Comma-based lattices: Difference between revisions
Wikispaces>MartinGough **Imported revision 441925792 - Original comment: ** |
Wikispaces>MartinGough **Imported revision 587448261 - 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> | : This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2016-07-21 17:30:45 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>587448261</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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<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 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. | 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 | 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 [[http:///file/view/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx/440072272/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx|spreadsheet]] and this [[file:Comma lattice (syntonic, schisma, kleisma) 3D.png|image]]. | ||
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]] | ||
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<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:13:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:13 --> </h1> | <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:13:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:13 --> </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 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.<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 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.<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/ | 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 class="wiki_link_ext" href="http:///file/view/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx/440072272/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29.xlsx" rel="nofollow">spreadsheet</a> and this <a href="/file/view/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29%203D.png/441925468/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29%203D.png" onclick="ws.common.trackFileLink('/file/view/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29%203D.png/441925468/Comma%20lattice%20%28syntonic%2C%20schisma%2C%20kleisma%29%203D.png');">image</a>.<br /> | ||
A just interval <strong><em>J</em></strong> is the product of a JI tuning vector and a monzo:<br /> | A just interval <strong><em>J</em></strong> is the product of a JI tuning vector and a monzo:<br /> | ||
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