Tp tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 348184558 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 350566972 - 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>2012- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-05 23:52:05 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>350566972</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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If q is any positive rational number, ||q||_p is the Lp norm defined by the monzo. | If q is any positive rational number, ||q||_p is the Lp norm defined by the monzo. | ||
For some just intonation group G, which is to say some finitely generated group of positive rational numbers which can be either a full prime-limit group or some subgroup of such a group, a regular temperament tuning T for an abstract temperament S is defined by a linear map from monzos belonging to G to a value in cents, such that T(c) = 0 for any comma c of the temperament. We define the error of the tuning on q, Err(q), as |T(q) - cents(q)|, and if q ≠ 1, the //Lp proportional error// is PEp(q) = Err(q)/||q||_p. For any tuning T of the temperament, the set of PEp(q) for all q ≠ 1 in G is bounded, and hence has a least upper bound, the supremum PEps(T). The set of values PEps(T) is bounded below, and by continuity achieves its minimum value, which is the Lp error Ep(S) of the abstract temperament S. Any tuning achieving this minimum, so that PEps(T) = Ep(S), is an Lp tuning. Usually this tuning is unique, but in the case p = 1, called the [[TOP tuning]], it may not be. In this case we can chose a TOP tuning canonically by setting it to the limit as p tends to 1 of the Lp tuning, thereby defining a unique tuning Lp(S) for any abstract temperament S on any group G. | For some just intonation group G, which is to say some finitely generated group of positive rational numbers which can be either a full prime-limit group or some subgroup of such a group, a regular temperament tuning T for an abstract temperament S is defined by a linear map from monzos belonging to G to a value in cents, such that T(c) = 0 for any comma c of the temperament. We define the error of the tuning on q, Err(q), as |T(q) - cents(q)|, and if q ≠ 1, the //Lp proportional error// is PEp(q) = Err(q)/||q||_p. For any tuning T of the temperament, the set of PEp(q) for all q ≠ 1 in G is bounded, and hence has a least upper bound, the supremum PEps(T). The set of values PEps(T) is bounded below, and by continuity achieves its minimum value, which is the Lp error Ep(S) of the abstract temperament S; if we measure in cents as we've defined above, Ep(S) has units of cents. Any tuning achieving this minimum, so that PEps(T) = Ep(S), is an Lp tuning. Usually this tuning is unique, but in the case p = 1, called the [[TOP tuning]], it may not be. In this case we can chose a TOP tuning canonically by setting it to the limit as p tends to 1 of the Lp tuning, thereby defining a unique tuning Lp(S) for any abstract temperament S on any group G. | ||
=Dual norm= | =Dual norm= | ||
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=L2 tuning= | =L2 tuning= | ||
In the special case where p = 2, the Lp norm becomes the L2 norm, which is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups. Starting from a [[gencom]] for the temperament, we may find the tuning by the following proceedure: | In the special case where p = 2, the Lp norm becomes the L2 norm, which is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups, and the TE error, which is E2(S) for the temperament S. Starting from a [[gencom]] for the temperament, we may find the tuning by the following proceedure: | ||
# Convert the gencom into monzo form. | # Convert the gencom into monzo form. | ||
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# Multiply each monzo by an indeterminate, creating a parametrized n-dimensional weighted monzo, where n is the number of primes in the prime limit; call that M. | # Multiply each monzo by an indeterminate, creating a parametrized n-dimensional weighted monzo, where n is the number of primes in the prime limit; call that M. | ||
# Take the dot product of M with a vector <x1 x2 ... xn| consisting of n new indeterminates x1, x2, ..., xn; call that X. | # Take the dot product of M with a vector <x1 x2 ... xn| consisting of n new indeterminates x1, x2, ..., xn; call that X. | ||
# Maximize X^2 subject to the constraint that the | # Maximize X^2 subject to the constraint that the M∙M, the dot product of M with itself, is equal to 1. This is maximinzing a quadric subject to a quadratic constraint, and so is easily done via Lagrange multipliers. | ||
# The result of the maximization process is a positive definite quadratic form Q(x1, x2, ..., xn) in n indeterminates x1, x2, ..., xn. | # The result of the maximization process is a positive definite quadratic form Q(x1, x2, ..., xn) in n indeterminates x1, x2, ..., xn. | ||
# Find the point T on the subspace of G-tuning space spanned by the [[Gencom|gencom mapping]] closest to the JIP using the distance function defined by Q. As usual the JIP is <1 1 1 ... 1|. Since Q is a quadratic polynomial, this minimal distance is easily found by calculus methods. | # Find the point T on the subspace of G-tuning space spanned by the [[Gencom|gencom mapping]] closest to the JIP using the distance function defined by Q. As usual the JIP is <1 1 1 ... 1|. Since Q is a quadratic polynomial, this minimal distance is easily found by calculus methods. | ||
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If q is any positive rational number, ||q||_p is the Lp norm defined by the monzo. <br /> | If q is any positive rational number, ||q||_p is the Lp norm defined by the monzo. <br /> | ||
<br /> | <br /> | ||
For some just intonation group G, which is to say some finitely generated group of positive rational numbers which can be either a full prime-limit group or some subgroup of such a group, a regular temperament tuning T for an abstract temperament S is defined by a linear map from monzos belonging to G to a value in cents, such that T(c) = 0 for any comma c of the temperament. We define the error of the tuning on q, Err(q), as |T(q) - cents(q)|, and if q ≠ 1, the <em>Lp proportional error</em> is PEp(q) = Err(q)/||q||_p. For any tuning T of the temperament, the set of PEp(q) for all q ≠ 1 in G is bounded, and hence has a least upper bound, the supremum PEps(T). The set of values PEps(T) is bounded below, and by continuity achieves its minimum value, which is the Lp error Ep(S) of the abstract temperament S. Any tuning achieving this minimum, so that PEps(T) = Ep(S), is an Lp tuning. Usually this tuning is unique, but in the case p = 1, called the <a class="wiki_link" href="/TOP%20tuning">TOP tuning</a>, it may not be. In this case we can chose a TOP tuning canonically by setting it to the limit as p tends to 1 of the Lp tuning, thereby defining a unique tuning Lp(S) for any abstract temperament S on any group G.<br /> | For some just intonation group G, which is to say some finitely generated group of positive rational numbers which can be either a full prime-limit group or some subgroup of such a group, a regular temperament tuning T for an abstract temperament S is defined by a linear map from monzos belonging to G to a value in cents, such that T(c) = 0 for any comma c of the temperament. We define the error of the tuning on q, Err(q), as |T(q) - cents(q)|, and if q ≠ 1, the <em>Lp proportional error</em> is PEp(q) = Err(q)/||q||_p. For any tuning T of the temperament, the set of PEp(q) for all q ≠ 1 in G is bounded, and hence has a least upper bound, the supremum PEps(T). The set of values PEps(T) is bounded below, and by continuity achieves its minimum value, which is the Lp error Ep(S) of the abstract temperament S; if we measure in cents as we've defined above, Ep(S) has units of cents. Any tuning achieving this minimum, so that PEps(T) = Ep(S), is an Lp tuning. Usually this tuning is unique, but in the case p = 1, called the <a class="wiki_link" href="/TOP%20tuning">TOP tuning</a>, it may not be. In this case we can chose a TOP tuning canonically by setting it to the limit as p tends to 1 of the Lp tuning, thereby defining a unique tuning Lp(S) for any abstract temperament S on any group G.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc1"><a name="Dual norm"></a><!-- ws:end:WikiTextHeadingRule:4 -->Dual norm</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc1"><a name="Dual norm"></a><!-- ws:end:WikiTextHeadingRule:4 -->Dual norm</h1> | ||
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc2"><a name="L2 tuning"></a><!-- ws:end:WikiTextHeadingRule:6 -->L2 tuning</h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc2"><a name="L2 tuning"></a><!-- ws:end:WikiTextHeadingRule:6 -->L2 tuning</h1> | ||
In the special case where p = 2, the Lp norm becomes the L2 norm, which is the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean</a> norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups. Starting from a <a class="wiki_link" href="/gencom">gencom</a> for the temperament, we may find the tuning by the following proceedure:<br /> | In the special case where p = 2, the Lp norm becomes the L2 norm, which is the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean</a> norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups, and the TE error, which is E2(S) for the temperament S. Starting from a <a class="wiki_link" href="/gencom">gencom</a> for the temperament, we may find the tuning by the following proceedure:<br /> | ||
<br /> | <br /> | ||
<ol><li>Convert the gencom into monzo form.</li><li>Convert the monzos to weighted coordinates.</li><li>Multiply each monzo by an indeterminate, creating a parametrized n-dimensional weighted monzo, where n is the number of primes in the prime limit; call that M.</li><li>Take the dot product of M with a vector &lt;x1 x2 ... xn| consisting of n new indeterminates x1, x2, ..., xn; call that X.</li><li>Maximize X^2 subject to the constraint that the | <ol><li>Convert the gencom into monzo form.</li><li>Convert the monzos to weighted coordinates.</li><li>Multiply each monzo by an indeterminate, creating a parametrized n-dimensional weighted monzo, where n is the number of primes in the prime limit; call that M.</li><li>Take the dot product of M with a vector &lt;x1 x2 ... xn| consisting of n new indeterminates x1, x2, ..., xn; call that X.</li><li>Maximize X^2 subject to the constraint that the M∙M, the dot product of M with itself, is equal to 1. This is maximinzing a quadric subject to a quadratic constraint, and so is easily done via Lagrange multipliers.</li><li>The result of the maximization process is a positive definite quadratic form Q(x1, x2, ..., xn) in n indeterminates x1, x2, ..., xn.</li><li>Find the point T on the subspace of G-tuning space spanned by the <a class="wiki_link" href="/Gencom">gencom mapping</a> closest to the JIP using the distance function defined by Q. As usual the JIP is &lt;1 1 1 ... 1|. Since Q is a quadratic polynomial, this minimal distance is easily found by calculus methods.</li><li>Since the primes may or may not be in the group G, the mapping of primes in T may or may not make sense by itself; however, now unweight T and apply it to the generators of the gencom, and obtain the TE tuning of those generators, which defines the TE tuning for the temperament defined by the gencom.</li></ol></body></html></pre></div> | ||