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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>
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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]]
[[Periodic scale copy]]
[[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=
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]]
[[Dual List]]
[[Meet and Join]]


**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]]
[[Recoverability]]
* [[Chord matrices]]
[[The dual]]
* [[Permutation product set]]
[[Interior product]]
* [[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]]
[[Comma-based lattices]]


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

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"