odd-limit, up to 20567edo (took a few minutes to generate):
+-odd-limit, up to 20567edo (outdated):
 {{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.}}
-Surprisingly, 20567edo itself isn't on the last list.
+For some reason, 45edo does extremely well in high limits?