Mathematical theory: 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:mbattaglia1|mbattaglia1]] and made on <tt>2012-07-31 03:16:59 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-07-31 03:38:19 UTC</tt>.<br>

: The original revision id was <tt>~~355646472~~</tt>.<br>

: The original revision id was <tt>355647886</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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<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">Tuning theory can become heavily mathematical, and here are links to some of the pages which wade into deeper waters.

== ==

==Scale ~~Foundations=~~=

=Scale Basics=

[[Periodic scale]]

[[Mathematics of MOS|MOS scales]]

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[[MODMOS scales]]

==~~Tuning Foundations==~~

=Regular Temperaments=

**Basics:**

[[Monzos and Interval Space]]

[[Vals and Tuning Space]]

[[Regular Temperaments]]

[[Saturation]]

~~==Regular Temperaments==~~

**Exterior algebra:**

~~[[Abstract regular temperament]]~~

[[Wedgies and Multivals]]

[[The wedgie]]

[[The dual]]

[[Interior product]]

[[~~The dual~~]]

[[~~Saturation~~]]

**Linear algebra:**

[[Mapping matrices]]

[[xenharmonic/Tmonzos and Tvals|Tmonzos and Tvals]]

[[Dual Transformations]]

**Temperament representations:**

[[Abstract regular temperament]]

[[Normal lists]]

[[~~Generator complexity~~]]

[[Gencom]]

**Other:**

[[Planar Temperament]]

[[Transversal generators]]

~~[[Gencom]]~~

==Subgroups==

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[[V-maps and M-maps]]

==~~Tuning~~ Optimization==

==Temperament Measures and Optimization==

[[Target tunings]]

[[TOP tuning]]

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[[Cangwu badness]]

[[Fractional monzos]]

[[Generator complexity]]

==Lattices and Scales==

[[Transversal]]

[[Minkowski blocks]]

~~[[Transversal]]~~

[[The Seven Limit Symmetrical Lattices]]

[[Tablets]]

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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>Mathematical Theory</title></head><body>Tuning theory can become heavily mathematical, and here are links to some of the pages which wade into deeper waters.<br />

<h2 id="toc0"> </h2>

<h2 id="toc1"><a name="x-Scale ~~Foundations~~"></a>Scale ~~Foundations~~</h2>

<h1 id="toc1"><a name="Scale Basics"></a>Scale Basics</h1>

<a class="wiki_link" href="/Periodic%20scale">Periodic scale</a><br />

<a class="wiki_link" href="/Mathematics%20of%20MOS">MOS scales</a><br />

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<a class="wiki_link" href="/MODMOS%20scales">MODMOS scales</a><br />

<br />

<h2 id="toc2"><a name="~~x-Tuning Foundations~~"></a>~~Tuning Foundations~~</h2>

<h1 id="toc2"><a name="Regular Temperaments"></a>Regular Temperaments</h1>

<a class="wiki_link" href="/Monzos%20and%20Interval%20Space">Monzos and Interval Space</a><br />

<strong>Basics:</strong><br />

<a class="wiki_link" href="/Monzos%20and%20Interval%20Space">Monzos and Interval Space</a><br />

<a class="wiki_link" href="/Vals%20and%20Tuning%20Space">Vals and Tuning Space</a><br />

<a class="wiki_link" href="/Regular%20Temperaments">Regular Temperaments</a><br />

<a class="wiki_link" href="/Saturation">Saturation</a><br />

<br />

<~~!-- ws:start:WikiTextHeadingRule:6:&lt;h2~~&~~amp;~~gt; ~~--><h2 id="toc3"><a name="x-Regular Temperaments"></a>Regular Temperaments</h2>~~

<strong>Exterior algebra:</strong><br />

~~<a class="wiki_link" href="/Abstract%20regular%20temperament">Abstract regular temperament~~</a><br />

<a class="wiki_link" href="/Wedgies%20and%20Multivals">Wedgies and Multivals</a><br />

<a class="wiki_link" href="/The%20wedgie">The wedgie</a><br />

<a class="wiki_link" href="/Interior%20product">Interior product</a><br />

<br />

<a class="wiki_link" href="/~~Saturation~~">~~Saturation~~</a><br />

<strong>Linear algebra:</strong><br />

<a class="wiki_link" href="/Mapping%20matrices">Mapping matrices</a><br />

<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tmonzos%20and%20Tvals">Tmonzos and Tvals</a><br />

<a class="wiki_link" href="/Dual%20Transformations">Dual Transformations</a><br />

<br />

<strong>Temperament representations:</strong><br />

<a class="wiki_link" href="/Abstract%20regular%20temperament">Abstract regular temperament</a><br />

<a class="wiki_link" href="/Normal%20lists">Normal lists</a><br />

<a class="wiki_link" href="/~~Generator%20complexity~~">~~Generator complexity~~</a><br />

<a class="wiki_link" href="/Gencom">Gencom</a><br />

<br />

<strong>Other:</strong><br />

<a class="wiki_link" href="/Planar%20Temperament">Planar Temperament</a><br />

<a class="wiki_link" href="/Transversal%20generators">Transversal generators</a><br />

~~<a class="wiki_link" href="/Gencom">Gencom</a><br />~~

<br />

<h2 id="~~toc4~~"><a name="x-Subgroups"></a>Subgroups</h2>

<h2 id="toc3"><a name="Regular Temperaments-Subgroups"></a>Subgroups</h2>

<a class="wiki_link" href="/Smonzos%20and%20Svals">Smonzos and Svals</a><br />

<br />

<h2 id="~~toc5~~"><a name="x-~~Tuning~~ Optimization"></a>~~Tuning~~ Optimization</h2>

<h2 id="toc4"><a name="Regular Temperaments-Temperament Measures and Optimization"></a>Temperament Measures and Optimization</h2>

<a class="wiki_link" href="/Target%20tunings">Target tunings</a><br />

<a class="wiki_link" href="/TOP%20tuning">TOP tuning</a><br />

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<a class="wiki_link" href="/Cangwu%20badness">Cangwu badness</a><br />

<a class="wiki_link" href="/Fractional%20monzos">Fractional monzos</a><br />

<a class="wiki_link" href="/Generator%20complexity">Generator complexity</a><br />

<br />

<h2 id="~~toc6~~"><a name="x-Lattices and Scales"></a>Lattices and Scales</h2>

<h2 id="toc5"><a name="Regular Temperaments-Lattices and Scales"></a>Lattices and Scales</h2>

<a class="wiki_link" href="/~~Minkowski%20blocks~~">~~Minkowski blocks~~</a><br />

<a class="wiki_link" href="/Transversal">Transversal</a><br />

<a class="wiki_link" href="/~~Transversal~~">~~Transversal~~</a><br />

<a class="wiki_link" href="/Minkowski%20blocks">Minkowski blocks</a><br />

<a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">The Seven Limit Symmetrical Lattices</a><br />

<a class="wiki_link" href="/Tablets">Tablets</a><br />

<a class="wiki_link" href="/Hahn%20distance">Hahn distance</a><br />

<br />

<h2 id="~~toc7~~"><a name="x-Other"></a>Other</h2>

<h2 id="toc6"><a name="Regular Temperaments-Other"></a>Other</h2>

<a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning">The Riemann Zeta Function and Tuning</a><br />

<a class="wiki_link" href="/Modulatory%20space">Modulatory space</a><br />

@@ 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:mbattaglia1|mbattaglia1]] and made on <tt>2012-07-31 03:16:59 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-07-31 03:38:19 UTC</tt>.<br>
-: The original revision id was <tt>355646472</tt>.<br>
+: The original revision id was <tt>355647886</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 8: / Line 8: @@
 <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">Tuning theory can become heavily mathematical, and here are links to some of the pages which wade into deeper waters.
 == ==
-==Scale Foundations==
+=Scale Basics=
 [[Periodic scale]]
 [[Mathematics of MOS|MOS scales]]
@@ Line 14: / Line 14: @@
 [[MODMOS scales]]
-==Tuning Foundations==
+=Regular Temperaments=
+**Basics:**
 [[Monzos and Interval Space]]
 [[Vals and Tuning Space]]
+[[Regular Temperaments]]
+[[Saturation]]
-==Regular Temperaments==
+**Exterior algebra:**
-[[Abstract regular temperament]]
 [[Wedgies and Multivals]]
 [[The wedgie]]
+[[The dual]]
 [[Interior product]]
-[[The dual]]
-[[Saturation]]
+**Linear algebra:**
+[[Mapping matrices]]
+[[xenharmonic/Tmonzos and Tvals|Tmonzos and Tvals]]
+[[Dual Transformations]]
+**Temperament representations:**
+[[Abstract regular temperament]]
 [[Normal lists]]
-[[Generator complexity]]
+[[Gencom]]
+**Other:**
 [[Planar Temperament]]
 [[Transversal generators]]
-[[Gencom]]
 ==Subgroups==
@@ Line 35: / Line 45: @@
 [[V-maps and M-maps]]
-==Tuning Optimization==
+==Temperament Measures and Optimization==
 [[Target tunings]]
 [[TOP tuning]]
@@ Line 44: / Line 54: @@
 [[Cangwu badness]]
 [[Fractional monzos]]
+[[Generator complexity]]
 ==Lattices and Scales==
+[[Transversal]]
 [[Minkowski blocks]]
-[[Transversal]]
 [[The Seven Limit Symmetrical Lattices]]
 [[Tablets]]
@@ Line 61: / Line 72: @@
 <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;Mathematical Theory&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Tuning theory can become heavily mathematical, and here are links to some of the pages which wade into deeper waters.&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt; &lt;/h2&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Scale Foundations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Scale Foundations&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Scale Basics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Scale Basics&lt;/h1&gt;
   &lt;a class="wiki_link" href="/Periodic%20scale"&gt;Periodic scale&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Mathematics%20of%20MOS"&gt;MOS scales&lt;/a&gt;&lt;br /&gt;
@@ Line 67: / Line 78: @@
 &lt;a class="wiki_link" href="/MODMOS%20scales"&gt;MODMOS scales&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Tuning Foundations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Tuning Foundations&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Regular Temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Regular Temperaments&lt;/h1&gt;
-  &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;Monzos and Interval Space&lt;/a&gt;&lt;br /&gt;
+  &lt;strong&gt;Basics:&lt;/strong&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;Monzos and Interval Space&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;Vals and Tuning Space&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;Regular Temperaments&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/Saturation"&gt;Saturation&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-Regular Temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Regular Temperaments&lt;/h2&gt;
+&lt;strong&gt;Exterior algebra:&lt;/strong&gt;&lt;br /&gt;
- &lt;a class="wiki_link" href="/Abstract%20regular%20temperament"&gt;Abstract regular temperament&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;Wedgies and Multivals&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/The%20wedgie"&gt;The wedgie&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/The%20dual"&gt;The dual&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Interior%20product"&gt;Interior product&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/The%20dual"&gt;The dual&lt;/a&gt;&lt;br /&gt;
+&lt;br /&gt;
-&lt;a class="wiki_link" href="/Saturation"&gt;Saturation&lt;/a&gt;&lt;br /&gt;
+&lt;strong&gt;Linear algebra:&lt;/strong&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/Mapping%20matrices"&gt;Mapping matrices&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tmonzos%20and%20Tvals"&gt;Tmonzos and Tvals&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/Dual%20Transformations"&gt;Dual Transformations&lt;/a&gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;strong&gt;Temperament representations:&lt;/strong&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/Abstract%20regular%20temperament"&gt;Abstract regular temperament&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Normal%20lists"&gt;Normal lists&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Generator%20complexity"&gt;Generator complexity&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/Gencom"&gt;Gencom&lt;/a&gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;strong&gt;Other:&lt;/strong&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Planar%20Temperament"&gt;Planar Temperament&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Transversal%20generators"&gt;Transversal generators&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Gencom"&gt;Gencom&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-Subgroups"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Subgroups&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Regular Temperaments-Subgroups"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Subgroups&lt;/h2&gt;
   &lt;a class="wiki_link" href="/Smonzos%20and%20Svals"&gt;Smonzos and Svals&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/V-maps%20and%20M-maps"&gt;V-maps and M-maps&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x-Tuning Optimization"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Tuning Optimization&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Regular Temperaments-Temperament Measures and Optimization"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Temperament Measures and Optimization&lt;/h2&gt;
   &lt;a class="wiki_link" href="/Target%20tunings"&gt;Target tunings&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/TOP%20tuning"&gt;TOP tuning&lt;/a&gt;&lt;br /&gt;
@@ Line 97: / Line 118: @@
 &lt;a class="wiki_link" href="/Cangwu%20badness"&gt;Cangwu badness&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;Fractional monzos&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/Generator%20complexity"&gt;Generator complexity&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x-Lattices and Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Lattices and Scales&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Regular Temperaments-Lattices and Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Lattices and Scales&lt;/h2&gt;
-  &lt;a class="wiki_link" href="/Minkowski%20blocks"&gt;Minkowski blocks&lt;/a&gt;&lt;br /&gt;
+  &lt;a class="wiki_link" href="/Transversal"&gt;Transversal&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Transversal"&gt;Transversal&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/Minkowski%20blocks"&gt;Minkowski blocks&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;The Seven Limit Symmetrical Lattices&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Tablets"&gt;Tablets&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Hahn%20distance"&gt;Hahn distance&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="x-Other"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Other&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Regular Temperaments-Other"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Other&lt;/h2&gt;
   &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning"&gt;The Riemann Zeta Function and Tuning&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Modulatory%20space"&gt;Modulatory space&lt;/a&gt;&lt;br /&gt;