Comma-based lattices: Difference between revisions

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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">= =  

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.

A just interval **//J//** is the product of a JI tuning vector and a monzo:

[[math]]

The standard monzos representing the intervals __2__, __3__ and __5__, which in the standard lattice are orthogonal, fold up under the transformation like the spokes of a collapsing umbrella until they lie almost in a straight line. Other simple intervals, being small integer combinations of these primitive intervals, also lie close to this line. At the same time smaller intervals, of a size and complexity commensurate with that of the basis commas, are pulled in radially and away from the JI zero plane to populate the space near the origin.

Viewed as a val, a rebased lattice point represents a tuning vector for an ET, and intervals in this ET (for a given monzo lattice point) are measured by distance in integer steps from a zero plane normal passing through the origin normal to the tuning vector. Since monzos for simple intervals now lie in a narrow cone of directions, their sizes in relation to the octave are approximated well by most vals, even if those vals have tuning vectors not closely aligned with the (rebased) JI tuning vector. The exceptions are those vals whose tuning vectors point nearly perpendicular to the octave monzo, and thus come close to tempering out the octave and other simple intervals.

[[math]]

\qquad \langle w \vert = \langle \, c \;\; \sigma \;\; k \, \vert

<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Comma-based lattices&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt; &lt;/h1&gt;

  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.&lt;br /&gt;

A just interval &lt;strong&gt;&lt;em&gt;J&lt;/em&gt;&lt;/strong&gt; is the product of a JI tuning vector and a monzo:&lt;br /&gt;

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The standard monzos representing the intervals &lt;u&gt;2&lt;/u&gt;, &lt;u&gt;3&lt;/u&gt; and &lt;u&gt;5&lt;/u&gt;, which in the standard lattice are orthogonal, fold up under the transformation like the spokes of a collapsing umbrella until they lie almost in a straight line. Other simple intervals, being small integer combinations of these primitive intervals, also lie close to this line. At the same time smaller intervals, of a size and complexity commensurate with that of the basis commas, are pulled in radially and away from the JI zero plane to populate the space near the origin.&lt;br /&gt;

Viewed as a val, a rebased lattice point represents a tuning vector for an ET, and intervals in this ET (for a given monzo lattice point) are measured by distance in integer steps from a zero plane normal passing through the origin normal to the tuning vector. Since monzos for simple intervals now lie in a narrow cone of directions, their sizes in relation to the octave are approximated well by most vals, even if those vals have tuning vectors not closely aligned with the (rebased) JI tuning vector. The exceptions are those vals whose tuning vectors point nearly perpendicular to the octave monzo, and thus come close to tempering out the octave and other simple intervals.&lt;br /&gt;

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[[math]]&amp;lt;br/&amp;gt;