Riemann zeta function: Difference between revisions

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From this we may deduce that {{nowrap|{{sfrac|θ(''t'')|π}} ≈ ''r'' ln(''r'') − ''r'' − {{sfrac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''t''|2π}} {{=}} {{sfrac|''x'' ln(2)}}}}; hence while x is the number of equal steps to an octave, ''r'' is the number of equal steps to an "''e''-tave", meaning the interval of ''e'', which is {{nowrap|{{sfrac|1200|ln(2)}} {{=}} 1731.234{{c}}}}.

From this we may deduce that {{nowrap|{{sfrac|θ(''t'')|π}} ≈ ''r'' ln(''r'') − ''r'' − {{sfrac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''t''|2π}} {{=}} {{sfrac|''x'' ln(2)}}}}; hence while x is the number of equal steps to an octave, ''r'' is the number of equal steps to an "''e''-tave", meaning the interval of ''e'', which is {{nowrap|{{sfrac|1200|ln(2)}} {{=}} 1731.234{{cent}}}}.

Recall that Gram points near to pure-octave edos, where ''x'' is an integer, can be expected to correspond to peak values of {{nowrap|{{!}}ζ{{!}} {{=}} {{!}}Z{{!}}}}. We can find these Gram points by Newton's method applied to the above formula. If {{nowrap|''r'' {{=}} {{sfrac|''x''|ln(2)}}}}, and if {{nowrap|''n'' {{=}} &lfloor;''r'' ln(''r'') − ''r'' + {{frac|3|8}}&rfloor;}} is the nearest integer to {{sfrac|θ(2π''r'')|π}}, then we may set {{nowrap|''r''<sup>+</sup> {{=}} {{sfrac|''r'' + ''n'' + {{sfrac|1|8}}|ln(''r'')}}}}. This is the first iteration of Newton's method, which we may repeat if we like, but in fact no more than one iteration is really required. This is the first black magic formula, giving an adjusted "Gram" tuning from the orginal one.

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-From this we may deduce that {{nowrap|{{sfrac|&theta;(''t'')|&pi;}} &#x2248; ''r'' ln(''r'') &minus; ''r'' &minus; {{sfrac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''t''|2&pi;}} {{=}} {{sfrac|''x'' ln(2)}}}}; hence while x is the number of equal steps to an octave, ''r'' is the number of equal steps to an "''e''-tave", meaning the interval of ''e'', which is {{nowrap|{{sfrac|1200|ln(2)}} {{=}} 1731.234{{c}}}}.
+From this we may deduce that {{nowrap|{{sfrac|&theta;(''t'')|&pi;}} &#x2248; ''r'' ln(''r'') &minus; ''r'' &minus; {{sfrac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''t''|2&pi;}} {{=}} {{sfrac|''x'' ln(2)}}}}; hence while x is the number of equal steps to an octave, ''r'' is the number of equal steps to an "''e''-tave", meaning the interval of ''e'', which is {{nowrap|{{sfrac|1200|ln(2)}} {{=}} 1731.234{{cent}}}}.
 Recall that Gram points near to pure-octave edos, where ''x'' is an integer, can be expected to correspond to peak values of {{nowrap|{{!}}&zeta;{{!}} {{=}} {{!}}Z{{!}}}}. We can find these Gram points by Newton's method applied to the above formula. If {{nowrap|''r'' {{=}} {{sfrac|''x''|ln(2)}}}}, and if {{nowrap|''n'' {{=}} &lfloor;''r'' ln(''r'') &minus; ''r'' + {{frac|3|8}}&rfloor;}} is the nearest integer to {{sfrac|&theta;(2&pi;''r'')|&pi;}}, then we may set {{nowrap|''r''<sup>+</sup> {{=}} {{sfrac|''r'' + ''n'' + {{sfrac|1|8}}|ln(''r'')}}}}. This is the first iteration of Newton's method, which we may repeat if we like, but in fact no more than one iteration is really required. This is the first black magic formula, giving an adjusted "Gram" tuning from the orginal one.