Generator complexity: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-07-14 20:17:43 UTC</tt>.<br>

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Suppose A = &lt;0 A₃ A₅ A₇ ... Ap| is the generator mapping val for a rank two temperament with P periods to the octave, and B = &lt;0 B₃ B₅ B₇ ... Bp| is the same val in weighted coordinates. For instance, &lt;0 1 -2 -2| is the generator mapping val for seven limit [[pajara]], and &lt;0 1/log2(3) -2/log2(5) -2/log2(7)| ≅ &lt;0 0.631 -0.831 -0.712| is the val in weighted coordinates. For any vector V, let max(V) - min(V) = span(V). The //generator complexity// of the temperament is P span(B). In the case of pajara, which has two periods to the octave, this would be 2*(0.631 - (-0.861)) = 2.984. This can also be described in terms of the wedgie W of the temperament, as span(2∨W), which is the span of 0 followed by the first n-1 elements of W, where n is the number of primes in the p-limit.

Generator complexity satisfies the inequality, for any p-limit interval I, G(I) ≤ C KE(I), where C is the generator complexity of the temperament, G(I) is the number of generator steps, times P, required to reach the tempered version of I, and KE(I) is the [[Kees height|Kees expressibility]] of I. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and KE(5/4) = log2(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4 (5/4 = (4/3)^2 * 45/64), and P = 2, so that G(5/4) = 2*2.

This inequality can be used to give an alternative definition of generator complexity: C = sup G(I)/KE(I) over non-octave intervals, where KE(I)&gt;0. A related definition can be extended to higher ranks: since the [[Tenney-Euclidean metrics#The OETES|OETES]] in the case of a rank two temperament is proportional (albeit with a different proportionality factor for each temperament) to G(I), we can define a complexity measure for any rank of temperament by C = sup OETES(I)/KE(I).

Generator complexity has the nice property that for any MOS of size N, floor(N/(C KE(I))) intervals with pitch class corresponding to I are guaranteed to exist in the MOS. Generator complexity is also useful in making complete searches using [[the wedgie]] for temperaments below a certain complexity and badness bounds, allowing for a more efficient search.

=Generator complexity and Kees expressibility=

The following proof is due to Mike Battaglia.

If m = |m2 m3 m5 ... mp&gt; is a vector with weighted coordinates in interval space, then KE(m), the Kees  expressibility of m, is (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2. The "2" coordinate, m2, plays  no role in Kees expressibility, so we may replace it with anything we choose. If we replace it with -e3 -e5-...-ep, we may define expressibility in terms of the L1 norm, as || |-e3-e5-...-ep  e3  e5 ... ep&gt;  ||/2.

For any vector space X with a subspace A, we may define a quotient space X/A as the equivalence classes of  vectors in X where two vectors are equivalent iff their difference lies in A. Then we have a short exact  sequence 0 → A → X → X/A → 0. Taking the duals of this gives us 0 → (X/A)* → X* → A* → 0. The annihilator  of A is the subspace A⁀ of X* consisting of those functionals f such that &lt;f|A&gt; equals 0; that is, it is the subspace of all the functionals f such that &lt;f|a&gt; equals 0 for every a in A. There is a natural isomorphism between the annihilator A⁀ of A and the dual of the quotient (X/A)*, and also between X*/A⁀ and A*.

Now suppose X is a finite dimensional real normed vector space. A is then also a finite dimensional real normed vector space, inheriting its norm from X, and X/A is a finite dimensional real normed vector space, with a norm given by, for an equivalence class [x], ||[x]|| equals inf {||x + a||, a∈ A}. Algebraically X is (noncanonically) isomorphic to X*, but in general they are no longer isomorphic as normed spaces. Instead, we have the [[dual norm]] on X*, defined by setting, over all nonzero x ∈ X, ||f||* = sup &lt;f|x&gt;/||x||. Under the dual norm X* is also a finite dimensional normed vector space, A⁀ is isometrically isomorphic to (X/A)*, and X*/A⁀ is isometrically isomorphic to A*.

In the situation which concerns us, X is the p-limit interval space of dimension n under a norm of one half times the L1 norm, A is a subspace of dimension n-1, whose coordinates sum to 0; hence A can be described as having the one-dimensional subspace A⁀ = {kJ}, where J is the [[JIP]], as its annihilator. X has a norm of half the L1 norm, and hence X* has a norm of twice the L∞ norm. The norm on A* is defined by its isomorphism with X*/A⁀; the minimum defining inf {||f + kJ||}  occurs for the value of k where the maximum of f + kJ and minus the minimum of f + kj are the same. In that case, 2||f + kJ||_L∞ = span(f), which is the generator complexity of f. Hence generator complexity is the dual norm for Kees expressibility as a norm on pitch classes.

If B = &lt;0 B₃ B₅ B₇ ... Bp| is the generator mapping val in weighted coordinates, and P is the period, then the //STD complexity// (a term due to Graham Breed) is P STD(B), where "STD" means the standard deviation. If μ(V) is the mean of the components of the vector V, and J is the [[JIP]] &lt;1 1 1 ... 1|, then  ₱(V) = V - μ(V)J is the projection of V onto the subspace of vectors with zero mean value. We have STD(V) = √( ₱(V) ∙ ₱(V) / dim(V)), where dim(V) is the dimension of V and the " ∙ " denotes the dot product. If M = [M0, M1] is the [[Temperament Mapping Matrices (M-maps)|mapping matrix]] in weighted coordinates in the standard [[Normal lists#x-Normal%20val%20lists|normal val list]] form, then we may express STD complexity as STDcom(M) = M0[1] STD(M1).

Associated to STD complexity is STD error. If S =  ₱(M0) ∧  ₱(M1), then STDerr(M) =  √(S ∙ S / dim(M1)*₱(M1) ∙  ₱(M1)).

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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Generator complexity&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:6:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt; | &lt;a href="#Generator complexity and Kees expressibility"&gt;Generator complexity and Kees expressibility&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt; | &lt;a href="#STD complexity"&gt;STD complexity&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definition&lt;/h1&gt;

Suppose A = &amp;lt;0 A₃ A₅ A₇ ... Ap| is the generator mapping val for a rank two temperament with P periods to the octave, and B = &amp;lt;0 B₃ B₅ B₇ ... Bp| is the same val in weighted coordinates. For instance, &amp;lt;0 1 -2 -2| is the generator mapping val for seven limit &lt;a class="wiki_link" href="/pajara"&gt;pajara&lt;/a&gt;, and &amp;lt;0 1/log2(3) -2/log2(5) -2/log2(7)| ≅ &amp;lt;0 0.631 -0.831 -0.712| is the val in weighted coordinates. For any vector V, let max(V) - min(V) = span(V). The &lt;em&gt;generator complexity&lt;/em&gt; of the temperament is P span(B). In the case of pajara, which has two periods to the octave, this would be 2*(0.631 - (-0.861)) = 2.984. This can also be described in terms of the wedgie W of the temperament, as span(2∨W), which is the span of 0 followed by the first n-1 elements of W, where n is the number of primes in the p-limit. &lt;br /&gt;

Generator complexity satisfies the inequality, for any p-limit interval I, G(I) ≤ C KE(I), where C is the generator complexity of the temperament, G(I) is the number of generator steps, times P, required to reach the tempered version of I, and KE(I) is the &lt;a class="wiki_link" href="/Kees%20height"&gt;Kees expressibility&lt;/a&gt; of I. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and KE(5/4) = log2(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4 (5/4 = (4/3)^2 * 45/64), and P = 2, so that G(5/4) = 2*2.&lt;br /&gt;

This inequality can be used to give an alternative definition of generator complexity: C = sup G(I)/KE(I) over non-octave intervals, where KE(I)&amp;gt;0. A related definition can be extended to higher ranks: since the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics#The OETES"&gt;OETES&lt;/a&gt; in the case of a rank two temperament is proportional (albeit with a different proportionality factor for each temperament) to G(I), we can define a complexity measure for any rank of temperament by C = sup OETES(I)/KE(I).&lt;br /&gt;

Generator complexity has the nice property that for any MOS of size N, floor(N/(C KE(I))) intervals with pitch class corresponding to I are guaranteed to exist in the MOS. Generator complexity is also useful in making complete searches using &lt;a class="wiki_link" href="/the%20wedgie"&gt;the wedgie&lt;/a&gt; for temperaments below a certain complexity and badness bounds, allowing for a more efficient search.&lt;br /&gt;