Metallic MOS: Difference between revisions

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different generators.

== -ishes ==

Some of the metallic means have special connections to each other. For example, the fourth metallic mean can be expressed in terms of the golden mean in a couple different ways:

<math>\qquad {}_{4}μ_{4} ≈ 4.236067978 = 2φ + 1 = φ^3

</math>

The eleventh metallic mean, too, can be expressed in terms of the golden mean:

<math>\qquad {}_{11}μ_{11} ≈ 11.0901699437 = 5φ + 3 = φ^5

</math>

In fact, every odd power of <math>φ</math> will be equivalent to a higher metallic mean. Whichever metallic mean that is will be the sum of two Fibonacci numbers <math>F_n</math> and <math>F_{n-2}</math> (e.g. 4 = 3 + 1, and 11 = 8 + 3). For the alternative expression of the higher metallic mean as a constant plus some coefficient on the golden mean, the values of the constant and the coefficient will also be values from the Fibonacci series (e.g. 1 & 2, 3 & 5).

We suggest referring to these particular metallic means as ''goldenish means''.

If we use <math>φ^2</math> as our generator, then all of the pitches in our scale will be goldenish means (relatively speaking; if we multiplied every one by <math>φ</math>, preserving their ratios, they would be). Equivalently, we could include goldenish means until we found scales with exactly two step sizes, then divide every pitch by their shared factor of <math>φ</math>.

In another fun bit of synergy, moments of symmetry for this generator will be found at cardinalities from the Fibonacci series. That is, this generator produces scales with cardinality sequences of 2, 3, 5, 8, 13, 21 ...

The same principles hold for every other metallic mean. For instance, the second silverish mean is the 14th metallic mean:

<math>\qquad {}_{14}μ_{14} ≈ 14.07106781 = 5δ_s + 2 = δ_s^3

</math>

and the third silverish mean is the 82nd metallic mean:

<math>\qquad {}_{82}μ_{82} ≈ 82.0121932454 = 29δ_s + 12 = δ_s^5

</math>

The pattern of powers remains the same for any metallic mean's -ishes: they are the odd powers. However, while the goldenish means draw their coefficients, constants, and cardinalities from the Fibonacci numbers, the silverish means draw theirs from their equivalent [[wikipedia:Recurrence_relation|recurrence relation]], the [[wikipedia:Pell_number|Pell numbers]] (2, 5, 12, 29, 70 ... )

So, if we use <math>δ_s^2</math> as our generator, then we get silverish scales, whose pitches are all silverish means.

And if we use <math>δ_b^2</math> as our generator, we can generate bronzish scales.

= Further reading =

@@ Line 19,154: / Line 19,154: @@
 different generators.
+== -ishes ==
+Some of the metallic means have special connections to each other. For example, the fourth metallic mean can be expressed in terms of the golden mean in a couple different ways:
+<math>\qquad {}_{4}μ_{4} ≈ 4.236067978 = 2φ + 1 = φ^3
+</math>
+The eleventh metallic mean, too, can be expressed in terms of the golden mean:
+<math>\qquad {}_{11}μ_{11} ≈ 11.0901699437 = 5φ + 3 = φ^5
+</math>
+In fact, every odd power of <span><math>φ</math></span> will be equivalent to a higher metallic mean. Whichever metallic mean that is will be the sum of two Fibonacci numbers <span><math>F_n</math></span> and <span><math>F_{n-2}</math></span> (e.g. 4 = 3 + 1, and 11 = 8 + 3). For the alternative expression of the higher metallic mean as a constant plus some coefficient on the golden mean, the values of the constant and the coefficient will also be values from the Fibonacci series (e.g. 1 & 2, 3 & 5).
+We suggest referring to these particular metallic means as ''goldenish means''.
+If we use <span><math>φ^2</math></span> as our generator, then all of the pitches in our scale will be goldenish means (relatively speaking; if we multiplied every one by <span><math>φ</math></span>, preserving their ratios, they would be). Equivalently, we could include goldenish means until we found scales with exactly two step sizes, then divide every pitch by their shared factor of <span><math>φ</math></span>.
+In another fun bit of synergy, moments of symmetry for this generator will be found at cardinalities from the Fibonacci series. That is, this generator produces scales with cardinality sequences of 2, 3, 5, 8, 13, 21 ...
+The same principles hold for every other metallic mean. For instance, the second silverish mean is the 14th metallic mean:
+<math>\qquad {}_{14}μ_{14} ≈ 14.07106781 = 5δ_s + 2 = δ_s^3
+</math>
+and the third silverish mean is the 82nd metallic mean:
+<math>\qquad {}_{82}μ_{82} ≈ 82.0121932454 = 29δ_s + 12 = δ_s^5
+</math>
+The pattern of powers remains the same for any metallic mean's -ishes: they are the odd powers. However, while the goldenish means draw their coefficients, constants, and cardinalities from the Fibonacci numbers, the silverish means draw theirs from their equivalent [[wikipedia:Recurrence_relation|recurrence relation]], the [[wikipedia:Pell_number|Pell numbers]] (2, 5, 12, 29, 70 ... )
+So, if we use <span><math>δ_s^2</math></span> as our generator, then we get silverish scales, whose pitches are all silverish means.
+And if we use <span><math>δ_b^2</math></span> as our generator, we can generate bronzish scales.
 = Further reading =