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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
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-17 13:30:03 UTC</tt>.<br>
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: The original revision id was <tt>358372154</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]]
[[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]]


**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;a class="wiki_link" href="/Graph-theoretic%20properties%20of%20scales"&gt;Graph-theoretic properties of scales&lt;/a&gt;&lt;br /&gt;
* [[Constrained tuning]]
&lt;br /&gt;
* [[Logarithmic approximants]]
&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;
=== Temperament classification ===
&lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;Monzos and Interval Space&lt;/a&gt;&lt;br /&gt;
* [[Rank-2 temperament]] (include linear temperaments)
&lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;Vals and Tuning Space&lt;/a&gt;&lt;br /&gt;
* [[Rank-3 temperament]] (include planar temperaments)
&lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;Regular Temperaments&lt;/a&gt;&lt;br /&gt;
(''Rank-1 temperaments are [[equal-step tuning]]s''.)
&lt;a class="wiki_link" href="/Saturation"&gt;Saturation&lt;/a&gt;&lt;br /&gt;
* [[Temperament families and clans]]
&lt;br /&gt;
* [[Cluster temperament]]
&lt;strong&gt;Linear algebra:&lt;/strong&gt;&lt;br /&gt;
* [[Associated 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;
* [[Microtemperament]] and [[exotemperament]]
&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tmonzos%20and%20Tvals"&gt;Tmonzos and Tvals&lt;/a&gt;&lt;br /&gt;
* [[Trivial temperament]]
&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;
* [[Sane and insane temperaments]]
&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;
== Scales ==
&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;
=== MOS scales ===
&lt;br /&gt;
* [[Periodic scale]]
&lt;strong&gt;Exterior algebra:&lt;/strong&gt;&lt;br /&gt;
* [[MOS scale]]
&lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;Wedgies and Multivals&lt;/a&gt;&lt;br /&gt;
* [[Mathematics of MOS]]
&lt;a class="wiki_link" href="/The%20wedgie"&gt;The wedgie&lt;/a&gt;&lt;br /&gt;
* [[MODMOS scale]]
&lt;a class="wiki_link" href="/The%20dual"&gt;The dual&lt;/a&gt;&lt;br /&gt;
* [[Modal UDP Notation]]
&lt;a class="wiki_link" href="/Interior%20product"&gt;Interior product&lt;/a&gt;&lt;br /&gt;
* [[Graph-theoretic properties of scales]]
&lt;br /&gt;
 
&lt;strong&gt;Abstract algebra:&lt;/strong&gt;&lt;br /&gt;
=== Scale measurements ===
&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;
* [[Consonant class scale]]
&lt;br /&gt;
* [[Distributional evenness]]
&lt;strong&gt;Temperament representations:&lt;/strong&gt;&lt;br /&gt;
* [[Maximum variety]]
&lt;a class="wiki_link" href="/Abstract%20regular%20temperament"&gt;Abstract regular temperament&lt;/a&gt;&lt;br /&gt;
* [[Lumma stability]]
&lt;a class="wiki_link" href="/Normal%20lists"&gt;Normal lists&lt;/a&gt;&lt;br /&gt;
* [[Structure metric]]
&lt;a class="wiki_link" href="/Gencom"&gt;Gencom&lt;/a&gt;&lt;br /&gt;
* [[Hahn distance]]
&lt;br /&gt;
 
&lt;strong&gt;Other:&lt;/strong&gt;&lt;br /&gt;
=== Lattices and scale shapes ===
&lt;a class="wiki_link" href="/Planar%20Temperament"&gt;Planar Temperament&lt;/a&gt;&lt;br /&gt;
* [[Projection pair]]
&lt;a class="wiki_link" href="/Transversal%20generators"&gt;Transversal generators&lt;/a&gt;&lt;br /&gt;
* [[Modulatory space]]
&lt;br /&gt;
* [[Detempering]]
&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;
* [[Generator preimage]]
&lt;a class="wiki_link" href="/Target%20tunings"&gt;Target tunings&lt;/a&gt;&lt;br /&gt;
* [[Fokker block]]
&lt;a class="wiki_link" href="/TOP%20tuning"&gt;TOP tuning&lt;/a&gt;&lt;br /&gt;
* [[Minkowski block]]
&lt;a class="wiki_link" href="/Tp%20tuning"&gt;Tp tuning&lt;/a&gt;&lt;br /&gt;
* [[Minkowski reduced bases for Fokker groups of certain vals]]
&lt;a class="wiki_link" href="/Tenney-Euclidean%20tuning"&gt;Tenney-Euclidean tuning&lt;/a&gt;&lt;br /&gt;
* [[7-limit symmetrical lattices]]
&lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures"&gt;Tenney-Euclidean temperament measures&lt;/a&gt;&lt;br /&gt;
* [[Comma-based lattices]]
&lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean metrics&lt;/a&gt;&lt;br /&gt;
=== Combinatorics on words ===
&lt;a class="wiki_link" href="/Cangwu%20badness"&gt;Cangwu badness&lt;/a&gt;&lt;br /&gt;
* [[Ternary scale theorems]]
&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;
== More pages ==
&lt;br /&gt;
* [[:Category: Theory]]
&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: Math]]
&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;
[[Category:Math]]
&lt;a class="wiki_link" href="/Weak%20Fokker%20blocks"&gt;Weak Fokker Blocks&lt;/a&gt;&lt;br /&gt;
[[Category:Theory]]
&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;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"