Tp tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 472561674 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 472562392 - 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>2013-11-26 18: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-11-26 18:59:52 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>472562392</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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=T2 tuning= | =T2 tuning= | ||
In the special case where p = 2, the Tp norm for the full prime limit becomes the | In the special case where p = 2, the Tp norm for the full prime limit becomes the T2 norm, which when divided by the square root of the number n of primes in the prime limit, is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, giving TE complexity. Associated to this norm is T2 tuning extended to arbitrary JI groups, and [[Tenney-Euclidean temperament measures#TE error|RMS error]], which for a tuning map T is ||(T - J)/n ||_2 = ||T - J||. | ||
For an example, consider [[Chromatic pairs#Indium|indium temperament]], with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the T2 (TE) tuning map is <1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting <1200 1200 1200 1200 1200| gives <-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, is 1.7414, and dividing by sqrt(5) gives the RMS error, 0.77879 cents.</pre></div> | For an example, consider [[Chromatic pairs#Indium|indium temperament]], with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the T2 (TE) tuning map is <1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting <1200 1200 1200 1200 1200| gives <-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, is 1.7414, and dividing by sqrt(5) gives the RMS error, 0.77879 cents.</pre></div> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc3"><a name="T2 tuning"></a><!-- ws:end:WikiTextHeadingRule:8 -->T2 tuning</h1> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc3"><a name="T2 tuning"></a><!-- ws:end:WikiTextHeadingRule:8 -->T2 tuning</h1> | ||
In the special case where p = 2, the Tp norm for the full prime limit becomes the | In the special case where p = 2, the Tp norm for the full prime limit becomes the T2 norm, which when divided by the square root of the number n of primes in the prime limit, is the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean</a> norm, giving TE complexity. Associated to this norm is T2 tuning extended to arbitrary JI groups, and <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE error">RMS error</a>, which for a tuning map T is ||(T - J)/n ||_2 = ||T - J||.<br /> | ||
<br /> | <br /> | ||
For an example, consider <a class="wiki_link" href="/Chromatic%20pairs#Indium">indium temperament</a>, with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the <a class="wiki_link" href="/Tenney-Euclidean%20tuning">usual methods</a>, in particular the pseudoinverse, we find that the T2 (TE) tuning map is &lt;1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting &lt;1200 1200 1200 1200 1200| gives &lt;-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, is 1.7414, and dividing by sqrt(5) gives the RMS error, 0.77879 cents.</body></html></pre></div> | For an example, consider <a class="wiki_link" href="/Chromatic%20pairs#Indium">indium temperament</a>, with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the <a class="wiki_link" href="/Tenney-Euclidean%20tuning">usual methods</a>, in particular the pseudoinverse, we find that the T2 (TE) tuning map is &lt;1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting &lt;1200 1200 1200 1200 1200| gives &lt;-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, is 1.7414, and dividing by sqrt(5) gives the RMS error, 0.77879 cents.</body></html></pre></div> | ||