TOP tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 548731802 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 548764630 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-04-26 16: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-04-26 23:16:14 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>548764630</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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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 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.</pre></div> | 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.</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>TOP tuning</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Proportional error"></a><!-- ws:end:WikiTextHeadingRule:0 -->Proportional error</h1> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>TOP tuning</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Proportional error"></a><!-- ws:end:WikiTextHeadingRule:0 -->Proportional error</h1> | ||
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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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name=" | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><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.</body></html></pre></div> | 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 = &lt;t₁ t₂ ... tₖ| then minimize E subject to nonnegativity and the linear constraints {tₙ/log₂(pₙ) - 1 ≤ E, 1 - tₙ/log₂(pₙ) ≤ E, &lt;T|cₖ&gt; = 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.</body></html></pre></div> | ||