Metallic MOS: Difference between revisions

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

The connections between these metallic means are explained by their ratio forms. All goldenish means can be expressed in the form:

<math>\qquad \frac{m + n\sqrt5}{2}

</math>

For the golden mean, <math>m = 1</math> and <math>n = 1</math>; for the fourth metallic mean, <math>m = 4</math> and <math>n = 2</math>.

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>.

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

</math>

The silverish means can also all be expressed in a ratio form, but where the golden mean uses <math>\sqrt5</math>, they use <math>\sqrt2</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 ... )

@@ Line 19,170: / Line 19,170: @@
 We suggest referring to these particular metallic means as ''goldenish means''.
+The connections between these metallic means are explained by their ratio forms. All goldenish means can be expressed in the form:
+<math>\qquad \frac{m + n\sqrt5}{2}
+</math>
+For the golden mean, <span><math>m = 1</math></span> and <span><math>n = 1</math></span>; for the fourth metallic mean, <span><math>m = 4</math></span> and <span><math>n = 2</math></span>.
 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>.
@@ Line 19,184: / Line 19,191: @@
 <math>\qquad {}_{82}μ_{82} ≈ 82.0121932454 = 29δ_s + 12 = δ_s^5
 </math>
+The silverish means can also all be expressed in a ratio form, but where the golden mean uses <span><math>\sqrt5</math></span>, they use <span><math>\sqrt2</math></span>.
 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 ... )