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<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.
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>

Revision as of 00:00, 17 July 2018

Español

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

Scale Basics

Periodic scale

MOS scales

Mathematics of MOS

MODMOS scales

Graph-theoretic properties of scales

Regular Temperaments

Basics:

Monzos and Interval Space

Vals and Tuning Space

Regular Temperaments

Saturation

Dual List

Meet and Join

Linear algebra:

Temperament Mapping Matrices (M-maps)

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:

Wedgies and Multivals

The wedgie

Recoverability

The dual

Interior product

Abstract algebra:

Groups, Lattices, Z-modules and Vector spaces

Temperament representations:

Abstract regular temperament

Normal lists

Gencom

Other:

Planar Temperament

Transversal generators

Temperament Measures and Optimization

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

Transversal

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

The Riemann Zeta Function and Tuning

Modulatory space

Chordal space

k*N subgroups

Logarithmic approximants

Don Page comma

ABC, High Quality Commas, and Epimericity

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