Mathematical theory: Difference between revisions

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<span style="display: block; text-align: right;">[[Teoría_Profunda|Español]]</span>
{{interwiki
Tuning theory can become heavily mathematical, and here are links to some of the pages which wade into deeper waters.
| en = Mathematical theory
| de =
| es = Teoría Profunda
| ja =
}}
Tuning theory can become heavily mathematical, and here are links to some of the pages which wade into deeper waters.  


== Scale Basics ==
It is recommended to read the pages listed on [[general theory]] first before exploring this page, as they lay the groundwork.
* [[Periodic scale]]
* [[MOS scales]]
* [[Mathematics of MOS]]
* [[MODMOS scales]]
* [[Graph-theoretic properties of scales]]


== Intervals ==
=== Interval metrics ===
* [[Height]]
* [[Benedetti height]]
* [[Tenney norm]]
* [[Kees seminorm]]
* [[Wilson norm]]
* [[Weil norm, Tenney–Weil norm, and TWp interval and tuning space|Weil norm and Tenney–Weil norm]]


== Regular Temperaments ==
=== Chords ===
'''Basics:'''
* [[Chordal space]]
* [[Chord complexity]]
* [[Chord matrices]]
* [[Permutation product set]]
* [[Tablet]]


* [[Monzos and Interval Space]]
=== Musical information theory ===
* [[Vals and Tuning Space]]
* [[Categorical entropy, mutual information, and channel capacity]]
* [[Regular Temperaments]]
* [[Harmonic entropy]]
* [[Saturation]]
* [[The Riemann zeta function and tuning]]
* [[Dual list]]
* [[Maximal dissonance tuning]]
* [[Meet and Join]]


'''Linear algebra:'''
=== Interval classification ===
* [[Delta-N ratio|Delta-''N'' ratio]]
* [[Non radical intervals with musical significance]]
* [[Abc, high quality commas, and epimericity|''abc'', high quality commas, and epimericity]]
* [[Don Page comma]]


* [[Temperament Mapping Matrices (M-maps)]]
== Regular temperaments ==
* [[Tmonzos and Tvals|Tempered Monzos ("tmonzos") and Tempered Vals ("tvals")]]
=== Foundational theory ===
* [[Subgroup Mapping Matrices (V-maps)]]
* [[Mathematical theory of regular temperaments]]
* [[Smonzos and Svals|Subgroup Monzos ("smonzos") and Subgroup Vals ("svals")]]
* [[Monzos and interval space]]
* [[Generalized Tenney Norms and Tp Interval Space]]
** [[Radical interval]]
* [[Generalized Tenney Dual Norms and Tp Tuning Space]]
* [[Vals and tuning space]]
* [[Temperament mapping matrix]]
* [[Dual list]]
* [[Mathematical theory of saturation]]
* [[Temperament merging]]
* [[Temperament addition]]


'''Exterior algebra:'''
=== Subgroup temperaments ===
* [[Subgroup temperament families, relationships, and genes]]
* [[Subgroup monzos and vals]]
* [[Tempered monzos and vals]]
* [[Subgroup basis matrix]]
* [[Meet and join]]
* [[k*N subgroups]]


* [[Wedgies and Multivals]]
=== Exterior algebra ===
* [[The wedgie]]
* [[Exterior algebra]]
* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]]
* [[Wedgie]] and [[Plücker coordinates]]
* [[Recoverability]]
* [[Recoverability]]
* [[The dual]]
* [[Hodge dual]]
* [[Interior product]]
* [[Interior product]]


'''Temperament representations:'''
=== Temperament measures and optimization ===
* [[Optimization]]
* [[Target tuning]]
* [[Cangwu badness]]
* [[Generator complexity]]
* [[TOP tuning]]
* [[BOP tuning]]
* [[Tenney–Euclidean tuning]]
* [[Tenney–Euclidean temperament measures]]
* [[Tenney–Euclidean metrics]]
* [[Generalized Tenney norms and Tp interval space|Generalized Tenney norms and T<sub>''p''</sub> interval space]]
* [[Generalized Tenney dual norms and Tp tuning space|Generalized Tenney dual norms and T<sub>''p''</sub> tuning space]]
* [[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]]
* [[Tp tuning|T<sub>''p''</sub> tuning]]
* [[Constrained tuning]]
* [[Logarithmic approximants]]


* [[Abstract regular temperament]]
=== Temperament classification ===
* [[Normal lists]]
* [[Rank-2 temperament]] (include linear temperaments)
* [[Gencom]]
* [[Rank-3 temperament]] (include planar temperaments)
(''Rank-1 temperaments are [[equal-step tuning]]s''.)
* [[Temperament families and clans]]
* [[Cluster temperament]]
* [[Associated temperament]]
* [[Microtemperament]] and [[exotemperament]]
* [[Trivial temperament]]
* [[Sane and insane temperaments]]


'''Other:'''
== Scales ==
=== MOS scales ===
* [[Periodic scale]]
* [[MOS scale]]
* [[Mathematics of MOS]]
* [[MODMOS scale]]
* [[Modal UDP Notation]]
* [[Graph-theoretic properties of scales]]


* [[Planar Temperament]]
=== Scale measurements ===
* [[Transversal generators]]
* [[Consonant class scale]]
* [[Distributional evenness]]
* [[Maximum variety]]
* [[Lumma stability]]
* [[Structure metric]]
* [[Hahn distance]]


=== Temperament Measures and Optimization ===
=== Lattices and scale shapes ===
* [[Target tunings]]
* [[Projection pair]]
* [[TOP tuning]]
* [[Modulatory space]]
* [[Tp tuning]]
* [[Detempering]]
* [[Tenney-Euclidean Tuning]]
* [[Generator preimage]]
* [[Tenney-Euclidean temperament measures]]
* [[Fokker block]]
* [[Tenney-Euclidean metrics]]
* [[Minkowski block]]
* [[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 reduced bases for Fokker groups of certain vals]]
* [[Minkowski blocks]]
* [[7-limit symmetrical lattices]]
* [[The Seven Limit Symmetrical Lattices]]
* [[Tablets]]
* [[Hahn distance]]
* [[Comma-based lattices]]
* [[Comma-based lattices]]
=== Combinatorics on words ===
* [[Ternary scale theorems]]


=== Information Theory ===
== More pages ==
* [[Harmonic Entropy]]
* [[:Category: Theory]]
* [[Categorical Entropy, Mutual Information, and Channel Capacity]]
* [[:Category: Math]]
 
=== Other ===
* [[The Riemann Zeta Function and Tuning]]
* [[Modulatory space]]
* [[Chordal space]]
* [[k*N subgroups]]
* [[Logarithmic approximants]]
* [[Don Page comma]]
* [[Sane and Insane Temperaments]]
* [[ABC, High Quality Commas, and Epimericity]]


[[Category:links]]
[[Category:Math]]
[[Category:math]]
[[Category:Theory]]
[[Category:overview]]
[[Category:theory]]

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

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