The Riemann zeta function and tuning: Difference between revisions

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For example, ⌊8.202⌉ would be 0.202, since it is the difference between 8.202 and the nearest integer, which is 8. Meanwhile, ⌊7.95⌉ would be 0.05, which is the difference between 7.95 and the nearest integer, which is 8. This represents the absolute relative error of the octave in equal tuning ''x'', or alternatively how much x is detuned from an edo.

For any value of ''x'', we can construct a ''p''-limit [[Patent_val|generalized patent val]]. We do so by rounding {{nowrap|''x'' log2(''q'')}} to the nearest integer for each prime ''q'' up to ''p''. For example, for x = 12, we find 2 at 12, 3 at 19, 5 at 28, etc. Now consider the function

For any value of ''x'', we can construct a ''p''-limit [[Patent_val|generalized patent val]]. We do so by rounding {{nowrap|''x'' log2(''q'')}} to the nearest integer for each prime ''q'' up to ''p''. For example, for {{nowrap|''x'' {{=}} 12}}, we find 2 at 12, 3 at 19, 5 at 28, etc. Now consider the function

<math>\displaystyle \xi_p(x) = \sum_{\substack{2 \leq q \leq p \\ q \text{ prime}}} \left(\frac{\rround{x \log_2 q}}{\log_2 q}\right)^2</math>

This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of ''x'' which are the [[Tenney-Euclidean_Tuning|Tenney-Euclidean tuning]]s of the octaves of the associated vals, while ξ''p'' for these minima is the square of the [[~~Tenney-Euclidean_metrics~~|~~Tenney-Euclidean~~ relative error]] of the val—equal to the TE error times the TE complexity, and sometimes known as "TE simple badness."

This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of ''x'' which are the [[Tenney-Euclidean_Tuning|Tenney-Euclidean tuning]]s of the octaves of the associated vals, while ξ''p'' for these minima is the square of the [[Tenney–Euclidean_metrics|Tenney–Euclidean relative error]] of the val—equal to the TE error times the TE complexity, and sometimes known as "TE simple badness."

Now suppose we don't want a formula for any specific prime limit, but which applies to all primes. We can't take the above sum to infinity, since it doesn't converge. However, we could change the weighting factor to a power so that it does converge:

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=== ''k''-ary-peak edos ===

{{Idiosyncratic terms|the term "''k''-ary-peak edos" itself, as well as the names for the different types of ''k''-ary-peak edos. Proposed by ~~[[User:Akselai~~|Akselai]] and [[Budjarn Lambeth]].}}

{{Idiosyncratic terms|the term "''k''-ary-peak edos" itself, as well as the names for the different types of ''k''-ary-peak edos. Proposed by {{u|Akselai}} and [[Budjarn Lambeth]].}}

If we want to find the second-best edos ranked by zeta peaks, then given a full list of zeta peaks, we can remove the successively higher peaks to get another sequence of succesively higher peaks, which correspond to edos called '''Parker edos'''.

@@ Line 21: / Line 21: @@
 For example, ⌊8.202⌉ would be 0.202, since it is the difference between 8.202 and the nearest integer, which is 8. Meanwhile, ⌊7.95⌉ would be 0.05, which is the difference between 7.95 and the nearest integer, which is 8. This represents the absolute relative error of the octave in equal tuning ''x'', or alternatively how much x is detuned from an edo.
-For any value of ''x'', we can construct a ''p''-limit [[Patent_val|generalized patent val]]. We do so by rounding {{nowrap|''x'' log<sub>2</sub>(''q'')}} to the nearest integer for each prime ''q'' up to ''p''. For example, for x = 12, we find 2 at 12, 3 at 19, 5 at 28, etc. Now consider the function
+For any value of ''x'', we can construct a ''p''-limit [[Patent_val|generalized patent val]]. We do so by rounding {{nowrap|''x'' log<sub>2</sub>(''q'')}} to the nearest integer for each prime ''q'' up to ''p''. For example, for {{nowrap|''x'' {{=}} 12}}, we find 2 at 12, 3 at 19, 5 at 28, etc. Now consider the function
 <math>\displaystyle \xi_p(x) = \sum_{\substack{2 \leq q \leq p \\ q \text{ prime}}} \left(\frac{\rround{x \log_2 q}}{\log_2 q}\right)^2</math>
-This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of ''x'' which are the [[Tenney-Euclidean_Tuning|Tenney-Euclidean tuning]]s of the octaves of the associated vals, while ξ<sub>''p''</sub> for these minima is the square of the [[Tenney-Euclidean_metrics|Tenney-Euclidean relative error]] of the val—equal to the TE error times the TE complexity, and sometimes known as "TE simple badness."
+This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of ''x'' which are the [[Tenney-Euclidean_Tuning|Tenney-Euclidean tuning]]s of the octaves of the associated vals, while ξ<sub>''p''</sub> for these minima is the square of the [[Tenney–Euclidean_metrics|Tenney–Euclidean relative error]] of the val—equal to the TE error times the TE complexity, and sometimes known as "TE simple badness."
 Now suppose we don't want a formula for any specific prime limit, but which applies to all primes. We can't take the above sum to infinity, since it doesn't converge. However, we could change the weighting factor to a power so that it does converge:
@@ Line 510: / Line 510: @@
 === ''k''-ary-peak edos ===
-{{Idiosyncratic terms|the term "''k''-ary-peak edos" itself, as well as the names for the different types of ''k''-ary-peak edos. Proposed by [[User:Akselai|Akselai]] and [[Budjarn Lambeth]].}}
+{{Idiosyncratic terms|the term "''k''-ary-peak edos" itself, as well as the names for the different types of ''k''-ary-peak edos. Proposed by {{u|Akselai}} and [[Budjarn Lambeth]].}}
 If we want to find the second-best edos ranked by zeta peaks, then given a full list of zeta peaks, we can remove the successively higher peaks to get another sequence of succesively higher peaks, which correspond to edos called '''Parker edos'''.