Metallic MOS: Difference between revisions
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The first metallic mean is by far the best known: the golden mean, | The first metallic mean is by far the best known: the golden mean, | ||
<math>\qquad φ ≈ 1.618034 | |||
</math> | |||
Scales based on this mean — or “phi”, as it is often called, after the Greek letter used to represent it — are relatively well-known, and these “golden” scales have been described by [[wikipedia:Erv_Wilson|Erv Wilson]], [[Kraig Grady|Kraig_Grady]], [http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm David J. Finnamore], [https://ia800908.us.archive.org/31/items/mosedo.html/mosedo.html Billy Stiltner], [[wikipedia:Paul_Erlich|Paul Erlich]], [[Graham Breed|Graham Breed]], [[Dave Keenan|Dave Keenan]], [[Margo Schulter|Margo Schulter]], [[Das Goldene Tonsystem|Thorvald Kornerup]] and many others. | |||
Scales based on this mean — or “phi”, as it is often called, after the Greek letter used to represent it — are relatively well-known, and these “golden” scales have been described by Erv Wilson | |||
But phi is only the first of an infinite continuum of such metallic means which can be used to generate scales offering interesting musical possibilities. And while some attention has been given to silver scales, what we seek to do here is centralize all met-MOS knowledge and generalize principles across all of the metallic means. | But phi is only the first of an infinite continuum of such metallic means which can be used to generate scales offering interesting musical possibilities. And while some attention has been given to silver scales, what we seek to do here is centralize all met-MOS knowledge and generalize principles across all of the metallic means. | ||
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MOS concepts are logarithmic, not acoustic. Frequency ratios related to metallic means, such as “acoustic phi” ≈ 833.09¢, have interesting properties too — creating recursive combination tones, for example — but these musical applications of metallic means are discussed elsewhere. | MOS concepts are logarithmic, not acoustic. Frequency ratios related to metallic means, such as “acoustic phi” ≈ 833.09¢, have interesting properties too — creating recursive combination tones, for example — but these musical applications of metallic means are discussed elsewhere. | ||
= | = Behavior = | ||
In the first part of this discussion we’ll go over basic met-MOS behavior, deferring mathematical explanations until later. | In the first part of this discussion we’ll go over basic met-MOS behavior, deferring mathematical explanations until later. | ||
== | == Generators == | ||
A natural topic to begin with is generators, since we need a generator before we can generate scales. | A natural topic to begin with is generators, since we need a generator before we can generate scales. | ||
=== | === Golden case === | ||
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''. | ||
=== | === Noble cases === | ||
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: | ||
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 | 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]. | ||
As for why the order of the segments is flipped here — well, that too will be explained later. | As for why the order of the segments is flipped here — well, that too will be explained later. | ||
=== | === Beyond golden cases === | ||
So far we’ve only considered the golden mean. We’ll next try the second metallic mean, the silver mean: | So far we’ve only considered the golden mean. We’ll next try the second metallic mean, the silver mean: | ||
<math>\qquad δ_s ≈ 2.414214 | |||
</math> | |||
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: | ||
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And from the bronze mean, | And from the bronze mean, | ||
<math>\qquad δ_b ≈ 3.302776 | |||
</math> | |||
we find the ''bronze generator'', approximately 0.232408: | we find the ''bronze generator'', approximately 0.232408: | ||
=== | === Isotopic cases === | ||
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 arithmetic progression (<nowiki>https://en.wikipedia.org/wiki/Arithmetic_progression</nowiki>) 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, | Values of the arithmetic progression (<nowiki>https://en.wikipedia.org/wiki/Arithmetic_progression</nowiki>) 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> | |||
finding a generator which is approximately 0.414214: | finding a generator which is approximately 0.414214: | ||
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Isotopes theoretically could be formed by adding 1 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 1 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. | ||
=== | === Aristocratic cases === | ||
We can find even more metallic generators by extending the concept of noble numbers to metallic means beyond the golden, as well as their isotopes. For example, we could choose the silver mean, and 0/1 to 1/2 as our interval, finding approximately 0.226541: | We can find even more metallic generators by extending the concept of noble numbers to metallic means beyond the golden, as well as their isotopes. For example, we could choose the silver mean, and 0/1 to 1/2 as our interval, finding approximately 0.226541: | ||
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== L:s sequences == | == L:s sequences == | ||
Each scale has exactly two step sizes: large and small, or L and s. We can refer to the ratio between these large and small steps as | Each scale has exactly two step sizes: large and small, or L and s. We can refer to the ratio between these large and small steps as | ||
<math>\qquad L:s | |||
</math> | |||
We’ll call the ordered set of scales a generator generates its ''scale sequence'', and the ordered set of L:s corresponding to these scales a generator’s ''L:s sequence''. | We’ll call the ordered set of scales a generator generates its ''scale sequence'', and the ordered set of L:s corresponding to these scales a generator’s ''L:s sequence''. | ||
=== | === Golden case === | ||
The golden generator’s L:s sequence is simple. Every L:s ratio is phi: | The golden generator’s L:s sequence is simple. Every L:s ratio is phi: | ||
<math>\qquad L:s = φ | |||
</math> | |||
=== noble cases === | === noble cases === | ||
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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: | ||
<math> | |||
\begin{equation} | |||
\begin{cases} | |||
δ_s \\ | |||
δ_s - 1 \\ | |||
\end{cases} | |||
\end{equation} | |||
</math> | |||
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: | ||