Comma-based lattices: Difference between revisions
Wikispaces>MartinGough **Imported revision 440072650 - Original comment: ** |
Wikispaces>MartinGough **Imported revision 440072748 - 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>2013-06-30 18: | : This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2013-06-30 18:36:28 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>440072748</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">= | <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">= = | ||
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]]. | ||
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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:12:&lt;h1&gt; --><h1 id="toc0" | <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:12:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:12 --> </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 J is the product of a JI tuning vector and a monzo:<br /> | ||