TOP tuning: Difference between revisions

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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<h4>Original Wikitext content:</h4>

=Proportional error=  

A //tuning// for a regular temperament is defined by a vector T in [[Vals and Tuning Space#Vals%20and%20Monzos|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 &lt;T|M&gt; 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).

=Maximal error semigroups=  

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 an example of how this works, in the 5 and 7 limits, the TOP comma for magic temperament is 3125/3072; in the 11-limit, |0 -11 15 0 -5&gt;; in the 13 limit, |0 0 46 0 -19 -11&gt;. Putting these in Graham's app will show how closely these are associated with magic. In many cases, the association is even more emphatic.</pre></div>

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&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#TOP tuning"&gt;TOP tuning&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Maximal error semigroups"&gt;Maximal error semigroups&lt;/a&gt;&lt;/div&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Maximal error semigroups"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Maximal error semigroups&lt;/h1&gt;

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For any regular temperament, we may define an &lt;em&gt;intrinsic prime&lt;/em&gt; 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 &lt;em&gt;intrinsic temperament&lt;/em&gt;. If the temperament is defined on a group generated by primes, then a prime which is not intrinsic is &lt;em&gt;extrinsic&lt;/em&gt;. 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 &lt;em&gt;potential TOP tuning&lt;/em&gt; 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.&lt;br /&gt;