Metallic MOS: Difference between revisions
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=== Noble cases === | === Noble cases === | ||
A noble generator’s L:s sequence is slightly more complex. Not every — but almost every — L:s is phi. Only the first few are not. | A noble generator’s L:s sequence is slightly more complex. Not every — but almost every — L:s is phi. Only the first few are not. | ||
=== Beyond golden cases === | === Beyond golden cases === | ||
Instead of every scale’s L:s equalling the same value, as is the case for the golden mean, the silver mean’s L:s sequence alternates between its isotopes that are greater than 1: | Instead of every scale’s L:s equalling the same value, as is the case for the golden mean, the silver mean’s L:s sequence alternates between its isotopes that are greater than 1: | ||
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And the bronze mean’s L:s sequence cycles through its isotopes that are greater than 1: | And the bronze mean’s L:s sequence cycles through its isotopes that are greater than 1: | ||
<math> | |||
\begin{equation} | |||
\begin{cases} | |||
δ_b \\ | |||
δ_b - 1 \\ | |||
δ_b - 2 \\ | |||
\end{cases} | |||
\end{equation} | |||
</math> | |||
Any n-metallic mean’s L:s sequence will cycle through its isotopes that are greater than 1. | Any n-metallic mean’s L:s sequence will cycle through its isotopes that are greater than 1. | ||
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Isotopic L:s sequences are just like those of their mean’s, but offset. | Isotopic L:s sequences are just like those of their mean’s, but offset. | ||
For example, the silver mean’s first isotope’s generator’s L:s sequence alternates between L:s = | For example, the silver mean’s first isotope’s generator’s L:s sequence alternates between L:s = δ_s and L:s = δ_s - 1, just like the silver generator’s, however — unlike the silver generator’s — it begins with L:s = δ_s - 1. | ||
=== Aristocratic cases === | === Aristocratic cases === | ||
Again, aristocratic scales synthesize both the complexities of noble scales and beyond golden scales. We’ll call the periodic part of an L:s sequence its ''L:s cycle''. So most of the L:s sequence will be the L:s cycle, with only the first few scales not being so. | Again, aristocratic scales synthesize both the complexities of noble scales and beyond golden scales. We’ll call the periodic part of an L:s sequence its ''L:s cycle''. So most of the L:s sequence will be the L:s cycle, with only the first few scales not being so. | ||
== Interval patterns == | == Interval patterns == | ||
As we’ve seen, the step sizes of metallic scales follow particular patterns. But not just the steps follow patterns — many other intervals of met-MOS scales do too. | As we’ve seen, the step sizes of metallic scales follow particular patterns. But not just the steps follow patterns — many other intervals of met-MOS scales do too. | ||
=== Golden case === | === Golden case === | ||
We know that for golden scales: | We know that for golden scales: | ||
L:s = φ | <math>\qquad L:s = φ | ||
</math> | |||
But that’s not all. Due to the mathemagic of phi, we also get a recursive interval relationship pattern: | But that’s not all. Due to the mathemagic of phi, we also get a recursive interval relationship pattern: | ||
<math>\qquad L:s = (L+s):L = (2L+s):(L+s) = (3L+2s):(2L+s) = (5L+3s):(3L+2s) = … = φ | |||
</math> | |||
Henceforth we’ll be referring to these types of recursive interval relationship patterns simply as ''interval patterns''. | Henceforth we’ll be referring to these types of recursive interval relationship patterns simply as ''interval patterns''. | ||
=== 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 L:s sequences only eventually exhibit L:s = φ). | Noble scales at first do not — but eventually do — reach a point where they start exhibiting this interval pattern (paralleling how their L:s sequences only eventually exhibit L:s = φ). | ||
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The silver generator, as it did for its L:s sequence, alternates in quality between its two > 1 isotopes for its intervals. For half of its scales, | The silver generator, as it did for its L:s sequence, alternates in quality between its two > 1 isotopes for its intervals. For half of its scales, | ||
L:s = (2L+s):L = (5L+2s):(2L+s) = (12L+5s):(5L+2s) = (29L+12s):(12L+5s) = … = | L:s = (2L+s):L = (5L+2s):(2L+s) = (12L+5s):(5L+2s) = (29L+12s):(12L+5s) = … = δ_s | ||
and the other half, | and the other half, | ||
<math>\qquad L:s = (L+2s):(L+s) = (3L+4s):(2L+3s) = (7L+10s):(5L+7s) = (17L+24s):(12L+17s) = ... = δ_s - 1 | |||
</math> | |||
L:s = (L+2s):(L+s) = (3L+4s):(2L+3s) = (7L+10s):(5L+7s) = (17L+24s):(12L+17s) = ... = | |||
Bronze’s scales cycle through three different interval patterns related to its respective > 1 isotopes. | Bronze’s scales cycle through three different interval patterns related to its respective > 1 isotopes. | ||
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=== Isotopic cases === | === Isotopic cases === | ||
As with L:s 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 L:s 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 === | ||
Predictably, the aristocratic case combines the noble case and the beyond golden cases: at first, the ratios do not exhibit interval patterns, but eventually they do, and when they start to, they follow the interval pattern cycle for the appropriate metallic mean or isotope thereof. | Predictably, the aristocratic case combines the noble case and the beyond golden cases: at first, the ratios do not exhibit interval patterns, but eventually they do, and when they start to, they follow the interval pattern cycle for the appropriate metallic mean or isotope thereof. | ||
= Mathematical explanations = | = Mathematical explanations = | ||
We’ll now start going through mathematical explanations for the behavior we’ve observed about met-MOS generators, L:s sequences, and interval patterns. | We’ll now start going through mathematical explanations for the behavior we’ve observed about met-MOS generators, L:s sequences, and interval patterns. | ||
== Infinite scale sequences == | == Infinite scale sequences == | ||
Every metallic generator generates an infinitely long scale sequence. | Every metallic generator generates an infinitely long scale sequence. | ||
This property is not unique to metallic generators, though — it is attributable to their being irrational numbers. A rational generator’s scale sequence eventually terminates, hitting bedrock when the period has been divided up into equal steps, i.e. where the notion of large steps and small steps no longer applies because L = s and L:s = 1. For example, the generator 5/12 = 0.41 | This property is not unique to metallic generators, though — it is attributable to their being irrational numbers. A rational generator’s scale sequence eventually terminates, hitting bedrock when the period has been divided up into equal steps, i.e. where the notion of large steps and small steps no longer applies because L = s and L:s = 1. For example, the generator | ||
<math>\qquad 5/12 = 0.41\overline{6} | |||
</math> | |||
generates scales with cardinality 2, 3, 5, 7, but when it reaches cardinality 12, it has generated 12edo. This occurs exactly at the moment when the generator has been repeated until it has returned exactly to from where it started (because 5/12 * 12 = 5, which is a multiple of the period, 1); if we repeated the generator any more, we’d just go over the ground we already trod. | |||
Irrational generators will never be able to return to exactly from where they started, so they will continue to divide the period up into smaller and smaller steps forever. At some point, however, the scales they generate will cease to be musically practical, because their steps will have become so small. | Irrational generators will never be able to return to exactly from where they started, so they will continue to divide the period up into smaller and smaller steps forever. At some point, however, the scales they generate will cease to be musically practical, because their steps will have become so small. | ||
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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 [[wikipedia: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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A similar case is true for the silver mean, except we have to subtract 1 from it twice before the resulting value’s reciprocal is equal to the silver mean. | A similar case is true for the silver mean, except we have to subtract 1 from it twice before the resulting value’s reciprocal is equal to the silver mean. | ||
[ math ] | [ math ] δ_s - 1 - 1 = 1 / δ_s | ||
And for the bronze ratio, we must subtract thrice: | And for the bronze ratio, we must subtract thrice: | ||
[ math ] | [ math ] δ_s - 1 - 1 - 1 = 1 / δ_s | ||
Basically, we are subtracting 1 from the means until only their decimal part remains. One way to describe the metallic means, then, would be the set of values for which the reciprocal of their decimal part equals themselves. | Basically, we are subtracting 1 from the means until only their decimal part remains. One way to describe the metallic means, then, would be the set of values for which the reciprocal of their decimal part equals themselves. | ||
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[ | [ | ||
[2; <s>2</s>] ≈ 2.414214 = | [2; <s>2</s>] ≈ 2.414214 = δ_s | ||
[1; <s>2</s>] ≈ 1.414214 = | [1; <s>2</s>] ≈ 1.414214 = δ_s - 1 | ||
... | ... | ||
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If the golden mean is the value for which a:b = (a+b):a, then the silver mean is the value for which | If the golden mean is the value for which a:b = (a+b):a, then the silver mean is the value for which | ||
[ math ] a:b = (2a+b):a = | [ math ] a:b = (2a+b):a = δ_s | ||
Following the same logic as we followed for the golden case, | Following the same logic as we followed for the golden case, | ||
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[ math ] L:s = (2L+s):L | [ math ] L:s = (2L+s):L | ||
So, wherever we have a scale where L:s = | So, wherever we have a scale where L:s = δ_s, we’ll also see the interval pattern | ||
[ math ] | [ math ] | ||
L:s = (2L+s):L = (5L+2s):(2L+s) = (12L+5s):(5L+2s) = (29L+12s):(12L+5s) = … = | L:s = (2L+s):L = (5L+2s):(2L+s) = (12L+5s):(5L+2s) = (29L+12s):(12L+5s) = … = δ_s | ||
Every other scale the silver generator generates has an L:s other than δ<sub>s</sub>, namely, its isotope, δ<sub>s</sub> - 1. These scales have a different pattern: L:s = (L+2s):(L+s) = δ<sub>s</sub> - 1. | Every other scale the silver generator generates has an L:s other than δ<sub>s</sub>, namely, its isotope, δ<sub>s</sub> - 1. These scales have a different pattern: L:s = (L+2s):(L+s) = δ<sub>s</sub> - 1. | ||
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'''[http://www.huygens-fokker.org/bpsite/833cent.html Bohlen's 833 cent scale]''' | '''[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 [[wikipedia:Combination_tone|combination tones]]. | ||
'''[http://tonalsoft.com/enc/b/brinko.aspx?fbclid=IwAR0Z5F4dXsUNo63TL1ukklQjIQQScDS2-NT61IJcqlXqcREgnKlcUl-pQ_4 Brinko]''' | '''[http://tonalsoft.com/enc/b/brinko.aspx?fbclid=IwAR0Z5F4dXsUNo63TL1ukklQjIQQScDS2-NT61IJcqlXqcREgnKlcUl-pQ_4 Brinko]''' | ||