Tenney–Euclidean temperament measures: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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>2013-11-26 11:43:22 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-11-26 12:28:25 UTC</tt>.<br>

: The original revision id was <tt>~~472438896~~</tt>.<br>

: The original revision id was <tt>472454310</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>

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\displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi

[[math]]

is an adjusted error which makes the error of a rank r temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament. ψ can be related to TE error as it appears on Graham Breed's [[http://x31eq.com/temper/|temperament finder]], which we will call ~~"Graham~~ error"; if G denotes ~~Graham~~ error for a temperament, then

is an adjusted error which makes the error of a rank r temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament. ψ can be related to TE error as it appears on Graham Breed's [[http://x31eq.com/temper/|temperament finder]], which we will call RMS error; if G denotes the RMS error for a temperament, then

[[math]]

\displaystyle G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi

[[math]]

~~Graham~~ error and ψ error both have the advantage that higher rank temperament error corresponds directly to rank one error, but the ~~Graham~~ normalization has the further advantage that in the rank one case, G = sin θ, where θ is the angle between J and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200 sin θ, TE error as it appears on the temperament finder pages. ~~Graham~~ error may also be found using the TE tuning map; if T is the tuning map, then

RMS error and ψ error both have the advantage that higher rank temperament error corresponds directly to rank one error, but the RMS normalization has the further advantage that in the rank one case, G = sin θ, where θ is the angle between J and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200 sin θ, TE error as it appears on the temperament finder pages. RMS error may also be found using the TE tuning map; if T is the tuning map, then

[[math]]

\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}}

\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}} = || T-J ||

[[math]]

where the dot represents the ordinary dot product. If T is denominated in cents, then J should be also, so that J = <1200 1200 ... 1200|.</pre></div>

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\displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi&lt;br/&gt;[[math]]

--><script type="math/tex">\displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi</script><br />

is an adjusted error which makes the error of a rank r temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament. ψ can be related to TE error as it appears on Graham Breed's <a class="wiki_link_ext" href="http://x31eq.com/temper/" rel="nofollow">temperament finder</a>, which we will call ~~&quot;Graham~~ error~~&quot;~~; if G denotes ~~Graham~~ error for a temperament, then<br />

is an adjusted error which makes the error of a rank r temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament. ψ can be related to TE error as it appears on Graham Breed's <a class="wiki_link_ext" href="http://x31eq.com/temper/" rel="nofollow">temperament finder</a>, which we will call RMS error; if G denotes the RMS error for a temperament, then<br />

<!-- ws:start:WikiTextMathRule:3:

[[math]]&lt;br/&gt;

\displaystyle G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi&lt;br/&gt;[[math]]

--><script type="math/tex">\displaystyle G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi</script><br />

~~Graham~~ error and ψ error both have the advantage that higher rank temperament error corresponds directly to rank one error, but the ~~Graham~~ normalization has the further advantage that in the rank one case, G = sin θ, where θ is the angle between J and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200 sin θ, TE error as it appears on the temperament finder pages. ~~Graham~~ error may also be found using the TE tuning map; if T is the tuning map, then<br />

RMS error and ψ error both have the advantage that higher rank temperament error corresponds directly to rank one error, but the RMS normalization has the further advantage that in the rank one case, G = sin θ, where θ is the angle between J and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200 sin θ, TE error as it appears on the temperament finder pages. RMS error may also be found using the TE tuning map; if T is the tuning map, then<br />

<!-- ws:start:WikiTextMathRule:4:

[[math]]&lt;br/&gt;

\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}}&lt;br/&gt;[[math]]

\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}} = || T-J ||&lt;br/&gt;[[math]]

--><script type="math/tex">\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}}</script><br />

--><script type="math/tex">\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}} = || T-J ||</script><br />

where the dot represents the ordinary dot product. If T is denominated in cents, then J should be also, so that J = &lt;1200 1200 ... 1200|.</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2013-11-26 11:43:22 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-11-26 12:28:25 UTC</tt>.<br>
-: The original revision id was <tt>472438896</tt>.<br>
+: The original revision id was <tt>472454310</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>
@@ Line 38: / Line 38: @@
 \displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi
 [[math]]
-is an adjusted error which makes the error of a rank r temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament. ψ can be related to TE error as it appears on Graham Breed's [[http://x31eq.com/temper/|temperament finder]], which we will call "Graham error"; if G denotes Graham error for a temperament, then
+is an adjusted error which makes the error of a rank r temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament. ψ can be related to TE error as it appears on Graham Breed's [[http://x31eq.com/temper/|temperament finder]], which we will call RMS error; if G denotes the RMS error for a temperament, then
 [[math]]
 \displaystyle G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi
 [[math]]
-Graham error and ψ error both have the advantage that higher rank temperament error corresponds directly to rank one error, but the Graham normalization has the further advantage that in the rank one case,  G = sin θ, where θ is the angle between J and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200 sin θ, TE error as it appears on the temperament finder pages. Graham error may also be found using the TE tuning map; if T is the tuning map, then
+RMS error and ψ error both have the advantage that higher rank temperament error corresponds directly to rank one error, but the RMS normalization has the further advantage that in the rank one case,  G = sin θ, where θ is the angle between J and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200 sin θ, TE error as it appears on the temperament finder pages. RMS error may also be found using the TE tuning map; if T is the tuning map, then
 [[math]]
-\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}}
+\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}} = || T-J ||
 [[math]]
 where the dot represents the ordinary dot product. If T is denominated in cents, then J should be also, so that J = &lt;1200 1200 ... 1200|.</pre></div>
@@ Line 84: / Line 84: @@
 \displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
-is an adjusted error which makes the error of a rank r temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament. ψ can be related to TE error as it appears on Graham Breed's &lt;a class="wiki_link_ext" href="http://x31eq.com/temper/" rel="nofollow"&gt;temperament finder&lt;/a&gt;, which we will call &amp;quot;Graham error&amp;quot;; if G denotes Graham error for a temperament, then&lt;br /&gt;
+is an adjusted error which makes the error of a rank r temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament. ψ can be related to TE error as it appears on Graham Breed's &lt;a class="wiki_link_ext" href="http://x31eq.com/temper/" rel="nofollow"&gt;temperament finder&lt;/a&gt;, which we will call RMS error; if G denotes the RMS error for a temperament, then&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:3:
 [[math]]&amp;lt;br/&amp;gt;
 \displaystyle G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\displaystyle G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:3 --&gt;&lt;br /&gt;
-Graham error and ψ error both have the advantage that higher rank temperament error corresponds directly to rank one error, but the Graham normalization has the further advantage that in the rank one case,  G = sin θ, where θ is the angle between J and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200 sin θ, TE error as it appears on the temperament finder pages. Graham error may also be found using the TE tuning map; if T is the tuning map, then&lt;br /&gt;
+RMS error and ψ error both have the advantage that higher rank temperament error corresponds directly to rank one error, but the RMS normalization has the further advantage that in the rank one case,  G = sin θ, where θ is the angle between J and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200 sin θ, TE error as it appears on the temperament finder pages. RMS error may also be found using the TE tuning map; if T is the tuning map, then&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:4:
 [[math]]&amp;lt;br/&amp;gt;
-\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}}&amp;lt;br/&amp;gt;[[math]]
+\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}} = || T-J ||&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:4 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}} = || T-J ||&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:4 --&gt;&lt;br /&gt;
 where the dot represents the ordinary dot product. If T is denominated in cents, then J should be also, so that J = &amp;lt;1200 1200 ... 1200|.&lt;/body&gt;&lt;/html&gt;</pre></div>

Tenney–Euclidean temperament measures: Difference between revisions

Wikispaces>genewardsmith