Tp tuning
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- This revision was by author genewardsmith and made on 2012-06-24 14:03:30 UTC.
- The original revision id was 347649984.
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Original Wikitext content:
**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. Note that if L(c) = 0, c belongs to the group G if and only if c = 1, so this is well-defined. 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.Original HTML content:
<html><head><title>Tp tuning</title></head><body><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]]<br/>
|| |b_2 \ b_3 \ ... \ b_k> ||_p = (|b_2|^p + |b_3|^p + ... + |b_k|^p)^{1/p}<br/>[[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]]<br/>
|| |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}<br/>[[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. Note that if L(c) = 0, c belongs to the group G if and only if c = 1, so this is well-defined. 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.</body></html>