The Riemann zeta function and tuning: Difference between revisions
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Let's take a breather and see what we've got. | Let's take a breather and see what we've got. | ||
=== Interpretation of results: " | === Interpretation of results: "Cosine relative error" === | ||
For every strictly positive rational n/d, there is a cosine with period 2π log<sub>2</sub>(n/d). This cosine peaks at {{nowrap|x {{=}} N/log<sub>2</sub>(n/d)}} for all integer N, or in other words, the Nth-equal division of the rational number n/d, and hits troughs midway between. | For every strictly positive rational n/d, there is a cosine with period 2π log<sub>2</sub>(n/d). This cosine peaks at {{nowrap|x {{=}} N/log<sub>2</sub>(n/d)}} for all integer N, or in other words, the Nth-equal division of the rational number n/d, and hits troughs midway between. | ||
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== Zeta edo lists == | == Zeta edo lists == | ||
=== Record edos === | === Record edos === | ||
The prime-approximating strength of an edo can be determined by the magnitude of Z(x). Since a higher |Z(x)| correlates to a stronger tuning, we would like to find a sequence with succesively larger |Z(x)|-associated values satisfying some property. | The prime-approximating strength of an edo can be determined by the magnitude of Z(x). Since a higher |Z(x)| correlates to a stronger tuning, we would like to find a sequence with succesively larger |Z(x)|-associated values satisfying some property. | ||
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If we want to find the second-best edos ranked by zeta peaks, then given a full list of zeta peaks, we can remove the successively higher peaks to get another sequence of succesively higher peaks, which correspond to edos called '''Parker edos'''. | If we want to find the second-best edos ranked by zeta peaks, then given a full list of zeta peaks, we can remove the successively higher peaks to get another sequence of succesively higher peaks, which correspond to edos called '''Parker edos'''. | ||
==== Parker edos ==== | |||
Non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak edo. Named after the Parker square in mathematics. A helpful list for finding an alternative to any given zeta peak edo of similar size and still-okay accuracy, but with different regular temperament properties (e.g. 9 as alternative to 10, 17 as alternative to 19). | Non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak edo. Named after the Parker square in mathematics. A helpful list for finding an alternative to any given zeta peak edo of similar size and still-okay accuracy, but with different regular temperament properties (e.g. 9 as alternative to 10, 17 as alternative to 19). | ||
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We can then remove those secondary peaks again to get '''tertiary-peak edos'''. | We can then remove those secondary peaks again to get '''tertiary-peak edos'''. | ||
==== Tertiary-peak edos ==== | |||
Non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak ''or'' Parker edo. | Non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak ''or'' Parker edo. | ||
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=== Non-record edos === | === Non-record edos === | ||
{{Idiosyncratic terms|the names for the different types of non-record edos. Proposed by [[Budjarn Lambeth]]}} | {{Idiosyncratic terms|the names for the different types of non-record edos. Proposed by [[Budjarn Lambeth]]}} | ||
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. | 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 ==== | |||
We may define ''local zeta'' edos as those that do not necessarily have successively higher zeta peaks but simply have a higher zeta peak than the edos on either side of them. This is 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). | |||
{{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, 80, 82, 84, 87, 89, 91, 94, 96, 99}}… | {{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, 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. Helpful for finding edos that force the use of methods other than traditional concordant harmony, or for composers seeking a challenge to inspire creativity. | Edos with a lower zeta peak than the edos on either side of them. Helpful 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}}… | {{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}}… | ||
==== Indecisive edos ==== | |||
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 many to promote indecision, but not so few as to promote frustration. | 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 many to promote indecision, but not so few as to promote frustration. | ||