The Riemann zeta function and tuning: Difference between revisions

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[[File:plot12.png|alt=plot12.png|plot12.png]]

The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an ~~edo~~. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/''x'', and hence the size of the octave in the zeta peak value tuning for ''N''edo is ''N''/''x''; if ''x'' is slightly larger than ''N'' as here with {{nowrap|''N'' {{=}} 12}}, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with ''x'' less than ''N'', we have stretched octaves.

The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an approximation of JI. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/''x'', and hence the size of the octave in the zeta peak value tuning for ''N''edo is ''N''/''x''; if ''x'' is slightly larger than ''N'' as here with {{nowrap|''N'' {{=}} 12}}, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with ''x'' less than ''N'', we have stretched octaves.

For larger edos, the width of the peak narrows, but for strong edos the height more than compensates, measured in terms of the area under the peak (the absolute value of the integral of Z between two zeros.) Note how 270 completely dominates its neighbors:

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 [[File:plot12.png|alt=plot12.png|plot12.png]]
-The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an edo. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/''x'', and hence the size of the octave in the zeta peak value tuning for ''N''edo is ''N''/''x''; if ''x'' is slightly larger than ''N'' as here with {{nowrap|''N'' {{=}} 12}}, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with ''x'' less than ''N'', we have stretched octaves.
+The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an approximation of JI. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/''x'', and hence the size of the octave in the zeta peak value tuning for ''N''edo is ''N''/''x''; if ''x'' is slightly larger than ''N'' as here with {{nowrap|''N'' {{=}} 12}}, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with ''x'' less than ''N'', we have stretched octaves.
 For larger edos, the width of the peak narrows, but for strong edos the height more than compensates, measured in terms of the area under the peak (the absolute value of the integral of Z between two zeros.) Note how 270 completely dominates its neighbors: