Tp tuning: Difference between revisions

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||\ |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}

[[math]]

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 //Tp 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 Tp 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 Tp 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 Tp tuning, thereby defining a unique tuning Tp(S) for any abstract temperament S on any group G. Given Tp(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 Tp(S)/Tp(S)(2). When p = 2, POL2 tuning generalizes POTE tuning.

||\ |m_2 \ m_3 \ ... \ m_k&amp;gt; ||_p = (|\log_2(2) m_2|^p + |\log_2(3)m_3|^p + ... + |\log_2(k) m_k|^p)^{1/p}&amp;lt;br/&amp;gt;[[math]]

  --&gt;&lt;script type="math/tex"&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;/script&gt;&lt;!-- ws:end:WikiTextMathRule:1 --&gt;&lt;br /&gt;

&lt;br /&gt;

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 &lt;em&gt;Tp proportional error&lt;/em&gt; 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 Tp 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 Tp tuning. Usually this tuning is unique, but in the case p = 1, called the &lt;a class="wiki_link" href="/TOP%20tuning"&gt;TOP tuning&lt;/a&gt;, 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 Tp tuning, thereby defining a unique tuning Tp(S) for any abstract temperament S on any group G. Given Tp(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 Tp(S)/Tp(S)(2). When p = 2, POL2 tuning generalizes POTE tuning.&lt;br /&gt;