Monzos and interval space: Difference between revisions

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The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of |4 -1 -1&gt;. In weighted coordinates, that becomes |4 -log2(3) -log2(5)&gt;, approximately |4 -1.585 -2.322&gt;. The TE norm is therefore sqrt(1^2 + log2(3)^2 + log2(5)^2) ~ sqrt(23.903) = 4.889.

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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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Monzos and Interval Space&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;p-limit&lt;/a&gt; rational number q can by definition be factored into primes of size less than or equal to p, giving q = 2^e2 3^e3 ... p^ep, where the exponents are integers (positive, negative, or zero.) This is often written in &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/Ket.html" rel="nofollow"&gt;ket vector&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Bra-ket_notation" rel="nofollow"&gt;wp&lt;/a&gt;) notation as |e2 e3 ... ep&amp;gt;, in which case it is called a &lt;strong&gt;monzo&lt;/strong&gt;, where the name refers to the enthusiastic advocacy of &lt;a class="wiki_link" href="/Joe%20Monzo"&gt;Joe Monzo&lt;/a&gt;.&lt;br /&gt;

The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of |4 -1 -1&amp;gt;. In weighted coordinates, that becomes |4 -log2(3) -log2(5)&amp;gt;, approximately |4 -1.585 -2.322&amp;gt;. The TE norm is therefore sqrt(1^2 + log2(3)^2 + log2(5)^2) ~ sqrt(23.903) = 4.889.&lt;br /&gt;

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