The Riemann zeta function and tuning: Difference between revisions

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Our mysterious substitution above was chosen to set the units for this up nicely. The variable x now happens to be measured in divisions of the octave. (The original variable ''t'', which was the imaginary part of the zeta argument ''s'', can be thought of as the number of divisions of the interval {{nowrap|''e''<sup>2π</sup> ≈ 535.49}}, or what [[Keenan_Pepper|Keenan Pepper]] has called the "natural interval.")

As mentioned in Gene's original zeta derivation, these cosine functions can be thought of as good approximations to the terms in the TE error computation, which are all the squared errors for the different primes. Rather than taking the square of the error, we instead put the error through the function {{sfrac~~|{{nowrap~~|1 − cos(''x'')}}|2}}, which is "close enough" for small values of ''x''. Since we are always rounding off to the best mapping, this error is never more 0.5 steps of the EDO, so since we have {{nowrap|−0.5 < x < 0.5}} we have a decent enough approximation.

As mentioned in Gene's original zeta derivation, these cosine functions can be thought of as good approximations to the terms in the TE error computation, which are all the squared errors for the different primes. Rather than taking the square of the error, we instead put the error through the function {{sfrac|1 − cos(''x'')|2}}, which is "close enough" for small values of ''x''. Since we are always rounding off to the best mapping, this error is never more 0.5 steps of the EDO, so since we have {{nowrap|−0.5 < x < 0.5}} we have a decent enough approximation.

We will call this '''cosine (relative) error''', by analogy with '''TE (relative) error'''. It is easy to see that the cosine error is approximately equal to the TE error when the error is small, and only diverges slightly for large errors.

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 Our mysterious substitution above was chosen to set the units for this up nicely. The variable x now happens to be measured in divisions of the octave. (The original variable ''t'', which was the imaginary part of the zeta argument ''s'', can be thought of as the number of divisions of the interval {{nowrap|''e''<sup>2&pi;</sup> &#8776; 535.49}}, or what [[Keenan_Pepper|Keenan Pepper]] has called the "natural interval.")
-As mentioned in Gene's original zeta derivation, these cosine functions can be thought of as good approximations to the terms in the TE error computation, which are all the squared errors for the different primes. Rather than taking the square of the error, we instead put the error through the function {{sfrac|{{nowrap|1 &minus; cos(''x'')}}|2}}, which is "close enough" for small values of ''x''. Since we are always rounding off to the best mapping, this error is never more 0.5 steps of the EDO, so since we have {{nowrap|&minus;0.5 &lt; x &lt; 0.5}} we have a decent enough approximation.
+As mentioned in Gene's original zeta derivation, these cosine functions can be thought of as good approximations to the terms in the TE error computation, which are all the squared errors for the different primes. Rather than taking the square of the error, we instead put the error through the function {{sfrac|1 &minus; cos(''x'')|2}}, which is "close enough" for small values of ''x''. Since we are always rounding off to the best mapping, this error is never more 0.5 steps of the EDO, so since we have {{nowrap|&minus;0.5 &lt; x &lt; 0.5}} we have a decent enough approximation.
 We will call this '''cosine (relative) error''', by analogy with '''TE (relative) error'''. It is easy to see that the cosine error is approximately equal to the TE error when the error is small, and only diverges slightly for large errors.