User:Contribution/Collection of tunings: Difference between revisions
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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. | 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. | 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. | ||
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 | 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 | ||
From the no-2 Riemann zeta: 39edt, 56edt, 69edt, 71edt, 75edt, 78edt, 82edt, 88edt, 99edt, 101edt, 105edt, 110edt, 131edt, 140edt, 144edt, 153edt, 170edt, 183edt, 185edt, 202edt, 209edt, 213edt, 215edt, 219edt, 245edt | From the no-2 Riemann zeta: 39edt, 56edt, 69edt, 71edt, 75edt, 78edt, 82edt, 88edt, 99edt, 101edt, 105edt, 110edt, 131edt, 140edt, 144edt, 153edt, 170edt, 183edt, 185edt, 202edt, 209edt, 213edt, 215edt, 219edt, 245edt | ||
From the no-3 Riemann zeta: (13edo), 16edo, 21edo, 25edo, (28edo), 35edo, 37edo, 43edo, 47edo, (52edo), 56edo, (66edo), 74edo, 78edo, 93edo, 109edo, 124edo | |||
From the no-2 no-3 Riemann zeta: | From the no-2 no-3 Riemann zeta: | ||
From the alpha-beta-gamma set: 5ed2/1, 7ed2/1, 12ed2/1, 7ed5/3, 9ed5/3, 16ed5/3, 9ed3/2, 11ed3/2, 20ed3/2, 11ed7/5, 13ed7/5, 24ed7/5, 13ed4/3, 15ed4/3, 28ed4/3 | |||
From the alpha-beta-gamma set: 7ed5/3, 9ed5/3, 16ed5/3, 9ed3/2, 11ed3/2, 20ed3/2, 11ed7/5, 13ed7/5, 24ed7/5, 13ed4/3, 15ed4/3, 28ed4/3 | |||
=== Equal divisions of a ratio & optimization === | === Equal divisions of a ratio & optimization === | ||