Tenney norm: Difference between revisions
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The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney norm. | The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney norm. | ||
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The Tenney norm of a ratio ''n''/''d'' is given by | The Tenney norm of a ratio ''n''/''d'' is given by | ||
$$\log_2 (nd) $$ | |||
=== Vector form === | === Vector form === | ||
The Tenney norm of a [[harmonic limit|''p''-limit]] [[monzo]] {{nowrap|'''m''' {{=}} {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }}}} (π being the {{w|prime-counting function}}) is given by | The Tenney norm of a [[harmonic limit|''p''-limit]] [[monzo]] {{nowrap|'''m''' {{=}} {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }}}} (π being the {{w|prime-counting function}}) is given by | ||
$$ | |||
\norm{H \vec m}_1 | \begin{align} | ||
= \abs{m_1} + \abs{m_2} | \norm{H \vec m}_1 &= \abs{m_1} + \abs{m_2} \log_2 (3) + \ldots + \abs{m_{\pi (p)}} \log_2 (p) \\ | ||
= \log_2\left(2^{\abs{m_1}} \cdot 3^{\abs{m_2}} \cdot \ldots \cdot p^{\abs{m_{\pi (p)}}}\right) | &= \log_2\left(2^{\abs{m_1}} \cdot 3^{\abs{m_2}} \cdot \ldots \cdot p^{\abs{m_{\pi (p)}}}\right) | ||
\end{align} | |||
$$ | |||
where ''H'' is the transformation matrix such that, for the prime basis {{nowrap|''Q'' {{=}} {{val| 2 3 5 … ''p'' }}}}, | where ''H'' is the transformation matrix such that, for the prime basis {{nowrap| ''Q'' {{=}} {{val| 2 3 5 … ''p'' }} }}, | ||
$$ H = \operatorname {diag} (\log_2 (Q)) $$ | |||
== Examples == | == Examples == | ||
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* [[Generalized Tenney norms and Tp interval space|Generalized Tenney norms and T<sub>''p''</sub> interval space]] | * [[Generalized Tenney norms and Tp interval space|Generalized Tenney norms and T<sub>''p''</sub> interval space]] | ||
== | == References == | ||
<references /> | <references /> | ||