Tp tuning

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[[toc|flat]]
=Definition= 
**Lp tuning** is a generalzation of [[TOP tuning|TOP]] and [[Tenney-Euclidean tuning|TE]] tuning. If p ≥ 1, define the Lp norm, which we may also call the Lp complexity, of any monzo in weighted coordinates b as
[[math]]
|| |b_2 \ b_3 \ ... \ b_k> ||_p = (|b_2|^p + |b_3|^p + ... + |b_k|^p)^{1/p}
[[math]]
where 2, 3, ... k are the primes up to k in order. In unweighted coordinates, this would be, for unweighted monzo m,
[[math]]|| |m_2 \ m_3 \ ... \ m_k> ||_p = (|\log_2(2) m_2|^p + |\log_2(3)m_3|^p + ... + |\log_2(k) m_k|^p)^{1/p}
[[math]]
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; 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. Given Lp(S) in a group G containing 2, we may define a coresponding pure-octaves tuning POLp(S) by dividing by the tuning of 2: POLp(S) = 1200 Lp(S)/Lp(S)(2). When p = 2, POL2 tuning generalizes POTE tuning.

=Dual norm= 
We can extend the Lp norm on monzos to a [[http://en.wikipedia.org/wiki/Normed_vector_space|vector space norm]] on [[Monzos and Interval Space|interval space]], thereby defining the real normed interval space Lp. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full p-limit will be the whole of Lp but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a [[http://en.wikipedia.org/wiki/Dual_norm|dual norm]]. If r1, r2, ... rn are a set of generators for G, which in particular could be a normal list and so define [[Smonzos and Svals|smonzos]] for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the normal G generator list, then <cents(r1) cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the Lp tuning Lp(S).

=Applying the Hahn-Banach theorem= 
Suppose T = Lp(S) is an Lp tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings of S since T is the Lp tuning. By the [[http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem|Hahn–Banach theorem]], Ɛ can be extended to an element Ƹ of the full p-limit tuning space with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. Additionally, due to a [[http://www.math.unl.edu/~s-bbockel1/928/node25.html|corollary of Hahn-Banach]], the set of such error maps valid for S can be extended to a larger set which is valid for an extended temperament S*; this temperament S* will be of rank greater than or equal to S, and will share the same kernel.

The norm of the full p-limit Ƹ, ||Ƹ||, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit JIP, must equal the Lp tuning for S*. Thus to find the Lp tuning of S for the group G, we may first find the Lp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.

Note that while the Hahn-Banach theorem is usually proven using the axiom of choice and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.

=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, and the L2 error, which is E2(S) for the temperament S, and which is approximately proportional to [[Tenney-Euclidean temperament measures#TE%20error|TE error]].

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 L2 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 POL2 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, gives an error of 1.7414 cents.

Original HTML content:

<html><head><title>Tp tuning</title></head><body><!-- ws:start:WikiTextTocRule:9:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#Dual norm">Dual norm</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Applying the Hahn-Banach theorem">Applying the Hahn-Banach theorem</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#L2 tuning">L2 tuning</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: -->
<!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:1 -->Definition</h1>
 <strong>Lp tuning</strong> is a generalzation of <a class="wiki_link" href="/TOP%20tuning">TOP</a> and <a class="wiki_link" href="/Tenney-Euclidean%20tuning">TE</a> tuning. If p ≥ 1, define the Lp norm, which we may also call the Lp complexity, of any monzo in weighted coordinates b as<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]&lt;br/&gt;
|| |b_2 \ b_3 \ ... \ b_k&gt; ||_p = (|b_2|^p + |b_3|^p + ... + |b_k|^p)^{1/p}&lt;br/&gt;[[math]]
 --><script type="math/tex">|| |b_2 \ b_3 \ ... \ b_k> ||_p = (|b_2|^p + |b_3|^p + ... + |b_k|^p)^{1/p}</script><!-- ws:end:WikiTextMathRule:0 --><br />
where 2, 3, ... k are the primes up to k in order. In unweighted coordinates, this would be, for unweighted monzo m,<br />
<a class="wiki_link" href="/math">math</a>|| |m_2 \ m_3 \ ... \ m_k&gt; ||_p = (|\log_2(2) m_2|^p + |\log_2(3)m_3|^p + ... + |\log_2(k) m_k|^p)^{1/p}<br />
<a class="wiki_link" href="/math">math</a><br />
If q is any positive rational number, ||q||_p is the Lp norm defined by the monzo.<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; 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. Given Lp(S) in a group G containing 2, we may define a coresponding pure-octaves tuning POLp(S) by dividing by the tuning of 2: POLp(S) = 1200 Lp(S)/Lp(S)(2). When p = 2, POL2 tuning generalizes POTE tuning.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="Dual norm"></a><!-- ws:end:WikiTextHeadingRule:3 -->Dual norm</h1>
 We can extend the Lp norm on monzos to a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow">vector space norm</a> on <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, thereby defining the real normed interval space Lp. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full p-limit will be the whole of Lp but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dual_norm" rel="nofollow">dual norm</a>. If r1, r2, ... rn are a set of generators for G, which in particular could be a normal list and so define <a class="wiki_link" href="/Smonzos%20and%20Svals">smonzos</a> for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the normal G generator list, then &lt;cents(r1) cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the Lp tuning Lp(S).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc2"><a name="Applying the Hahn-Banach theorem"></a><!-- ws:end:WikiTextHeadingRule:5 -->Applying the Hahn-Banach theorem</h1>
 Suppose T = Lp(S) is an Lp tuning for the temperament S, and J is the JI tuning. These are both elements of G-tuning space, which are linear functionals on G-interval space, and hence the error map Ɛ = T - J is also. The norm ||Ɛ|| of Ɛ is minimal among all error maps for tunings of S since T is the Lp tuning. By the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem" rel="nofollow">Hahn–Banach theorem</a>, Ɛ can be extended to an element Ƹ of the full p-limit tuning space with the same norm; that is, so that ||Ɛ|| = ||Ƹ||. Additionally, due to a <a class="wiki_link_ext" href="http://www.math.unl.edu/~s-bbockel1/928/node25.html" rel="nofollow">corollary of Hahn-Banach</a>, the set of such error maps valid for S can be extended to a larger set which is valid for an extended temperament S*; this temperament S* will be of rank greater than or equal to S, and will share the same kernel.<br />
<br />
The norm of the full p-limit Ƹ, ||Ƹ||, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit JIP, must equal the Lp tuning for S*. Thus to find the Lp tuning of S for the group G, we may first find the Lp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.<br />
<br />
Note that while the Hahn-Banach theorem is usually proven using the axiom of choice and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:7:&lt;h1&gt; --><h1 id="toc3"><a name="L2 tuning"></a><!-- ws:end:WikiTextHeadingRule:7 -->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, and the L2 error, which is E2(S) for the temperament S, and which is approximately proportional to <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE%20error">TE error</a>.<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 L2 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 POL2 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, gives an error of 1.7414 cents.</body></html>