7-limit symmetrical lattices: Difference between revisions

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<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:genewardsmith|genewardsmith]] and made on <tt>2011-09-14 21:33:07 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-15 14:34:30 UTC</tt>.

: The original revision id was <tt>~~254179268~~</tt>.

: The original revision id was <tt>254459368</tt>.

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

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In the two dimensional case of the 5-limit, this gives the plane lattice of equilateral triangles, called A2 or the hexagonal lattice (since the Voroni cells, regions of points closer to a given lattice point than any other, are hexagons.) The higher dimensional versions of this are called An, in n dimensions, so the 7-limit lattice is the A3 lattice. However, the 7-limit is unique in that there is another family of lattices, called Dn, to which it also belongs as D3, the [[http://en.wikipedia.org/wiki/Crystal_structure|face-centered cubic lattice]].

The 4:5:6:7 major tetrad consists of the notes |* 0 0 0>, |* 1 0 0>, |* 0 1 0>, and |* 0 0 1>; the centroid of this is |* 1/2 1/2 1/2>; similarly the centroid of 1/4:1/5:1/6:1/7 is |* -1/2 -1/2 -1/2>. If we shift the origin to |* 1/2 1/2 1/2>, major tetrads correspond to [a b c], a+b+c even, and minor tetrads to [a-1 b-1 c-1], a+b+c even, which is the same as saying [a b c], a+b+c odd. Hence the 7-limit tetrads form the simplest kind of lattice, the cubic or grid lattice consisting of triples of integers with the ordinary Euclidean distance. This~~, once again,~~ is a unique feature of the 7-limit; in no other limit do the complete utonalities and otonalities form a lattice.

The 4:5:6:7 major tetrad consists of the notes |* 0 0 0>, |* 1 0 0>, |* 0 1 0>, and |* 0 0 1>; the centroid of this is |* 1/2 1/2 1/2>; similarly the centroid of 1/4:1/5:1/6:1/7 is |* -1/2 -1/2 -1/2>. If we shift the origin to |* 1/2 1/2 1/2>, major tetrads correspond to [a b c], a+b+c even, and minor tetrads to [a-1 b-1 c-1], a+b+c even, which is the same as saying [a b c], a+b+c odd.

Suppose m0, m1, m2 and m3 are four monzos denoting four notes of a 7-limit tetrad, either otonal or utonal, with distinct pitch classes, so that all four chord elements are represented. The product of the four notes is represented by the monzo m = m0+m1+m2+m3, and the pitch class for m, which ignores the first coefficient m[1] defining octaves, is represented by the other three, m[2], m[3] and m[4]. This pitch class is uniquely associated to the tetrad, and can be used to name it. If the tetrad is otonal with the root being |* e3 e5 e7>, where the asterisk can be any integer value, then m = |* 4e3+1 4e5+1 4e7+1>. on the other hand, if it is utonal, with the fifth of the chord |* e3 e5 e7> then m = |* 4e3-1 4e5-1 4e7-1>. If we denote the 3-tuple [(m[3]+m[4]-2)/2 (m[2]+m[4]-2)/2 (m[2]+m[3]-2)/2] by [a b c], then if the tetrad is otonal, [a b c] = [e5+e7 e3+e7 e3+e5], whereas if it is utonal [a b c] = [e5+e7-1 e3+e7-1 e3+e5-1]. From this it follows that [a b c] is a triple of integers, and that a+b+c is even if the tetrad is otonal, and odd if the tetrad is utonal.

Hence every triple of integers refers uniquely to a 7-limit tetrad; this is the lattice of 7-limit tetrads. 7-limit tetrads form the simplest kind of lattice, the cubic or grid lattice consisting of triples of integers with the ordinary Euclidean distance. This is a unique feature of the 7-limit; in no other limit do the complete utonalities and otonalities form a lattice.

If [a b c] is any triple of integers, then it represents the major tetrad with root 3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+b-c)/2) if a+b+c is even, and the minor tetrad with root 3^((-1-a+b+c)/2) 5^((1+a-b+c)/2 7^((1+a+b-c)/2) if a+b+c is odd. Each unit cube corresponds to a [[http://tonalsoft.com/enc/stellat.htm|stellated hexany]], or tetradekany, or dekatesserany, though chord cube would be less of a mouthful.

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In the two dimensional case of the 5-limit, this gives the plane lattice of equilateral triangles, called A2 or the hexagonal lattice (since the Voroni cells, regions of points closer to a given lattice point than any other, are hexagons.) The higher dimensional versions of this are called An, in n dimensions, so the 7-limit lattice is the A3 lattice. However, the 7-limit is unique in that there is another family of lattices, called Dn, to which it also belongs as D3, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Crystal_structure" rel="nofollow">face-centered cubic lattice</a>.

The 4:5:6:7 major tetrad consists of the notes |* 0 0 0&gt;, |* 1 0 0&gt;, |* 0 1 0&gt;, and |* 0 0 1&gt;; the centroid of this is |* 1/2 1/2 1/2&gt;; similarly the centroid of 1/4:1/5:1/6:1/7 is |* -1/2 -1/2 -1/2&gt;. If we shift the origin to |* 1/2 1/2 1/2&gt;, major tetrads correspond to [a b c], a+b+c even, and minor tetrads to [a-1 b-1 c-1], a+b+c even, which is the same as saying [a b c], a+b+c odd. Hence the 7-limit tetrads form the simplest kind of lattice, the cubic or grid lattice consisting of triples of integers with the ordinary Euclidean distance. This~~, once again,~~ is a unique feature of the 7-limit; in no other limit do the complete utonalities and otonalities form a lattice.

The 4:5:6:7 major tetrad consists of the notes |* 0 0 0&gt;, |* 1 0 0&gt;, |* 0 1 0&gt;, and |* 0 0 1&gt;; the centroid of this is |* 1/2 1/2 1/2&gt;; similarly the centroid of 1/4:1/5:1/6:1/7 is |* -1/2 -1/2 -1/2&gt;. If we shift the origin to |* 1/2 1/2 1/2&gt;, major tetrads correspond to [a b c], a+b+c even, and minor tetrads to [a-1 b-1 c-1], a+b+c even, which is the same as saying [a b c], a+b+c odd.

Suppose m0, m1, m2 and m3 are four monzos denoting four notes of a 7-limit tetrad, either otonal or utonal, with distinct pitch classes, so that all four chord elements are represented. The product of the four notes is represented by the monzo m = m0+m1+m2+m3, and the pitch class for m, which ignores the first coefficient m[1] defining octaves, is represented by the other three, m[2], m[3] and m[4]. This pitch class is uniquely associated to the tetrad, and can be used to name it. If the tetrad is otonal with the root being |* e3 e5 e7&gt;, where the asterisk can be any integer value, then m = |* 4e3+1 4e5+1 4e7+1&gt;. on the other hand, if it is utonal, with the fifth of the chord |* e3 e5 e7&gt; then m = |* 4e3-1 4e5-1 4e7-1&gt;. If we denote the 3-tuple [(m[3]+m[4]-2)/2 (m[2]+m[4]-2)/2 (m[2]+m[3]-2)/2] by [a b c], then if the tetrad is otonal, [a b c] = [e5+e7 e3+e7 e3+e5], whereas if it is utonal [a b c] = [e5+e7-1 e3+e7-1 e3+e5-1]. From this it follows that [a b c] is a triple of integers, and that a+b+c is even if the tetrad is otonal, and odd if the tetrad is utonal.

Hence every triple of integers refers uniquely to a 7-limit tetrad; this is the lattice of 7-limit tetrads. 7-limit tetrads form the simplest kind of lattice, the cubic or grid lattice consisting of triples of integers with the ordinary Euclidean distance. This is a unique feature of the 7-limit; in no other limit do the complete utonalities and otonalities form a lattice.

If [a b c] is any triple of integers, then it represents the major tetrad with root 3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+b-c)/2) if a+b+c is even, and the minor tetrad with root 3^((-1-a+b+c)/2) 5^((1+a-b+c)/2 7^((1+a+b-c)/2) if a+b+c is odd. Each unit cube corresponds to a <a class="wiki_link_ext" href="http://tonalsoft.com/enc/stellat.htm" rel="nofollow">stellated hexany</a>, or tetradekany, or dekatesserany, though chord cube would be less of a mouthful.

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2011-09-14 21:33:07 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-15 14:34:30 UTC</tt>.<br>
-: The original revision id was <tt>254179268</tt>.<br>
+: The original revision id was <tt>254459368</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>
@@ Line 18: / Line 18: @@
 In the two dimensional case of the 5-limit, this gives the plane lattice of equilateral triangles, called A2 or the hexagonal lattice (since the Voroni cells, regions of points closer to a given lattice point than any other, are hexagons.) The higher dimensional versions of this are called An, in n dimensions, so the 7-limit lattice is the A3 lattice. However, the 7-limit is unique in that there is another family of lattices, called Dn, to which it also belongs as D3, the [[http://en.wikipedia.org/wiki/Crystal_structure|face-centered cubic lattice]].
-The 4:5:6:7 major tetrad consists of the notes |* 0 0 0&gt;, |* 1 0 0&gt;, |* 0 1 0&gt;, and |* 0 0 1&gt;; the centroid of this is |* 1/2 1/2 1/2&gt;; similarly the centroid of 1/4:1/5:1/6:1/7 is |* -1/2 -1/2 -1/2&gt;. If we shift the origin to |* 1/2 1/2 1/2&gt;, major tetrads correspond to [a b c], a+b+c even, and minor tetrads to [a-1 b-1 c-1], a+b+c even, which is the same as saying [a b c], a+b+c odd. Hence the 7-limit tetrads form the simplest kind of lattice, the cubic or grid lattice consisting of triples of integers with the ordinary Euclidean distance. This, once again, is a unique feature of the 7-limit; in no other limit do the complete utonalities and otonalities form a lattice.
+The 4:5:6:7 major tetrad consists of the notes |* 0 0 0&gt;, |* 1 0 0&gt;, |* 0 1 0&gt;, and |* 0 0 1&gt;; the centroid of this is |* 1/2 1/2 1/2&gt;; similarly the centroid of 1/4:1/5:1/6:1/7 is |* -1/2 -1/2 -1/2&gt;. If we shift the origin to |* 1/2 1/2 1/2&gt;, major tetrads correspond to [a b c], a+b+c even, and minor tetrads to [a-1 b-1 c-1], a+b+c even, which is the same as saying [a b c], a+b+c odd.
+Suppose m0, m1, m2 and m3 are four monzos denoting four notes of a 7-limit tetrad, either otonal or utonal, with distinct pitch classes, so that all four chord elements are represented. The product of the four notes is represented by the monzo m = m0+m1+m2+m3, and the pitch class for m, which ignores the first coefficient m[1] defining octaves, is represented by the other three, m[2], m[3] and m[4]. This pitch class is uniquely associated to the tetrad, and can be used to name it. If the tetrad is otonal with the root being |* e3 e5 e7&gt;, where the asterisk can be any integer value, then m = |* 4e3+1 4e5+1 4e7+1&gt;. on the other hand, if it is utonal, with the fifth of the chord |* e3 e5 e7&gt; then m = |* 4e3-1 4e5-1 4e7-1&gt;. If we denote the 3-tuple  [(m[3]+m[4]-2)/2 (m[2]+m[4]-2)/2 (m[2]+m[3]-2)/2] by [a b c], then if the tetrad is otonal, [a b c] = [e5+e7 e3+e7 e3+e5], whereas if it is utonal [a b c] = [e5+e7-1 e3+e7-1 e3+e5-1]. From this it follows that [a b c] is a triple of integers, and that a+b+c is even if the tetrad is otonal, and odd if the tetrad is utonal.
+Hence every triple of integers refers uniquely to a 7-limit tetrad; this is the lattice of 7-limit tetrads. 7-limit tetrads form the simplest kind of lattice, the cubic or grid lattice consisting of triples of integers with the ordinary Euclidean distance. This is a unique feature of the 7-limit; in no other limit do the complete utonalities and otonalities form a lattice.
 If [a b c] is any triple of integers, then it represents the major tetrad with root 3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+b-c)/2) if a+b+c is even, and the minor tetrad with root 3^((-1-a+b+c)/2) 5^((1+a-b+c)/2 7^((1+a+b-c)/2) if a+b+c is odd. Each unit cube corresponds to a [[http://tonalsoft.com/enc/stellat.htm|stellated hexany]], or tetradekany, or dekatesserany, though chord cube would be less of a mouthful.
@@ Line 49: / Line 53: @@
 In the two dimensional case of the 5-limit, this gives the plane lattice of equilateral triangles, called A2 or the hexagonal lattice (since the Voroni cells, regions of points closer to a given lattice point than any other, are hexagons.) The higher dimensional versions of this are called An, in n dimensions, so the 7-limit lattice is the A3 lattice. However, the 7-limit is unique in that there is another family of lattices, called Dn, to which it also belongs as D3, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Crystal_structure" rel="nofollow"&gt;face-centered cubic lattice&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-The 4:5:6:7 major tetrad consists of the notes |* 0 0 0&amp;gt;, |* 1 0 0&amp;gt;, |* 0 1 0&amp;gt;, and |* 0 0 1&amp;gt;; the centroid of this is |* 1/2 1/2 1/2&amp;gt;; similarly the centroid of 1/4:1/5:1/6:1/7 is |* -1/2 -1/2 -1/2&amp;gt;. If we shift the origin to |* 1/2 1/2 1/2&amp;gt;, major tetrads correspond to [a b c], a+b+c even, and minor tetrads to [a-1 b-1 c-1], a+b+c even, which is the same as saying [a b c], a+b+c odd. Hence the 7-limit tetrads form the simplest kind of lattice, the cubic or grid lattice consisting of triples of integers with the ordinary Euclidean distance. This, once again, is a unique feature of the 7-limit; in no other limit do the complete utonalities and otonalities form a lattice.&lt;br /&gt;
+The 4:5:6:7 major tetrad consists of the notes |* 0 0 0&amp;gt;, |* 1 0 0&amp;gt;, |* 0 1 0&amp;gt;, and |* 0 0 1&amp;gt;; the centroid of this is |* 1/2 1/2 1/2&amp;gt;; similarly the centroid of 1/4:1/5:1/6:1/7 is |* -1/2 -1/2 -1/2&amp;gt;. If we shift the origin to |* 1/2 1/2 1/2&amp;gt;, major tetrads correspond to [a b c], a+b+c even, and minor tetrads to [a-1 b-1 c-1], a+b+c even, which is the same as saying [a b c], a+b+c odd. &lt;br /&gt;
+&lt;br /&gt;
+Suppose m0, m1, m2 and m3 are four monzos denoting four notes of a 7-limit tetrad, either otonal or utonal, with distinct pitch classes, so that all four chord elements are represented. The product of the four notes is represented by the monzo m = m0+m1+m2+m3, and the pitch class for m, which ignores the first coefficient m[1] defining octaves, is represented by the other three, m[2], m[3] and m[4]. This pitch class is uniquely associated to the tetrad, and can be used to name it. If the tetrad is otonal with the root being |* e3 e5 e7&amp;gt;, where the asterisk can be any integer value, then m = |* 4e3+1 4e5+1 4e7+1&amp;gt;. on the other hand, if it is utonal, with the fifth of the chord |* e3 e5 e7&amp;gt; then m = |* 4e3-1 4e5-1 4e7-1&amp;gt;. If we denote the 3-tuple  [(m[3]+m[4]-2)/2 (m[2]+m[4]-2)/2 (m[2]+m[3]-2)/2] by [a b c], then if the tetrad is otonal, [a b c] = [e5+e7 e3+e7 e3+e5], whereas if it is utonal [a b c] = [e5+e7-1 e3+e7-1 e3+e5-1]. From this it follows that [a b c] is a triple of integers, and that a+b+c is even if the tetrad is otonal, and odd if the tetrad is utonal. &lt;br /&gt;
+&lt;br /&gt;
+Hence every triple of integers refers uniquely to a 7-limit tetrad; this is the lattice of 7-limit tetrads. 7-limit tetrads form the simplest kind of lattice, the cubic or grid lattice consisting of triples of integers with the ordinary Euclidean distance. This is a unique feature of the 7-limit; in no other limit do the complete utonalities and otonalities form a lattice.&lt;br /&gt;
 &lt;br /&gt;
 If [a b c] is any triple of integers, then it represents the major tetrad with root 3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+b-c)/2) if a+b+c is even, and the minor tetrad with root 3^((-1-a+b+c)/2) 5^((1+a-b+c)/2 7^((1+a+b-c)/2) if a+b+c is odd. Each unit cube corresponds to a &lt;a class="wiki_link_ext" href="http://tonalsoft.com/enc/stellat.htm" rel="nofollow"&gt;stellated hexany&lt;/a&gt;, or tetradekany, or dekatesserany, though chord cube would be less of a mouthful.&lt;br /&gt;