Metallic MOS: Difference between revisions
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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. <span><math>δ_s - 1</math></span> 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. <span><math>δ_s - 1</math></span> is the first isotope of the silver mean (and we’ll make no claim as to the scientific appropriateness of this analogy). | ||
Only isotopes greater than <span><math>1</math></span> find new generators; more on this later. | Only isotopes greater than <span><math>1</math></span> find new generators; more on this later. | ||
Isotopes theoretically could be formed by adding <span><math>1</math></span> repeatedly to each mean, instead of subtracting, but these also do not find new generators, and for simplicity we’ll not be considering these to be isotopes at all for our purposes here. | Isotopes theoretically could be formed by adding <span><math>1</math></span> repeatedly to each mean, instead of subtracting, but these also do not find new generators, and for simplicity we’ll not be considering these to be isotopes at all for our purposes here. | ||
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== <span><math>L{:}s</math></span> sequences == | ==<span><math>L{:}s</math></span> sequences== | ||
Each scale has exactly two step sizes: large and small, or <span><math>L</math></span> and <span><math>s</math></span>. We can refer to the ratio between these large and small steps as | Each scale has exactly two step sizes: large and small, or <span><math>L</math></span> and <span><math>s</math></span>. We can refer to the ratio between these large and small steps as | ||
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=== Beyond golden cases === | === Beyond golden cases === | ||
Instead of every scale’s <span><math>L{:}s</math></span> equaling the same value, as is the case for the golden mean, the silver mean’s <span><math>L{:}s</math></span> sequence alternates between its isotopes that are greater than 1: | Instead of every scale’s <span><math>L{:}s</math></span> equaling the same value, as is the case for the golden mean, the silver mean’s <span><math>L{:}s</math></span> sequence alternates between its isotopes that are greater than 1: | ||
<math> | <math> | ||
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</math> | </math> | ||
And the bronze mean’s <span><math>L{:}s</math></span> sequence cycles through its isotopes that are greater than 1: | And the bronze mean’s <span><math>L{:}s</math></span> sequence cycles through its isotopes that are greater than 1: | ||
<math> | <math> | ||
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=== Isotopic cases === | === Isotopic cases === | ||
Isotopic <span><math>L{:}s</math></span> sequences are just like those of their mean’s, but offset. | Isotopic <span><math>L{:}s</math></span> sequences are just like those of their mean’s, but offset. | ||
For example, the silver mean’s first isotope’s generator’s <span><math>L{:}s</math></span> sequence alternates between <span><math>L{:}s = δ_s</math></span> and <span><math>L{:}s = δ_s - 1</math></span>, just like the silver generator’s, however — unlike the silver generator’s — it begins with <span><math>L{:}s = δ_s - 1</math></span>. | For example, the silver mean’s first isotope’s generator’s <span><math>L{:}s</math></span> sequence alternates between <span><math>L{:}s = δ_s</math></span> and <span><math>L{:}s = δ_s - 1</math></span>, just like the silver generator’s, however — unlike the silver generator’s — it begins with <span><math>L{:}s = δ_s - 1</math></span>. | ||
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=== Noble cases === | === Noble cases === | ||
Noble scales at first do not — but eventually do — reach a point where they start exhibiting this interval pattern (paralleling how their <span><math>L{:}s</math></span> sequences only eventually exhibit <span><math>L{:}s = φ</math></span>). | Noble scales at first do not — but eventually do — reach a point where they start exhibiting this interval pattern (paralleling how their <span><math>L{:}s</math></span> sequences only eventually exhibit <span><math>L{:}s = φ</math></span>). | ||
Once we’ve iterated past the point that our scale exhibits <span><math>L{:}s = φ</math></span>, some of the smaller intervals will begin to be related by <span><math>φ</math></span>, but its larger intervals will never be related by <span><math>φ</math></span>. | Once we’ve iterated past the point that our scale exhibits <span><math>L{:}s = φ</math></span>, some of the smaller intervals will begin to be related by <span><math>φ</math></span>, but its larger intervals will never be related by <span><math>φ</math></span>. | ||
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</math> | </math> | ||
Bronze’s scales cycle through three different interval patterns related to its respective <span><math>>1</math></span> isotopes. | Bronze’s scales cycle through three different interval patterns related to its respective <span><math>>1</math></span> isotopes. | ||
This pattern continues for other metallic means. As another entry to our family of sequence terms (along with scale sequence and <span><math>L{:}s</math></span> sequence) we shall use the term ''interval pattern sequence'', and for the periodic part at the end, the ''interval pattern cycle''. | This pattern continues for other metallic means. As another entry to our family of sequence terms (along with scale sequence and <span><math>L{:}s</math></span> sequence) we shall use the term ''interval pattern sequence'', and for the periodic part at the end, the ''interval pattern cycle''. | ||
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=== Isotopic cases === | === Isotopic cases === | ||
As with <span><math>L{:}s</math></span> sequences, isotopic interval pattern sequences are identical to their metallic mean’s, cycling through a set of interval patterns from the beginning, except starting at a different position in that cycle. | As with <span><math>L{:}s</math></span> sequences, isotopic interval pattern sequences are identical to their metallic mean’s, cycling through a set of interval patterns from the beginning, except starting at a different position in that cycle. | ||
=== Aristocratic cases === | === Aristocratic cases === | ||
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</math> | </math> | ||
This alone would not suffice to explain how the <span><math>L{:}s</math></span> sequences lock into a cycle of isotopes. But here’s where the magic of the metallic means comes into play. <span><math>φ</math></span> has the property that | This alone would not suffice to explain how the <span><math>L{:}s</math></span> sequences lock into a cycle of isotopes. But here’s where the magic of the metallic means comes into play. <span><math>φ</math></span> has the property that | ||
<math>\qquad φ - 1 = 1 / φ | <math>\qquad φ - 1 = 1 / φ | ||
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Crossing nobles with beyond golden cases results in continued fractions which can start with anything but eventually settle on all 2’s, 3’s, or n if we base our noble on the n<sup>th</sup> metallic mean. For example, our earlier example 0.226541 is [0; 4, <span style="text-decoration:overline">2</span>] | Crossing nobles with beyond golden cases results in continued fractions which can start with anything but eventually settle on all 2’s, 3’s, or n if we base our noble on the n<sup>th</sup> metallic mean. For example, our earlier example 0.226541 is [0; 4, <span style="text-decoration:overline">2</span>] | ||
=== Application: <span><math>L{:}s</math></span> sequences === | === Application: <span><math>L{:}s</math></span> sequences=== | ||
Continued fractions can compute <span><math>L{:}s</math></span> sequences by repeatedly depleting the terms of the continued fraction for the generator. For example, we’ll look at the <span><math>L{:}s</math></span> sequence for g = [0; 2, 2, <span style="text-decoration:overline">1</span>] ≈ 0.419821. | Continued fractions can compute <span><math>L{:}s</math></span> sequences by repeatedly depleting the terms of the continued fraction for the generator. For example, we’ll look at the <span><math>L{:}s</math></span> sequence for g = [0; 2, 2, <span style="text-decoration:overline">1</span>] ≈ 0.419821. | ||
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=== Application: finding generator === | === Application: finding generator === | ||
To compute the <span><math>L{:}s</math></span> sequence, we depleted terms of the generator’s continued fraction. By doing the opposite — gradually building up the generator’s continued fraction by incrementing terms — we can determine the path our generator takes through the Stern-Brocot tree. | To compute the <span><math>L{:}s</math></span> sequence, we depleted terms of the generator’s continued fraction. By doing the opposite — gradually building up the generator’s continued fraction by incrementing terms — we can determine the path our generator takes through the Stern-Brocot tree. | ||
We’ll use the example of the golden generator, with continued fraction [0; 2, 1] : | We’ll use the example of the golden generator, with continued fraction [0; 2, 1] : | ||
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</math> | </math> | ||
Every other scale the silver generator generates has an <span><math>L{:}s</math></span> other than <span><math>δ_s</math></span>, namely, its isotope, <span><math>δ_s - 1</math></span>. These scales have a different pattern: | Every other scale the silver generator generates has an <span><math>L{:}s</math></span> other than <span><math>δ_s</math></span>, namely, its isotope, <span><math>δ_s - 1</math></span>. These scales have a different pattern: | ||
<math> L{:}s = (L+2s):(L+s) = δ_s - 1 | <math> L{:}s = (L+2s):(L+s) = δ_s - 1 | ||
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And now we’ll look not at the odd, but at the even iterations, rings 2, 4, 6, 8, etc. to see the pattern visualized. | And now we’ll look not at the odd, but at the even iterations, rings 2, 4, 6, 8, etc. to see the pattern visualized. | ||
There’s something a bit different about the interval pattern for <span><math>δ_s - 1</math></span> from the other two we’ve looked at so far. The interval patterns for <span><math>δ_s</math></span> and <span><math>φ</math></span> exhibited overlap, i.e. we saw something like | There’s something a bit different about the interval pattern for <span><math>δ_s - 1</math></span> from the other two we’ve looked at so far. The interval patterns for <span><math>δ_s</math></span> and <span><math>φ</math></span> exhibited overlap, i.e. we saw something like | ||
<math> a:b = b:c = c:d = … = δ_s | <math> a:b = b:c = c:d = … = δ_s | ||
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== Wilson/Pepper Fifth Tuning == | == Wilson/Pepper Fifth Tuning == | ||
The thinking behind this tuning is similar, except that the two steps in the ratio of <span><math>φ</math></span> with each other are the tone and the chromatic semitone. | The thinking behind this tuning is similar, except that the two steps in the ratio of <span><math>φ</math></span> with each other are the tone and the chromatic semitone. | ||
Abstractly speaking, Wilson/Pepper Fifth Tuning’s generator would be the noble generator weighted by <span><math>φ</math></span> from <span><math>\frac 37</math></span> to <span><math>\frac 25</math></span>, <span><math>≈ 0.413254</math></span>. | Abstractly speaking, Wilson/Pepper Fifth Tuning’s generator would be the noble generator weighted by <span><math>φ</math></span> from <span><math>\frac 37</math></span> to <span><math>\frac 25</math></span>, <span><math>≈ 0.413254</math></span>. | ||
== Argent Temperament == | == Argent Temperament == | ||
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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]''' | * '''[http://tonalsoft.com/enc/b/brinko.aspx?fbclid=IwAR0Z5F4dXsUNo63TL1ukklQjIQQScDS2-NT61IJcqlXqcREgnKlcUl-pQ_4 Brinko]:''' 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]''' | |||
Acoustic phi features prominently in this scale, giving rise to recursive stacks of [[wikipedia:Combination_tone|combination tones]]. | * '''[http://dkeenan.com/Music/NobleMediant.txt?fbclid=IwAR1kgaKREuE1eULDyAfWrVjntO1eGzmdYkIjGZvlycM5uLni_UETdF2wuX0 The Noble Mediant: Complex ratios and metastable musical intervals]''' | ||
'''[http://tonalsoft.com/enc/b/brinko.aspx?fbclid=IwAR0Z5F4dXsUNo63TL1ukklQjIQQScDS2-NT61IJcqlXqcREgnKlcUl-pQ_4 Brinko]''' | |||
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|Golden Ratio on the Xen Wiki]]:''' Contains a bunch of interesting links. | |||
'''[[Golden Ratio|Golden Ratio on the Xen Wiki]]''' | * '''[http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm Elven Minstrel]:''' A fun and helpful take on noble MOS scales. | ||
* '''[[Logarithmic approximants|Logarithmic Approximants]]:''' This article covers some fascinating ideas, and toward the end touches upon Golden Meantone and Argent Temperament, with thought-provoking visualizations. | |||
Contains a bunch of interesting links. | * '''[http://www.anaphoria.com/meruthree.pdf another Wilson document]:''' This document includes 2-Zig/2-Zag. | ||
* '''[https://sevish.com/2017/golden-ratio-music-interval/ Sevish’s phi scale]:''' Synthesizes acoustic and logarithmic phi. | |||
'''[http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm Elven Minstrel]''' | * '''[https://musical-patterns.douglasblumeyer.com Musical Patterns] - MetMOS:''' Author’s site. Contains a rudimentary interactive MetMOS generator. | ||
A fun and helpful take on noble MOS scales. | |||
'''[[Logarithmic approximants|Logarithmic Approximants]]''' | |||
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. | |||
'''[https://sevish.com/2017/golden-ratio-music-interval/ Sevish’s phi scale]''' | |||
Synthesizes acoustic and logarithmic phi. | |||
'''[https://musical-patterns.douglasblumeyer.com Musical Patterns] - MetMOS''' | |||
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. | |||
'''[https://untwelve.org/static/javascript_demos/MOSring.html MOS generator]''' | |||
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]''' | * '''[http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html Discussion about Continued fractions]:''' 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. | |||
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. | |||
== Stern-Brocot tree == | == Stern-Brocot tree == | ||
* '''[[wikipedia:Farey_sequence|Farey Sequence]]:''' Another way of slicing and dicing the texture of rationals. | |||
'''[[wikipedia:Farey_sequence|Farey Sequence]]''' | |||
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. | |||
'''[http://www.mi.sanu.ac.rs/vismath/spinadel/ The Family of Metallic Means]''' | * '''[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. | ||
* '''[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. | |||
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. | |||
'''[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. | |||
= Glossary = | = Glossary = | ||
* '''cycle, interval pattern:''' The periodic part at the end of an interval pattern sequence. | |||
'''cycle, interval pattern''' | * '''<span><math>L{:}s</math></span>''' '''cycle, '''The periodic part at the end of an sequence, cycling through the isotopes of the given metallic mean. | ||
<span><math>L{:}s</math></span>* '''generator, aristocratic:''' A generator which is found using the weighted mediant formula on a non-period Stern-Brocot tree interval with a beyond golden mean. | |||
The periodic part at the end of an interval pattern sequence. | * '''generator, bronze:''' The generator found using the weighted mediant formula on the period interval with the golden mean, equal to . | ||
<span><math>[0; 4, \overline{3}] ≈ 0.232408</math></span>* '''generator, complement:''' A generator ’s complement generator is equal to . | |||
''' | <span><math>g</math></span><span><math>1 - g</math></span>* '''generator, golden:''' The generator found using the weighted mediant formula on the period interval with the golden mean, equal to . | ||
<span><math>[0; 2, \overline{1}] ≈ 0.381966</math></span>* '''generator, isotopic:''' A generator found using the weighted mediant formula on the period interval with any isotope of a beyond golden mean. | |||
The periodic part at the end of an <span><math>L{:}s</math></span> | * '''generator, isotopic aristocratic:''' A generator found using the weighted mediant formula on a non-period Stern-Brocot tree interval with any isotope of a beyond golden mean. | ||
* '''generator, metallic:''' A generator found using the weighted mediant formula on any interval from the Stern-Brocot tree (including the period interval) with any metallic mean or isotope. | |||
'''generator, aristocratic''' | * '''generator, noble:''' A generator which is found using the weighted mediant formula on a non-period Stern-Brocot tree interval with the golden mean. | ||
* '''generator, reduced:''' Of the generator and its complement, the lesser of the two; the one less than 0.5. | |||
A generator which is found using the weighted mediant formula on a non-period Stern-Brocot tree interval with a beyond golden mean. | * '''generator, silver:''' The generator found using the weighted mediant formula on the period interval with the silver mean, equal to ≈ 0.292893 | ||
<span><math>[0; 3, \overline{2}]</math></span>* '''interval pattern:''' A recursive interval relationship pattern exhibited by met-MOS scales. | |||
'''generator, bronze''' | * '''level:''' A measure of depth in the Stern-Brocot tree, with the root being 1. | ||
* '''lean:''' Of a generator, its weighted mediant formula used to find it, which of two bounding ratios received the weight: the parent ratio, or the child ratio. | |||
The generator found using the weighted mediant formula on the period interval with the golden mean, equal to <span><math>[0; 4, \overline{3}] ≈ 0.232408</math></span> | * '''lean, parentward:''' The weighted mediant formula received the weight on the parent ratio. | ||
* '''lean, childward:''' The weighted mediant formula received the weight on the child ratio. | |||
'''generator, complement''' | * '''sequence, cardinality:''' For a given scale sequence, the corresponding sequence of their cardinalities. | ||
* '''sequence, interval pattern:''' For a given scale sequence, the corresponding sequence of their interval patterns. | |||
A generator <span><math>g</math></span> | * '''<span><math>L{:}s</math></span>''' '''sequence, '''For a given scale sequence, the corresponding sequence of their ratios. | ||
<span><math>L{:}s</math></span>* '''sequence, scale:''' The ordered sequence of scales a generator generates, each one containing the previous, strictly adding new pitches to it. | |||
'''generator, golden''' | |||
The generator found using the weighted mediant formula on the period interval with the golden mean, equal to <span><math>[0; 2, \overline{1}] ≈ 0.381966</math></span> | |||
'''generator, isotopic''' | |||
A generator found using the weighted mediant formula on the period interval with any isotope of a beyond golden mean. | |||
'''generator, isotopic aristocratic''' | |||
A generator found using the weighted mediant formula on a non-period Stern-Brocot tree interval with any isotope of a beyond golden mean. | |||
'''generator, metallic''' | |||
A generator found using the weighted mediant formula on any interval from the Stern-Brocot tree (including the period interval) with any metallic mean or isotope. | |||
'''generator, noble''' | |||
A generator which is found using the weighted mediant formula on a non-period Stern-Brocot tree interval with the golden mean. | |||
'''generator, reduced''' | |||
Of the generator and its complement, the lesser of the two; the one less than 0.5. | |||
'''generator, silver''' | |||
The generator found using the weighted mediant formula on the period interval with the silver mean, equal to <span><math>[0; 3, \overline{2}]</math></span> | |||
'''interval pattern''' | |||
A recursive interval relationship pattern exhibited by met-MOS scales. | |||
'''level''' | |||
A measure of depth in the Stern-Brocot tree, with the root being 1. | |||
'''lean''' | |||
Of a generator, its weighted mediant formula used to find it, which of two bounding ratios received the weight: the parent ratio, or the child ratio. | |||
'''lean, parentward''' | |||
The weighted mediant formula received the weight on the parent ratio. | |||
'''lean, childward''' | |||
The weighted mediant formula received the weight on the child ratio. | |||
'''sequence, cardinality''' | |||
For a given scale sequence, the corresponding sequence of their cardinalities. | |||
'''sequence, interval pattern''' | |||
For a given scale sequence, the corresponding sequence of their interval patterns. | |||
''' | |||
For a given scale sequence, the corresponding sequence of their <span><math>L{:}s</math></span> | |||
'''sequence, scale''' | |||
The ordered sequence of scales a generator generates, each one containing the previous, strictly adding new pitches to it. | |||