Metallic MOS: Difference between revisions
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So if we want a golden scale, and we also happen to want a generator near <span><math>0.276393</math></span>, then we’re in luck. But if we want a golden generator that is close to <span><math>0.275267</math></span>, we may be disappointed to hear that it is not “golden” enough for us. | So if we want a golden scale, and we also happen to want a generator near <span><math>0.276393</math></span>, then we’re in luck. But if we want a golden generator that is close to <span><math>0.275267</math></span>, we may be disappointed to hear that it is not “golden” enough for us. | ||
Wilson [http://www.anaphoria.com/hrgm.PDF documented noble scale sequences through the sixth level of the Stern-Brocot tree] (or as he called it, the “scale tree” or “Peirce Series”), totalling 32 noble generators. He also [http://anaphoria.com/sctree.pdf | Wilson [http://www.anaphoria.com/hrgm.PDF documented noble scale sequences through the sixth level of the Stern-Brocot tree] (or as he called it, the “scale tree” or “Peirce Series”), totalling 32 noble generators. He also [http://anaphoria.com/sctree.pdf recorded just the generator values down to the eleventh level] for a total of 1024 generators. Exploring generators beyond that was probably just not worth it, because their metallicity levels are too low. We will cut ourselves off at the seventh level in our scale trees, as we depict generators for the silver and bronze means, and their isotopes too. | ||
=== Naming generators === | === Naming generators === | ||
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Abstractly speaking, Golden Meantone’s generator is a noble generator weighted by <span><math>φ</math></span> from <span><math>\frac 13</math></span> toward <span><math>\frac 12</math></span>, <span><math>≈ 0.419821</math></span>, which by design was the one chosen for all noble generator examples in this discussion. | Abstractly speaking, Golden Meantone’s generator is a noble generator weighted by <span><math>φ</math></span> from <span><math>\frac 13</math></span> toward <span><math>\frac 12</math></span>, <span><math>≈ 0.419821</math></span>, which by design was the one chosen for all noble generator examples in this discussion. | ||
Wilson called this scale [http://anaphoria.com/kornerup.pdf | Wilson called this scale [http://anaphoria.com/kornerup.pdf Kornerup], after Thorvald Kornerup, who was an early explorer of golden scales. | ||
== Wilson/Pepper Fifth Tuning == | == Wilson/Pepper Fifth Tuning == | ||
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== Other metallic xenharmonic but not met-MOS == | == Other metallic xenharmonic but not met-MOS == | ||
* '''[http://www.huygens-fokker.org/bpsite/833cent.html Bohlen's 833 cent scale]:''' Acoustic phi features prominently in this scale, giving rise to recursive stacks of [[wikipedia:Combination_tone|combination tones]]. | * '''[http://www.huygens-fokker.org/bpsite/833cent.html Bohlen's 833 cent scale]:''' Acoustic phi features prominently in this scale, giving rise to recursive stacks of [[wikipedia:Combination_tone|combination tones]]. | ||
* '''[http://tonalsoft.com/enc/b/brinko.aspx | * '''[http://tonalsoft.com/enc/b/brinko.aspx Brinko]:''' Also see [http://tonalsoft.com/enc/m/mars-pyramid.aspx this]. | ||
* '''[https://soundcloud.com/cmloegcmluin/metallic-harmonic-series-first-four-octaves Metallic Harmonic Series]''' | * '''[https://soundcloud.com/cmloegcmluin/metallic-harmonic-series-first-four-octaves Metallic Harmonic Series]''' | ||
* '''[http://dkeenan.com/Music/NobleMediant.txt | * '''[http://dkeenan.com/Music/NobleMediant.txt The Noble Mediant: Complex ratios and metastable musical intervals]''' | ||
== Other met-MOS == | == Other met-MOS == | ||
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== Metallic means == | == Metallic means == | ||
* '''[http://www.mi.sanu.ac.rs/vismath/spinadel/ The Family of Metallic Means]:''' Generalizes patterns of the metallic means even beyond the ones used in this discussion. | * '''[http://www.mi.sanu.ac.rs/vismath/spinadel/ The Family of Metallic Means]:''' Generalizes patterns of the metallic means even beyond the ones used in this discussion. | ||
* '''[http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibrab.html | * '''[http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibrab.html rabbit sequences]:''' Follow the same pattern as the horogram for the golden generator. | ||
* '''[https://m.youtube.com/playlist?list=PLt5AfwLFPxWKMXtxxL5qm9AcarCzNJDM0 Numberphile’s playlist of videos related to the Golden Ratio]:''' Some fun and informative videos on metallic means. | * '''[https://m.youtube.com/playlist?list=PLt5AfwLFPxWKMXtxxL5qm9AcarCzNJDM0 Numberphile’s playlist of videos related to the Golden Ratio]:''' Some fun and informative videos on metallic means. | ||