Tenney–Euclidean tuning: Difference between revisions

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If T = JP is the TE tuning map, then a corresponding pure-octaves map can be found by [[http://en.wikipedia.org/wiki/Scalar_multiplication|scalar multiplication]], T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J. We might call pure-octaves Tenney-Euclidean tuning the POTE tuning.

We may also do the same things starting from unweighted vals. This leads to a different tuning, the [[Fractional monzos|Frobenius tuning]], which is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are [[Fractional monzos|fractional monzos]]. The projection map therefore, like the [[Wedgies and Multivals|wedgie]], defines a completely canonical object not depending on any arbitrary definition (eg how Hermite normal form or LLL reduction is specifically defined) which corresponds 1-1 with temperaments, and which automatically takes care of "torsion problems". It also may be found starting either from a set of vals or a set of commas, since if Q is the projection map found by treating monzos in the same way as vals, P = I-Q is the same projection map as would be found if starting from a set of vals defining the same temperament.

If T = JP is the TE tuning map, then a corresponding pure-octaves map can be found by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Scalar_multiplication" rel="nofollow"&gt;scalar multiplication&lt;/a&gt;, T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J. We might call pure-octaves Tenney-Euclidean tuning the POTE tuning.&lt;br /&gt;

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We may also do the same things starting from unweighted vals. This leads to a different tuning, the &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;Frobenius tuning&lt;/a&gt;, which is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;fractional monzos&lt;/a&gt;. The projection map therefore, like the &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt;, defines a completely canonical object not depending on any arbitrary definition (eg how Hermite normal form or LLL reduction is specifically defined) which corresponds 1-1 with temperaments, and which automatically takes care of &amp;quot;torsion problems&amp;quot;. It also may be found starting either from a set of vals or a set of commas, since if Q is the projection map found by treating monzos in the same way as vals, P = I-Q is the same projection map as would be found if starting from a set of vals defining the same temperament.&lt;br /&gt;

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