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. | |||
== Computation == | |||
=== Ratio form === | |||
The Tenney norm of a ratio ''n''/''d'' is given by | |||
$$\log_2 (nd) $$ | |||
< | |||
=== 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 | |||
$$ | |||
\begin{align} | |||
\norm{H \vec m}_1 &= \abs{m_1} + \abs{m_2} \log_2 (3) + \ldots + \abs{m_{\pi (p)}} \log_2 (p) \\ | |||
&= \log_2\left(2^{\abs{m_1}} \cdot 3^{\abs{m_2}} \cdot \ldots \cdot p^{\abs{m_{\pi (p)}}}\right) | |||
\end{align} | |||
$$ | |||
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 == | |||
{| class="wikitable center-2" | |||
|- | |||
! Interval name | |||
! Ratio (''n''/''d'') | |||
! Monzo | |||
! Tenney norm | |||
|- | |||
| Unison | |||
| [[1/1]] | |||
| {{Monzo| 0 }} | |||
| 0 | |||
|- | |||
| Octave | |||
| [[2/1]] | |||
| {{Monzo| 1 }} | |||
| 1 | |||
|- | |||
| Just perfect fifth | |||
| [[3/2]] | |||
| {{Monzo| -1 1 }} | |||
| 2.585 | |||
|- | |||
| Just major third | |||
| [[5/4]] | |||
| {{Monzo| -2 0 1 }} | |||
| 4.322 | |||
|- | |||
| Harmonic seventh | |||
| [[7/4]] | |||
| {{Monzo| -2 0 0 1 }} | |||
| 4.807 | |||
|} | |||
== History and terminology == | |||
In general mathematics, this measurement is known as ''log-product complexity''. With respect to microtonal tuning, this measurement was first described by [[James Tenney]], who himself called it ''harmonic distance''.<ref>[https://www.plainsound.org/pdfs/JC&ToH.pdf ''John Cage and the Theory of Harmony'']. James Tenney. </ref><ref>[https://zh.booksc.eu/book/68954431/f87a1d ''On the Conception and Measure of Consonance'']. Alex Wand. </ref><ref>[https://scholar.sun.ac.za/bitstream/handle/10019.1/98644/brand_signal_2016.pdf?sequence=2&isAllowed=y ''A Signal-Based Model of Teleology in Tonal Music'']. Mark André Brand. p. 28. "Tenney's measure of ''harmonic distance'' (Hd) is thus singled out as perhaps his most 'crucial development', affording him the means towards 'compactness'. His is a Manhattan, rather than Euclidean metric, defined as {{nowrap|Hd(''a''/''b'') {{=}} ''k'' log(''ab'')}}, with ''a''/''b'' the maximally reduced ratio representing the frequency difference, and {{nowrap|''k'' {{=}} 1}} indicating measure in octaves."</ref> This terminology was also used in [[Paul Erlich]]'s paper [[A Middle Path]]<ref>Wherein Erlich writes: "This is why, in Tenney’s terminology, the taxicab distance an interval traverses in his lattice is the 'Harmonic Distance' of that interval."</ref>. | |||
== See also == | |||
* [[Generalized Tenney norms and Tp interval space|Generalized Tenney norms and T<sub>''p''</sub> interval space]] | |||
== References == | |||
<references /> | |||
[[Category:Consonance and dissonance]] | |||
[[Category:Harmonic entropy]] | |||
[[Category:Interval complexity measures]] | |||
[[Category:Tenney-weighted measures]] | |||