Tablet: 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:<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-15 ~~14:47~~:39 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-16 12:15:43 UTC</tt>.<br>

: The original revision id was <tt>~~254465024~~</tt>.<br>

: The original revision id was <tt>254787718</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>

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=The 5-limit 3et tablet=

If we want a way to uniquely denote the octave-equivalent class of a 5-limit major triad, we can simply give the root of the chord, ignoring octaves. In terms of monzos that means a major triad would be denoted <* e3 e5 e7|, where the asterisk can be anything. Since the asterisk is not being used to impart information, we can use it to distinguish major from minor triads. So we might decide that <r e3 e5 e7| defines a major triad with root given by |* e3 e5 e7> when r is even, and a minor triad when r is odd.

If r is even, therefore, we will regard the 4-tuple [r e3 e5] as denoting a major triad, and if [n, [r e3 e5]] is the tablet we wish to define, we first calculate u = n - 5e3 - 7e5. Then if r is even, we have

* If u mod 3 = 0, then

note(n, [r e3 e5]) = |u/3 e3 e5>

* If u mod 3 = 1, then

note(n, [r e3 e5]) = |(u-7)/3 e3 e5+1>

* If u mod 3 = 2, then

note(n, [r e3 e5]) = |(u-5)/3 e3+1 e5>

On the other hand, if r is odd then

* If u mod 3 = 0, then

note(n, [r e3 e5]) = |u/3 e3 e3-e5>

* If u mod 3 = 1, then

note(n, [r e3 e5]) = |(u-7)/3+3 e3+1 e3-e5-1>

* If u mod 3 = 2, then

note(n, [r e3 e5]) = |(u-5)/3 e3+1 e5>

Then if c is a tablet, note(n, c) gives the note belonging to the triad denoted by c, satisfying <3 5 7|note(n, c)> = n.

=The 7-limit 4et tablet=

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note(t) = |(u-11)/4 (-a+b+c)/2 (a-b+c)/2 1+(a+b-c)/2>

If a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). Then note(t) is the note defined by the tablet t. If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w)] is a 7-limit tetrad.

If a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). Then note(t) is the note defined by the tablet t. If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w)] is a 7-limit tetrad, and <4 6 9 11|note(n, t)| = n.

=The 7-limit 5et tablet=

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note(t) = |(u-14)/5 (-a+b+c)/2 (a-b+c)/2 1+(a+b-c)/2>

Once again, if a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w), note(n+4), w)] is a complete 9-odd-limit quintad.

Once again, if a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w), note(n+4), w)] is a complete 9-odd-limit quintad, where <5 8 12 14|note(n, t)> = n.

=The 13-limit 7et tablet=

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<br />

<h1 id="toc2"><a name="The 5-limit 3et tablet"></a>The 5-limit 3et tablet</h1>

If we want a way to uniquely denote the octave-equivalent class of a 5-limit major triad, we can simply give the root of the chord, ignoring octaves. In terms of monzos that means a major triad would be denoted &lt;* e3 e5 e7|, where the asterisk can be anything. Since the asterisk is not being used to impart information, we can use it to distinguish major from minor triads. So we might decide that &lt;r e3 e5 e7| defines a major triad with root given by |* e3 e5 e7&gt; when r is even, and a minor triad when r is odd.<br />

<br />

If r is even, therefore, we will regard the 4-tuple [r e3 e5] as denoting a major triad, and if [n, [r e3 e5]] is the tablet we wish to define, we first calculate u = n - 5e3 - 7e5. Then if r is even, we have<br />

<br />

<br />

<br />

<br />

On the other hand, if r is odd then<br />

<br />

<br />

<br />

<br />

Then if c is a tablet, note(n, c) gives the note belonging to the triad denoted by c, satisfying &lt;3 5 7|note(n, c)&gt; = n.<br />

<br />

<h1 id="toc3"><a name="The 7-limit 4et tablet"></a>The 7-limit 4et tablet</h1>

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<br />

If a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). Then note(t) is the note defined by the tablet t. If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w)] is a 7-limit tetrad.<br />

If a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). Then note(t) is the note defined by the tablet t. If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w)] is a 7-limit tetrad, and &lt;4 6 9 11|note(n, t)| = n.<br />

<br />

<h1 id="toc4"><a name="The 7-limit 5et tablet"></a>The 7-limit 5et tablet</h1>

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<br />

Once again, if a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w), note(n+4), w)] is a complete 9-odd-limit quintad.<br />

Once again, if a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w), note(n+4), w)] is a complete 9-odd-limit quintad, where &lt;5 8 12 14|note(n, t)&gt; = n.<br />

<br />

<h1 id="toc5"><a name="The 13-limit 7et tablet"></a>The 13-limit 7et tablet</h1>

</body></html></pre></div>

@@ 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>2011-09-15 14:47:39 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-16 12:15:43 UTC</tt>.<br>
-: The original revision id was <tt>254465024</tt>.<br>
+: The original revision id was <tt>254787718</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 16: / Line 16: @@
 =The 5-limit 3et tablet=
+If we want a way to uniquely denote the octave-equivalent class of a 5-limit major triad, we can simply give the root of the chord, ignoring octaves. In terms of monzos that means a major triad would be denoted &lt;* e3 e5 e7|, where the asterisk can be anything. Since the asterisk is not being used to impart information, we can use it to distinguish major from minor triads. So we might decide that &lt;r e3 e5 e7| defines a major triad with root given by |* e3 e5 e7&gt; when r is even, and a minor triad when r is odd.
+If r is even, therefore, we will regard the 4-tuple [r e3 e5] as denoting a major triad, and if [n, [r e3 e5]] is the tablet we wish to define, we first calculate u = n - 5e3 - 7e5. Then if r is even, we have
+* If u mod 3 = 0, then
+note(n, [r e3 e5]) = |u/3 e3 e5&gt;
+* If u mod 3 = 1, then
+note(n, [r e3 e5]) = |(u-7)/3 e3 e5+1&gt;
+* If u mod 3 = 2, then
+note(n, [r e3 e5]) = |(u-5)/3 e3+1 e5&gt;
+On the other hand, if r is odd then
+* If u mod 3 = 0, then
+note(n, [r e3 e5]) = |u/3 e3 e3-e5&gt;
+* If u mod 3 = 1, then
+note(n, [r e3 e5]) = |(u-7)/3+3 e3+1 e3-e5-1&gt;
+* If u mod 3 = 2, then
+note(n, [r e3 e5]) = |(u-5)/3 e3+1 e5&gt;
+Then if c is a tablet, note(n, c) gives the note belonging to the triad denoted by c, satisfying &lt;3 5 7|note(n, c)&gt; = n.
 =The 7-limit 4et tablet=
@@ Line 31: / Line 56: @@
 note(t) = |(u-11)/4  (-a+b+c)/2  (a-b+c)/2  1+(a+b-c)/2&gt;
-If a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). Then note(t) is the note defined by the tablet t. If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w)] is a 7-limit tetrad.
+If a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). Then note(t) is the note defined by the tablet t. If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w)] is a 7-limit tetrad, and &lt;4 6 9 11|note(n, t)| = n.
 =The 7-limit 5et tablet=
@@ Line 48: / Line 73: @@
 note(t) = |(u-14)/5  (-a+b+c)/2  (a-b+c)/2  1+(a+b-c)/2&gt;
-Once again, if a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w), note(n+4), w)] is a complete 9-odd-limit quintad.
+Once again, if a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w), note(n+4), w)] is a complete 9-odd-limit quintad, where &lt;5 8 12 14|note(n, t)&gt; = n.
 =The 13-limit 7et tablet=
@@ Line 63: / Line 88: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="The 5-limit 3et tablet"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;The 5-limit 3et tablet&lt;/h1&gt;
+If we want a way to uniquely denote the octave-equivalent class of a 5-limit major triad, we can simply give the root of the chord, ignoring octaves. In terms of monzos that means a major triad would be denoted &amp;lt;* e3 e5 e7|, where the asterisk can be anything. Since the asterisk is not being used to impart information, we can use it to distinguish major from minor triads. So we might decide that &amp;lt;r e3 e5 e7| defines a major triad with root given by |* e3 e5 e7&amp;gt; when r is even, and a minor triad when r is odd.&lt;br /&gt;
+&lt;br /&gt;
+If r is even, therefore, we will regard the 4-tuple [r e3 e5] as denoting a major triad, and if [n, [r e3 e5]] is the tablet we wish to define, we first calculate u = n - 5e3 - 7e5. Then if r is even, we have&lt;br /&gt;
+&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;If u mod 3 = 0, then&lt;/li&gt;&lt;/ul&gt;note(n, [r e3 e5]) = |u/3 e3 e5&amp;gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;If u mod 3 = 1, then&lt;/li&gt;&lt;/ul&gt;note(n, [r e3 e5]) = |(u-7)/3 e3 e5+1&amp;gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;If u mod 3 = 2, then&lt;/li&gt;&lt;/ul&gt;note(n, [r e3 e5]) = |(u-5)/3 e3+1 e5&amp;gt;&lt;br /&gt;
+&lt;br /&gt;
+On the other hand, if r is odd then&lt;br /&gt;
+&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;If u mod 3 = 0, then&lt;/li&gt;&lt;/ul&gt;note(n, [r e3 e5]) = |u/3 e3 e3-e5&amp;gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;If u mod 3 = 1, then&lt;/li&gt;&lt;/ul&gt;note(n, [r e3 e5]) = |(u-7)/3+3 e3+1 e3-e5-1&amp;gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;If u mod 3 = 2, then&lt;/li&gt;&lt;/ul&gt;note(n, [r e3 e5]) = |(u-5)/3 e3+1 e5&amp;gt;&lt;br /&gt;
+&lt;br /&gt;
+Then if c is a tablet, note(n, c) gives the note belonging to the triad denoted by c, satisfying &amp;lt;3 5 7|note(n, c)&amp;gt; = n.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="The 7-limit 4et tablet"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;The 7-limit 4et tablet&lt;/h1&gt;
@@ Line 74: / Line 118: @@
 &lt;ul&gt;&lt;li&gt;If u mod 4 = 3, then&lt;/li&gt;&lt;/ul&gt;note(t) = |(u-11)/4  (-a+b+c)/2  (a-b+c)/2  1+(a+b-c)/2&amp;gt;&lt;br /&gt;
 &lt;br /&gt;
-If a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). Then note(t) is the note defined by the tablet t. If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w)] is a 7-limit tetrad.&lt;br /&gt;
+If a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). Then note(t) is the note defined by the tablet t. If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w)] is a 7-limit tetrad, and &amp;lt;4 6 9 11|note(n, t)| = n.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="The 7-limit 5et tablet"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;The 7-limit 5et tablet&lt;/h1&gt;
@@ Line 86: / Line 130: @@
 &lt;ul&gt;&lt;li&gt;If u mod 5 = 4, then&lt;/li&gt;&lt;/ul&gt;note(t) = |(u-14)/5  (-a+b+c)/2  (a-b+c)/2  1+(a+b-c)/2&amp;gt;&lt;br /&gt;
 &lt;br /&gt;
-Once again, if a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w), note(n+4), w)] is a complete 9-odd-limit quintad.&lt;br /&gt;
+Once again, if a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w), note(n+4), w)] is a complete 9-odd-limit quintad, where &amp;lt;5 8 12 14|note(n, t)&amp;gt; = n.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="The 13-limit 7et tablet"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;The 13-limit 7et tablet&lt;/h1&gt;
 &lt;/body&gt;&lt;/html&gt;</pre></div>