Comma-based lattices: Difference between revisions

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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 has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a '''comma-based lattice''' 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 has the effect of drawing a set of similar-sized commas into 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 ]]and this [[:File:Comma_lattice_(syntonic,_schisma,_kleisma)_3D.png|image]].

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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:

~~<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.

* meantone temperaments: horizontal lines

* schismic temperaments: vertical lines

* diaschismic temperaments: leading diagonals

* aristoxenean temperaments: trailing diagonals

* 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.

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.

[[Category:31edo]]

[[Category:~~comma~~]]

[[Category:Comma]]

[[Category:~~lattice~~]]

[[Category:Lattice]]

[[Category:~~what_is]]~~

[[Category:What is]]

~~[[Category:wiki~~]]

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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 has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a '''comma-based lattice''' 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 has the effect of drawing a set of similar-sized commas into 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 ]]and this [[:File:Comma_lattice_(syntonic,_schisma,_kleisma)_3D.png|image]].
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 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:
-<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.
+* meantone temperaments: horizontal lines
+* schismic temperaments: vertical lines
+* diaschismic temperaments: leading diagonals
+* aristoxenean temperaments: trailing diagonals
+* 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.
 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&gt; |raider&gt; |atom&gt; ], turns up the magnification to focus on schismina-sized commas and the associated temperaments.
 [[Category:31edo]]
-[[Category:comma]]
+[[Category:Comma]]
-[[Category:lattice]]
+[[Category:Lattice]]
-[[Category:what_is]]
+[[Category:What is]]
-[[Category:wiki]]