The Riemann zeta function and tuning: Difference between revisions
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=== ''k''-ary peak edos === | === ''k''-ary peak edos === | ||
{{Idiosyncratic terms|the term "k-ary peak edos" itself, as well as the names for the different types of k-ary peak edos. Proposed by [[Akselai]] and [[Budjarn Lambeth]].}} | |||
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''' | '''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 '''Grothendieck edos'''. | We can then remove those secondary peaks again to get '''Grothendieck edos'''. | ||
'''Grothendieck edos''' | '''Grothendieck edos''' | ||
Non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak ''or'' Parker edo. Named after the "Grothendieck prime" (the number 57), another reference to the "almost" nature of these edos. | Non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak ''or'' Parker edo. Named after the "Grothendieck prime" (the number 57), another reference to the "almost" nature of these edos. | ||
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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]]}} | |||
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''' | '''Local zeta edos''' | ||
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). | 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). | ||
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'''Anti-zeta edos''' | '''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. | ||
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'''Indecisive edos''' | '''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. | ||