Metallic MOS: Difference between revisions
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=== As interval finder === | === As interval finder === | ||
We’ve left still another part about our instructions for finding metallic generators vague: the “an interval” part. It turns out that not just ''any'' interval can yield a metallic generator — only those whose bounding ratios are sequential convergents (or semiconvergents) of a continued fraction | We’ve left still another part about our instructions for finding metallic generators vague: the “an interval” part. It turns out that not just ''any'' interval can yield a metallic generator — only those whose bounding ratios are sequential convergents (or semiconvergents) of a [[wikipedia:Continued_fraction|continued fraction]]. | ||
Now that’s a mouthful, to be sure — but we don’t need to understand what that entails at this point, because fortunately for us, an algorithm called the Stern-Brocot tree | Now that’s a mouthful, to be sure — but we don’t need to understand what that entails at this point, because fortunately for us, an algorithm called the [[wikipedia:Stern–Brocot_tree|Stern-Brocot tree]] graphs all of these intervals for us: | ||
[[File:1200px-SternBrocotTree.svg.png|618x618px]] | |||
As we can see, Stern-Brocot tree actually covers ratios greater than 1, but for met-MOS purposes, we only need to consider the left half of it (in fact, we only need to consider the left quarter of it, since generators greater than 1/2 are complements of those less than 1/2 and generate the same scales; but more on that later). | As we can see, Stern-Brocot tree actually covers ratios greater than 1, but for met-MOS purposes, we only need to consider the left half of it (in fact, we only need to consider the left quarter of it, since generators greater than 1/2 are complements of those less than 1/2 and generate the same scales; but more on that later). | ||
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So if we want a golden scale, and we also happen to want a generator near 0.276393, then we’re in luck. But if we want a golden generator that is close to 0.275267, 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 0.276393, then we’re in luck. But if we want a golden generator that is close to 0.275267, we may be disappointed to hear that it is not “golden” enough for us. | ||
Wilson | 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?fbclid=IwAR2PahVuZJ18faXQA_IggdD52y9PWP4uyEeQALE8Q73MhIlploPYDinbAAk 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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== Golden Meantone == | == Golden Meantone == | ||
The thinking behind | The thinking behind [[Golden Meantone]] is to put the whole step and half step into the ratio of phi with each other. Most discussion of Golden Meantone assumes a twelve-note scale that spans an octave. | ||
Abstractly speaking, Golden Meantone’s generator is a noble generator weighted by phi from 1/3 toward 1/2, ≈ 0.419821, 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 phi from 1/3 toward 1/2, ≈ 0.419821, which by design was the one chosen for all noble generator examples in this discussion. | ||
Wilson called this scale | Wilson called this scale [http://anaphoria.com/kornerup.pdf?fbclid=IwAR3nAiMo-W51E5IT5vXxmMFBIsSocPu6tAr4ETbQF-ITrxvoakAR11DpM_4 Kornerup], after Thorvald Kornerup, who was an early explorer of golden scales. | ||
== Wilson/Pepper Fifth Tuning == | == Wilson/Pepper Fifth Tuning == | ||
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|Fibonacci | |Fibonacci | ||
|0.381966 | |0.381966 | ||
|cardinality sequence is the Fibonacci numbers | |cardinality sequence is the [[wikipedia:Fibonacci_number|Fibonacci numbers]] | ||
|Golden Father | |Golden Father | ||
|- | |- | ||
|Lucas | |Lucas | ||
|0.276393 | |0.276393 | ||
|cardinality sequence is the Lucas numbers | |cardinality sequence is the [[wikipedia:Lucas_number|Lucas numbers]] | ||
|<nowiki>-</nowiki> | |<nowiki>-</nowiki> | ||
|- | |- | ||
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This article covers some fascinating ideas, and toward the end touches upon Golden Meantone and Argent Temperament, with thought-provoking visualizations. | This article covers some fascinating ideas, and toward the end touches upon Golden Meantone and Argent Temperament, with thought-provoking visualizations. | ||
''' | '''[http://www.anaphoria.com/meruthree.pdf another Wilson document]''' | ||
This document includes 2-Zig/2-Zag. | This document includes 2-Zig/2-Zag. | ||
''' | '''[https://sevish.com/2017/golden-ratio-music-interval/ Sevish’s phi scale]''' | ||
Synthesizes acoustic and logarithmic phi. | Synthesizes acoustic and logarithmic phi. | ||
''' | '''[https://musical-patterns.douglasblumeyer.com Musical Patterns] - MetMOS''' | ||
Author’s site. Contains a rudimentary interactive MetMOS generator. | Author’s site. Contains a rudimentary interactive MetMOS generator. | ||
== Other MOS == | == Other MOS == | ||
''' | '''[https://untwelve.org/static/javascript_demos/MOSring.html MOS generator]''' | ||
Helpful tool for generating horograms and quickly finding cardinality sequences. | Helpful tool for generating horograms and quickly finding cardinality sequences. | ||
== Continued fractions == | == Continued fractions == | ||
''' | '''[http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html Discussion about Continued fractions]''' | ||
Very detailed. | Very detailed. | ||
''' | '''[http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfCALC.html Continued Fraction Calculator]''' | ||
An indispensable tool for calculating continued fractions, from the same source. | An indispensable tool for calculating continued fractions, from the same source. | ||
== Stern-Brocot tree == | == Stern-Brocot tree == | ||
'''Farey Sequence | '''[[wikipedia:Farey_sequence|Farey Sequence]]''' | ||
Another way of slicing and dicing the texture of rationals. | Another way of slicing and dicing the texture of rationals. | ||
== 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. | 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?fbclid=IwAR2T2MW82SRxixVcHwe0R288FhhAhzixrenoPALzgaGWrEO1Kp-Fv5tCQyA#section1.1 rabbit sequences]''' | ||
Follow the same pattern as the horogram for the golden generator. | 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. | Some fun and informative videos on metallic means. | ||