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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | A '''rank-3 temperament''' is a [[regular temperament]] with three [[generator]]s. If one of the generators can be an [[2/1|octave]], it is called a '''planar temperament''', though the word is sometimes applied to any rank-3 temperament. There are two interpretations for the name ''planar temperament'': first, the octave classes of notes of a planar temperament can be embedded in a plane as a [[lattice]]; and second, the set of all possible tunings of such a temperament is represented by a plane in a [[projective tuning space]] of three or more dimensions. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-11 06:53:13 UTC</tt>.<br>
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| : The original revision id was <tt>141052599</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">
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| A rank three temperament is a <A HREF="regular.html"><TT>regular temperament</TT></A><FONT | |
| COLOR="#C00000"><TT> with three generators. If one of the generators can be an octave, it is called a planar temperament, though the word is sometimes applied to any rank three temperament. The octave classes of notes of a planar temperament can be embedded in a plane as a lattice, hence the name.</TT></FONT>
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| <FONT COLOR="#C00000"><TT>The most elegant way to put a Euclidean metric, and hence a lattice structure, on
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| the pitch classes of a planar temperament is to start from a lattice of pitch classes for just intonation, and orthogonally project onto the subspace perpendicular to the space determined by the commas of the temperament. | |
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| For instance, 7-limit just intonation has a </TT></FONT><A HREF="sevlat.htm"><TT>symmetrical lattice structure</TT></A><FONT COLOR="#C00000"><TT>and a 7-limit planar temperament is defined by a single comma. If u = |* a b c> is
| | See [[Tour of regular temperaments #Rank-3 temperaments]] for a list of families, clans, and collections of rank-3 temperaments. |
| the octave class of this comma, then v = u/||u|| is the corresponding unit vector. Then if g1 and g2 are the two
| | |
| generators, j1 = g1 - (g1.v)v and j2 = g2 - (g2.v)v will be projected versions of the generators, and (j1.j1)a^2
| | == Euclidean metric on the lattice == |
| + 2(j1.j2)ab + (j2.j2)b^2 the projected quadratic form defining the metric stucture of the planar temperament lattice.
| | The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the [[comma]]s of the temperament. To do this we need a Euclidean metric on the space in which ''p''-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be [[Euclidean interval space]]. |
| Here the dot product is defined by the </TT></FONT><A HREF="http://mathworld.wolfram.com/SymmetricBilinearForm.html"><TT>bilinear
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| form</TT></A><FONT COLOR="#C00000"><TT> giving the metric structure. One good, and canonical, choice for generators
| | === Example === |
| are the generators found by using </TT></FONT><A HREF="http://mathworld.wolfram.com/HermiteNormalForm.html"><TT>Hermite
| | 7-limit [[marvel]] temperament is defined by [[tempering out]] a single comma, [[225/224]]. If we convert that to a weighted [[monzo]] '''m''' = {{monzo| -5 3.17 4.64 -2.81 }} and call the weighted monzo {{monzo| 1 0 0 0 }} for 2 "'''t'''", then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2×4 matrix {{nowrap| ''U'' {{=}} ['''t''', '''m'''] }}. If ''U''<sup>+</sup> denotes the [[pseudoinverse]] of ''U'', then letting {{nowrap| ''Q'' {{=}} ''U''<sup>+</sup>''U'' }} take {{nowrap| ''P'' {{=}} ''I'' - ''Q'' }}, where ''I'' is the identity matrix. ''P'' is the [[projection matrix]] that maps from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that '''m'''''P'' and '''t'''''P'' are the zero vector {{monzo| 0 0 0 0 }} representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the [[secor]], 16/15 and 15/14, and the second closest independent point the [[3/2|fifth]] (or alternatively, fourth). The secor and the fifth give a [[Minkowski basis]] for the lattice, but we could also use the [[5/4|major third]] and fifth as a basis. The secor and fifth are at an angle of 106.96, and the major third is angled 129.84 to the fifth. |
| reduction</TT></A><FONT COLOR="#C00000"><TT> with the proviso that if the generators so obtained are less than
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| 1, we take their reciprocal.</TT></FONT>
| | If we list 2 first in the list of commas, the matrix ''P'' for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for ''P'', by monzo-weighting the columns of ''P'', which is to say, scalar multiplying the successive rows by log<sub>2</sub>(''q'') for each of the primes ''q'' up to ''p'', which allows us to project unweighted monzos without first transforming coordinates. |
| <FONT COLOR="#C00000"><TT>The Hermite generators for marvel (225/224 planar) temperament are {2,3,5}, and the
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| projected lattice strucuture is defined by the norm sqrt(11a^2-14ab+11b^2), where "a" is the exponent
| | == See also == |
| of 3 and "b" of five. The Hermite generators for breed (2401/2400 planar) temperament are {2,49/40,10/7},
| | * [[:Category: Rank-3 temperaments]] |
| and the projected lattice structure has 49/40 perpendicular to 10/7, so it has orthogonal axes, with a norm given
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| by sqrt(11a^2+8b^2), where now "a" is the exponent of 49/40, and "b" the exponent of 10/7.</TT></FONT><P ALIGN="CENTER"><A HREF="index.html"><FONT FACE="Courier New"><TT>home</TT></FONT></A>
| | == External links == |
| <FONT COLOR="#C00000"><TT> </TT></FONT></BODY></HTML></pre></div>
| | * [http://lumma.org/tuning/gws/planar.htm Xenharmony | ''Planar Temperaments''] |
| <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>Planar Temperament</title></head><body><br />
| | [[Category:Regular temperament theory]] |
| A rank three temperament is a &lt;A HREF=&quot;regular.html&quot;&gt;&lt;TT&gt;regular temperament&lt;/TT&gt;&lt;/A&gt;&lt;FONT <br />
| | [[Category:Rank 3| ]] <!-- main article --> |
| COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt; with three generators. If one of the generators can be an octave, it is called a planar temperament, though the word is sometimes applied to any rank three temperament. The octave classes of notes of a planar temperament can be embedded in a plane as a lattice, hence the name.&lt;/TT&gt;&lt;/FONT&gt;<br />
| | [[Category:Math]] |
| &lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt;The most elegant way to put a Euclidean metric, and hence a lattice structure, on <br />
| |
| the pitch classes of a planar temperament is to start from a lattice of pitch classes for just intonation, and orthogonally project onto the subspace perpendicular to the space determined by the commas of the temperament. <br /> | |
| <br />
| |
| For instance, 7-limit just intonation has a &lt;/TT&gt;&lt;/FONT&gt;&lt;A HREF=&quot;sevlat.htm&quot;&gt;&lt;TT&gt;symmetrical lattice structure&lt;/TT&gt;&lt;/A&gt;&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt;and a 7-limit planar temperament is defined by a single comma. If u = |* a b c&gt; is <br />
| |
| the octave class of this comma, then v = u/||u|| is the corresponding unit vector. Then if g1 and g2 are the two <br /> | |
| generators, j1 = g1 - (g1.v)v and j2 = g2 - (g2.v)v will be projected versions of the generators, and (j1.j1)a^2 <br />
| |
| + 2(j1.j2)ab + (j2.j2)b^2 the projected quadratic form defining the metric stucture of the planar temperament lattice. <br />
| |
| Here the dot product is defined by the &lt;/TT&gt;&lt;/FONT&gt;&lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule:20:http://mathworld.wolfram.com/SymmetricBilinearForm.html --><a class="wiki_link_ext" href="http://mathworld.wolfram.com/SymmetricBilinearForm.html" rel="nofollow">http://mathworld.wolfram.com/SymmetricBilinearForm.html</a><!-- ws:end:WikiTextUrlRule:20 -->&quot;&gt;&lt;TT&gt;bilinear <br />
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| form&lt;/TT&gt;&lt;/A&gt;&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt; giving the metric structure. One good, and canonical, choice for generators <br />
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| are the generators found by using &lt;/TT&gt;&lt;/FONT&gt;&lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule:21:http://mathworld.wolfram.com/HermiteNormalForm.html --><a class="wiki_link_ext" href="http://mathworld.wolfram.com/HermiteNormalForm.html" rel="nofollow">http://mathworld.wolfram.com/HermiteNormalForm.html</a><!-- ws:end:WikiTextUrlRule:21 -->&quot;&gt;&lt;TT&gt;Hermite <br />
| |
| reduction&lt;/TT&gt;&lt;/A&gt;&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt; with the proviso that if the generators so obtained are less than <br />
| |
| 1, we take their reciprocal.&lt;/TT&gt;&lt;/FONT&gt;<br />
| |
| &lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt;The Hermite generators for marvel (225/224 planar) temperament are {2,3,5}, and the <br />
| |
| projected lattice strucuture is defined by the norm sqrt(11a^2-14ab+11b^2), where &quot;a&quot; is the exponent <br />
| |
| of 3 and &quot;b&quot; of five. The Hermite generators for breed (2401/2400 planar) temperament are {2,49/40,10/7}, <br />
| |
| and the projected lattice structure has 49/40 perpendicular to 10/7, so it has orthogonal axes, with a norm given <br />
| |
| by sqrt(11a^2+8b^2), where now &quot;a&quot; is the exponent of 49/40, and &quot;b&quot; the exponent of 10/7.&lt;/TT&gt;&lt;/FONT&gt;&lt;P ALIGN=&quot;CENTER&quot;&gt;&lt;A HREF=&quot;index.html&quot;&gt;&lt;FONT FACE=&quot;Courier New&quot;&gt;&lt;TT&gt;home&lt;/TT&gt;&lt;/FONT&gt;&lt;/A&gt;<br />
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| &lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt; &lt;/TT&gt;&lt;/FONT&gt;&lt;/BODY&gt;&lt;/HTML&gt;</body></html></pre></div>
| |
A rank-3 temperament is a regular temperament with three generators. If one of the generators can be an octave, it is called a planar temperament, though the word is sometimes applied to any rank-3 temperament. There are two interpretations for the name planar temperament: first, the octave classes of notes of a planar temperament can be embedded in a plane as a lattice; and second, the set of all possible tunings of such a temperament is represented by a plane in a projective tuning space of three or more dimensions.
See Tour of regular temperaments #Rank-3 temperaments for a list of families, clans, and collections of rank-3 temperaments.
Euclidean metric on the lattice
The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the commas of the temperament. To do this we need a Euclidean metric on the space in which p-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be Euclidean interval space.
Example
7-limit marvel temperament is defined by tempering out a single comma, 225/224. If we convert that to a weighted monzo m = [-5 3.17 4.64 -2.81⟩ and call the weighted monzo [1 0 0 0⟩ for 2 "t", then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2×4 matrix U = [t, m]. If U+ denotes the pseudoinverse of U, then letting Q = U+U take P = I - Q, where I is the identity matrix. P is the projection matrix that maps from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that mP and tP are the zero vector [0 0 0 0⟩ representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the secor, 16/15 and 15/14, and the second closest independent point the fifth (or alternatively, fourth). The secor and the fifth give a Minkowski basis for the lattice, but we could also use the major third and fifth as a basis. The secor and fifth are at an angle of 106.96, and the major third is angled 129.84 to the fifth.
If we list 2 first in the list of commas, the matrix P for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for P, by monzo-weighting the columns of P, which is to say, scalar multiplying the successive rows by log2(q) for each of the primes q up to p, which allows us to project unweighted monzos without first transforming coordinates.
See also
External links