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=== μ notation ===
=== μ notation ===


A number of names and symbols have historically been used to denote metallic means. But many of them are ambiguous, or outright conflict with each other, and unfortunately none of them are optimal for met-MOS purposes. We’ve gotten by alright so far using traditional names and symbols in this discussion, but for the master charts we’re going to need to break from tradition in order to most clearly convey the patterns therein. So, we here propose a new notation using the Greek letter μ, or “mu” (μ because “m” figures so prominently in this domain: “m” for metallic, mean, or moment).  
A number of names and symbols have historically been used to denote metallic means. But many of them are ambiguous, or outright conflict with each other, and unfortunately none of them are optimal for met-MOS purposes. We’ve gotten by alright so far using traditional names and symbols in this discussion, but for the master charts we’re going to need to break from tradition in order to most clearly convey the patterns therein. So, we here propose a new notation using the Greek letter <span><math>μ</math></span>, read “mu” (<span><math>μ</math></span> because “m” figures so prominently in this domain: “m” for metallic, mean, or moment).
 
This <span><math>μ</math></span> notation is a direct mapping of the continued fraction for the metallic mean or isotope, as can clearly be seen in the following chart.


This μ notation is a direct mapping of the continued fraction for the metallic mean or isotope, as can clearly be seen in the following chart.
{| class="wikitable"
{| class="wikitable"
|'''our name'''
|'''our name'''
Line 640: Line 641:
|<nowiki>-</nowiki>
|<nowiki>-</nowiki>
|1
|1
|[1; 0]
|<math>[1; \overline{0}]</math>
|<sub>1</sub>μ<sub>0</sub>
|<math>{}_{1}μ_{0}</math>
|σ<sub>1</sub><sup>0</sup>
|<math>σ_1^0</math>
|0 + √4
|<math>\frac{0 + \sqrt{4}}{2}</math>
 
2
|-
|-
|golden mean
|golden mean
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|Φ, φ, τ, σ<sub>g</sub>
|Φ, φ, τ, σ<sub>g</sub>
|1.618034…
|1.618034…
|[1; 1]
|<math>[1; \overline{1}]</math>
|<sub>1</sub>μ<sub>1</sub>
|<math>{}_{1}μ_{1}</math>
|σ<sub>1</sub><sup>1</sup>
|<math>σ_1^1</math>
|1 + √5
|<math>\frac{1 + \sqrt{5}}{2}</math>
 
2
|-
|-
|golden mean's isotope
|golden mean's isotope
Line 662: Line 659:
|φ
|φ
|0.618034…
|0.618034…
|[0; 1]
|<math>[0; \overline{1}]</math>
|<sub>0</sub>μ<sub>1</sub>
|<math>{}_{0}μ_{1}</math>
|σ<sub>-1</sub><sup>1</sup>
|<math>σ_{-1}^1</math>
|<nowiki>-1 + √5</nowiki>
|<math>\frac{-1 + \sqrt{5}}{2}</math>
 
2
|-
|-
|silver mean
|silver mean
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|δ<sub>s</sub>, σ<sub>s</sub>
|δ<sub>s</sub>, σ<sub>s</sub>
|2.414214…
|2.414214…
|[2; 2]
|<math>[2; \overline{2}]</math>
|<sub>2</sub>μ<sub>2</sub>
|<math>{}_{2}μ_{2}</math>
|σ<sub>2</sub><sup>1</sup>
|<math>σ_2^1</math>
|2 + √8
|<math>\frac{2 + \sqrt{8}}{2}</math>
 
2
|-
|-
|silver mean's first isotope  
|silver mean's first isotope  
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|√2
|√2
|1.414214…
|1.414214…
|[1; 2]
|<math>[1; \overline{2}]</math>
|<sub>1</sub>μ<sub>2</sub>
|<math>{}_{1}μ_{2}</math>
|σ<sub>0</sub><sup>2</sup>
|<math>σ_0^2</math>
|0 + √8
|<math>\frac{0 + \sqrt{8}}{2}</math>
 
2
|-
|-
|silver mean's second isotope
|silver mean's second isotope
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|<nowiki>-</nowiki>
|<nowiki>-</nowiki>
|0.414214…
|0.414214…
|[0; 2]
|<math>[0; \overline{2}]</math>
|<sub>0</sub>μ<sub>2</sub>
|<math>{}_{0}μ_{2}</math>
|σ<sub>-2</sub><sup>1</sup>
|<math>σ_{-2}^1</math>
|<nowiki>-2 + √8</nowiki>
|<math>\frac{-2 + \sqrt{8}}{2}</math>
 
2
|-
|-
|bronze mean
|bronze mean
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|δ<sub>b</sub>, σ<sub>b</sub>
|δ<sub>b</sub>, σ<sub>b</sub>
|3.302776…
|3.302776…
|[3; 3]
|<math>[3; \overline{3}]</math>
|<sub>3</sub>μ<sub>3</sub>
|<math>{}_{3}μ_{3}</math>
|σ<sub>3</sub><sup>1</sup>
|<math>σ_3^1</math>
|3 + √13
|<math>\frac{3 + \sqrt{13}}{2}</math>
 
2
|-
|-
|bronze mean's first isotope
|bronze mean's first isotope
Line 719: Line 706:
|<nowiki>-</nowiki>
|<nowiki>-</nowiki>
|2.302776…
|2.302776…
|[2; 3]
|<math>[2; \overline{3}]</math>
|<sub>2</sub>μ<sub>3</sub>
|<math>{}_{2}μ_{3}</math>
|σ<sub>1</sub><sup>3</sup>
|<math>σ_1^3</math>
|1 + √13
|<math>\frac{1 + \sqrt{13}}{2}</math>
 
2
|-
|-
|bronze mean's second isotope
|bronze mean's second isotope
Line 730: Line 715:
|<nowiki>-</nowiki>
|<nowiki>-</nowiki>
|1.302776…
|1.302776…
|[1; 3]
|<math>[1; \overline{3}]</math>
|<sub>1</sub>μ<sub>3</sub>
|<math>{}_{1}μ_{3}</math>
|σ<sub>-1</sub><sup>3</sup>
|<math>σ_{-1}^3</math>
|<nowiki>-1 + √13</nowiki>
|<math>\frac{-1 + \sqrt{13}}{2}</math>
 
2
|-
|-
|bronze mean's third isotope
|bronze mean's third isotope
Line 741: Line 724:
|<nowiki>-</nowiki>
|<nowiki>-</nowiki>
|0.302776…
|0.302776…
|[0; 3]
|<math>[0; \overline{3}]</math>
|<sub>0</sub>μ<sub>3</sub>
|<math>{}_{0}μ_{3}</math>
|σ<sub>-3</sub><sup>1</sup>
|<math>σ_{-3}^1</math>
|<nowiki>-3 + √13</nowiki>
|<math>\frac{-3 + \sqrt{13}}{2}</math>
 
2
|}
|}


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