Benedetti height: Difference between revisions
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The '''Benedetti height''' is a simple [[height]] function which measures the [[complexity]] of a [[JI]] [[interval]]. The Benedetti height of a positive rational number ''n''/''d'' reduced to lowest terms (no common factor between ''n'' and ''d'') is equal to ''nd'', the product of the numerator and denominator. In general mathematics it is known as ''product complexity''. | The '''Benedetti height''' is a simple [[height]] function which measures the [[complexity]] of a [[JI]] [[interval]]. The Benedetti height of a positive rational number ''n''/''d'' reduced to lowest terms (no common factor between ''n'' and ''d'') is equal to ''nd'', the product of the numerator and denominator. In general mathematics it is known as ''product complexity''. | ||
The [[logarithm base two]] of the Benedetti height is the | The [[logarithm base two]] of the Benedetti height is the Tenney height, or [[Tenney norm]]. | ||
== Computation == | == Computation == | ||
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The Benedetti height of a ratio ''n''/''d'' is given by | The Benedetti height of a ratio ''n''/''d'' is given by | ||
$$ nd $$ | |||
=== Vector form === | === Vector form === | ||
The Benedetti height of a [[Harmonic limit|''p''-limit]] [[monzo]] m = {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }} (π being the [[Wikipedia: prime-counting function|prime-counting function]]) is given by | The Benedetti height of a [[Harmonic limit|''p''-limit]] [[monzo]] '''m''' = {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }} (π being the [[Wikipedia: prime-counting function|prime-counting function]]) is given by | ||
$$ 2^{\lVert H \vec m \rVert_1} = 2^{|m_1|} \cdot 3^{|m_2|} \cdot \ldots \cdot p^{|m_{\pi (p)}|} $$ | |||
= 2^{|m_1|} \cdot 3^{|m_2|} \cdot \ldots \cdot p^{|m_{\pi (p)}|} | |||
where H is the transformation matrix such that, for the prime basis Q = {{val| 2 3 5 … ''p'' }}, | where ''H'' is the transformation matrix such that, for the prime basis ''Q'' = {{val| 2 3 5 … ''p'' }}, | ||
$$ H = \operatorname {diag} (\log_2 (Q)) $$ | |||
== Examples == | == Examples == | ||
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|- | |- | ||
| [[1/1]] | | [[1/1]] | ||
| {{ | | {{Monzo| 0 }} | ||
| 1 | | 1 | ||
|- | |- | ||
| [[2/1]] | | [[2/1]] | ||
| {{ | | {{Monzo| 1 }} | ||
| 2 | | 2 | ||
|- | |- | ||
| [[3/2]] | | [[3/2]] | ||
| {{ | | {{Monzo| -1 1 }} | ||
| 6 | | 6 | ||
|- | |- | ||
| [[6/5]] | | [[6/5]] | ||
| {{ | | {{Monzo| 1 1 -1 }} | ||
| 30 | | 30 | ||
|- | |- | ||
| [[9/7]] | | [[9/7]] | ||
| {{ | | {{Monzo| 0 2 0 -1 }} | ||
| 63 | | 63 | ||
|- | |- | ||
| [[13/11]] | | [[13/11]] | ||
| {{ | | {{Monzo| 0 0 0 0 -1 1 }} | ||
| 143 | | 143 | ||
|} | |} | ||
| Line 63: | Line 62: | ||
[[Category:Interval complexity measures]] | [[Category:Interval complexity measures]] | ||
[[Category:Tenney-weighted measures]] | |||
[[Category:Tenney]] | |||