The Riemann zeta function and tuning: Difference between revisions

Line 300:

We may define the ''strict zeta edos'' to be the edos that are in all four of the above lists. The list of strict zeta edos begins {{EDOs|2, 5, 7, 12, 19, 31, 53, 270, 1395, 1578, 8539, 14348, 58973}}... .

=== ~~Local property~~ edos ===

=== Non-record edos ===

The following ~~list~~ of edos are not determined by successively large measured values, they are edos that ~~purely~~ satisfy some ~~local~~ property relating to zeta peaks instead.

The following lists of edos are not determined by successively large measured values, they are edos that satisfy some other property relating to zeta peaks instead.

'''Local zeta edos'''{{idiosyncratic}}

Edos with a higher zeta peak than the edos on either side of them. A helpful list for finding edos that approximate primes well in size ranges that lack any record-holding zeta edos (e.g. between 60 and 70 tones).

~~==== Local zeta edos ====~~

~~Edos with a higher zeta peak than the edos on either side of them:~~

{{EDOs|5, 7, 10, 12, 15, 17, 19, 22, 24, 27, 29, 31, 34, 36, 38, 41, 43, 46, 48, 50, 53, 56, 58, 60, 63, 65, 68, 72, 75, 77, 79, 80, 82, 84, 87, 89, 91, 94, 96, 99}}…

~~====~~ Anti-zeta edos ~~====~~

Edos with a lower zeta peak than the edos on either side of them:

'''Anti-zeta edos'''{{idiosyncratic}}

Edos with a lower zeta peak than the edos on either side of them. A helpful list for finding edos that force the use of methods other than traditional concordant harmony, or for composers seeking a challenge to inspire creativity.

{{EDOs|6, 8, 11, 13, 16, 18, 20, 23, 25, 28, 30, 33, 35, 37, 40, 42, 44, 47, 49, 52, 54, 57, 59, 61, 64, 66, 69, 71, 73, 76, 78, 81, 83, 86, 88, 90, 92, 95, 97}}…

~~==== Sloped-zeta~~ edos ~~====~~

Edos which are neither local zeta edos, nor anti-zeta edos:

'''Indecisive edos'''{{idiosyncratic}}

Edos which are neither local zeta edos, nor anti-zeta edos. Helpful for finding edos that are more restrictive than local zeta edos, but not as far off the deep end as anti-zeta edos. They might narrow down the range of compositional choices available so as to be not so much to promote indecision, but not so few as to promote frustration.

{{EDOs|1, 2, 3, 4, 9, 14, 21, 26, 32, 39, 45, 51, 54, 62, 67, 70, 74, 85, 93, 98}}…

'''Parker edos'''{{idiosyncratic}}

Those non-Zeta-peak edos with a higher Zeta peak than any other non-Zeta-peak edo so far. Named after the Parker square in mathematics. A helpful list for finding an alternative to any given Zeta peak edo of similar size and accuracy but with different regular temperament properties (e.g. 9 as alternative to 10, 17 as alternative to 19).

{{EDOs|6, 8, 9, 14, 15, 17, 24, 34, 46, 58, 65, 77, 87}}…

== Optimal octave stretch ==

@@ Line 300: / Line 300: @@
 We may define the ''strict zeta edos'' to be the edos that are in all four of the above lists. The list of strict zeta edos begins {{EDOs|2, 5, 7, 12, 19, 31, 53, 270, 1395, 1578, 8539, 14348, 58973}}... .
-=== Local property edos ===
+=== Non-record edos ===
-The following list of edos are not determined by successively large measured values, they are edos that purely satisfy some local property relating to zeta peaks instead.
+The following lists of edos are not determined by successively large measured values, they are edos that satisfy some other property relating to zeta peaks instead.
+'''Local zeta edos'''{{idiosyncratic}}
+Edos with a higher zeta peak than the edos on either side of them. A helpful list for finding edos that approximate primes well in size ranges that lack any record-holding zeta edos (e.g. between 60 and 70 tones).
-==== Local zeta edos ====
-Edos with a higher zeta peak than the edos on either side of them:
 {{EDOs|5, 7, 10, 12, 15, 17, 19, 22, 24, 27, 29, 31, 34, 36, 38, 41, 43, 46, 48, 50, 53, 56, 58, 60, 63, 65, 68, 72, 75, 77, 79, 80, 82, 84, 87, 89, 91, 94, 96, 99}}…
-==== Anti-zeta edos ====
-Edos with a lower zeta peak than the edos on either side of them:
+'''Anti-zeta edos'''{{idiosyncratic}}
+Edos with a lower zeta peak than the edos on either side of them. A helpful list for finding edos that force the use of methods other than traditional concordant harmony, or for composers seeking a challenge to inspire creativity.
 {{EDOs|6, 8, 11, 13, 16, 18, 20, 23, 25, 28, 30, 33, 35, 37, 40, 42, 44, 47, 49, 52, 54, 57, 59, 61, 64, 66, 69, 71, 73, 76, 78, 81, 83, 86, 88, 90, 92, 95, 97}}…
-==== Sloped-zeta edos ====
-Edos which are neither local zeta edos, nor anti-zeta edos:
+'''Indecisive edos'''{{idiosyncratic}}
+Edos which are neither local zeta edos, nor anti-zeta edos. Helpful for finding edos that are more restrictive than local zeta edos, but not as far off the deep end as anti-zeta edos. They might narrow down the range of compositional choices available so as to be not so much to promote indecision, but not so few as to promote frustration.
 {{EDOs|1, 2, 3, 4, 9, 14, 21, 26, 32, 39, 45, 51, 54, 62, 67, 70, 74, 85, 93, 98}}…
+'''Parker edos'''{{idiosyncratic}}
+Those non-Zeta-peak edos with a higher Zeta peak than any other non-Zeta-peak edo so far. Named after the Parker square in mathematics. A helpful list for finding an alternative to any given Zeta peak edo of similar size and accuracy but with different regular temperament properties (e.g. 9 as alternative to 10, 17 as alternative to 19).
+{{EDOs|6, 8, 9, 14, 15, 17, 24, 34, 46, 58, 65, 77, 87}}…
 == Optimal octave stretch ==