Comma-based lattices: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2013-07-~~01 18~~:02:43 UTC</tt>.

: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2013-07-03 12:24:12 UTC</tt>.

: The original revision id was <tt>~~440174940~~</tt>.

: The original revision id was <tt>440369388</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

<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">= =

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 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]].

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]]

\qquad J = \langle v \vert m \rangle

Line 18:

[[math]]

is the JI tuning vector, here expressed in a convenient shorthand in which an underscore denotes a suitable logarithm function. (The choice of logarithmic base is arbitrary and sets the unit for interval measurement.)

This standard coordinate system can be transformed into a rebased system using a square unimodular matrix **//~~~~W~~~~//** and its inverse **//~~~~N~~~~//**:

This standard coordinate system can be transformed into a rebased system using a square unimodular matrix **//W//** and its inverse **//N//**:

[[math]]

\qquad \langle w \vert = \langle v \vert W \qquad \qquad \vert n \rangle = N \vert m \rangle \\

Line 24:

\qquad WN = NW = I \qquad \vert W \vert = \vert N \vert = \pm 1

[[math]]

where

[[math]]

\qquad \langle w \vert = \langle w_1 \; w_2 \; w_3 \vert

Line 33:

[[math]]

is the rebased monzo.

The columns of **~~~~//W// ~~~~**are the standard monzos for the new basis intervals and the columns of **~~~~//N// ~~~~**are rebased monzos for the standard basis intervals.

The columns of **//W//** are the standard monzos for the new basis intervals and the columns of **//N//** are rebased monzos for the standard basis intervals.

Evaluation of an interval in the rebased system follows the usual procedure:

[[math]]

Line 42:

\qquad J' = \langle v' \vert m \rangle = \langle w' \vert n \rangle

[[math]]

where **//~~~~J’~~~~//** is the number of steps representing interval **//~~~~J~~~~//** in an equal temperament, and

where **//J’//** is the number of steps representing interval **//J//** in an equal temperament, and

[[math]]

\qquad \langle v’ \vert = \langle \underline{2}’ \; \underline{3}’ \; \underline{5}’ \vert \\

Line 53:

\qquad NW = I

[[math]]

can be read as a statement that the rows of //~~~~**N**~~~~// are standard vals representing a set of ETs which temper the basis commas (whose monzos are the columns of W) to <1 0 0|, <0 1 0| and <0 0 1|, respectively. Each of these ETs sets one of the basis commas to a single step of the temperament while tempering out all the others.

can be read as a statement that the rows of //**N**// are standard vals representing a set of ETs which temper the basis commas (whose monzos are the columns of W) to <1 0 0|, <0 1 0| and <0 0 1|, respectively. Each of these ETs sets one of the basis commas to a single step of the temperament while tempering out all the others.

Every point on the rebased lattice is both a monzo (interval) and a val (ET). This is of course also true for the standard lattice, but in the rebased system every lattice point near the origin (as well as more distant points near the JI zero plane) is the monzo of a comma-sized interval, and every lattice point which is not too close to a certain plane is the val for an ET approximating JI.

Expressed in the standard basis the monzos are

Line 65:

where //w//1’, //w//2’ and //w//3’ are the coordinates of a general lattice point representing the sizes of the commas //w//1, //w//2 and //w//3 in steps of the val’s ET.

This lattice of commas/ETs has several interesting properties.

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 ~~comma~~ size and ~~above~~, are pulled in radially and away from the JI zero plane to populate the space near the origin~~. Musically interesting commas (those with low complexity and size) cluster around the origin and spread out more sparsely along the JI zero plane~~.

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.

With a suitable choice of basis intervals the rebased lattice can provide a framework for cataloguing both commas and ETs. In the 5-limit the following basis set (represented here by its rebased tuning vector) proves useful and is illustrated in the linked [[file:Comma lattice (syntonic, schisma, kleisma).xlsx|file]]:

Line 74:

Its change-of-basis matrices are

[[math]]

\qquad W =

\left[ \; \vert c \rangle \; \vert \sigma \rangle \; \vert k \rangle \; \right] =

\left[ \begin{array}{rr}

-4 & -15 & -6\\

Line 106:

<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><h1 id="toc0"> </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.

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 <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>.

A just interval <em~~>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:

<!-- ws:start:WikiTextMathRule:0:

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

Line 119:

--><script type="math/tex">\qquad \langle v \vert = \langle \underline{2} \; \underline{3} \; \underline{5} \vert</script>

is the JI tuning vector, here expressed in a convenient shorthand in which an underscore denotes a suitable logarithm function. (The choice of logarithmic base is arbitrary and sets the unit for interval measurement.)

This standard coordinate system can be transformed into a rebased system using a square unimodular matrix <em~~>W~~~~ and its inverse <em~~>N~~~~:

This standard coordinate system can be transformed into a rebased system using a square unimodular matrix W and its inverse N:

<!-- ws:start:WikiTextMathRule:2:

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

Line 128:

\qquad \langle v \vert = \langle w \vert N \qquad \qquad \, \vert m \rangle = W \vert n \rangle \\

\qquad WN = NW = I \qquad \vert W \vert = \vert N \vert = \pm 1</script>

where

where

<!-- ws:start:WikiTextMathRule:3:

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

Line 139:

--><script type="math/tex">\qquad \vert n \rangle = \vert n_1 \; n_2 \; n_3 \rangle</script>

is the rebased monzo.

The columns of <strong~~>W</em~~> </span~~>are the standard monzos for the new basis intervals and the columns of <strong~~>N</em~~> </span~~>are rebased monzos for the standard basis intervals.

The columns of W are the standard monzos for the new basis intervals and the columns of N are rebased monzos for the standard basis intervals.

Evaluation of an interval in the rebased system follows the usual procedure:

<!-- ws:start:WikiTextMathRule:5:

Line 150:

\qquad J' = \langle v' \vert m \rangle = \langle w' \vert n \rangle&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad J' = \langle v' \vert m \rangle = \langle w' \vert n \rangle</script>

where <em~~>J’~~~~ is the number of steps representing interval <em~~>J~~~~ in an equal temperament, and

where J’ is the number of steps representing interval J in an equal temperament, and

<!-- ws:start:WikiTextMathRule:7:

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

Line 164:

\qquad NW = I&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad NW = I</script>

can be read as a statement that the rows of <em~~>N</strong~~></span> are standard vals representing a set of ETs which temper the basis commas (whose monzos are the columns of W) to &lt;1 0 0|, &lt;0 1 0| and &lt;0 0 1|, respectively. Each of these ETs sets one of the basis commas to a single step of the temperament while tempering out all the others.

can be read as a statement that the rows of N are standard vals representing a set of ETs which temper the basis commas (whose monzos are the columns of W) to &lt;1 0 0|, &lt;0 1 0| and &lt;0 0 1|, respectively. Each of these ETs sets one of the basis commas to a single step of the temperament while tempering out all the others.

Every point on the rebased lattice is both a monzo (interval) and a val (ET). This is of course also true for the standard lattice, but in the rebased system every lattice point near the origin (as well as more distant points near the JI zero plane) is the monzo of a comma-sized interval, and every lattice point which is not too close to a certain plane is the val for an ET approximating JI.

Expressed in the standard basis the monzos are

Line 178:

where w1’, w2’ and w3’ are the coordinates of a general lattice point representing the sizes of the commas w1, w2 and w3 in steps of the val’s ET.

This lattice of commas/ETs has several interesting properties.

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 ~~comma~~ size and ~~above~~, are pulled in radially and away from the JI zero plane to populate the space near the origin~~. Musically interesting commas (those with low complexity and size) cluster around the origin and spread out more sparsely along the JI zero plane~~.

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.

With a suitable choice of basis intervals the rebased lattice can provide a framework for cataloguing both commas and ETs. In the 5-limit the following basis set (represented here by its rebased tuning vector) proves useful and is illustrated in the linked <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');">file</a>:

Line 189:

<!-- ws:start:WikiTextMathRule:12:

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

\qquad W = &lt;br /&gt;

\qquad W =&lt;br /&gt;

\left[ \; \vert c \rangle \; \vert \sigma \rangle \; \vert k \rangle \; \right] = &lt;br /&gt;

\left[ \; \vert c \rangle \; \vert \sigma \rangle \; \vert k \rangle \; \right] =&lt;br /&gt;

\left[ \begin{array}{rr}&lt;br /&gt;

-4 &amp; -15 &amp; -6\\&lt;br /&gt;

Line 201:

12 &amp; 19 &amp; 28\\&lt;br /&gt;

\end{array} \right]&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad W =

\left[ \; \vert c \rangle \; \vert \sigma \rangle \; \vert k \rangle \; \right] =

\left[ \begin{array}{rr}

-4 & -15 & -6\\

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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 18:02:43 UTC</tt>.<br>
+: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2013-07-03 12:24:12 UTC</tt>.<br>
-: The original revision id was <tt>440174940</tt>.<br>
+: The original revision id was <tt>440369388</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>
 <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">= =
-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 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]].
-A just interval **//&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;J&lt;/span&gt;//** 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]]
 \qquad J = \langle v \vert m \rangle
@@ Line 18: / Line 18: @@
 [[math]]
 is the JI tuning vector, here expressed in a convenient shorthand in which an underscore denotes a suitable logarithm function. (The choice of logarithmic base is arbitrary and sets the unit for interval measurement.)
-This standard coordinate system can be transformed into a rebased system using a square unimodular matrix **//&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;W&lt;/span&gt;//** and its inverse **//&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;N&lt;/span&gt;//**:
+This standard coordinate system can be transformed into a rebased system using a square unimodular matrix **//W//** and its inverse **//N//**:
 [[math]]
 \qquad \langle w \vert = \langle v \vert W \qquad \qquad \vert n \rangle = N \vert m \rangle \\
@@ Line 24: / Line 24: @@
 \qquad WN = NW = I \qquad \vert W \vert = \vert N \vert = \pm 1
 [[math]]
 where
 [[math]]
 \qquad \langle w \vert = \langle w_1 \; w_2 \; w_3 \vert
@@ Line 33: / Line 33: @@
 [[math]]
 is the rebased monzo.
-The columns of **&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;//W// &lt;/span&gt;**are the standard monzos for the new basis intervals and the columns of **&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;//N// &lt;/span&gt;**are rebased monzos for the standard basis intervals.
+The columns of **//W//** are the standard monzos for the new basis intervals and the columns of **//N//** are rebased monzos for the standard basis intervals.
 Evaluation of an interval in the rebased system follows the usual procedure:
 [[math]]
@@ Line 42: / Line 42: @@
 \qquad J' = \langle v' \vert m \rangle = \langle w' \vert n \rangle
 [[math]]
-where **//&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;J’&lt;/span&gt;//** is the number of steps representing interval **//&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;J&lt;/span&gt;//** in an equal temperament, and
+where **//J’//** is the number of steps representing interval **//J//** in an equal temperament, and
 [[math]]
 \qquad \langle v’ \vert = \langle \underline{2}’ \; \underline{3}’ \; \underline{5}’ \vert \\
@@ Line 53: / Line 53: @@
 \qquad NW = I
 [[math]]
-can be read as a statement that the rows of //&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;**N**&lt;/span&gt;// are standard vals representing a set of ETs which temper the basis commas (whose monzos are the columns of W) to &lt;1 0 0|, &lt;0 1 0| and &lt;0 0 1|, respectively. Each of these ETs sets one of the basis commas to a single step of the temperament while tempering out all the others.
+can be read as a statement that the rows of //**N**// are standard vals representing a set of ETs which temper the basis commas (whose monzos are the columns of W) to &lt;1 0 0|, &lt;0 1 0| and &lt;0 0 1|, respectively. Each of these ETs sets one of the basis commas to a single step of the temperament while tempering out all the others.
 Every point on the rebased lattice is both a monzo (interval) and a val (ET). This is of course also true for the standard lattice, but in the rebased system every lattice point near the origin (as well as more distant points near the JI zero plane) is the monzo of a comma-sized interval, and every lattice point which is not too close to a certain plane is the val for an ET approximating JI.
 Expressed in the standard basis the monzos are
@@ Line 65: / Line 65: @@
 where //w//&lt;span style="font-size: 10.06px;"&gt;1&lt;/span&gt;’, //w//&lt;span style="font-size: 10.06px;"&gt;2&lt;/span&gt;’ and //w//&lt;span style="font-size: 10.06px;"&gt;3&lt;/span&gt;’ are the coordinates of a general lattice point representing the sizes of the commas //w//&lt;span style="font-size: 10.06px;"&gt;1&lt;/span&gt;, //w//&lt;span style="font-size: 10.06px;"&gt;2&lt;/span&gt; and //w//&lt;span style="font-size: 10.06px;"&gt;3&lt;/span&gt; in steps of the val’s ET.
 This lattice of commas/ETs has several interesting properties.
-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 comma size and above, are pulled in radially and away from the JI zero plane to populate the space near the origin. Musically interesting commas (those with low complexity and size) cluster around the origin and spread out more sparsely along the JI zero plane.
+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.
 With a suitable choice of basis intervals the rebased lattice can provide a framework for cataloguing both commas and ETs. In the 5-limit the following basis set (represented here by its rebased tuning vector) proves useful and is illustrated in the linked [[file:Comma lattice (syntonic, schisma, kleisma).xlsx|file]]:
@@ Line 74: / Line 74: @@
 Its change-of-basis matrices are
 [[math]]
 \qquad W =
 \left[ \; \vert c \rangle \; \vert \sigma \rangle \; \vert k \rangle \; \right] =
 \left[ \begin{array}{rr}
 -4 &amp; -15 &amp; -6\\
@@ Line 106: / Line 106: @@
 <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">&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 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.&lt;br /&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;
 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 &lt;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');"&gt;spreadsheet&lt;/a&gt;.&lt;br /&gt;
-A just interval &lt;strong&gt;&lt;em&gt;&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;J&lt;/span&gt;&lt;/em&gt;&lt;/strong&gt; is the product of a JI tuning vector and a monzo:&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;
 &lt;!-- ws:start:WikiTextMathRule:0:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 119: / Line 119: @@
   --&gt;&lt;script type="math/tex"&gt;\qquad \langle v \vert = \langle \underline{2} \; \underline{3} \; \underline{5} \vert&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:1 --&gt;&lt;br /&gt;
 is the JI tuning vector, here expressed in a convenient shorthand in which an underscore denotes a suitable logarithm function. (The choice of logarithmic base is arbitrary and sets the unit for interval measurement.)&lt;br /&gt;
-This standard coordinate system can be transformed into a rebased system using a square unimodular matrix &lt;strong&gt;&lt;em&gt;&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;W&lt;/span&gt;&lt;/em&gt;&lt;/strong&gt; and its inverse &lt;strong&gt;&lt;em&gt;&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;N&lt;/span&gt;&lt;/em&gt;&lt;/strong&gt;:&lt;br /&gt;
+This standard coordinate system can be transformed into a rebased system using a square unimodular matrix &lt;strong&gt;&lt;em&gt;W&lt;/em&gt;&lt;/strong&gt; and its inverse &lt;strong&gt;&lt;em&gt;N&lt;/em&gt;&lt;/strong&gt;:&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:2:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 128: / Line 128: @@
 \qquad \langle v \vert = \langle w \vert N \qquad \qquad \, \vert m \rangle = W \vert n \rangle \\
 \qquad WN = NW = I \qquad \vert W \vert = \vert N \vert = \pm 1&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
-where &lt;br /&gt;
+where&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:3:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 139: / Line 139: @@
   --&gt;&lt;script type="math/tex"&gt;\qquad \vert n \rangle = \vert n_1 \; n_2 \; n_3 \rangle&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:4 --&gt;&lt;br /&gt;
 is the rebased monzo.&lt;br /&gt;
-The columns of &lt;strong&gt;&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;&lt;em&gt;W&lt;/em&gt; &lt;/span&gt;&lt;/strong&gt;are the standard monzos for the new basis intervals and the columns of &lt;strong&gt;&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;&lt;em&gt;N&lt;/em&gt; &lt;/span&gt;&lt;/strong&gt;are rebased monzos for the standard basis intervals.&lt;br /&gt;
+The columns of &lt;strong&gt;&lt;em&gt;W&lt;/em&gt;&lt;/strong&gt; are the standard monzos for the new basis intervals and the columns of &lt;strong&gt;&lt;em&gt;N&lt;/em&gt;&lt;/strong&gt; are rebased monzos for the standard basis intervals.&lt;br /&gt;
 Evaluation of an interval in the rebased system follows the usual procedure:&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:5:
@@ Line 150: / Line 150: @@
 \qquad J' = \langle v' \vert m \rangle = \langle w' \vert n \rangle&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad J' = \langle v' \vert m \rangle = \langle w' \vert n \rangle&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:6 --&gt;&lt;br /&gt;
-where &lt;strong&gt;&lt;em&gt;&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;J’&lt;/span&gt;&lt;/em&gt;&lt;/strong&gt; is the number of steps representing interval &lt;strong&gt;&lt;em&gt;&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;J&lt;/span&gt;&lt;/em&gt;&lt;/strong&gt; in an equal temperament, and&lt;br /&gt;
+where &lt;strong&gt;&lt;em&gt;J’&lt;/em&gt;&lt;/strong&gt; is the number of steps representing interval &lt;strong&gt;&lt;em&gt;J&lt;/em&gt;&lt;/strong&gt; in an equal temperament, and&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:7:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 164: / Line 164: @@
 \qquad NW = I&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad NW = I&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:8 --&gt;&lt;br /&gt;
-can be read as a statement that the rows of &lt;em&gt;&lt;span style="font-family: "Times New Roman",Times,serif;"&gt;&lt;strong&gt;N&lt;/strong&gt;&lt;/span&gt;&lt;/em&gt; are standard vals representing a set of ETs which temper the basis commas (whose monzos are the columns of W) to &amp;lt;1 0 0|, &amp;lt;0 1 0| and &amp;lt;0 0 1|, respectively. Each of these ETs sets one of the basis commas to a single step of the temperament while tempering out all the others.&lt;br /&gt;
+can be read as a statement that the rows of &lt;em&gt;&lt;strong&gt;N&lt;/strong&gt;&lt;/em&gt; are standard vals representing a set of ETs which temper the basis commas (whose monzos are the columns of W) to &amp;lt;1 0 0|, &amp;lt;0 1 0| and &amp;lt;0 0 1|, respectively. Each of these ETs sets one of the basis commas to a single step of the temperament while tempering out all the others.&lt;br /&gt;
 Every point on the rebased lattice is both a monzo (interval) and a val (ET). This is of course also true for the standard lattice, but in the rebased system every lattice point near the origin (as well as more distant points near the JI zero plane) is the monzo of a comma-sized interval, and every lattice point which is not too close to a certain plane is the val for an ET approximating JI.&lt;br /&gt;
 Expressed in the standard basis the monzos are&lt;br /&gt;
@@ Line 178: / Line 178: @@
 where &lt;em&gt;w&lt;/em&gt;&lt;span style="font-size: 10.06px;"&gt;1&lt;/span&gt;’, &lt;em&gt;w&lt;/em&gt;&lt;span style="font-size: 10.06px;"&gt;2&lt;/span&gt;’ and &lt;em&gt;w&lt;/em&gt;&lt;span style="font-size: 10.06px;"&gt;3&lt;/span&gt;’ are the coordinates of a general lattice point representing the sizes of the commas &lt;em&gt;w&lt;/em&gt;&lt;span style="font-size: 10.06px;"&gt;1&lt;/span&gt;, &lt;em&gt;w&lt;/em&gt;&lt;span style="font-size: 10.06px;"&gt;2&lt;/span&gt; and &lt;em&gt;w&lt;/em&gt;&lt;span style="font-size: 10.06px;"&gt;3&lt;/span&gt; in steps of the val’s ET.&lt;br /&gt;
 This lattice of commas/ETs has several interesting properties.&lt;br /&gt;
-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 comma size and above, are pulled in radially and away from the JI zero plane to populate the space near the origin. Musically interesting commas (those with low complexity and size) cluster around the origin and spread out more sparsely along the JI zero plane.&lt;br /&gt;
+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;
 With a suitable choice of basis intervals the rebased lattice can provide a framework for cataloguing both commas and ETs. In the 5-limit the following basis set (represented here by its rebased tuning vector) proves useful and is illustrated in the linked &lt;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');"&gt;file&lt;/a&gt;:&lt;br /&gt;
@@ Line 189: / Line 189: @@
 &lt;!-- ws:start:WikiTextMathRule:12:
 [[math]]&amp;lt;br/&amp;gt;
-\qquad W = &amp;lt;br /&amp;gt;
+\qquad W =&amp;lt;br /&amp;gt;
-\left[ \; \vert c \rangle \; \vert \sigma \rangle \; \vert k \rangle \; \right] = &amp;lt;br /&amp;gt;
+\left[ \; \vert c \rangle \; \vert \sigma \rangle \; \vert k \rangle \; \right] =&amp;lt;br /&amp;gt;
 \left[ \begin{array}{rr}&amp;lt;br /&amp;gt;
 -4 &amp;amp; -15 &amp;amp; -6\\&amp;lt;br /&amp;gt;
@@ Line 201: / Line 201: @@
 &amp;amp; 19 &amp;amp; 28\\&amp;lt;br /&amp;gt;
 \end{array} \right]&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad W =
 \left[ \; \vert c \rangle \; \vert \sigma \rangle \; \vert k \rangle \; \right] =
 \left[ \begin{array}{rr}
 -4 &amp; -15 &amp; -6\\