Mathematical theory: Difference between revisions

Line 1:

<~~h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

<span style="display: block; text-align: right;">[[Teoría_Profunda|Español]]</span>

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

~~: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2016-11-09 07:26:51 UTC</tt>.<br>~~

~~: The original revision id was <tt>598979688</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"><span style="display: block; text-align: right;"~~>~~[[~~Teoría Profunda~~|Español]]

~~<~~/span~~>~~

Tuning theory can become heavily mathematical, and here are links to some of the pages which wade into deeper waters.

~~== ==~~

~~=Scale Basics=~~

~~[[Periodic scale]]~~

~~[[Mathematics of MOS|MOS scales]]~~

~~<span class="PageTitle">[[Mathematics of MOS]]</span>~~

~~[[MODMOS scales]]~~

~~[[Graph-theoretic properties of scales]]~~

=Regular Temperaments=

== ==

**Basics:**

[[Monzos and Interval Space]]

=Scale Basics=

[[Vals and Tuning Space]]

[[Periodic_scale|Periodic scale]]

[[Regular Temperaments]]

[[Saturation]]

[[Mathematics_of_MOS|MOS scales]]

[[Dual List]]

[[Meet and Join]]

<span style="">[[Mathematics_of_MOS|Mathematics of MOS]]</span>

[[MODMOS_Scales|MODMOS scales]]

[[Graph-theoretic_properties_of_scales|Graph-theoretic properties of scales]]

=Regular Temperaments=

'''Basics:'''

[[Monzos_and_Interval_Space|Monzos and Interval Space]]

[[Vals_and_Tuning_Space|Vals and Tuning Space]]

[[Regular_Temperaments|Regular Temperaments]]

[[Saturation|Saturation]]

[[dual_list|Dual List]]

[[Meet_and_Join|Meet and Join]]

'''Linear algebra:'''

[[Temperament_Mapping_Matrices_(M-maps)|Temperament Mapping Matrices (M-maps)]]

[[Tmonzos_and_Tvals|Tmonzos and Tvals]]

[[Subgroup_Mapping_Matrices_(V-maps)|Subgroup Mapping Matrices (V-maps)]]

[[Smonzos_and_Svals|Smonzos and Svals]]

[[Generalized_Tenney_Norms_and_Tp_Interval_Space|Generalized Tenney Norms and Tp Interval Space]]

[[Generalized_Tenney_Dual_Norms_and_Tp_Tuning_Space|Generalized Tenney Dual Norms and Tp Tuning Space]]

'''Exterior algebra:'''

[[Wedgies_and_Multivals|Wedgies and Multivals]]

[[The_wedgie|The wedgie]]

[[Recoverability|Recoverability]]

[[The_dual|The dual]]

[[Interior_product|Interior product]]

'''Abstract algebra:'''

[[Groups,_Lattices,_Z-modules,_and_Vector_Spaces|Groups, Lattices, Z-modules and Vector spaces]]

'''Temperament representations:'''

[[Abstract_regular_temperament|Abstract regular temperament]]

[[Normal_lists|Normal lists]]

[[Gencom|Gencom]]

'''Other:'''

[[Planar_Temperament|Planar Temperament]]

[[Transversal_generators|Transversal generators]]

==Temperament Measures and Optimization==

[[Target_tunings|Target tunings]]

[[TOP_tuning|TOP tuning]]

[[Tp_tuning|Tp tuning]]

[[Tenney-Euclidean_Tuning|Tenney-Euclidean tuning]]

[[Tenney-Euclidean_temperament_measures|Tenney-Euclidean temperament measures]]

[[Tenney-Euclidean_metrics|Tenney-Euclidean metrics]]

[[Cangwu_badness|Cangwu badness]]

[[Fractional_monzos|Fractional monzos]]

[[Generator_complexity|Generator complexity]]

==Lattices and Scales==

[[Transversal|Transversal]]

[[Fokker_blocks|Fokker blocks]]

[[Weak_Fokker_blocks|Weak Fokker Blocks]]

[[Minkowski_reduced_bases_for_Fokker_groups_of_certain_vals|Minkowski reduced bases for Fokker groups of certain vals]]

[[Minkowski_blocks|Minkowski blocks]]

[[The_Seven_Limit_Symmetrical_Lattices|The Seven Limit Symmetrical Lattices]]

[[Tablets|Tablets]]

[[Hahn_distance|Hahn distance]]

~~**Linear algebra:**~~

[[Comma-based_lattices|Comma-based lattices]]

[[~~Temperament Mapping Matrices (M~~-~~maps)]]~~

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

~~[[Subgroup Mapping Matrices (V~~-~~maps)]]~~

~~[[Smonzos and Svals]]~~

~~[[Generalized Tenney Norms and Tp Interval Space]]~~

~~[[Generalized Tenney Dual Norms and Tp Tuning Space~~]]

~~**Exterior algebra:**~~

==Other==

[[~~Wedgies~~ and ~~Multivals]]~~

[[The_Riemann_Zeta_Function_and_Tuning|The Riemann Zeta Function and Tuning]]

~~[[The wedgie]]~~

~~[[Recoverability]]~~

~~[[The dual]]~~

~~[[Interior product~~]]

~~**Abstract algebra:**~~

[[Modulatory_space|Modulatory space]]

[[~~Groups, Lattices, Z-modules, and Vector Spaces~~|~~Groups, Lattices, Z-modules and Vector spaces~~]]

~~**Temperament representations:**~~

[[Chordal_space|Chordal space]]

[[~~Abstract regular temperament]]~~

~~[[Normal lists]]~~

~~[[Gencom~~]]

**Other:**

[[k*N_subgroups|k*N subgroups]]

~~[[Planar Temperament]]~~

~~[[Transversal generators~~]]

~~==Temperament Measures and Optimization==~~

[[Logarithmic_approximants|Logarithmic approximants]]

[[~~Target tunings]]~~

~~[[TOP tuning]]~~

~~[[Tp tuning]]~~

~~[[Tenney-Euclidean tuning]]~~

~~[[Tenney-Euclidean temperament measures]]~~

~~[[Tenney-Euclidean metrics]]~~

~~[[Cangwu badness]]~~

~~[[Fractional monzos]]~~

~~[[Generator complexity~~]]

~~==Lattices and Scales==~~

[[Don_Page_comma|Don Page comma]]

[[~~Transversal]]~~

~~[[Fokker blocks]]~~

~~[[Weak Fokker blocks~~|~~Weak Fokker Blocks]]~~

~~[[Minkowski reduced bases for Fokker groups of certain vals]]~~

~~[[Minkowski blocks]]~~

~~[[The Seven Limit Symmetrical Lattices]]~~

~~[[Tablets]]~~

~~[[Hahn distance]]~~

~~[[Comma-based lattices~~]]

~~==Other==~~

[[ABC,_High_Quality_Commas,_and_Epimericity|ABC, High Quality Commas, and Epimericity]] [[Category:links]]

[[~~The Riemann Zeta Function~~ and ~~Tuning~~]]

[[Category:math]]

[[~~Modulatory space~~]]

[[Category:overview]]

[[~~Chordal space~~]]

[[Category:theory]]

[[~~k*N subgroups~~]]

[[~~Logarithmic approximants]]~~

~~[[Don Page comma~~]]

~~[[xenharmonic/ABC, High Quality Commas, and Epimericity|ABC, High Quality Commas, and Epimericity]]</pre></div>~~

~~<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>Mathematical Theory</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="/Teor%C3%ADa%20Profunda">Español</a><br />

~~</span><br />~~

~~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>~~

~~<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 />~~

~~<span class="PageTitle"><a class="wiki_link" href="/Mathematics%20of%20MOS">Mathematics of MOS</a></span><br />~~

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

~~<a class="wiki_link" href="/Graph-theoretic%20properties%20of%20scales">Graph-theoretic properties of scales</a><br />~~

~~<br />~~

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

~~<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 />~~

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

~~<a class="wiki_link" href="/Meet%20and%20Join">Meet and Join</a><br />~~

~~<br />~~

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

~~<a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29">Temperament Mapping Matrices (M-maps)</a><br />~~

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

~~<a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29">Subgroup Mapping Matrices (V-maps)</a><br />~~

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

~~<a class="wiki_link" href="/Generalized%20Tenney%20Norms%20and%20Tp%20Interval%20Space">Generalized Tenney Norms and Tp Interval Space</a><br />~~

~~<a class="wiki_link" href="/Generalized%20Tenney%20Dual%20Norms%20and%20Tp%20Tuning%20Space">Generalized Tenney Dual Norms and Tp Tuning Space</a><br />~~

~~<br />~~

~~<strong>Exterior algebra:</strong><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="/Recoverability">Recoverability</a><br />~~

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

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

~~<br />~~

~~<strong>Abstract algebra:</strong><br />~~

~~<a class="wiki_link" href="/Groups%2C%20Lattices%2C%20Z-modules%2C%20and%20Vector%20Spaces">Groups, Lattices, Z-modules and Vector spaces</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="/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 />~~

~~<br />~~

<h2 id="toc3"><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 />~~

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

~~<a class="wiki_link" href="/Tenney-Euclidean%20tuning">Tenney-Euclidean tuning</a><br />~~

~~<a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures">Tenney-Euclidean temperament measures</a><br />~~

~~<a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean metrics</a><br />~~

~~<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="toc4"><a name="Regular Temperaments-Lattices and Scales"></a>Lattices and Scales</h2>

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

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

~~<a class="wiki_link" href="/Weak%20Fokker%20blocks">Weak Fokker Blocks</a><br />~~

~~<a class="wiki_link" href="/Minkowski%20reduced%20bases%20for%20Fokker%20groups%20of%20certain%20vals">Minkowski reduced bases for Fokker groups of certain vals</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 />~~

~~<a class="wiki_link" href="/Comma-based%20lattices">Comma-based lattices</a><br />~~

~~<br />~~

~~<h2 id="toc5"><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 />~~

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

~~<a class="wiki_link" href="/k%2AN%20subgroups">k*N subgroups</a><br />~~

~~<a class="wiki_link" href="/Logarithmic%20approximants">Logarithmic approximants</a><br />~~

~~<a class="wiki_link" href="/Don%20Page%20comma">Don Page comma</a><br />~~

<a class="wiki_link" href="http://xenharmonic.wikispaces.com/ABC%2C%20High%20Quality%20Commas%2C%20and%20Epimericity">ABC, High Quality Commas, and Epimericity</a></body></html></pre></div>

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+<span style="display: block; text-align: right;">[[Teoría_Profunda|Español]]</span>
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2016-11-09 07:26:51 UTC</tt>.<br>
-: The original revision id was <tt>598979688</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">&lt;span style="display: block; text-align: right;"&gt;[[Teoría Profunda|Español]]
-&lt;/span&gt;
 Tuning theory can become heavily mathematical, and here are links to some of the pages which wade into deeper waters.
-== ==
-=Scale Basics=
-[[Periodic scale]]
-[[Mathematics of MOS|MOS scales]]
-&lt;span class="PageTitle"&gt;[[Mathematics of MOS]]&lt;/span&gt;
-[[MODMOS scales]]
-[[Graph-theoretic properties of scales]]
-=Regular Temperaments=
+== ==
-**Basics:**
-[[Monzos and Interval Space]]
+=Scale Basics=
-[[Vals and Tuning Space]]
+[[Periodic_scale|Periodic scale]]
-[[Regular Temperaments]]
-[[Saturation]]
+[[Mathematics_of_MOS|MOS scales]]
-[[Dual List]]
-[[Meet and Join]]
+<span style="">[[Mathematics_of_MOS|Mathematics of MOS]]</span>
+[[MODMOS_Scales|MODMOS scales]]
+[[Graph-theoretic_properties_of_scales|Graph-theoretic properties of scales]]
+=Regular Temperaments=
+'''Basics:'''
+[[Monzos_and_Interval_Space|Monzos and Interval Space]]
+[[Vals_and_Tuning_Space|Vals and Tuning Space]]
+[[Regular_Temperaments|Regular Temperaments]]
+[[Saturation|Saturation]]
+[[dual_list|Dual List]]
+[[Meet_and_Join|Meet and Join]]
+'''Linear algebra:'''
+[[Temperament_Mapping_Matrices_(M-maps)|Temperament Mapping Matrices (M-maps)]]
+[[Tmonzos_and_Tvals|Tmonzos and Tvals]]
+[[Subgroup_Mapping_Matrices_(V-maps)|Subgroup Mapping Matrices (V-maps)]]
+[[Smonzos_and_Svals|Smonzos and Svals]]
+[[Generalized_Tenney_Norms_and_Tp_Interval_Space|Generalized Tenney Norms and Tp Interval Space]]
+[[Generalized_Tenney_Dual_Norms_and_Tp_Tuning_Space|Generalized Tenney Dual Norms and Tp Tuning Space]]
+'''Exterior algebra:'''
+[[Wedgies_and_Multivals|Wedgies and Multivals]]
+[[The_wedgie|The wedgie]]
+[[Recoverability|Recoverability]]
+[[The_dual|The dual]]
+[[Interior_product|Interior product]]
+'''Abstract algebra:'''
+[[Groups,_Lattices,_Z-modules,_and_Vector_Spaces|Groups, Lattices, Z-modules and Vector spaces]]
+'''Temperament representations:'''
+[[Abstract_regular_temperament|Abstract regular temperament]]
+[[Normal_lists|Normal lists]]
+[[Gencom|Gencom]]
+'''Other:'''
+[[Planar_Temperament|Planar Temperament]]
+[[Transversal_generators|Transversal generators]]
+==Temperament Measures and Optimization==
+[[Target_tunings|Target tunings]]
+[[TOP_tuning|TOP tuning]]
+[[Tp_tuning|Tp tuning]]
+[[Tenney-Euclidean_Tuning|Tenney-Euclidean tuning]]
+[[Tenney-Euclidean_temperament_measures|Tenney-Euclidean temperament measures]]
+[[Tenney-Euclidean_metrics|Tenney-Euclidean metrics]]
+[[Cangwu_badness|Cangwu badness]]
+[[Fractional_monzos|Fractional monzos]]
+[[Generator_complexity|Generator complexity]]
+==Lattices and Scales==
+[[Transversal|Transversal]]
+[[Fokker_blocks|Fokker blocks]]
+[[Weak_Fokker_blocks|Weak Fokker Blocks]]
+[[Minkowski_reduced_bases_for_Fokker_groups_of_certain_vals|Minkowski reduced bases for Fokker groups of certain vals]]
+[[Minkowski_blocks|Minkowski blocks]]
+[[The_Seven_Limit_Symmetrical_Lattices|The Seven Limit Symmetrical Lattices]]
+[[Tablets|Tablets]]
+[[Hahn_distance|Hahn distance]]
-**Linear algebra:**
+[[Comma-based_lattices|Comma-based lattices]]
-[[Temperament Mapping Matrices (M-maps)]]
-[[xenharmonic/Tmonzos and Tvals|Tmonzos and Tvals]]
-[[Subgroup Mapping Matrices (V-maps)]]
-[[Smonzos and Svals]]
-[[Generalized Tenney Norms and Tp Interval Space]]
-[[Generalized Tenney Dual Norms and Tp Tuning Space]]
-**Exterior algebra:**
+==Other==
-[[Wedgies and Multivals]]
+[[The_Riemann_Zeta_Function_and_Tuning|The Riemann Zeta Function and Tuning]]
-[[The wedgie]]
-[[Recoverability]]
-[[The dual]]
-[[Interior product]]
-**Abstract algebra:**
+[[Modulatory_space|Modulatory space]]
-[[Groups, Lattices, Z-modules, and Vector Spaces|Groups, Lattices, Z-modules and Vector spaces]]
-**Temperament representations:**
+[[Chordal_space|Chordal space]]
-[[Abstract regular temperament]]
-[[Normal lists]]
-[[Gencom]]
-**Other:**
+[[k*N_subgroups|k*N subgroups]]
-[[Planar Temperament]]
-[[Transversal generators]]
-==Temperament Measures and Optimization==
+[[Logarithmic_approximants|Logarithmic approximants]]
-[[Target tunings]]
-[[TOP tuning]]
-[[Tp tuning]]
-[[Tenney-Euclidean tuning]]
-[[Tenney-Euclidean temperament measures]]
-[[Tenney-Euclidean metrics]]
-[[Cangwu badness]]
-[[Fractional monzos]]
-[[Generator complexity]]
-==Lattices and Scales==
+[[Don_Page_comma|Don Page comma]]
-[[Transversal]]
-[[Fokker blocks]]
-[[Weak Fokker blocks|Weak Fokker Blocks]]
-[[Minkowski reduced bases for Fokker groups of certain vals]]
-[[Minkowski blocks]]
-[[The Seven Limit Symmetrical Lattices]]
-[[Tablets]]
-[[Hahn distance]]
-[[Comma-based lattices]]
-==Other==
+[[ABC,_High_Quality_Commas,_and_Epimericity|ABC, High Quality Commas, and Epimericity]]      [[Category:links]]
-[[The Riemann Zeta Function and Tuning]]
+[[Category:math]]
-[[Modulatory space]]
+[[Category:overview]]
-[[Chordal space]]
+[[Category:theory]]
-[[k*N subgroups]]
-[[Logarithmic approximants]]
-[[Don Page comma]]
-[[xenharmonic/ABC, High Quality Commas, and Epimericity|ABC, High Quality Commas, and Epimericity]]</pre></div>
-<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;Mathematical Theory&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; text-align: right;"&gt;&lt;a class="wiki_link" href="/Teor%C3%ADa%20Profunda"&gt;Español&lt;/a&gt;&lt;br /&gt;
-&lt;/span&gt;&lt;br /&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;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;
-&lt;span class="PageTitle"&gt;&lt;a class="wiki_link" href="/Mathematics%20of%20MOS"&gt;Mathematics of MOS&lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/MODMOS%20scales"&gt;MODMOS scales&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Graph-theoretic%20properties%20of%20scales"&gt;Graph-theoretic properties of scales&lt;/a&gt;&lt;br /&gt;
-&lt;br /&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;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;a class="wiki_link" href="/Dual%20List"&gt;Dual List&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Meet%20and%20Join"&gt;Meet and Join&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;strong&gt;Linear algebra:&lt;/strong&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29"&gt;Temperament Mapping Matrices (M-maps)&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="/Subgroup%20Mapping%20Matrices%20%28V-maps%29"&gt;Subgroup Mapping Matrices (V-maps)&lt;/a&gt;&lt;br /&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="/Generalized%20Tenney%20Norms%20and%20Tp%20Interval%20Space"&gt;Generalized Tenney Norms and Tp Interval Space&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Generalized%20Tenney%20Dual%20Norms%20and%20Tp%20Tuning%20Space"&gt;Generalized Tenney Dual Norms and Tp Tuning Space&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;strong&gt;Exterior algebra:&lt;/strong&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="/Recoverability"&gt;Recoverability&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;br /&gt;
-&lt;strong&gt;Abstract algebra:&lt;/strong&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Groups%2C%20Lattices%2C%20Z-modules%2C%20and%20Vector%20Spaces"&gt;Groups, Lattices, Z-modules and Vector spaces&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="/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;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Regular Temperaments-Temperament Measures and Optimization"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&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;
-&lt;a class="wiki_link" href="/Tp%20tuning"&gt;Tp tuning&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Tenney-Euclidean%20tuning"&gt;Tenney-Euclidean tuning&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures"&gt;Tenney-Euclidean temperament measures&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean metrics&lt;/a&gt;&lt;br /&gt;
-&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:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Regular Temperaments-Lattices and Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Lattices and Scales&lt;/h2&gt;
- &lt;a class="wiki_link" href="/Transversal"&gt;Transversal&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Fokker%20blocks"&gt;Fokker blocks&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Weak%20Fokker%20blocks"&gt;Weak Fokker Blocks&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Minkowski%20reduced%20bases%20for%20Fokker%20groups%20of%20certain%20vals"&gt;Minkowski reduced bases for Fokker groups of certain vals&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;a class="wiki_link" href="/Comma-based%20lattices"&gt;Comma-based lattices&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="Regular Temperaments-Other"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&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;
-&lt;a class="wiki_link" href="/Chordal%20space"&gt;Chordal space&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/k%2AN%20subgroups"&gt;k*N subgroups&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Logarithmic%20approximants"&gt;Logarithmic approximants&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Don%20Page%20comma"&gt;Don Page comma&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/ABC%2C%20High%20Quality%20Commas%2C%20and%20Epimericity"&gt;ABC, High Quality Commas, and Epimericity&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>