Comma-based lattices: Difference between revisions
Wikispaces>MartinGough **Imported revision 440072748 - Original comment: ** |
Wikispaces>MartinGough **Imported revision 440110628 - 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:MartinGough|MartinGough]] and made on <tt>2013- | : This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2013-07-01 04:21:19 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>440110628</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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When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size concentrates commas in the region near the origin, where their interrelationships become apparent. The dual of such a lattice of commas is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas. | When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size concentrates commas in the region near the origin, where their interrelationships become apparent. The dual of such a lattice of commas is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas. | ||
The theory behind this technique is set out below, illustrated for the 5-limit but extending in a straightforward way to any prime limit. An example of its application in the 5-limit can be viewed in this [[file:Comma lattice (syntonic, schisma, kleisma).xlsx|spreadsheet]]. | The theory behind this technique is set out below, illustrated for the 5-limit but extending in a straightforward way to any prime limit. An example of its application in the 5-limit can be viewed in this [[file:Comma lattice (syntonic, schisma, kleisma).xlsx|spreadsheet]]. | ||
A just interval J is the product of a JI tuning vector and a monzo: | A just interval //J// is the product of a JI tuning vector and a monzo: | ||
[[math]] | [[math]] | ||
\qquad J = \langle v \vert m \rangle | \qquad J = \langle v \vert m \rangle | ||
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This standard coordinate system can be transformed into a rebased system using a square unimodular matrix //W// and its inverse //N//: | This standard coordinate system can be transformed into a rebased system using a square unimodular matrix //W// and its inverse //N//: | ||
[[math]] | [[math]] | ||
\qquad \langle w \vert = \langle v \vert W \qquad \qquad \ | \qquad \langle w \vert = \langle v \vert W \qquad \qquad \vert n \rangle = N \vert m \rangle \\ | ||
\qquad \langle v \vert = \langle w \vert N \qquad \qquad \, \ | \qquad \langle v \vert = \langle w \vert N \qquad \qquad \, \vert m \rangle = W \vert n \rangle \\ | ||
\qquad WN = NW = I \qquad \vert W \vert = \vert N \vert = \pm 1 | \qquad WN = NW = I \qquad \vert W \vert = \vert N \vert = \pm 1 | ||
[[math]] | [[math]] | ||
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\qquad J' = \langle v' \vert m \rangle = \langle v' \vert W N \vert m \rangle = \langle w' \vert n \rangle | \qquad J' = \langle v' \vert m \rangle = \langle v' \vert W N \vert m \rangle = \langle w' \vert n \rangle | ||
[[math]] | [[math]] | ||
where //J//’ is the number of steps representing interval //J// in an equal temperament, | where //J//’ is the number of steps representing interval //J// in an equal temperament, and | ||
and | |||
[[math]] | [[math]] | ||
\qquad \langle v’ \vert = \langle \underline{2}’ \; \underline{3}’ \; \underline{5}’ \vert \\ | |||
\qquad \langle w’ \vert = \langle w_1’ \; w_2’ \; w_3’ \vert | \qquad \langle w’ \vert = \langle w_1’ \; w_2’ \; w_3’ \vert | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
can be read as a statement that the rows of //N// are standard vals representing a set of ETs which temper the basis commas (whose monzos are the columns of W) to <1 0 0|, <0 1 0| and <0 0 1|, respectively. Each of these ETs sets one of the basis commas to a single step of the temperament while tempering out all the others. | can be read as a statement that the rows of //N// are standard vals representing a set of ETs which temper the basis commas (whose monzos are the columns of W) to <1 0 0|, <0 1 0| and <0 0 1|, respectively. Each of these ETs sets one of the basis commas to a single step of the temperament while tempering out all the others. | ||
Every point on the rebased lattice is both a monzo (interval) and a val (ET). This is of course also true for the standard lattice, but in the rebased system every lattice point near the origin (as well as more distant points near the JI zero plane) is the monzo of a comma-sized interval, and | Every point on the rebased lattice is both a monzo (interval) and a val (ET). This is of course also true for the standard lattice, but in the rebased system every lattice point near the origin (as well as more distant points near the JI zero plane) is the monzo of a comma-sized interval, and every lattice point which is not too close to a certain plane is the val for an ET approximating JI. | ||
Expressed in the standard basis the monzos are | Expressed in the standard basis the monzos are | ||
[[math]] | [[math]] | ||
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Another basis set, [ |monzisma> |raider> |atom> ], turns up the magnification to focus on schismina-sized commas and the associated temperaments.</pre></div> | Another basis set, [ |monzisma> |raider> |atom> ], turns up the magnification to focus on schismina-sized commas and the associated temperaments.</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>Comma-based lattices</title></head><body><!-- ws:start:WikiTextHeadingRule: | <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>Comma-based lattices</title></head><body><!-- ws:start:WikiTextHeadingRule:11:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:11 --> </h1> | ||
When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size concentrates commas in the region near the origin, where their interrelationships become apparent. The dual of such a lattice of commas is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.<br /> | When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size concentrates commas in the region near the origin, where their interrelationships become apparent. The dual of such a lattice of commas is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.<br /> | ||
The theory behind this technique is set out below, illustrated for the 5-limit but extending in a straightforward way to any prime limit. An example of its application in the 5-limit can be viewed in this <a href="/file/view/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx/587448909/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx" onclick="ws.common.trackFileLink('/file/view/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx/587448909/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx');">spreadsheet</a>.<br /> | The theory behind this technique is set out below, illustrated for the 5-limit but extending in a straightforward way to any prime limit. An example of its application in the 5-limit can be viewed in this <a href="/file/view/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx/587448909/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx" onclick="ws.common.trackFileLink('/file/view/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx/587448909/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx');">spreadsheet</a>.<br /> | ||
A just interval J is the product of a JI tuning vector and a monzo:<br /> | A just interval <em>J</em> is the product of a JI tuning vector and a monzo:<br /> | ||
<!-- ws:start:WikiTextMathRule:0: | <!-- ws:start:WikiTextMathRule:0: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\qquad \langle w \vert = \langle v \vert W \qquad \qquad \ | \qquad \langle w \vert = \langle v \vert W \qquad \qquad \vert n \rangle = N \vert m \rangle \\&lt;br /&gt; | ||
\qquad \langle v \vert = \langle w \vert N \qquad \qquad \, \ | \qquad \langle v \vert = \langle w \vert N \qquad \qquad \, \vert m \rangle = W \vert n \rangle \\&lt;br /&gt; | ||
\qquad WN = NW = I \qquad \vert W \vert = \vert N \vert = \pm 1&lt;br/&gt;[[math]] | \qquad WN = NW = I \qquad \vert W \vert = \vert N \vert = \pm 1&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\qquad \langle w \vert = \langle v \vert W \qquad \qquad \ | --><script type="math/tex">\qquad \langle w \vert = \langle v \vert W \qquad \qquad \vert n \rangle = N \vert m \rangle \\ | ||
\qquad \langle v \vert = \langle w \vert N \qquad \qquad \, \ | \qquad \langle v \vert = \langle w \vert N \qquad \qquad \, \vert m \rangle = W \vert n \rangle \\ | ||
\qquad WN = NW = I \qquad \vert W \vert = \vert N \vert = \pm 1</script><!-- ws:end:WikiTextMathRule:2 --><br /> | \qquad WN = NW = I \qquad \vert W \vert = \vert N \vert = \pm 1</script><!-- ws:end:WikiTextMathRule:2 --><br /> | ||
where &lt;<em>w</em>| is the rebased tuning vector and |<em>n</em>&gt; is the rebased monzo.<br /> | where &lt;<em>w</em>| is the rebased tuning vector and |<em>n</em>&gt; is the rebased monzo.<br /> | ||
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\qquad J' = \langle v' \vert m \rangle = \langle v' \vert W N \vert m \rangle = \langle w' \vert n \rangle&lt;br/&gt;[[math]] | \qquad J' = \langle v' \vert m \rangle = \langle v' \vert W N \vert m \rangle = \langle w' \vert n \rangle&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\qquad J' = \langle v' \vert m \rangle = \langle v' \vert W N \vert m \rangle = \langle w' \vert n \rangle</script><!-- ws:end:WikiTextMathRule:4 --><br /> | --><script type="math/tex">\qquad J' = \langle v' \vert m \rangle = \langle v' \vert W N \vert m \rangle = \langle w' \vert n \rangle</script><!-- ws:end:WikiTextMathRule:4 --><br /> | ||
where <em>J</em>’ is the number of steps representing interval <em>J</em> in an equal temperament,<br /> | where <em>J</em>’ is the number of steps representing interval <em>J</em> in an equal temperament, and<br /> | ||
<!-- ws:start:WikiTextMathRule:5: | <!-- ws:start:WikiTextMathRule:5: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\qquad \langle v’ \vert = \langle \underline{2}’ \; \underline{3}’ \; \underline{5}’ \vert | \qquad \langle v’ \vert = \langle \underline{2}’ \; \underline{3}’ \; \underline{5}’ \vert \\&lt;br /&gt; | ||
\qquad \langle w’ \vert = \langle w_1’ \; w_2’ \; w_3’ \vert&lt;br/&gt;[[math]] | \qquad \langle w’ \vert = \langle w_1’ \; w_2’ \; w_3’ \vert&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\qquad \langle w’ \vert = \langle w_1’ \; w_2’ \; w_3’ \vert</script><!-- ws:end:WikiTextMathRule: | --><script type="math/tex">\qquad \langle v’ \vert = \langle \underline{2}’ \; \underline{3}’ \; \underline{5}’ \vert \\ | ||
\qquad \langle w’ \vert = \langle w_1’ \; w_2’ \; w_3’ \vert</script><!-- ws:end:WikiTextMathRule:5 --><br /> | |||
(with integer elements) are the standard and rebased vals for that temperament.<br /> | (with integer elements) are the standard and rebased vals for that temperament.<br /> | ||
When changing the basis in this way the unimodular property can be preserved by proceeding from the identity matrix in a series of steps in which a multiple of one column (basis interval) is subtracted from another.<br /> | When changing the basis in this way the unimodular property can be preserved by proceeding from the identity matrix in a series of steps in which a multiple of one column (basis interval) is subtracted from another.<br /> | ||
The interesting situation is when all the basis intervals are commas. In this case the equation<br /> | The interesting situation is when all the basis intervals are commas. In this case the equation<br /> | ||
<!-- ws:start:WikiTextMathRule: | <!-- ws:start:WikiTextMathRule:6: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\qquad NW = I&lt;br/&gt;[[math]] | \qquad NW = I&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\qquad NW = I</script><!-- ws:end:WikiTextMathRule: | --><script type="math/tex">\qquad NW = I</script><!-- ws:end:WikiTextMathRule:6 --><br /> | ||
can be read as a statement that the rows of <em>N</em> are standard vals representing a set of ETs which temper the basis commas (whose monzos are the columns of W) to &lt;1 0 0|, &lt;0 1 0| and &lt;0 0 1|, respectively. Each of these ETs sets one of the basis commas to a single step of the temperament while tempering out all the others.<br /> | can be read as a statement that the rows of <em>N</em> are standard vals representing a set of ETs which temper the basis commas (whose monzos are the columns of W) to &lt;1 0 0|, &lt;0 1 0| and &lt;0 0 1|, respectively. Each of these ETs sets one of the basis commas to a single step of the temperament while tempering out all the others.<br /> | ||
Every point on the rebased lattice is both a monzo (interval) and a val (ET). This is of course also true for the standard lattice, but in the rebased system every lattice point near the origin (as well as more distant points near the JI zero plane) is the monzo of a comma-sized interval, and | Every point on the rebased lattice is both a monzo (interval) and a val (ET). This is of course also true for the standard lattice, but in the rebased system every lattice point near the origin (as well as more distant points near the JI zero plane) is the monzo of a comma-sized interval, and every lattice point which is not too close to a certain plane is the val for an ET approximating JI.<br /> | ||
Expressed in the standard basis the monzos are<br /> | Expressed in the standard basis the monzos are<br /> | ||
<!-- ws:start:WikiTextMathRule: | <!-- ws:start:WikiTextMathRule:7: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\qquad \vert m \rangle = \vert m \negthinspace \underline{_2} \; m \negthinspace \underline{_3} \; m \negthinspace \underline{_5} \rangle = W \vert n_1 \; n_2 \; n_3 \rangle&lt;br/&gt;[[math]] | \qquad \vert m \rangle = \vert m \negthinspace \underline{_2} \; m \negthinspace \underline{_3} \; m \negthinspace \underline{_5} \rangle = W \vert n_1 \; n_2 \; n_3 \rangle&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\qquad \vert m \rangle = \vert m \negthinspace \underline{_2} \; m \negthinspace \underline{_3} \; m \negthinspace \underline{_5} \rangle = W \vert n_1 \; n_2 \; n_3 \rangle</script><!-- ws:end:WikiTextMathRule: | --><script type="math/tex">\qquad \vert m \rangle = \vert m \negthinspace \underline{_2} \; m \negthinspace \underline{_3} \; m \negthinspace \underline{_5} \rangle = W \vert n_1 \; n_2 \; n_3 \rangle</script><!-- ws:end:WikiTextMathRule:7 --><br /> | ||
where <em>n</em><span style="font-size: 10.06px;">1</span>, <em>n</em><span style="font-size: 10.06px;">2</span> and <em>n</em><span style="font-size: 10.06px;">3</span> are the coordinates of a general lattice point representing the coefficients of each basis comma in the monzo, and the vals are<br /> | where <em>n</em><span style="font-size: 10.06px;">1</span>, <em>n</em><span style="font-size: 10.06px;">2</span> and <em>n</em><span style="font-size: 10.06px;">3</span> are the coordinates of a general lattice point representing the coefficients of each basis comma in the monzo, and the vals are<br /> | ||
<!-- ws:start:WikiTextMathRule: | <!-- ws:start:WikiTextMathRule:8: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\qquad \langle v’ \vert = \langle \underline{2}’ \; \underline{3}’ \; \underline{5}’ \vert = \langle w_1’ \; w_2’ \; w_3’ \vert N&lt;br/&gt;[[math]] | \qquad \langle v’ \vert = \langle \underline{2}’ \; \underline{3}’ \; \underline{5}’ \vert = \langle w_1’ \; w_2’ \; w_3’ \vert N&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\qquad \langle v’ \vert = \langle \underline{2}’ \; \underline{3}’ \; \underline{5}’ \vert = \langle w_1’ \; w_2’ \; w_3’ \vert N</script><!-- ws:end:WikiTextMathRule: | --><script type="math/tex">\qquad \langle v’ \vert = \langle \underline{2}’ \; \underline{3}’ \; \underline{5}’ \vert = \langle w_1’ \; w_2’ \; w_3’ \vert N</script><!-- ws:end:WikiTextMathRule:8 --><br /> | ||
where <em>w</em><span style="font-size: 10.06px;">1</span>’, <em>w</em><span style="font-size: 10.06px;">2</span>’ and <em>w</em><span style="font-size: 10.06px;">3</span>’ are the coordinates of a general lattice point representing the sizes of the commas <em>w</em><span style="font-size: 10.06px;">1</span>, <em>w</em><span style="font-size: 10.06px;">2</span> and <em>w</em><span style="font-size: 10.06px;">3</span> in steps of the val’s ET.<br /> | where <em>w</em><span style="font-size: 10.06px;">1</span>’, <em>w</em><span style="font-size: 10.06px;">2</span>’ and <em>w</em><span style="font-size: 10.06px;">3</span>’ are the coordinates of a general lattice point representing the sizes of the commas <em>w</em><span style="font-size: 10.06px;">1</span>, <em>w</em><span style="font-size: 10.06px;">2</span> and <em>w</em><span style="font-size: 10.06px;">3</span> in steps of the val’s ET.<br /> | ||
This lattice of commas/ETs has several interesting properties.<br /> | This lattice of commas/ETs has several interesting properties.<br /> | ||
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Viewed as a val, a rebased lattice point represents a tuning vector for an ET, and intervals in this ET (for a given monzo lattice point) are measured by distance in integer steps from a zero plane normal passing through the origin normal to the tuning vector. Since monzos for simple intervals now lie in a narrow cone of directions, their sizes in relation to the octave are approximated well by most vals, even if those vals have tuning vectors not closely aligned with the (rebased) JI tuning vector. The exceptions are those vals whose tuning vectors point nearly perpendicular to the octave monzo, and thus come close to tempering out the octave and other simple intervals.<br /> | Viewed as a val, a rebased lattice point represents a tuning vector for an ET, and intervals in this ET (for a given monzo lattice point) are measured by distance in integer steps from a zero plane normal passing through the origin normal to the tuning vector. Since monzos for simple intervals now lie in a narrow cone of directions, their sizes in relation to the octave are approximated well by most vals, even if those vals have tuning vectors not closely aligned with the (rebased) JI tuning vector. The exceptions are those vals whose tuning vectors point nearly perpendicular to the octave monzo, and thus come close to tempering out the octave and other simple intervals.<br /> | ||
With a suitable choice of basis intervals the rebased lattice can provide a framework for cataloguing both commas and ETs. In the 5-limit the following basis set (one of many possibilities) proves useful and is illustrated in the linked <a href="/file/view/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx/587448909/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx" onclick="ws.common.trackFileLink('/file/view/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx/587448909/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx');">file</a>:<br /> | With a suitable choice of basis intervals the rebased lattice can provide a framework for cataloguing both commas and ETs. In the 5-limit the following basis set (one of many possibilities) proves useful and is illustrated in the linked <a href="/file/view/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx/587448909/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx" onclick="ws.common.trackFileLink('/file/view/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx/587448909/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx');">file</a>:<br /> | ||
<!-- ws:start:WikiTextMathRule: | <!-- ws:start:WikiTextMathRule:9: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\qquad W = [\; \vert c \rangle \; \vert \sigma \rangle \; \vert k \rangle \; ]&lt;br/&gt;[[math]] | \qquad W = [\; \vert c \rangle \; \vert \sigma \rangle \; \vert k \rangle \; ]&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\qquad W = [\; \vert c \rangle \; \vert \sigma \rangle \; \vert k \rangle \; ]</script><!-- ws:end:WikiTextMathRule: | --><script type="math/tex">\qquad W = [\; \vert c \rangle \; \vert \sigma \rangle \; \vert k \rangle \; ]</script><!-- ws:end:WikiTextMathRule:9 --><br /> | ||
where <em>c</em> = syntonic comma, <em>σ</em> = schisma, <em>k</em> = kleisma.<br /> | where <em>c</em> = syntonic comma, <em>σ</em> = schisma, <em>k</em> = kleisma.<br /> | ||
Its change-of-basis matrices are<br /> | Its change-of-basis matrices are<br /> | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\qquad W = \left[ \begin{array}{rr}&lt;br /&gt; | \qquad W = \left[ \begin{array}{rr}&lt;br /&gt; | ||
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-19 & -30 & -44\\ | -19 & -30 & -44\\ | ||
12 & 19 & 28\\ | 12 & 19 & 28\\ | ||
\end{array} \right]</script><!-- ws:end:WikiTextMathRule: | \end{array} \right]</script><!-- ws:end:WikiTextMathRule:10 --><br /> | ||
from which it can be seen that the associated basis ETs are 53edo, 19edo and 12edo. (The negative signs in the 19edo val are the result of the JI tuning vector falling in a different quadrant from the octave monzo under this transformation.)<br /> | from which it can be seen that the associated basis ETs are 53edo, 19edo and 12edo. (The negative signs in the 19edo val are the result of the JI tuning vector falling in a different quadrant from the octave monzo under this transformation.)<br /> | ||
The commas can be conveniently displayed in layers of the lattice with specified values of <em>n</em>k.<br /> | The commas can be conveniently displayed in layers of the lattice with specified values of <em>n</em>k.<br /> | ||