Metallic MOS: Difference between revisions
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The simplest ''metallic generator'' splits the period into two segments which are related by phi. Wilson named this generator “Fibonacci”, after the famous recurrence relation associated with phi, but for consistency here we’ll be calling this the ''golden generator''. | The simplest ''metallic generator'' splits the period into two segments which are related by phi. Wilson named this generator “Fibonacci”, after the famous recurrence relation associated with phi, but for consistency here we’ll be calling this the ''golden generator''. | ||
[[File:Golden generator.png| | [[File:Golden generator.png|618x618px]] | ||
=== Noble cases === | === Noble cases === | ||
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Another way to think about the period is the interval from 0/1 to 1/1. We can find slightly more complex metallic generators by choosing an interval other than the entire period to split into two segments related by phi. For example, we could pick 1/3 to 1/2, giving us approximately 0.419821: | Another way to think about the period is the interval from 0/1 to 1/1. We can find slightly more complex metallic generators by choosing an interval other than the entire period to split into two segments related by phi. For example, we could pick 1/3 to 1/2, giving us approximately 0.419821: | ||
[[File:Noble generator.png| | [[File:Noble generator.png|618x618px]] | ||
Disclaimer: while these two segments are indeed related by phi, it is not simply by their lengths, as it may appear in the diagram. For now, let it suffice to say that extensions of the golden mean such as this are known as [http://mathworld.wolfram.com/NobleNumber.html noble numbers]. | Disclaimer: while these two segments are indeed related by phi, it is not simply by their lengths, as it may appear in the diagram. For now, let it suffice to say that extensions of the golden mean such as this are known as [http://mathworld.wolfram.com/NobleNumber.html noble numbers]. | ||
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The ''silver generator'' splits the entire period into two segments related by the silver mean, and is approximately equal to 0.292894: | The ''silver generator'' splits the entire period into two segments related by the silver mean, and is approximately equal to 0.292894: | ||
[[File:Silver generator.png| | [[File:Silver generator.png|618x618px]] | ||
And from the bronze mean, | And from the bronze mean, | ||
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we find the ''bronze generator'', approximately 0.232408: | we find the ''bronze generator'', approximately 0.232408: | ||
[[File:Bronze generator.png| | [[File:Bronze generator.png|618x618px]] | ||
=== Isotopic cases === | === Isotopic cases === | ||
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For metallic means beyond golden, a new category of generator becomes available. | For metallic means beyond golden, a new category of generator becomes available. | ||
Values of the | Values of the [[Https://en.wikipedia.org/wiki/Arithmetic progression|arithmetic progression]] from the metallic mean downwards by 1 also impart metallic effects when used to split the period to find a generator. The simplest example uses the silver mean minus one, | ||
<math>\qquad δ_s - 1 ≈ 1.414214 | <math>\qquad δ_s - 1 ≈ 1.414214 | ||
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finding a generator which is approximately 0.414214: | finding a generator which is approximately 0.414214: | ||
[[File:Isotopic generator.png| | [[File:Isotopic generator.png|618x618px]] | ||
We’ll be returning to these values regularly, so for convenience, we’ll refer to them as ''isotopes'' of their respective metallic mean, e.g. δ<sub>s</sub> - 1 is the first isotope of the silver mean (and we’ll make no claim as to the scientific appropriateness of this analogy). | We’ll be returning to these values regularly, so for convenience, we’ll refer to them as ''isotopes'' of their respective metallic mean, e.g. δ<sub>s</sub> - 1 is the first isotope of the silver mean (and we’ll make no claim as to the scientific appropriateness of this analogy). | ||
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However enticingly simple this definition may be, it unfortunately does not work in general. In fact, it ''only'' gives the correct value when the interval being split is the entire period. | However enticingly simple this definition may be, it unfortunately does not work in general. In fact, it ''only'' gives the correct value when the interval being split is the entire period. | ||
The correct general definition of a metallic generator is actually a | The correct general definition of a metallic generator is actually a [[Https://en.m.wikipedia.org/wiki/Mediant (mathematics)|mediant]] of the two ratios which bound the interval. | ||
But even then it’s not quite that simple, because it’s not a ''simple'' mediant, which would look like this: | But even then it’s not quite that simple, because it’s not a ''simple'' mediant, which would look like this: | ||
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[ math ] (num1 + num2) / (den1 + den2) | [ math ] (num1 + num2) / (den1 + den2) | ||
Rather, it’s a | Rather, it’s a [https://www.mathpages.com/home/kmath055/kmath055.htm ''weighted'' mediant], which looks like this: | ||
[ math ] (num1 * weight_num + num2 * weight_den) / (den1 * weight_num + den2 * weight_den) | [ math ] (num1 * weight_num + num2 * weight_den) / (den1 * weight_num + den2 * weight_den) | ||
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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 | Acoustic phi features prominently in this scale, giving rise to recursive stacks of [[Https://en.wikipedia.org/wiki/Combination tone|combination tones]]. | ||
''' | '''[http://tonalsoft.com/enc/b/brinko.aspx?fbclid=IwAR0Z5F4dXsUNo63TL1ukklQjIQQScDS2-NT61IJcqlXqcREgnKlcUl-pQ_4 Brinko]''' | ||
Also see | Also see [http://tonalsoft.com/enc/m/mars-pyramid.aspx?fbclid=IwAR3FAchzsbteCt4A1qbVS3aGrdgjeFf5YMbn7RrITTb0dM_DI3VqSpxMJSc this]. | ||
''' | '''[https://soundcloud.com/cmloegcmluin/metallic-harmonic-series-first-four-octaves Metallic Harmonic Series]''' | ||
''' | '''[http://dkeenan.com/Music/NobleMediant.txt?fbclid=IwAR1kgaKREuE1eULDyAfWrVjntO1eGzmdYkIjGZvlycM5uLni_UETdF2wuX0 The Noble Mediant: Complex ratios and metastable musical intervals]''' | ||
== other met-MOS == | == other met-MOS == | ||
'''Golden Ratio on the Xen Wiki | '''[[Golden Ratio|Golden Ratio on the Xen Wiki]]''' | ||
Contains a bunch of interesting links. | Contains a bunch of interesting links. | ||
''' | '''[http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm Elven Minstrel]''' | ||
A fun and helpful take on noble MOS scales. | A fun and helpful take on noble MOS scales. | ||
''' | '''[[Logarithmic Approximants]]''' | ||
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. | ||