User:Contribution/Collection of tunings: Difference between revisions

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== Equal-step tunings ==

{{todo|Add other EDRs|inline=1|comment=add more EDOs from the no-3 Riemann zeta, <s>the no-5 Riemann zeta, the no-3 no-5 Riemann zeta,</s> add more EDTs from the no-2 Riemann zeta, <s>the no-2 no-5 Riemann zeta,</s> add more ED5 from the no-2 no-3 Riemann zeta, <s>add more ED7 from the no-2 no-3 no-5 Riemann zeta</s> (327ed7 is enough for ED7s). Add ED4s, <s>add ED6s (detect those with a bad 2 and a bad 3, but with good results on other primes), <s>add ED8s.</s>}}

=== About this list ===

The table that follows is '''not a “best-of” roster but a modest snapshot of equal-step tunings that happen to score highly under a few specific mathematical lenses'''. In particular, it gathers:

* '''Prominent peak counts from the classic Riemann zeta function'''

* '''Prominent peaks after removing the prime 2 from the zeta product'''

* '''Prominent peaks after removing the prime 3'''

* '''Prominent peaks after simultaneously removing the primes 2 and 3'''

* '''The α–β–γ family, with an equave sliding from 2/1 down to 4/3'''

These tunings earn the label “optimized” only relative to the limited set of zeta-derived functions explored here. In practice, you can make convincing music with ''any'' equal-step interval, every real-number step size repeated ad infinitum forms its own viable lattice. When you layer many differently pruned zeta functions in a tool such as Wolfram Mathematica, striking peaks emerge almost everywhere; the peaks simply shift as each combination of omitted primes reshapes the landscape. That ubiquity means there is no absolute “good” or “bad” equal-step tuning, only different alignments of primes that reveal different musical affordances.

Consequently, the list below is inherently '''biased toward a handful of functions''' and can only hint at the boundless diversity of xenharmonic equal-step systems. Treat it as a useful starting palette, not a definitive canon.{{todo|Add other EDRs|inline=1|comment=add more EDOs from the no-3 Riemann zeta, <s>the no-5 Riemann zeta, the no-3 no-5 Riemann zeta,</s> add more EDTs from the no-2 Riemann zeta, <s>the no-2 no-5 Riemann zeta,</s> add more ED5 from the no-2 no-3 Riemann zeta, <s>add more ED7 from the no-2 no-3 no-5 Riemann zeta</s> (327ed7 is enough for ED7s). Add ED4s, <s>add ED6s (detect those with a bad 2 and a bad 3, but with good results on other primes), <s>add ED8s.</s>}}

From the original Riemann zeta: 12edo, 14edo, 15edo, 17edo, 19edo, 22edo, 24edo, 26edo, 27edo, 29edo, 31edo, 34edo, 36edo, 41edo, 46edo, 50edo, 53edo, 58edo, 60edo, 65edo, 68edo, 72edo, 77edo, 80edo, 84edo, 87edo, 94edo, 99edo, 103edo, 111edo, 118edo, 130edo, 140edo, 152edo, 171edo, 270edo, 311edo

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 == Equal-step tunings ==
-{{todo|Add other EDRs|inline=1|comment=add more EDOs from the no-3 Riemann zeta, <s>the no-5 Riemann zeta, the no-3 no-5 Riemann zeta,</s> add more EDTs from the no-2 Riemann zeta, <s>the no-2 no-5 Riemann zeta,</s> add more ED5 from the no-2 no-3 Riemann zeta, <s>add more ED7 from the no-2 no-3 no-5 Riemann zeta</s> (327ed7 is enough for ED7s). Add ED4s, <s>add ED6s (detect those with a bad 2 and a bad 3, but with good results on other primes), <s>add ED8s.</s>}}
+=== About this list ===
+The table that follows is '''not a “best-of” roster but a modest snapshot of equal-step tunings that happen to score highly under a few specific mathematical lenses'''. In particular, it gathers:
+* '''Prominent peak counts from the classic Riemann zeta function'''
+* '''Prominent peaks after removing the prime 2 from the zeta product'''
+* '''Prominent peaks after removing the prime 3'''
+* '''Prominent peaks after simultaneously removing the primes 2 and 3'''
+* '''The α–β–γ family, with an equave sliding from 2/1 down to 4/3'''
+These tunings earn the label “optimized” only relative to the limited set of zeta-derived functions explored here. In practice, you can make convincing music with ''any'' equal-step interval, every real-number step size repeated ad infinitum forms its own viable lattice. When you layer many differently pruned zeta functions in a tool such as Wolfram Mathematica, striking peaks emerge almost everywhere; the peaks simply shift as each combination of omitted primes reshapes the landscape. That ubiquity means there is no absolute “good” or “bad” equal-step tuning, only different alignments of primes that reveal different musical affordances.
+Consequently, the list below is inherently '''biased toward a handful of functions''' and can only hint at the boundless diversity of xenharmonic equal-step systems. Treat it as a useful starting palette, not a definitive canon.{{todo|Add other EDRs|inline=1|comment=add more EDOs from the no-3 Riemann zeta, <s>the no-5 Riemann zeta, the no-3 no-5 Riemann zeta,</s> add more EDTs from the no-2 Riemann zeta, <s>the no-2 no-5 Riemann zeta,</s> add more ED5 from the no-2 no-3 Riemann zeta, <s>add more ED7 from the no-2 no-3 no-5 Riemann zeta</s> (327ed7 is enough for ED7s). Add ED4s, <s>add ED6s (detect those with a bad 2 and a bad 3, but with good results on other primes), <s>add ED8s.</s>}}
 From the original Riemann zeta: 12edo, 14edo, 15edo, 17edo, 19edo, 22edo, 24edo, 26edo, 27edo, 29edo, 31edo, 34edo, 36edo, 41edo, 46edo, 50edo, 53edo, 58edo, 60edo, 65edo, 68edo, 72edo, 77edo, 80edo, 84edo, 87edo, 94edo, 99edo, 103edo, 111edo, 118edo, 130edo, 140edo, 152edo, 171edo, 270edo, 311edo