User:Overthink/Asymptotic consistency score: Difference between revisions

Line 18:

== 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 ~~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/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:

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

{{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~~:

And in the 127-odd-limit:

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

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

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

~~{{Edos|(1, 2, 3, 4,~~ 5,) 10, 15, 22, 34, 37, 41, 46, 53, ~~58, 80~~, 183~~, 217~~, 270, 311.}}

63-odd-limit:

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

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

31-odd-limit:

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

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

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

~~63-odd-limit~~, ~~up to 20567edo:~~

{{Edos|(1, 3, 4,) 5, 10, 15, 22, 24, 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~~.}}

Surprisingly, 20567edo itself isn't on the last list. We will now look at alternative lists where we choose the second closest mapping of an odd harmonic when both it and its closest mapping have the same number of additional inconsistencies.

@@ Line 18: / Line 18: @@
 == 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 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 255, here is a sequence of EDOs that have better consistency scores:
-{{edos|(1, 2, 3, 4, 5,) 10, 15, 22, 34, 37, 41, 46, 53, 58, 80, 183, 217, 270, 311.}}
+{{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:
+And in the 127-odd-limit:
-{{edos|(1, 2, 3, 4, 5,) 10, 15, 22, 31, 34, 37, 41, 46, 53, 58, 80, 121, 183, 217, 270, 311.}}
+{{Edos|(1, 3, 4,) 5, 10, 15, 22, 29, 31, 41, 46, 53, 87, 183, 270, 311.}}
-In the 127-odd-limit:
-{{Edos|(1, 2, 3, 4, 5,) 10, 15, 22, 34, 37, 41, 46, 53, 58, 80, 183, 217, 270, 311.}}
 -odd-limit:
-{{Edos|(1, 2, 3, 4, 5,) 10, 15, 22, 31, 34, 37, 41, 58, 80, 183, 217, 270, 311.}}
+{{Edos|(1, 3, 4,) 5, 10, 15, 22, 24, 29, 31, 41, 87, 159, 183, 270, 311.}}
 -odd-limit:
-{{Edos|(1, 3, 4, 5,) 10, 15, 22, 41, 58, 80, 159, 217, 282, 311.}}
+{{Edos|(1, 3, 4,) 5, 10, 15, 22, 29, 41, 87, 159, 217, 282, 311.}}
+-odd-limit, up to 20567edo (took a few minutes to generate):
--odd-limit, up to 20567edo:
+{{Edos|(1, 3, 4,) 5, 10, 15, 22, 24, 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.}}
+Surprisingly, 20567edo itself isn't on the last list. We will now look at alternative lists where we choose the second closest mapping of an odd harmonic when both it and its closest mapping have the same number of additional inconsistencies.