Comma-based lattices: Difference between revisions
Wikispaces>MartinGough **Imported revision 440110628 - Original comment: ** |
Wikispaces>MartinGough **Imported revision 440110836 - 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>2013-07-01 04: | : This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2013-07-01 04:25:49 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>440110836</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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* –mercator, 41-tone, sublimma, 17-tone, limma, apotome... | * –mercator, 41-tone, sublimma, 17-tone, limma, apotome... | ||
Any comma in the //n//k = 1 plane can substitute for the kleisma as the third basis comma to form an alternative lattice. | Any comma in the //n//k = 1 plane can substitute for the kleisma as the third basis comma to form an alternative lattice. | ||
The 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 | The 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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<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>Any comma in the <em>n</em>k = 1 plane can substitute for the kleisma as the third basis comma to form an alternative lattice.<br /> | <ul><li>–mercator, 41-tone, sublimma, 17-tone, limma, apotome...</li></ul>Any comma in the <em>n</em>k = 1 plane can substitute for the kleisma as the third basis comma to form an alternative lattice.<br /> | ||
The 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 | The 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 /> | ||
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> | ||