Tp tuning
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author genewardsmith and made on 2012-06-23 23:24:26 UTC.
- The original revision id was 347567292.
- The revision comment was:
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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) b_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 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 derfine the error of the tuning on q, Err(q), as |T(q) - cents(q)}, and if q ≠ 1, the //Lp proportional error// as PEp(q) = Err(q)/||q||_p. 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) b_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) b_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 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 derfine the error of the tuning on q, Err(q), as |T(q) - cents(q)}, and if q ≠ 1, the <em>Lp proportional error</em> as PEp(q) = Err(q)/||q||_p.</body></html>