Metallic MOS: Difference between revisions

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The golden generator is essentially the noble generator for the interval <math>\frac 01</math> to <math>\frac 11</math>, which — being the root of the Stern-Brocot tree — is as golden as we can get: we see <math>L{:}s = φ</math> and <math>φ</math>'s distinctive interval pattern from the very start.

And the noble generator between 0/1 and 1/3

And the noble generator between <math>\frac 01</math> and <math>\frac 13</math>

<math>\qquad \frac{1φ + 0}{3φ + 1} ≈ 0.276393

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On the other hand, the noble generator equal to <math>0.275267</math> — while only a smidgen off from the other noble generator we just looked at — necessitates iterating ''six'' times before attaining <math>L{:}s = φ</math>. This corresponds to it being the noble generator between <math>\frac {5}{18}</math> and <math>\frac {3}{11}</math>, an interval which lies five levels deeper in the Stern-Brocot tree than the interval from <math>\frac 01</math> to <math>\frac 13</math>.

So if we want a golden scale, and we also happen to want a generator near <math>0.276393</~~span~~></~~math~~>, then we’re in luck. But if we want a golden generator that is close to <math>0.275267</~~span~~></~~math~~>, we may be disappointed to hear that it is not “golden” enough for us.

So if we want a golden scale, and we also happen to want a generator near <math>0.276393</math>, then we’re in luck. But if we want a golden generator that is close to <math>0.275267</math>, we may be disappointed to hear that it is not “golden” enough for us.

Wilson [http://www.anaphoria.com/hrgm.PDF documented noble scale sequences through the sixth level of the Stern-Brocot tree] (or as he called it, the “scale tree” or “Peirce Series”), totalling 32 noble generators. He also [http://anaphoria.com/sctree.pdf?fbclid=IwAR2PahVuZJ18faXQA_IggdD52y9PWP4uyEeQALE8Q73MhIlploPYDinbAAk recorded just the generator values down to the eleventh level] for a total of 1024 generators. Exploring generators beyond that was probably just not worth it, because their metallicity levels are too low. We will cut ourselves off at the seventh level in our scale trees, as we depict generators for the silver and bronze means, and their isotopes too.

@@ Line 402: / Line 402: @@
 The golden generator is essentially the noble generator for the interval <span><math>\frac 01</math></span> to <span><math>\frac 11</math></span>, which — being the root of the Stern-Brocot tree — is as golden as we can get: we see <span><math>L{:}s = φ</math></span> and <span><math>φ</math></span>'s distinctive interval pattern from the very start.
-And the noble generator between 0/1 and 1/3
+And the noble generator between <span><math>\frac 01</math></span> and <span><math>\frac 13</math></span>
 <math>\qquad \frac{1φ + 0}{3φ + 1} ≈ 0.276393
@@ Line 411: / Line 411: @@
 On the other hand, the noble generator equal to <span><math>0.275267</math></span> — while only a smidgen off from the other noble generator we just looked at — necessitates iterating ''six'' times before attaining <span><math>L{:}s = φ</math></span>. This corresponds to it being the noble generator between <span><math>\frac {5}{18}</math></span> and <span><math>\frac {3}{11}</math></span>, an interval which lies five levels deeper in the Stern-Brocot tree than the interval from <span><math>\frac 01</math></span> to <span><math>\frac 13</math></span>.
-So if we want a golden scale, and we also happen to want a generator near <span><math>0.276393</span></math>, then we’re in luck. But if we want a golden generator that is close to <span><math>0.275267</span></math>, we may be disappointed to hear that it is not “golden” enough for us.
+So if we want a golden scale, and we also happen to want a generator near <span><math>0.276393</math></span>, then we’re in luck. But if we want a golden generator that is close to <span><math>0.275267</math></span>, we may be disappointed to hear that it is not “golden” enough for us.
 Wilson [http://www.anaphoria.com/hrgm.PDF documented noble scale sequences through the sixth level of the Stern-Brocot tree] (or as he called it, the “scale tree” or “Peirce Series”), totalling 32 noble generators. He also [http://anaphoria.com/sctree.pdf?fbclid=IwAR2PahVuZJ18faXQA_IggdD52y9PWP4uyEeQALE8Q73MhIlploPYDinbAAk recorded just the generator values down to the eleventh level] for a total of 1024 generators. Exploring generators beyond that was probably just not worth it, because their metallicity levels are too low. We will cut ourselves off at the seventh level in our scale trees, as we depict generators for the silver and bronze means, and their isotopes too.