TOP tuning
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=Proportional error= A //tuning// for a regular temperament is defined by a vector T in [[Vals and Tuning Space#Vals and Monzos|Tenney tuning space]] whose entries are the size of the interval, in cents, which the k generators of the regular temperament (often the first k primes) are mapped to. T is denoted by a [[http://en.wikipedia.org/wiki/Bra-ket_notation|bra vector]], and if M is a monzo then <T|M> is the size, in cents, of the interval defined by M in the tuning T. If q is the rational number which M represents, then we may also write this quantity as T(q). Given a tuning T and a rational number q in the domain of T, the //signed error// of T on q is defined as Err(q) = T(q) - cents(q). The //absolute error// Arr(q) = |Err(q)| is the absolute value of the difference between the value in cents T assigns to q and the actual size in cents of q. The //absolute proportional error// is defined as 0 when q equals 1 and otherwise APE(q) = Arr(q)/cents(Ben(q)), where Ben(q) is the [[Benedetti height]], the product of the numerator and denominator of q. Similarly, the //proprotional error// PE(q) = Err(q)/cents(Ben(q)). While these definition used cents to define proportional error, any logarithm base will lead to the same result, so that the definition is not in fact based on cents. =TOP tuning= For any tuning T, we may define the absolute proportional error APE(T) of T as the [[http://mathworld.wolfram.com/Supremum.html|supremum]] (maximum) of the absolute proportional errors of all q belonging to the domain of T; that is, for which T provides a value. A **TOP tuning** for a regular temperament is a tuning supporting the temperament (ie, one which sends commas of the temperament to 0) with minimal APE. This minimal proportional error is a measure of the error of the temperament, which we might call the TOP error. There is always at least one TOP tuning, and may be only one, but in general the set of TOP tunings is a convex region in Tenney tuning space. This region has a [[http://en.wikipedia.org/wiki/Centroid|centroid]], which is one way to define a canonical TOP tuning. Another choice for a canonical TOP tuning is the limit of the [[Lp tuning]] as p tends to 1, which is sometimes called TIPTOP. It has the advantage that after minimizing the maximum error, it goes on if possible to minimize the second maximum, and so forth, so long as this can be done. It should be noted that the definition works as well for any [[Just intonation subgroups|subgroup temperament]] as it does for a full prime limit temperament. The concept of a TOP tuning was first suggested by [[Paul Erlich]], who gave it its name, which stands for both Tenney OPtimal and Tempered Octaves Please, the latter due to the fact that usually the octaves are tempered. =Maximal error semigroups== For a tuning T and absolute proportional error E = APE(T), consider the set S of all rational q>0 such that PE(q) = E. If a and b are elements of s, then PE(ab) = E. Hence S is a semigroup under multiplcation, with the structure of a finitely generated free abelian semigroup. A minimal set of generators consists of a finite set of primes or the inverses of primes, where one or the other is chosen so they are tuned sharply, which entails that PE(q)>1 in each case. This is the //sharp semigroup//; inverting the elements of S leads to a mirror image flat semigroup. This has the consequence that the tuning of S is defined entirely by the tuning of the primes in the sharp group S or in the corresponding flat group. From this we may conclude that E is the minimal weighted L-inf error and TOP tuning may also be defined as the minimal weighted L-inf error tuning.
Original HTML content:
<html><head><title>TOP tuning</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Proportional error"></a><!-- ws:end:WikiTextHeadingRule:0 -->Proportional error</h1> A <em>tuning</em> for a regular temperament is defined by a vector T in <a class="wiki_link" href="/Vals%20and%20Tuning%20Space#Vals and Monzos">Tenney tuning space</a> whose entries are the size of the interval, in cents, which the k generators of the regular temperament (often the first k primes) are mapped to. T is denoted by a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Bra-ket_notation" rel="nofollow">bra vector</a>, and if M is a monzo then <T|M> is the size, in cents, of the interval defined by M in the tuning T. If q is the rational number which M represents, then we may also write this quantity as T(q). <br /> <br /> Given a tuning T and a rational number q in the domain of T, the <em>signed error</em> of T on q is defined as Err(q) = T(q) - cents(q). The <em>absolute error</em> Arr(q) = |Err(q)| is the absolute value of the difference between the value in cents T assigns to q and the actual size in cents of q. The <em>absolute proportional error</em> is defined as 0 when q equals 1 and otherwise APE(q) = Arr(q)/cents(Ben(q)), where Ben(q) is the <a class="wiki_link" href="/Benedetti%20height">Benedetti height</a>, the product of the numerator and denominator of q. Similarly, the <em>proprotional error</em> PE(q) = Err(q)/cents(Ben(q)). While these definition used cents to define proportional error, any logarithm base will lead to the same result, so that the definition is not in fact based on cents.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="TOP tuning"></a><!-- ws:end:WikiTextHeadingRule:2 -->TOP tuning</h1> For any tuning T, we may define the absolute proportional error APE(T) of T as the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Supremum.html" rel="nofollow">supremum</a> (maximum) of the absolute proportional errors of all q belonging to the domain of T; that is, for which T provides a value. A <strong>TOP tuning</strong> for a regular temperament is a tuning supporting the temperament (ie, one which sends commas of the temperament to 0) with minimal APE. This minimal proportional error is a measure of the error of the temperament, which we might call the TOP error. There is always at least one TOP tuning, and may be only one, but in general the set of TOP tunings is a convex region in Tenney tuning space. This region has a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Centroid" rel="nofollow">centroid</a>, which is one way to define a canonical TOP tuning. Another choice for a canonical TOP tuning is the limit of the <a class="wiki_link" href="/Lp%20tuning">Lp tuning</a> as p tends to 1, which is sometimes called TIPTOP. It has the advantage that after minimizing the maximum error, it goes on if possible to minimize the second maximum, and so forth, so long as this can be done. It should be noted that the definition works as well for any <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup temperament</a> as it does for a full prime limit temperament.<br /> <br /> The concept of a TOP tuning was first suggested by <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a>, who gave it its name, which stands for both Tenney OPtimal and Tempered Octaves Please, the latter due to the fact that usually the octaves are tempered.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Maximal error semigroups="></a><!-- ws:end:WikiTextHeadingRule:4 -->Maximal error semigroups=</h1> For a tuning T and absolute proportional error E = APE(T), consider the set S of all rational q>0 such that PE(q) = E. If a and b are elements of s, then PE(ab) = E. Hence S is a semigroup under multiplcation, with the structure of a finitely generated free abelian semigroup. A minimal set of generators consists of a finite set of primes or the inverses of primes, where one or the other is chosen so they are tuned sharply, which entails that PE(q)>1 in each case. This is the <em>sharp semigroup</em>; inverting the elements of S leads to a mirror image flat semigroup. This has the consequence that the tuning of S is defined entirely by the tuning of the primes in the sharp group S or in the corresponding flat group. From this we may conclude that E is the minimal weighted L-inf error and TOP tuning may also be defined as the minimal weighted L-inf error tuning.</body></html>