Vals and tuning space: Difference between revisions
Wikispaces>genewardsmith **Imported revision 385525542 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 515965054 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt> | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-07-10 12:55:09 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>515965054</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | 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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It useful to renormalize to the RMS (root mean square) instead, which requires dividing the above by sqrt(n), where n = pi(p) is the number of primes up to p. This is the TE, or Tenney-Euclidean, norm. | It useful to renormalize to the RMS (root mean square) instead, which requires dividing the above by sqrt(n), where n = pi(p) is the number of primes up to p. This is the TE, or Tenney-Euclidean, norm. | ||
It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or JIP, which in weighted coordinates is <1 1 1 ... 1|. It has the property that if M is a monzo in weighted coordinates, then <JIP|M>, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = <1 log2(3) ... log2(p)|, and applied to a monzo this gives the log base two of the corresponding interval. | It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or [[JIP]], which in weighted coordinates is <1 1 1 ... 1|. It has the property that if M is a monzo in weighted coordinates, then <JIP|M>, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = <1 log2(3) ... log2(p)|, and applied to a monzo this gives the log base two of the corresponding interval. | ||
=Example= | =Example= | ||
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It useful to renormalize to the RMS (root mean square) instead, which requires dividing the above by sqrt(n), where n = pi(p) is the number of primes up to p. This is the TE, or Tenney-Euclidean, norm.<br /> | It useful to renormalize to the RMS (root mean square) instead, which requires dividing the above by sqrt(n), where n = pi(p) is the number of primes up to p. This is the TE, or Tenney-Euclidean, norm.<br /> | ||
<br /> | <br /> | ||
It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or JIP, which in weighted coordinates is &lt;1 1 1 ... 1|. It has the property that if M is a monzo in weighted coordinates, then &lt;JIP|M&gt;, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = &lt;1 log2(3) ... log2(p)|, and applied to a monzo this gives the log base two of the corresponding interval.<br /> | It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or <a class="wiki_link" href="/JIP">JIP</a>, which in weighted coordinates is &lt;1 1 1 ... 1|. It has the property that if M is a monzo in weighted coordinates, then &lt;JIP|M&gt;, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = &lt;1 log2(3) ... log2(p)|, and applied to a monzo this gives the log base two of the corresponding interval.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc3"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:8 -->Example</h1> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc3"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:8 -->Example</h1> | ||