Comma-based lattices: Difference between revisions
Wikispaces>MartinGough **Imported revision 440369388 - Original comment: ** |
Wikispaces>MartinGough **Imported revision 440630296 - 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>2013-07- | : This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2013-07-06 18:32:51 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>440630296</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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[[math]] | [[math]] | ||
is the JI tuning vector, here expressed in a convenient shorthand in which an underscore denotes a suitable logarithm function. (The choice of logarithmic base is arbitrary and sets the unit for interval measurement.) | is the JI tuning vector, here expressed in a convenient shorthand in which an underscore denotes a suitable logarithm function. (The choice of logarithmic base is arbitrary and sets the unit for interval measurement.) | ||
This standard coordinate system can be transformed into a rebased system using a | This standard coordinate system can be transformed into a rebased system using a unimodular matrix **//W//** and its inverse **//N//**: | ||
[[math]] | [[math]] | ||
\qquad \langle w \vert = \langle v \vert W \qquad \qquad \vert n \rangle = N \vert m \rangle \\ | \qquad \langle w \vert = \langle v \vert W \qquad \qquad \vert n \rangle = N \vert m \rangle \\ | ||
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which links up with a diagonal sequence of Pythagorean intervals: | which links up with a diagonal sequence of Pythagorean intervals: | ||
* –mercator, 41-tone, sublimma, 17-tone, limma, apotome... | * –mercator, 41-tone, sublimma, 17-tone, limma, apotome... | ||
This comma lattice provides a framework for displaying ETs approximating the 5-limit. In the rebased lattice simple sub-octave intervals are lattice points lying close to the main diagonal of a rectilinear ‘loaf’ having the octave (with coordinates |53 -19 12>) at one corner. Slicing the loaf parallel to its three axes yields 53edo, 19edo and 12edo, while angled cuts give other ETs. | |||
The zero planes for ETs tempering out a particular comma form sheaves of planes radiating from that comma’s monzo vector. They appear as lines marking the intersection of their zero planes with the //n//k = 1 plane, and fall into family groups including: | The zero planes for ETs tempering out a particular comma form sheaves of planes radiating from that comma’s monzo vector. They appear as lines marking the intersection of their zero planes with the //n//k = 1 plane, and fall into family groups including: | ||
* meantone temperaments: horizontal lines | * meantone temperaments: horizontal lines | ||
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* misty temperaments: lines with gradient -2 | * misty temperaments: lines with gradient -2 | ||
Other temperament families (such as kleismic) can be plotted as lines radiating from the tempered-out comma. Regular temperaments such as quarter-comma meantone can also be plotted, and the graphic has a number of other nice features. | Other temperament families (such as kleismic) can be plotted as lines radiating from the tempered-out comma. Regular temperaments such as quarter-comma meantone can also be plotted, and the graphic has a number of other nice features. | ||
In a given prime limit there is an infinite number of comma basis sets to choose from, and any such set can be transformed in simple ways to generate others. For example, starting from the set described above, any comma in the //n//k = 1 plane can substitute for the kleisma as the third basis comma. | |||
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> | ||
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--><script type="math/tex">\qquad \langle v \vert = \langle \underline{2} \; \underline{3} \; \underline{5} \vert</script><!-- ws:end:WikiTextMathRule:1 --><br /> | --><script type="math/tex">\qquad \langle v \vert = \langle \underline{2} \; \underline{3} \; \underline{5} \vert</script><!-- ws:end:WikiTextMathRule:1 --><br /> | ||
is the JI tuning vector, here expressed in a convenient shorthand in which an underscore denotes a suitable logarithm function. (The choice of logarithmic base is arbitrary and sets the unit for interval measurement.)<br /> | is the JI tuning vector, here expressed in a convenient shorthand in which an underscore denotes a suitable logarithm function. (The choice of logarithmic base is arbitrary and sets the unit for interval measurement.)<br /> | ||
This standard coordinate system can be transformed into a rebased system using a | This standard coordinate system can be transformed into a rebased system using a unimodular matrix <strong><em>W</em></strong> and its inverse <strong><em>N</em></strong>:<br /> | ||
<!-- ws:start:WikiTextMathRule:2: | <!-- ws:start:WikiTextMathRule:2: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
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The <em>n</em>k = 1 plane contains all the intervals (including commas) tempered to plus or minus one 12edo step. This group contains more than 30 named commas, including a vertical sequence of schisma-separated commas which are all tempered out by 53edo:<br /> | The <em>n</em>k = 1 plane contains all the intervals (including commas) tempered to plus or minus one 12edo step. This group contains more than 30 named commas, including a vertical sequence of schisma-separated commas which are all tempered out by 53edo:<br /> | ||
<ul><li>semicomma, kleisma, amity, vulture, tricot, monzisma, –counterschisma, –mercator</li></ul>which links up with a diagonal sequence of Pythagorean intervals:<br /> | <ul><li>semicomma, kleisma, amity, vulture, tricot, monzisma, –counterschisma, –mercator</li></ul>which links up with a diagonal sequence of Pythagorean intervals:<br /> | ||
<ul><li>–mercator, 41-tone, sublimma, 17-tone, limma, apotome...</li></ul> | <ul><li>–mercator, 41-tone, sublimma, 17-tone, limma, apotome...</li></ul>This comma lattice provides a framework for displaying ETs approximating the 5-limit. In the rebased lattice simple sub-octave intervals are lattice points lying close to the main diagonal of a rectilinear ‘loaf’ having the octave (with coordinates |53 -19 12&gt;) at one corner. Slicing the loaf parallel to its three axes yields 53edo, 19edo and 12edo, while angled cuts give other ETs.<br /> | ||
The zero planes for ETs tempering out a particular comma form sheaves of planes radiating from that comma’s monzo vector. They appear as lines marking the intersection of their zero planes with the <em>n</em>k = 1 plane, and fall into family groups including:<br /> | The zero planes for ETs tempering out a particular comma form sheaves of planes radiating from that comma’s monzo vector. They appear as lines marking the intersection of their zero planes with the <em>n</em>k = 1 plane, and fall into family groups including:<br /> | ||
<ul><li>meantone temperaments: horizontal lines</li><li>schismic temperaments: vertical lines</li><li>diaschismic temperaments: leading diagonals</li><li>aristoxenean temperaments: trailing diagonals</li><li>misty temperaments: lines with gradient -2</li></ul>Other temperament families (such as kleismic) can be plotted as lines radiating from the tempered-out comma. Regular temperaments such as quarter-comma meantone can also be plotted, and the graphic has a number of other nice features.<br /> | <ul><li>meantone temperaments: horizontal lines</li><li>schismic temperaments: vertical lines</li><li>diaschismic temperaments: leading diagonals</li><li>aristoxenean temperaments: trailing diagonals</li><li>misty temperaments: lines with gradient -2</li></ul>Other temperament families (such as kleismic) can be plotted as lines radiating from the tempered-out comma. Regular temperaments such as quarter-comma meantone can also be plotted, and the graphic has a number of other nice features.<br /> | ||
In a given prime limit there is an infinite number of comma basis sets to choose from, and any such set can be transformed in simple ways to generate others. For example, starting from the set described above, any comma in the <em>n</em>k = 1 plane can substitute for the kleisma as the third basis comma.<br /> | |||
Another basis set, [ |monzisma&gt; |raider&gt; |atom&gt; ], turns up the magnification to focus on schismina-sized commas and the associated temperaments.</body></html></pre></div> | Another basis set, [ |monzisma&gt; |raider&gt; |atom&gt; ], turns up the magnification to focus on schismina-sized commas and the associated temperaments.</body></html></pre></div> | ||