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]]


=Regular Temperaments=
== Intervals ==
'''Basics:'''
=== 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]]


* [[Monzos and Interval Space]]
=== Chords ===
* [[Vals and Tuning Space]]
* [[Chordal space]]
* [[Regular Temperaments]]
* [[Chord complexity]]
* [[Saturation]]
* [[Chord matrices]]
* [[Dual list]]
* [[Permutation product set]]
* [[Meet and Join]]
* [[Tablet]]


'''Linear algebra:'''
=== Musical information theory ===
* [[Categorical entropy, mutual information, and channel capacity]]
* [[Harmonic entropy]]
* [[The Riemann zeta function and tuning]]
* [[Maximal dissonance tuning]]


* [[Temperament Mapping Matrices (M-maps)]]
=== Interval classification ===
* [[Tmonzos and Tvals]]
* [[Delta-N ratio|Delta-''N'' ratio]]
* [[Subgroup Mapping Matrices (V-maps)]]
* [[Non radical intervals with musical significance]]
* [[Smonzos and Svals]]
* [[Abc, high quality commas, and epimericity|''abc'', high quality commas, and epimericity]]
* [[Generalized Tenney Norms and Tp Interval Space]]
* [[Don Page comma]]
* [[Generalized Tenney Dual Norms and Tp Tuning Space]]


'''Exterior algebra:'''
== Regular temperaments ==
=== Foundational theory ===
* [[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]]


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


'''Abstract algebra:'''
=== 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]]


* [[Groups, Lattices, Z-modules and Vector spaces]]
=== Temperament classification ===
* [[Rank-2 temperament]] (include linear temperaments)
* [[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]]


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


* [[Abstract regular temperament]]
=== Scale measurements ===
* [[Normal lists]]
* [[Consonant class scale]]
* [[Gencom]]
* [[Distributional evenness]]
* [[Maximum variety]]
* [[Lumma stability]]
* [[Structure metric]]
* [[Hahn distance]]


'''Other:'''
=== Lattices and scale shapes ===
 
* [[Projection pair]]
* [[Planar Temperament]]
* [[Modulatory space]]
* [[Transversal generators]]
* [[Detempering]]
 
* [[Generator preimage]]
==Temperament Measures and Optimization==
* [[Fokker block]]
* [[Target tunings]]
* [[Minkowski block]]
* [[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 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]]


==Other==
== More pages ==
* [[The Riemann Zeta Function and Tuning]]
* [[:Category: Theory]]
* [[Modulatory space]]
* [[:Category: Math]]
* [[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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