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.
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 semigroup S or in the corresponding flat semigroup. 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.
For any regular temperament, we may define an //intrinsic prime// to be a prime dividing the numerator or denominator of some comma of the temperament. If the set of intrinsic primes generates the group on which the temperament is defined, we may call the temperament an //intrinsic temperament//. If the temperament is defined on a group generated by primes, then a prime which is not intrinsic is //extrinsic//. Consider a TOP tuning of an intrinsic temperament. As with any regular temperament there is a group of commas, the kernel, of rank n-k, where n is the dimension of the temperament and k is the rank of the temperament; n-k is the corank of the temperament. Since the TOP tuning minimizes the maximum sharp error, the rank of the sharp semigroup needs to be as large as possible, and this rank is k+1. If the set of sharp semigroup generators is {s₀, s₁, ...,sₖ} then {PE(s₀) - PE(s₁), PE(s₀) - PE(x₂), ..., PE(s₀) - PE(sxₖ)} is a set of k linear equations, which added to the n-k linear equations denoting setting the tempered commas to zero, gives n equations in n unknowns; the most we can set. Hence an intrinsic temperament has a unique kernel of rank n-k and a unique TOP tuning sharp semigroup of rank k+1. Conversely, if we add to the set of n-k comma generators a subset of k+1 elements of the set of primes and inverted primes of the intrinsic temperament, where for each prime either the prime or the inverse prime is selected, but not both, we obtain a //potential TOP tuning// on solving the n equations in n unknowns consisting of the n-k kernel equations and the k equations {PE(s₀) - PE(s₁), PE(s₀) - PE(x₂), ..., PE(s₀) - PE(sxₖ)} . Since the set of potential TOP tunings is finite, one way of finding the (unique) TOP tuning is simply to check all of them.
=Finding the tuning=
For a temperament with both intrinsic and extrinsic primes, we may find the set of TOP tunings by first computing the tuning of the intrinsic primes. Then if r is an extrinsic prime, the tuning may be anything in the range where APE(r) ≤ E. The limit of the [[Lp tuning]] as p tends to 1 and the centroid of the region of TOP tunings both lead to choosing the JI tuning for r. This produces the canonical TOP tuning, called TIPTOP. To find the TIPTOP tuning one method is to solve for all the potential TOP tunings of the intrinsic primes, find the tuning with least error, and then tune all the extrinsic primes purely. An alternative method is to first set up a linear programming problem; if T is a val with indeterminate coefficients T = <t₁ t₂ ... tₖ| then minimize E subject to nonnegativity and the linear constraints {tₙ/log₂(pₙ) - 1 ≤ E, 1 - tₙ/log₂(pₙ) ≤ E, <T|cₖ> = 0} where the pₙ are the primes of the temperament, and the cₖ are the commas. We then may replace the tuning of all of the extrinsic primes with the pure JI tuning to get TIPTOP.
We may solve the sharp semigroup equations exactly to obtain solutions in the transcendental extension Q(log₂(q₁), log₂(t₂), ..., log₂(qₖ)) where the qₙ are the intrinsic primes other than 2. For example, take 5-limit meantone. Since 2 and 5 divide 80 and 3 divides 81. this is an intrinsic temperament. Solving for the TOP tuning either by linear programing or checking all the potential TOP tunings, we find the sharp semigroup is generated by {2, 1/3, 5}. Solving the sharp semigroup equations gives us a TOP tuning T = <3q₃/log₂(6480) (8q₃ + 2q₃q₅)/log₂(6480) 8q₃q₅/log₂(6480)|. Here q₃ = log₂(3), q₅ = log₂(5), and the denominator can also be written 4 + 4q₃ + q₅. A more complex example including an extrinsic prime is 13-limit [[Rastmic temperaments#Parahemif|parahemif temperament]]. Setting D = 22 + q₁₁ + 5q₁₃, we have T = <(2q₁₁ +10q₁₃)/D (18q₁₁ + 2q₁₃)/D q₅ (-102q₁₁ + 62q₁₃)/D 44q₁₁/D 44q₁₃/D|.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 />
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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 />
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<!-- 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.<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 semigroup S or in the corresponding flat semigroup. 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.<br />
<br />
For any regular temperament, we may define an <em>intrinsic prime</em> to be a prime dividing the numerator or denominator of some comma of the temperament. If the set of intrinsic primes generates the group on which the temperament is defined, we may call the temperament an <em>intrinsic temperament</em>. If the temperament is defined on a group generated by primes, then a prime which is not intrinsic is <em>extrinsic</em>. Consider a TOP tuning of an intrinsic temperament. As with any regular temperament there is a group of commas, the kernel, of rank n-k, where n is the dimension of the temperament and k is the rank of the temperament; n-k is the corank of the temperament. Since the TOP tuning minimizes the maximum sharp error, the rank of the sharp semigroup needs to be as large as possible, and this rank is k+1. If the set of sharp semigroup generators is {s₀, s₁, ...,sₖ} then {PE(s₀) - PE(s₁), PE(s₀) - PE(x₂), ..., PE(s₀) - PE(sxₖ)} is a set of k linear equations, which added to the n-k linear equations denoting setting the tempered commas to zero, gives n equations in n unknowns; the most we can set. Hence an intrinsic temperament has a unique kernel of rank n-k and a unique TOP tuning sharp semigroup of rank k+1. Conversely, if we add to the set of n-k comma generators a subset of k+1 elements of the set of primes and inverted primes of the intrinsic temperament, where for each prime either the prime or the inverse prime is selected, but not both, we obtain a <em>potential TOP tuning</em> on solving the n equations in n unknowns consisting of the n-k kernel equations and the k equations {PE(s₀) - PE(s₁), PE(s₀) - PE(x₂), ..., PE(s₀) - PE(sxₖ)} . Since the set of potential TOP tunings is finite, one way of finding the (unique) TOP tuning is simply to check all of them.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Finding the tuning"></a><!-- ws:end:WikiTextHeadingRule:6 -->Finding the tuning</h1>
For a temperament with both intrinsic and extrinsic primes, we may find the set of TOP tunings by first computing the tuning of the intrinsic primes. Then if r is an extrinsic prime, the tuning may be anything in the range where APE(r) ≤ E. The limit of the <a class="wiki_link" href="/Lp%20tuning">Lp tuning</a> as p tends to 1 and the centroid of the region of TOP tunings both lead to choosing the JI tuning for r. This produces the canonical TOP tuning, called TIPTOP. To find the TIPTOP tuning one method is to solve for all the potential TOP tunings of the intrinsic primes, find the tuning with least error, and then tune all the extrinsic primes purely. An alternative method is to first set up a linear programming problem; if T is a val with indeterminate coefficients T = <t₁ t₂ ... tₖ| then minimize E subject to nonnegativity and the linear constraints {tₙ/log₂(pₙ) - 1 ≤ E, 1 - tₙ/log₂(pₙ) ≤ E, <T|cₖ> = 0} where the pₙ are the primes of the temperament, and the cₖ are the commas. We then may replace the tuning of all of the extrinsic primes with the pure JI tuning to get TIPTOP.<br />
<br />
We may solve the sharp semigroup equations exactly to obtain solutions in the transcendental extension Q(log₂(q₁), log₂(t₂), ..., log₂(qₖ)) where the qₙ are the intrinsic primes other than 2. For example, take 5-limit meantone. Since 2 and 5 divide 80 and 3 divides 81. this is an intrinsic temperament. Solving for the TOP tuning either by linear programing or checking all the potential TOP tunings, we find the sharp semigroup is generated by {2, 1/3, 5}. Solving the sharp semigroup equations gives us a TOP tuning T = <3q₃/log₂(6480) (8q₃ + 2q₃q₅)/log₂(6480) 8q₃q₅/log₂(6480)|. Here q₃ = log₂(3), q₅ = log₂(5), and the denominator can also be written 4 + 4q₃ + q₅. A more complex example including an extrinsic prime is 13-limit <a class="wiki_link" href="/Rastmic%20temperaments#Parahemif">parahemif temperament</a>. Setting D = 22 + q₁₁ + 5q₁₃, we have T = <(2q₁₁ +10q₁₃)/D (18q₁₁ + 2q₁₃)/D q₅ (-102q₁₁ + 62q₁₃)/D 44q₁₁/D 44q₁₃/D|.</body></html>