TOP tuning: Difference between revisions

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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. (In the product ab both errors and Tenney heights add; there can be no cancellation of prime factors since that would imply PE(ab) > E, contra hypothesis.) 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)>0 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.

For any regular temperament, we may define an ''intrinsic prime'' to be a prime dividing the numerator or denominator of some comma that vanishes in 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=

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 For a tuning T and absolute proportional error E = APE(T), consider the set S of all rational q&gt;0 such that PE(q) = E. If a and b are elements of S, then PE(ab) = E. (In the product ab both errors and Tenney heights add; there can be no cancellation of prime factors since that would imply PE(ab) &gt; E, contra hypothesis.) 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)&gt;0 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.
+For any regular temperament, we may define an ''intrinsic prime'' to be a prime dividing the numerator or denominator of some comma that vanishes in 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=