Tenney norm: Difference between revisions
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→Computation: Tenney weighter should be defined on Tenney height, not the other way around |
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The Tenney height of a [[Harmonic limit|''p''-limit]] [[monzo]] b = {{monzo| ''b''<sub>1</sub> ''b''<sub>2</sub> … ''b''<sub>π (''p'')</sub> }} (π being the [[Wikipedia: prime-counting function|prime-counting function]]) is given by | The Tenney height of a [[Harmonic limit|''p''-limit]] [[monzo]] b = {{monzo| ''b''<sub>1</sub> ''b''<sub>2</sub> … ''b''<sub>π (''p'')</sub> }} (π being the [[Wikipedia: prime-counting function|prime-counting function]]) is given by | ||
<math>\lVert | <math>\lVert H \vec b \rVert_1 \\ | ||
= \vert b_1 \vert + \vert b_2 \vert \log_2 (3) + \ldots + \vert b_{\pi (p)} \vert \log_2 (p) \\ | = \vert b_1 \vert + \vert b_2 \vert \log_2 (3) + \ldots + \vert b_{\pi (p)} \vert \log_2 (p) \\ | ||
= \log_2 (2^{|b_1|} \cdot 3^{|b_2|} \cdot \ldots \cdot p^{|b_{\pi (p)}|})</math> | = \log_2 (2^{|b_1|} \cdot 3^{|b_2|} \cdot \ldots \cdot p^{|b_{\pi (p)}|})</math> | ||
where | where H is the transformation matrix such that, for the prime basis Q = {{val| 2 3 5 … ''p'' }}, | ||
<math> | <math>H = \operatorname {diag} (\log_2 (Q))</math> | ||
== Examples == | == Examples == | ||