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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | | de = Nichtganzzahlige Intervallvektoren |
| : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-24 05:00:23 UTC</tt>.<br>
| | | en = Radical interval |
| : The original revision id was <tt>238544521</tt>.<br>
| | | es = |
| : The revision comment was: <tt></tt><br>
| | | ja = |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| | }} |
| <h4>Original Wikitext content:</h4>
| | A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. |
| <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;white-space: pre-wrap ! important" class="old-revision-html">A //fractional monzo// is like an ordinary [[Monzos and Interval Space|monzo]] except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep> is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |1/13 -1/13 7/26> represents the interval 2^(1/13) 3^(-1/13) 5^(7/26). By taking the [[Least common multiple|least common multiple]] of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (312500/9)^(1/26). By taking a dot product with <cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (1/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 696.1648 cents.
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| Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it.
| | === Fractional monzos === |
| | For the sake of clarity, monzos representing radical intervals are called '''fractional monzos'''. Mathematically, fractional monzos behave the same as ordinary [[monzo]]s, except that elements have been extended to allow them to be rational numbers. If {{monzo| ''e''<sub>2</sub> ''e''<sub>3</sub> … ''e''<sub>p</sub> }} is a fractional monzo, then it represents 2<sup>''e''<sub>2</sub></sup> 3<sup>''e''<sub>3</sub></sup> … ''p''<sup>''e''<sub>''p''</sub></sup> just as with an ordinary monzo. Hence, for instance, {{monzo| 1/13 -1/13 7/26 }} represents the interval 2<sup>1/13</sup> 3<sup>-1/13</sup> 5<sup>7/26</sup>. By taking the [[least common multiple]] of the denominators, intervals represented by a fractional monzo can always be written as an ''n''-th root of a positive rational number; for instance from our example, (312500/9)<sup>1/26</sup>, which may also be written as 1\26ed312500/9. |
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| ===Fractional projection maps===
| | By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}. |
| A square matrix P is a [[http://en.wikipedia.org/wiki/Projection_%28linear_algebra%29|projection]] if P^2 = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has [[http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace|eigenvalues]] of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted [[Monzos and Interval Space|interval space]], then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v.
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| In particular, this is true of matrices with rows consisting of fractional monzos. This is of interest since several of the most important tunings, in particular minimax and least squares, have tuning values which can be expressed as fractional monzos. For example, the fractional monzo we have used as an example is the tuning for a fifth in the 7/26-comma Woolhouse meantone. Indeed, any meantone whose tuning is expressed as a fraction of a comma has an associated 3x3 projection matrix defining the tuning.
| | Vectors in [[monzos and interval space|interval space]], where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one ''n''-th root of a positive rational number which corresponds to it. |
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| ===Tunings in terms of fractional monzos===
| | == Tunings in terms of radical intervals == |
| If n is the number of primes pi(p) less than or equal to p, we may define a unique nxn projection matrix by means of a list of n-r commas c and r //eigenmonzos// e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection map P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left [[http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace|eigenvector]] for the eigenvalue 1.
| | Any number that can be expressed as a root corresponds to a radical interval, so radical intervals can be used to express the degrees of equal tunings. For example, [[12edo]]'s fifth can be expressed as {{monzo| 7/12 }} or 2<sup>7/12</sup>, and the [[Bohlen–Pierce]] supermajor third may be expressed as {{monzo| 0 3/13 }} or 3<sup>3/13</sup>. |
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| There are various [[http://en.wikipedia.org/wiki/Matrix_norm|matrix norms]], and given a matrix norm, we can define an associated tuning of a regular temperament by taking the matrix with the minimum norm subject to the condition that commas of the temperament are left eigenvectors with eigenvalue 0 (that is, elements of the null space) while vals of the temperament are right eigenvectors with eigenvalue 1. Perhaps the easiest norm to work with is the Frobenius norm, which simply treats a matrix like a vector and takes the square root of the sum of squares of the coefficients of the matrix. The associated tuning is the [[Tenney-Euclidean tuning|Frobenius tuning]], which is the same as the unweighted rms tuning which can be found using the [[pseudoinverse]]. If r is the rank of the temperament, the Frobenius norm of the Frobenius tuning is sqrt(r), which is minimal; this follows from the [[http://en.wikipedia.org/wiki/Singular_value_decomposition|singular value decomposition]].
| | What this additionally unlocks is the ability to stack intervals from multiple edo systems. For example, one could define intervals in the 12edo.13edt subgroup by specifying their monzo, for example a subminor third of about 261 cents can be generated by {{monzo| 7/12 -3/13 }}. This also introduces the potential for dividing intervals outside of pure edo systems: one method of building scales can be to divide just intervals into portions. This is similar to temperaments like [[slendric]], and is identical to defining an [[eigenmonzo]] or rational comma-fraction tuning of these temperaments, except that while those temperaments are sometimes understood through a 2-step process of (1) equally dividing a just interval and (2) assigning the divisions to another just interval, radical intervals provide a framework for skipping the second step (if you deem it unnecessary). In fact, the structure of slendric can be described as equating {{monzo| 3 0 0 -1 }} and {{monzo| -1/3 1/3 }}. |
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| The projection matrix of the Frobenius tuning is symmetrical. Because of this, r vals spanning the subspace of the temperament can be taken to be r eigenmonzos instead. The same considerations apply to the projection matrix of RMS-TOP tuning in weighted coordinates. This projection matrix can be transformed to a matrix in unweighted coordinates by monzo weighting the columns and val weighting the rows, but the resulting matrix is no longer symmetrical. However, a val (in unweighted coordinates) can be converted to a left eigenvalue with eigenvalue 1 by doubly val weighing it; that is, val weighting it, and then val weighting it again. By finding fractional monzos which approximate r such left eigenvalues, a projection map with fractional monzos as rows which approximates TOP-RMS tuning may be obtained. An easy way to do this is to use an equal temperament val to approximate val weighting.
| | === Radical subgroups === |
| | A radical subgroup may be notated in the same manner as a normal subgroup, except where the elements are names of equal tunings. For example, quarter-comma meantone intervals can be considered to be radical intervals in the 2.4ed5 subgroup. |
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| For instance, if we take <19 30 44 53| and divide each coordinate twice by the corresponding coordinate of <31 49 72 87| we obtain a fractional eigenmonzo <19/961 30/2401 11/1296 53/7569|, and similarly from the 31et val we have <1/31 1/49 1/72 1/87|. Using these as fractional eigenmonzos gives us a tuning which is already very close (less than 0.01 cents error for the primes) to TOP-RMS. Using 171et instead, the error is less than 0.0002 cents.
| | == Fractional monzos in projection matrices == |
| | {{Main| Projection matrices }} |
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| These methods suggest a wide range of tuning possibilities. For instance, Frobenius tuning uses unweighted eigenmonzos, whereas RMS-TOP doubly weights the vals to make left eigenvectors of them. Intermediate to the two is the tuning which val weights the vals and uses them as left eigenvectors; this gives a tuning which is less biased towards smaller primes than TOP-RMS. For rank two temperaments, using 2 and the product W of odd primes to p as eigenmonzos gives a tuning which has a simple expression in terms of roots of products of 2 and W, and where the errors in the tunings of the odd primes must cancel since W is tuned justly.
| | [[Category:Regular temperament theory]] |
| | | [[Category:Math]] |
| While minimax tuning for a list of target interval (such as an odd limit tonality diamond) can be found by setting it up as a linear programming problem, this is not necessary and in important respects not desirable. The problem is simple enough that it can be solved by brute force, which makes it easy to find exact answers in the form of fractional monzos. It can happen that the minimax solution is not unique, and the brute force approach allows ties to be broken using eg minimum squared error.
| | [[Category:Tuning]] |
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| If the target set is a q limit diamond, eigenmonzos in the range 1<x<sqrt(2) may be found by simply finding all of the sets of r-1 q-limit diamond intervals in that range which together with 2 define an independent set of intervals, and computing the corresponding projection matrix. The matrix leading to the least maximum error on elements of the diamond will be the minimax tuning. If there is a tie or ties, it may be broken by choosing the tuning with the smallest sum of squares of the error.
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| ===Algebraic considerations===
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| For the mathematically inclined (other people may want to skip this paragraph) we note that monzos are elements of a [[http://en.wikipedia.org/wiki/Free_abelian_group|free abelian group]] (or equivalently, Z-module) of rank n equal to the number of primes less than or equal to p for the p-limit in question. Fractional monzos do not define a free group but rather a [[http://en.wikipedia.org/wiki/Divisible_group|divisible group]], meaning any element may be divided by any nonzero integer. They are Z-modules, but more than that also Q-modules, or stated equivalently, elements in a [[http://en.wikipedia.org/wiki/Vector_space|vector space]] (of dimension n) over the rational numbers. They are also torsion-free (equivalently, [[http://en.wikipedia.org/wiki/Flat_module|flat]]) abelian groups, and are the [[http://en.wikipedia.org/wiki/Injective_hull|injective hulls]] of the corresponding monzos.</pre></div>
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| <h4>Original HTML content:</h4>
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| <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>Fractional monzos</title></head><body>A <em>fractional monzo</em> is like an ordinary <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep&gt; is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |1/13 -1/13 7/26&gt; represents the interval 2^(1/13) 3^(-1/13) 5^(7/26). By taking the <a class="wiki_link" href="/Least%20common%20multiple">least common multiple</a> of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (312500/9)^(1/26). By taking a dot product with &lt;cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (1/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 696.1648 cents.<br />
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| <br />
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| Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it. <br />
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--Fractional projection maps"></a><!-- ws:end:WikiTextHeadingRule:0 -->Fractional projection maps</h3>
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| A square matrix P is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Projection_%28linear_algebra%29" rel="nofollow">projection</a> if P^2 = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace" rel="nofollow">eigenvalues</a> of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v. <br />
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| <br />
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| In particular, this is true of matrices with rows consisting of fractional monzos. This is of interest since several of the most important tunings, in particular minimax and least squares, have tuning values which can be expressed as fractional monzos. For example, the fractional monzo we have used as an example is the tuning for a fifth in the 7/26-comma Woolhouse meantone. Indeed, any meantone whose tuning is expressed as a fraction of a comma has an associated 3x3 projection matrix defining the tuning.<br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x--Tunings in terms of fractional monzos"></a><!-- ws:end:WikiTextHeadingRule:2 -->Tunings in terms of fractional monzos</h3>
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| If n is the number of primes pi(p) less than or equal to p, we may define a unique nxn projection matrix by means of a list of n-r commas c and r <em>eigenmonzos</em> e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection map P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace" rel="nofollow">eigenvector</a> for the eigenvalue 1.<br />
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| <br />
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| There are various <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Matrix_norm" rel="nofollow">matrix norms</a>, and given a matrix norm, we can define an associated tuning of a regular temperament by taking the matrix with the minimum norm subject to the condition that commas of the temperament are left eigenvectors with eigenvalue 0 (that is, elements of the null space) while vals of the temperament are right eigenvectors with eigenvalue 1. Perhaps the easiest norm to work with is the Frobenius norm, which simply treats a matrix like a vector and takes the square root of the sum of squares of the coefficients of the matrix. The associated tuning is the <a class="wiki_link" href="/Tenney-Euclidean%20tuning">Frobenius tuning</a>, which is the same as the unweighted rms tuning which can be found using the <a class="wiki_link" href="/pseudoinverse">pseudoinverse</a>. If r is the rank of the temperament, the Frobenius norm of the Frobenius tuning is sqrt(r), which is minimal; this follows from the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Singular_value_decomposition" rel="nofollow">singular value decomposition</a>.<br />
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| <br />
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| The projection matrix of the Frobenius tuning is symmetrical. Because of this, r vals spanning the subspace of the temperament can be taken to be r eigenmonzos instead. The same considerations apply to the projection matrix of RMS-TOP tuning in weighted coordinates. This projection matrix can be transformed to a matrix in unweighted coordinates by monzo weighting the columns and val weighting the rows, but the resulting matrix is no longer symmetrical. However, a val (in unweighted coordinates) can be converted to a left eigenvalue with eigenvalue 1 by doubly val weighing it; that is, val weighting it, and then val weighting it again. By finding fractional monzos which approximate r such left eigenvalues, a projection map with fractional monzos as rows which approximates TOP-RMS tuning may be obtained. An easy way to do this is to use an equal temperament val to approximate val weighting.<br />
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| <br />
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| For instance, if we take &lt;19 30 44 53| and divide each coordinate twice by the corresponding coordinate of &lt;31 49 72 87| we obtain a fractional eigenmonzo &lt;19/961 30/2401 11/1296 53/7569|, and similarly from the 31et val we have &lt;1/31 1/49 1/72 1/87|. Using these as fractional eigenmonzos gives us a tuning which is already very close (less than 0.01 cents error for the primes) to TOP-RMS. Using 171et instead, the error is less than 0.0002 cents.<br />
| |
| <br />
| |
| These methods suggest a wide range of tuning possibilities. For instance, Frobenius tuning uses unweighted eigenmonzos, whereas RMS-TOP doubly weights the vals to make left eigenvectors of them. Intermediate to the two is the tuning which val weights the vals and uses them as left eigenvectors; this gives a tuning which is less biased towards smaller primes than TOP-RMS. For rank two temperaments, using 2 and the product W of odd primes to p as eigenmonzos gives a tuning which has a simple expression in terms of roots of products of 2 and W, and where the errors in the tunings of the odd primes must cancel since W is tuned justly.<br />
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| <br />
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| While minimax tuning for a list of target interval (such as an odd limit tonality diamond) can be found by setting it up as a linear programming problem, this is not necessary and in important respects not desirable. The problem is simple enough that it can be solved by brute force, which makes it easy to find exact answers in the form of fractional monzos. It can happen that the minimax solution is not unique, and the brute force approach allows ties to be broken using eg minimum squared error.<br />
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| <br />
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| If the target set is a q limit diamond, eigenmonzos in the range 1&lt;x&lt;sqrt(2) may be found by simply finding all of the sets of r-1 q-limit diamond intervals in that range which together with 2 define an independent set of intervals, and computing the corresponding projection matrix. The matrix leading to the least maximum error on elements of the diamond will be the minimax tuning. If there is a tie or ties, it may be broken by choosing the tuning with the smallest sum of squares of the error.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x--Algebraic considerations"></a><!-- ws:end:WikiTextHeadingRule:4 -->Algebraic considerations</h3>
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| For the mathematically inclined (other people may want to skip this paragraph) we note that monzos are elements of a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">free abelian group</a> (or equivalently, Z-module) of rank n equal to the number of primes less than or equal to p for the p-limit in question. Fractional monzos do not define a free group but rather a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Divisible_group" rel="nofollow">divisible group</a>, meaning any element may be divided by any nonzero integer. They are Z-modules, but more than that also Q-modules, or stated equivalently, elements in a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow">vector space</a> (of dimension n) over the rational numbers. They are also torsion-free (equivalently, <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Flat_module" rel="nofollow">flat</a>) abelian groups, and are the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Injective_hull" rel="nofollow">injective hulls</a> of the corresponding monzos.</body></html></pre></div>
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