Riemann zeta function: Difference between revisions

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== Gene Smith's original derivation ==

Suppose ''x'' is a variable representing some equal division of the octave. For example, if {{nowrap|''x'' {{=}} 80}}, ''x'' ~~reflects~~ 80edo with a step size of 15 ~~cents~~ and with pure octaves. Suppose that ''x'' can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well. The [[Bohlen–Pierce scale]], 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of {{nowrap|''x'' {{=}} 8.202}}.

Suppose ''x'' is a variable representing some equal division of the octave. For example, if {{nowrap|''x'' {{=}} 80}}, ''x'' corresponds to [[80edo]] with a step size of 15 [[cent]]s and with pure octaves. Suppose that ''x'' can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well, so that it ranges over all possible [[equal temperament]]s. The [[Bohlen–Pierce scale]], 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of {{nowrap|''x'' {{=}} 8.202}}.

Now suppose that [https://www.desmos.com/calculator/krigk43int {{rr|''x''}}] denotes the difference between ''x'' and the integer nearest to ''x'':

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For example, {{nowrap|{{rr|8.202}} {{=}} 0.202}}, since it is the difference between 8.202 and the nearest integer, which is 8. Meanwhile, {{nowrap|{{rr|7.95}} {{=}} 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 {{nowrap|''x'' {{=}} 12}}, we find 2 at 12, 3 at 19, 5 at 28, etc. Now consider [https://www.desmos.com/calculator/4uamhon9tt the function]

For any value of ''x'', we can construct a ''p''-[[limit]] [[val]] by rounding {{nowrap|''x'' log2(''q'')}} to the nearest integer for each prime ''q'' up to ''p''. (More technically, this corresponds to what's known as a [[generalized patent val]].) For example, for {{nowrap|''x'' {{=}} 12}}, we find 2 at 12, 3 at 19, 5 at 28, etc. Now consider [https://www.desmos.com/calculator/4uamhon9tt the function]

<math>

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</math>

Here, the numerator represents the relative error on each prime, and the denominator represents a weight factor of the logarithm of each prime, so the function represents a p-limit badness metric.

Here, the numerator represents the relative error on each prime, and the denominator represents a weight factor of the logarithm of each prime*, so the function represents a p-limit badness metric.

(<nowiki>*</nowiki> Specifically, the reason we use the weighting 1/log2(''p'') is because of certain desirable properties it has that singles it out as of unique interest: if the complexity of a prime ''p'' is {{nowrap| log2(''p'') }}, then ''p''n is ''n'' times as complex as ''p''. Using the {{nowrap| log2(''p'') }} weighting also means that the complexity of a harmonic, according to its prime factorization, exactly matches where it's found in the harmonic series, so that e.g. {{nowrap| 25 {{=}} 5 × 5 }} is slightly less complex than {{nowrap| 26 {{=}} 2 × 13 }} is slightly less complex than {{nowrap| 27 {{=}} 3 × 3 × 3 }}. Therefore, the {{nowrap| 1/log2(''p'') }} is a kind of natural inverse-complexity weighting, that is, a simplicity weighting.)

(Also, for those unfamiliar, squaring the error is commonly done because it solves the flaws of two alternative ways of measuring error. Specifically, if you look only at the maximum error, you miss opportunities to make the tuning much better ''overall'' by allowing slightly more damage on the most damaged intervals, while if you look only at the average error, then it may be that you are unnecessarily damaging a few intervals a lot just to get intervals that are already in-tune slightly more in-tune, so both extremes have pathological behaviours, and using the squared error counters both of these behaviours so that it represents a more balanced approach to optimization that is used in a variety of disciplines.)

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 relative error]] of the val—equal to the TE error times the TE complexity, and sometimes known as "TE simple badness."

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<math>\displaystyle \xi_\infty(x) = \sum_{\substack{q \geq 2 \\ q \text{ prime}}} \frac{\lfloor x \log_2 q \rceil^2}{q^s}</math>

Importantly, when ''s'' = 1, the weighting is 1/''p'' for a prime ''p'', so that it's very similar to a 1/(''p'' - 1) weighting. This latter weighting is equal to the average number of times prime ''p'' occurs as a factor in the harmonic series, counting repetition, so is of interest because it represents how many harmonics will feel damage from this prime being mistuned. Therefore, the weighting {{nowrap| 1/''p'' }} corresponds to under-prioritizing small primes slightly (with the effect being less slight the smaller the prime, so that at the most extreme, at prime 2 we have 1/2 instead of 1/1 weighting and at prime 3 we have 1/3 instead of 1/2 weighting).

Seeing that we notate the power as ''s'', it might become apparent where the Riemann zeta function will eventually show up.

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so that we see that the absolute value of the zeta function serves to measure the relative error of an equal division.

There is an open question as to whether the analytic continuation preserves the properties we are interested in, though empirically it seems to. However, if you do not want to trust the analytic continuation, then the {{nowrap| ''s'' {{=}} 1 }} line is of interest for the reasons discussed, as though the infinite sum does not converge, every sum for {{nowrap| ''s'' > 1 }} ''does'' converge, so that we can consider the line at {{nowrap| ''s'' {{=}} 1 }} as being the "limit" as ''s'' approaches 1 from above. This can also be used to sanity-check results at {{nowrap| ''s'' {{=}} 1/2 }} by seeing where they agree, e.g. as done in the section on [[#Absolute zeta peak edos]], which looks at getting more practical tuning information out of the zeta function via an adjustment for considering absolute mistuning rather than relative error.

== Mike Battaglia's expanded results ==

@@ Line 62: / Line 62: @@
 == Gene Smith's original derivation ==
-Suppose ''x'' is a variable representing some equal division of the octave. For example, if {{nowrap|''x'' {{=}} 80}}, ''x'' reflects 80edo with a step size of 15 cents and with pure octaves. Suppose that ''x'' can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well. The [[Bohlen–Pierce scale]], 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of {{nowrap|''x'' {{=}} 8.202}}.
+Suppose ''x'' is a variable representing some equal division of the octave. For example, if {{nowrap|''x'' {{=}} 80}}, ''x'' corresponds to [[80edo]] with a step size of 15 [[cent]]s and with pure octaves. Suppose that ''x'' can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well, so that it ranges over all possible [[equal temperament]]s. The [[Bohlen–Pierce scale]], 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of {{nowrap|''x'' {{=}} 8.202}}.
 Now suppose that [https://www.desmos.com/calculator/krigk43int {{rr|''x''}}] denotes the difference between ''x'' and the integer nearest to ''x'':
@@ Line 72: / Line 72: @@
 For example, {{nowrap|{{rr|8.202}} {{=}} 0.202}}, since it is the difference between 8.202 and the nearest integer, which is 8. Meanwhile, {{nowrap|{{rr|7.95}} {{=}} 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 {{nowrap|''x'' {{=}} 12}}, we find 2 at 12, 3 at 19, 5 at 28, etc. Now consider [https://www.desmos.com/calculator/4uamhon9tt the function]
+For any value of ''x'', we can construct a ''p''-[[limit]] [[val]] by rounding {{nowrap|''x'' log<sub>2</sub>(''q'')}} to the nearest integer for each prime ''q'' up to ''p''. (More technically, this corresponds to what's known as a [[generalized patent val]].) For example, for {{nowrap|''x'' {{=}} 12}}, we find 2 at 12, 3 at 19, 5 at 28, etc. Now consider [https://www.desmos.com/calculator/4uamhon9tt the function]
 <math>
@@ Line 79: / Line 79: @@
 </math>
-Here, the numerator represents the relative error on each prime, and the denominator represents a weight factor of the logarithm of each prime, so the function represents a p-limit badness metric.
+Here, the numerator represents the relative error on each prime, and the denominator represents a weight factor of the logarithm of each prime*, so the function represents a p-limit badness metric.
+(<nowiki>*</nowiki> Specifically, the reason we use the weighting 1/log<sub>2(''p'') is because of certain desirable properties it has that singles it out as of unique interest: if the complexity of a prime ''p'' is {{nowrap| log<sub>2</sub>(''p'') }}, then ''p''<sup>n</sup> is ''n'' times as complex as ''p''. Using the {{nowrap| log<sub>2</sub>(''p'') }} weighting also means that the complexity of a harmonic, according to its prime factorization, exactly matches where it's found in the harmonic series, so that e.g. {{nowrap| 25 {{=}} 5 × 5 }} is slightly less complex than {{nowrap| 26 {{=}} 2 × 13 }} is slightly less complex than {{nowrap| 27 {{=}} 3 × 3 × 3 }}. Therefore, the {{nowrap| 1/log<sub>2</sub>(''p'') }} is a kind of natural inverse-complexity weighting, that is, a simplicity weighting.)
+(Also, for those unfamiliar, squaring the error is commonly done because it solves the flaws of two alternative ways of measuring error. Specifically, if you look only at the maximum error, you miss opportunities to make the tuning much better ''overall'' by allowing slightly more damage on the most damaged intervals, while if you look only at the average error, then it may be that you are unnecessarily damaging a few intervals a lot just to get intervals that are already in-tune slightly more in-tune, so both extremes have pathological behaviours, and using the squared error counters both of these behaviours so that it represents a more balanced approach to optimization that is used in a variety of disciplines.)
 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 relative error]] of the val—equal to the TE error times the TE complexity, and sometimes known as "TE simple badness."
@@ Line 86: / Line 90: @@
 <math>\displaystyle \xi_\infty(x) = \sum_{\substack{q \geq 2 \\ q \text{ prime}}} \frac{\lfloor x \log_2 q \rceil^2}{q^s}</math>
+Importantly, when ''s'' = 1, the weighting is 1/''p'' for a prime ''p'', so that it's very similar to a 1/(''p'' - 1) weighting. This latter weighting is equal to the average number of times prime ''p'' occurs as a factor in the harmonic series, counting repetition, so is of interest because it represents how many harmonics will feel damage from this prime being mistuned. Therefore, the weighting {{nowrap| 1/''p''  }} corresponds to under-prioritizing small primes slightly (with the effect being less slight the smaller the prime, so that at the most extreme, at prime 2 we have 1/2 instead of 1/1 weighting and at prime 3 we have 1/3 instead of 1/2 weighting).
 Seeing that we notate the power as ''s'', it might become apparent where the Riemann zeta function will eventually show up.
@@ Line 134: / Line 140: @@
 so that we see that the absolute value of the zeta function serves to measure the relative error of an equal division.
+There is an open question as to whether the analytic continuation preserves the properties we are interested in, though empirically it seems to. However, if you do not want to trust the analytic continuation, then the {{nowrap| ''s'' {{=}} 1 }} line is of interest for the reasons discussed, as though the infinite sum does not converge, every sum for {{nowrap| ''s'' > 1 }} ''does'' converge, so that we can consider the line at {{nowrap| ''s'' {{=}} 1 }} as being the "limit" as ''s'' approaches 1 from above. This can also be used to sanity-check results at {{nowrap| ''s'' {{=}} 1/2 }} by seeing where they agree, e.g. as done in the section on [[#Absolute zeta peak edos]], which looks at getting more practical tuning information out of the zeta function via an adjustment for considering absolute mistuning rather than relative error.
 == Mike Battaglia's expanded results ==