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. 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=
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).

=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:

# Convert the gencom into monzo form.
# Convert the monzos to weighted coordinates.
# 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 consisting of n new indeterminates x1, x2, ..., xn, and take the dot product of that with M; call that X.
# 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 this 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.
# Find the point T on the subspace of G-tuning space spanned by the [[Gencom|gencom mapping]] closest to the JIP, which as usual is <1 1 1 ... 1|.
# 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.

Original HTML content:

<html><head><title>Tp tuning</title></head><body><!-- ws:start:WikiTextTocRule:8:&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:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#Dual norm">Dual norm</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#L2 tuning">L2 tuning</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: -->
<!-- ws:end:WikiTextTocRule:12 --><br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:2 -->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 />
<!-- ws:start:WikiTextMathRule:1:
[[math]]&lt;br/&gt;
|| |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}&lt;br/&gt;[[math]]
 --><script type="math/tex">|| |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}</script><!-- ws:end:WikiTextMathRule:1 --><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. 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 />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc1"><a name="Dual norm"></a><!-- ws:end:WikiTextHeadingRule:4 -->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: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 />
<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 consisting of n new indeterminates x1, x2, ..., xn, and take the dot product of that with M; 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 this 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, which as usual is &lt;1 1 1 ... 1|.</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>