The Riemann zeta function and tuning: Difference between revisions

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==== Zeta peak integer edos ====

Alternatively (as [[groundfault]] has found), if we do not allow octave detuning and instead look at only the record {{nowrap|{{!}}Z(''x''){{!}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs| 1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}} … of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}} … and edos present in the previous list but not present here include {{EDOs| 4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}} … with 72's removal perhaps being the most surprising, showing the strength of 53 in that 72 does not improve on 53's peak. This definition may be better for measuring how accurate EDOs are without stretched or compressed octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos."

~~Similarly, we can look at pure-tritave EDTs, etc.~~

==== Zeta integral edos ====

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Multiplying the Z-function by this factor of adjustment gives a Z-function with the prime ''p'' removed from consideration. Zeta peak and zeta integral tunings may then be found as before.

Removing 2 leads to increasing adjusted peak values corresponding to ~~the~~ division of 3 ~~(the~~ "tritave") into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 45, 52, 56, 71, 75, 88, 131, 245, 316,}} … parts. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen-Pierce]] division of the tritave, but the multiples 26, 39, and 52 also.

Removing 2 leads to increasing adjusted peak values corresponding to EDTs (division of the [[3/1|{{ordinal|3}}]] harmonic, or "tritave") into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 45, 52, 56, 71, 75, 88, 131, 245, 316,}} … parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] EDOs. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen-Pierce]] division of the tritave, but the multiples 26, 39, and 52 also.

=== Black magic formulas ===

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 ==== Zeta peak integer edos ====
 Alternatively (as [[groundfault]] has found), if we do not allow octave detuning and instead look at only the record {{nowrap|{{!}}Z(''x''){{!}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs| 1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}} … of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}} … and edos present in the previous list but not present here include {{EDOs| 4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}} … with 72's removal perhaps being the most surprising, showing the strength of 53 in that 72 does not improve on 53's peak. This definition may be better for measuring how accurate EDOs are without stretched or compressed octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos."
-Similarly, we can look at pure-tritave EDTs, etc.
 ==== Zeta integral edos ====
@@ Line 635: / Line 633: @@
 Multiplying the Z-function by this factor of adjustment gives a Z-function with the prime ''p'' removed from consideration. Zeta peak and zeta integral tunings may then be found as before.
-Removing 2 leads to increasing adjusted peak values corresponding to the division of 3 (the "tritave") into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 45, 52, 56, 71, 75, 88, 131, 245, 316,}} … parts. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen-Pierce]] division of the tritave, but the multiples 26, 39, and 52 also.
+Removing 2 leads to increasing adjusted peak values corresponding to EDTs (division of the [[3/1|{{ordinal|3}}]] harmonic, or "tritave") into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 45, 52, 56, 71, 75, 88, 131, 245, 316,}} … parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] EDOs. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen-Pierce]] division of the tritave, but the multiples 26, 39, and 52 also.
 === Black magic formulas ===