The Riemann zeta function and tuning: 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}}}}.

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'' + {{~~sfrac~~|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.

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.

For an example, consider {{nowrap|''x'' {{=}} 12}}, so that {{nowrap|''r'' {{=}} {{sfrac|12|ln(2)}} {{=}} 17.312}}. Then {{nowrap|''r'' ln(''r'') − ''r'' − {{sfrac|1|8}} {{=}} 31.927}}, which rounded to the nearest integer is 32, so {{nowrap|''n'' {{=}} 32}}. Then {{nowrap|{{sfrac|''r'' + ''n'' + {{sfrac|1|8}}|ln(''r'')}} {{=}} 17.338}}, corresponding to {{nowrap|''x'' {{=}} 12.0176}}, which means a single step is 99.853 cents and the octave is tempered to twelve of these, which is 1198.238 cents.

The fact that ''x'' is slightly greater than 12 means 12 has an overall sharp quality. We may also find this out by looking at the value we computed for {{nowrap|θ(2π''r'') / π}}, which was 31.927. Then {{nowrap|32 − 31.927 {{=}} 0.0726}}, which is positive but not too large; this is the second black magic formula, evaluating the nature of an edo ''x'' by computing {{nowrap|&lfloor;''r'' ln(''r'') − ''r'' + {{~~sfrac~~|3|8}}&rfloor; − ''r'' ln(''r'') + ''r'' + {{~~sfrac~~|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''x''|ln(2)}}}}. This works more often than not on the clearcut cases, but when ''x'' is extreme it may not; 49 is very sharp in tendency, for example, but this method calls it as flat; similarly it counts 45 as sharp.

The fact that ''x'' is slightly greater than 12 means 12 has an overall sharp quality. We may also find this out by looking at the value we computed for {{nowrap|θ(2π''r'') / π}}, which was 31.927. Then {{nowrap|32 − 31.927 {{=}} 0.0726}}, which is positive but not too large; this is the second black magic formula, evaluating the nature of an edo ''x'' by computing {{nowrap|&lfloor;''r'' ln(''r'') − ''r'' + {{frac|3|8}}&rfloor; − ''r'' ln(''r'') + ''r'' + {{frac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''x''|ln(2)}}}}. This works more often than not on the clearcut cases, but when ''x'' is extreme it may not; 49 is very sharp in tendency, for example, but this method calls it as flat; similarly it counts 45 as sharp.

== Computing zeta ==

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 From this we may deduce that {{nowrap|{{sfrac|&theta;(''t'')|&pi;}} &#8776; ''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}}}}.
-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'' + {{sfrac|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.
+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.
 For an example, consider {{nowrap|''x'' {{=}} 12}}, so that {{nowrap|''r'' {{=}} {{sfrac|12|ln(2)}} {{=}} 17.312}}. Then {{nowrap|''r'' ln(''r'') &minus; ''r'' &minus; {{sfrac|1|8}} {{=}} 31.927}}, which rounded to the nearest integer is 32, so {{nowrap|''n'' {{=}} 32}}. Then {{nowrap|{{sfrac|''r'' + ''n'' + {{sfrac|1|8}}|ln(''r'')}} {{=}} 17.338}}, corresponding to {{nowrap|''x'' {{=}} 12.0176}}, which means a single step is 99.853 cents and the octave is tempered to twelve of these, which is 1198.238 cents.
-The fact that ''x'' is slightly greater than 12 means 12 has an overall sharp quality. We may also find this out by looking at the value we computed for {{nowrap|&theta;(2&pi;''r'') / &pi;}}, which was 31.927. Then {{nowrap|32 &minus; 31.927 {{=}} 0.0726}}, which is positive but not too large; this is the second black magic formula, evaluating the nature of an edo ''x'' by computing {{nowrap|&lfloor;''r'' ln(''r'') &minus; ''r'' + {{sfrac|3|8}}&rfloor; &minus; ''r'' ln(''r'') + ''r'' + {{sfrac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''x''|ln(2)}}}}. This works more often than not on the clearcut cases, but when ''x'' is extreme it may not; 49 is very sharp in tendency, for example, but this method calls it as flat; similarly it counts 45 as sharp.
+The fact that ''x'' is slightly greater than 12 means 12 has an overall sharp quality. We may also find this out by looking at the value we computed for {{nowrap|&theta;(2&pi;''r'') / &pi;}}, which was 31.927. Then {{nowrap|32 &minus; 31.927 {{=}} 0.0726}}, which is positive but not too large; this is the second black magic formula, evaluating the nature of an edo ''x'' by computing {{nowrap|&lfloor;''r'' ln(''r'') &minus; ''r'' + {{frac|3|8}}&rfloor; &minus; ''r'' ln(''r'') + ''r'' + {{frac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''x''|ln(2)}}}}. This works more often than not on the clearcut cases, but when ''x'' is extreme it may not; 49 is very sharp in tendency, for example, but this method calls it as flat; similarly it counts 45 as sharp.
 == Computing zeta ==