The Riemann zeta function and tuning: Difference between revisions
→Gene Smith's original derivation: explain motivations & increase accessibility. also note the theoretical significance of s=1 |
m →Gene Smith's original derivation: fix formatting error |
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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.) | (<nowiki>*</nowiki> Specifically, the reason we use the weighting 1/log<sub>2</sub>(''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.) | (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.) | ||