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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | 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. |
| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-01 07:42:43 UTC</tt>.<br>
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| : The original revision id was <tt>146185801</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">If p/q is a positive rational number reduced to its lowest terms, then the **Tenney height**, named for [[James Tenney]] who proposed it, is the integer pq. Often it is more convenient instead to take the logarithm (usually base 2) of the height. In either form it is widely used as a measure of inharmonicity and/or complexity for intervals.
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| The [[Tenney Height|Tenney height]] of a [[monzo]] is given by | | == Computation == |
| [[code type=text]]
| | === Ratio form === |
| || |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p) |ep|
| | The Tenney norm of a ratio ''n''/''d'' is given by |
| [[code]]
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| ==Examples==
| | $$\log_2 (nd) $$ |
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| || **interval name** || **frequency ratio** || **ket vector** || **Tenney Height** ||
| | === Vector form === |
| || prime || 1/1 || |0> || 0 ||
| | 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 |
| || octave || 2/1 || |1> || 1 ||
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| || just perfect fifth || 3/2 || |-1 1> || log2(6) = 2.585 ||
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| || just major third || 5/4 || |-2 0 1> || log2(20) = 4.322 ||
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| || harmonic seventh || 7/4 || |-2 0 0 1> || log2(28) = 4.807 ||
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| | $$ |
| | \begin{align} |
| | \norm{H \vec m}_1 &= \abs{m_1} + \abs{m_2} \log_2 (3) + \ldots + \abs{m_{\pi (p)}} \log_2 (p) \\ |
| | &= \log_2\left(2^{\abs{m_1}} \cdot 3^{\abs{m_2}} \cdot \ldots \cdot p^{\abs{m_{\pi (p)}}}\right) |
| | \end{align} |
| | $$ |
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| //see also discussion on http://lumma.org/music/theory/TuningFAQ.txt section CONSONANCE AND DISSONANCE//
| | where ''H'' is the transformation matrix such that, for the prime basis {{nowrap| ''Q'' {{=}} {{val| 2 3 5 … ''p'' }} }}, |
| </pre></div>
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| <h4>Original HTML content:</h4>
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| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Tenney Height</title></head><body>If p/q is a positive rational number reduced to its lowest terms, then the <strong>Tenney height</strong>, named for <a class="wiki_link" href="/James%20Tenney">James Tenney</a> who proposed it, is the integer pq. Often it is more convenient instead to take the logarithm (usually base 2) of the height. In either form it is widely used as a measure of inharmonicity and/or complexity for intervals.<br />
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| <br />
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| The <a class="wiki_link" href="/Tenney%20Height">Tenney height</a> of a <a class="wiki_link" href="/monzo">monzo</a> is given by<br />
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| <!-- ws:start:WikiTextCodeRule:0:
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| &lt;pre class=&quot;text&quot;&gt;|| |e2 e3 ... ep&amp;gt; || = |e2| + log2(3)|e3| + ... + log2(p) |ep|&lt;/pre&gt;
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| * GeSHi (C) 2004 - 2007 Nigel McNie, 2007 - 2008 Benny Baumann
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| | $$ H = \operatorname {diag} (\log_2 (Q)) $$ |
| </style><pre class="text">|| |e2 e3 ... ep&gt; || = |e2| + log2(3)|e3| + ... + log2(p) |ep|</pre>
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| <!-- ws:end:WikiTextCodeRule:0 --><br />
| | == Examples == |
| <!-- ws:start:WikiTextHeadingRule:1:&lt;h2&gt; --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:1 -->Examples</h2>
| | {| class="wikitable center-2" |
| <br />
| | |- |
| | ! 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 |
| | |} |
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| | == 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>. |
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| <table class="wiki_table">
| | == See also == |
| <tr>
| | * [[Generalized Tenney norms and Tp interval space|Generalized Tenney norms and T<sub>''p''</sub> interval space]] |
| <td><strong>interval name</strong><br />
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| </td>
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| <td><strong>frequency ratio</strong><br />
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| </td>
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| <td><strong>ket vector</strong><br />
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| </td>
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| <td><strong>Tenney Height</strong><br />
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| </td>
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| </tr>
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| <tr>
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| <td>prime<br />
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| </td>
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| <td>1/1<br />
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| </td>
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| <td>|0&gt;<br />
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| </td>
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| <td>0<br />
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| </td>
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| </tr>
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| <tr>
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| <td>octave<br />
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| </td>
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| <td>2/1<br />
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| </td>
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| <td>|1&gt;<br />
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| </td>
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| <td>1<br />
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| </td>
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| </tr>
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| <tr>
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| <td>just perfect fifth<br />
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| </td>
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| <td>3/2<br />
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| </td>
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| <td>|-1 1&gt;<br />
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| </td>
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| <td>log2(6) = 2.585<br />
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| </td>
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| </tr>
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| <tr>
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| <td>just major third<br />
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| </td>
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| <td>5/4<br />
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| </td>
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| <td>|-2 0 1&gt;<br />
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| </td>
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| <td>log2(20) = 4.322<br />
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| </td>
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| </tr>
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| <tr>
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| <td>harmonic seventh<br />
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| </td>
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| <td>7/4<br />
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| </td>
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| <td>|-2 0 0 1&gt;<br />
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| </td>
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| <td>log2(28) = 4.807<br />
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| </td>
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| </tr>
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| </table>
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| <br />
| | == References == |
| <br />
| | <references /> |
| <em>see also discussion on <!-- ws:start:WikiTextUrlRule:99:http://lumma.org/music/theory/TuningFAQ.txt --><a class="wiki_link_ext" href="http://lumma.org/music/theory/TuningFAQ.txt" rel="nofollow">http://lumma.org/music/theory/TuningFAQ.txt</a><!-- ws:end:WikiTextUrlRule:99 --> section CONSONANCE AND DISSONANCE</em></body></html></pre></div>
| | |
| | [[Category:Consonance and dissonance]] |
| | [[Category:Harmonic entropy]] |
| | [[Category:Interval complexity measures]] |
| | [[Category:Tenney-weighted measures]] |