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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| en = Mathematical theory
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: The original revision id was <tt>357976422</tt>.<br>
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<h4>Original Wikitext content:</h4>
Tuning theory can become heavily mathematical, and here are links to some of the pages which wade into deeper waters.  
<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 Basics=
[[Periodic scale]]
[[Mathematics of MOS|MOS scales]]
&lt;span class="PageTitle"&gt;[[Mathematics of MOS]]&lt;/span&gt;
[[MODMOS scales]]


=Regular Temperaments=
It is recommended to read the pages listed on [[general theory]] first before exploring this page, as they lay the groundwork.
**Basics:**
[[Monzos and Interval Space]]
[[Vals and Tuning Space]]
[[Regular Temperaments]]
[[Saturation]]


**Linear algebra:**
== Intervals ==
[[Temperament Mapping Matrices (M-maps)]]
=== Interval metrics ===
[[xenharmonic/Tmonzos and Tvals|Tmonzos and Tvals]]
* [[Height]]
[[Subgroup Mapping Matrices (V-maps)]]
* [[Benedetti height]]
[[Smonzos and Svals]]
* [[Tenney norm]]
[[Generalized Tenney Norms and Tp Interval Space]]
* [[Kees seminorm]]
[[Generalized Tenney Dual Norms and Tp Tuning Space]]
* [[Wilson norm]]
* [[Weil norm, Tenney–Weil norm, and TWp interval and tuning space|Weil norm and Tenney–Weil norm]]


**Exterior algebra:**
=== Chords ===
[[Wedgies and Multivals]]
* [[Chordal space]]
[[The wedgie]]
* [[Chord complexity]]
[[The dual]]
* [[Chord matrices]]
[[Interior product]]
* [[Permutation product set]]
* [[Tablet]]


**Abstract algebra:**
=== Musical information theory ===
[[Groups, Lattices, Z-modules, and Vector Spaces|Groups, Lattices, Z-modules and Vector spaces]]
* [[Categorical entropy, mutual information, and channel capacity]]
* [[Harmonic entropy]]
* [[The Riemann zeta function and tuning]]
* [[Maximal dissonance tuning]]


**Temperament representations:**
=== Interval classification ===
[[Abstract regular temperament]]
* [[Delta-N ratio|Delta-''N'' ratio]]
[[Normal lists]]
* [[Non radical intervals with musical significance]]
[[Gencom]]
* [[Abc, high quality commas, and epimericity|''abc'', high quality commas, and epimericity]]
* [[Don Page comma]]


**Other:**
== Regular temperaments ==
[[Planar Temperament]]
=== Foundational theory ===
[[Transversal generators]]
* [[Mathematical theory of regular temperaments]]
* [[Monzos and interval space]]
** [[Radical interval]]
* [[Vals and tuning space]]
* [[Temperament mapping matrix]]
* [[Dual list]]
* [[Mathematical theory of saturation]]
* [[Temperament merging]]
* [[Temperament addition]]


==Temperament Measures and Optimization==  
=== Subgroup temperaments ===
[[Target tunings]]
* [[Subgroup temperament families, relationships, and genes]]
[[TOP tuning]]
* [[Subgroup monzos and vals]]
[[Tp tuning]]
* [[Tempered monzos and vals]]
[[Tenney-Euclidean tuning]]
* [[Subgroup basis matrix]]
[[Tenney-Euclidean temperament measures]]
* [[Meet and join]]
[[Tenney-Euclidean metrics]]
* [[k*N subgroups]]
[[Cangwu badness]]
[[Fractional monzos]]
[[Generator complexity]]


==Lattices and Scales==  
=== Exterior algebra ===
[[Transversal]]
* [[Exterior algebra]]
[[Fokker blocks]]
* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]]
[[Weak Fokker blocks|Weak Fokker Blocks]]
* [[Wedgie]] and [[Plücker coordinates]]
[[Minkowski reduced bases for Fokker groups of certain vals]]
* [[Recoverability]]
[[Minkowski blocks]]
* [[Hodge dual]]
[[The Seven Limit Symmetrical Lattices]]
* [[Interior product]]
[[Tablets]]
[[Hahn distance]]


==Other==  
=== Temperament measures and optimization ===
[[The Riemann Zeta Function and Tuning]]
* [[Optimization]]
[[Modulatory space]]
* [[Target tuning]]
[[Chordal space]]
* [[Cangwu badness]]
[[k*N subgroups]]
* [[Generator complexity]]
[[Don Page comma]]</pre></div>
* [[TOP tuning]]
<h4>Original HTML content:</h4>
* [[BOP tuning]]
<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;
* [[Tenney–Euclidean tuning]]
&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;
* [[Tenney–Euclidean temperament measures]]
&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;
* [[Tenney–Euclidean metrics]]
&lt;a class="wiki_link" href="/Periodic%20scale"&gt;Periodic scale&lt;/a&gt;&lt;br /&gt;
* [[Generalized Tenney norms and Tp interval space|Generalized Tenney norms and T<sub>''p''</sub> interval space]]
&lt;a class="wiki_link" href="/Mathematics%20of%20MOS"&gt;MOS scales&lt;/a&gt;&lt;br /&gt;
* [[Generalized Tenney dual norms and Tp tuning space|Generalized Tenney dual norms and T<sub>''p''</sub> tuning space]]
&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;
* [[Weil norm, Tenney–Weil norm, and TWp interval and tuning space|Weil norm, Tenney–Weil norm, and TW<sub>''p''</sub> interval and tuning space]]
&lt;a class="wiki_link" href="/MODMOS%20scales"&gt;MODMOS scales&lt;/a&gt;&lt;br /&gt;
* [[Tp tuning|T<sub>''p''</sub> tuning]]
&lt;br /&gt;
* [[Constrained tuning]]
&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;
* [[Logarithmic approximants]]
&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;
=== Temperament classification ===
&lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;Vals and Tuning Space&lt;/a&gt;&lt;br /&gt;
* [[Rank-2 temperament]] (include linear temperaments)
&lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;Regular Temperaments&lt;/a&gt;&lt;br /&gt;
* [[Rank-3 temperament]] (include planar temperaments)
&lt;a class="wiki_link" href="/Saturation"&gt;Saturation&lt;/a&gt;&lt;br /&gt;
(''Rank-1 temperaments are [[equal-step tuning]]s''.)
&lt;br /&gt;
* [[Temperament families and clans]]
&lt;strong&gt;Linear algebra:&lt;/strong&gt;&lt;br /&gt;
* [[Cluster temperament]]
&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;
* [[Associated temperament]]
&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tmonzos%20and%20Tvals"&gt;Tmonzos and Tvals&lt;/a&gt;&lt;br /&gt;
* [[Microtemperament]] and [[exotemperament]]
&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;
* [[Trivial temperament]]
&lt;a class="wiki_link" href="/Smonzos%20and%20Svals"&gt;Smonzos and Svals&lt;/a&gt;&lt;br /&gt;
* [[Sane and insane temperaments]]
&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;
== Scales ==
&lt;br /&gt;
=== MOS scales ===
&lt;strong&gt;Exterior algebra:&lt;/strong&gt;&lt;br /&gt;
* [[Periodic scale]]
&lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;Wedgies and Multivals&lt;/a&gt;&lt;br /&gt;
* [[MOS scale]]
&lt;a class="wiki_link" href="/The%20wedgie"&gt;The wedgie&lt;/a&gt;&lt;br /&gt;
* [[Mathematics of MOS]]
&lt;a class="wiki_link" href="/The%20dual"&gt;The dual&lt;/a&gt;&lt;br /&gt;
* [[MODMOS scale]]
&lt;a class="wiki_link" href="/Interior%20product"&gt;Interior product&lt;/a&gt;&lt;br /&gt;
* [[Modal UDP Notation]]
&lt;br /&gt;
* [[Graph-theoretic properties of scales]]
&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;
=== Scale measurements ===
&lt;br /&gt;
* [[Consonant class scale]]
&lt;strong&gt;Temperament representations:&lt;/strong&gt;&lt;br /&gt;
* [[Distributional evenness]]
&lt;a class="wiki_link" href="/Abstract%20regular%20temperament"&gt;Abstract regular temperament&lt;/a&gt;&lt;br /&gt;
* [[Maximum variety]]
&lt;a class="wiki_link" href="/Normal%20lists"&gt;Normal lists&lt;/a&gt;&lt;br /&gt;
* [[Lumma stability]]
&lt;a class="wiki_link" href="/Gencom"&gt;Gencom&lt;/a&gt;&lt;br /&gt;
* [[Structure metric]]
&lt;br /&gt;
* [[Hahn distance]]
&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;
=== Lattices and scale shapes ===
&lt;a class="wiki_link" href="/Transversal%20generators"&gt;Transversal generators&lt;/a&gt;&lt;br /&gt;
* [[Projection pair]]
&lt;br /&gt;
* [[Modulatory space]]
&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;
* [[Detempering]]
&lt;a class="wiki_link" href="/Target%20tunings"&gt;Target tunings&lt;/a&gt;&lt;br /&gt;
* [[Generator preimage]]
&lt;a class="wiki_link" href="/TOP%20tuning"&gt;TOP tuning&lt;/a&gt;&lt;br /&gt;
* [[Fokker block]]
&lt;a class="wiki_link" href="/Tp%20tuning"&gt;Tp tuning&lt;/a&gt;&lt;br /&gt;
* [[Minkowski block]]
&lt;a class="wiki_link" href="/Tenney-Euclidean%20tuning"&gt;Tenney-Euclidean tuning&lt;/a&gt;&lt;br /&gt;
* [[Minkowski reduced bases for Fokker groups of certain vals]]
&lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures"&gt;Tenney-Euclidean temperament measures&lt;/a&gt;&lt;br /&gt;
* [[7-limit symmetrical lattices]]
&lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean metrics&lt;/a&gt;&lt;br /&gt;
* [[Comma-based lattices]]
&lt;a class="wiki_link" href="/Cangwu%20badness"&gt;Cangwu badness&lt;/a&gt;&lt;br /&gt;
=== Combinatorics on words ===
&lt;a class="wiki_link" href="/Fractional%20monzos"&gt;Fractional monzos&lt;/a&gt;&lt;br /&gt;
* [[Ternary scale theorems]]
&lt;a class="wiki_link" href="/Generator%20complexity"&gt;Generator complexity&lt;/a&gt;&lt;br /&gt;
 
&lt;br /&gt;
== More pages ==
&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;
* [[:Category: Theory]]
&lt;a class="wiki_link" href="/Transversal"&gt;Transversal&lt;/a&gt;&lt;br /&gt;
* [[:Category: Math]]
&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;
[[Category:Math]]
&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;
[[Category:Theory]]
&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: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="/Don%20Page%20comma"&gt;Don Page comma&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 16:07, 6 December 2025

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

It is recommended to read the pages listed on general theory first before exploring this page, as they lay the groundwork.

Intervals

Interval metrics

Chords

Musical information theory

Interval classification

Regular temperaments

Foundational theory

Subgroup temperaments

Exterior algebra

Temperament measures and optimization

Temperament classification

(Rank-1 temperaments are equal-step tunings.)

Scales

MOS scales

Scale measurements

Lattices and scale shapes

Combinatorics on words

More pages

Retrieved from "https://en.xen.wiki/index.php?title=Mathematical_theory&oldid=218680"