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 ===
* [[Monzos and interval space]]
* [[Vals and tuning space]]
* [[Temperament mapping matrices]]
* [[Dual list]]
* [[Mathematical theory of regular temperaments]]
* [[Mathematical theory of saturation]]
 
=== Interval Metrics ===
* [[Height]]
* [[Height]]
* [[Benedetti height]]
* [[Benedetti height]]
* [[Tenney height]]
* [[Tenney norm]]
* [[Kees height]]
* [[Kees seminorm]]
* [[Wilson height]]
* [[Wilson norm]]
* [[Weil height and Tenney-Weil height]]
* [[Weil norm, Tenney–Weil norm, and TWp interval and tuning space|Weil norm and Tenney–Weil norm]]


=== Chords ===
=== Chords ===
* [[Chordal space]]
* [[Chord complexity]]
* [[Chord complexity]]
* [[Chord matrices]]
* [[Chord matrices]]
* [[Permutation product set]]
* [[Tablet]]
=== Musical information theory ===
* [[Categorical entropy, mutual information, and channel capacity]]
* [[Harmonic entropy]]
* [[The Riemann zeta function and tuning]]
* [[Maximal dissonance tuning]]
=== 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]]


=== Subgroup Temperaments ===
== Regular temperaments ==
* [[Subgroup Temperament Families, Relationships, and Genes]]
=== Foundational theory ===
* [[Smonzos and svals|Subgroup monzos ("smonzos") and subgroup vals ("svals")]]
* [[Mathematical theory of regular temperaments]]
* [[Tmonzos and tvals|Tempered monzos ("tmonzos") and tempered vals ("tvals")]]
* [[Monzos and interval space]]
* [[Subgroup basis matrices]]
** [[Radical interval]]
* [[Vals and tuning space]]
* [[Temperament mapping matrix]]
* [[Dual list]]
* [[Mathematical theory of saturation]]
* [[Temperament merging]]
* [[Temperament addition]]
 
=== Subgroup temperaments ===
* [[Subgroup temperament families, relationships, and genes]]
* [[Subgroup monzos and vals]]
* [[Tempered monzos and vals]]
* [[Subgroup basis matrix]]
* [[Meet and join]]
* [[Meet and join]]
* [[Sane and insane temperaments]]
* [[k*N subgroups]]
* [[k*N subgroups]]


=== Exterior algebra ===
=== Exterior algebra ===
* [[Wedgies and multivals]]
* [[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]]
* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]]
** ''Mike's note: Douglas and Dave, having their own interpretation of RTT, sometimes use different terminology than the rest of the stuff here, but this page still has some very good insights and is a worthwhile read.''


=== Temperament Measures and Optimization ===
=== Temperament measures and optimization ===
* [[Optimization]]
* [[Optimization]]
* [[Target tuning]]
* [[Cangwu badness]]
* [[Generator complexity]]
* [[TOP tuning]]
* [[TOP tuning]]
* [[Tenney-Euclidean tuning]]
* [[Tenney-Euclidean temperament measures]]
* [[Tenney-Euclidean metrics]]
* [[BOP tuning]]
* [[BOP tuning]]
* [[Generalized Tenney norms and Tp interval space]]
* [[Tenney–Euclidean tuning]]
* [[Generalized Tenney dual norms and Tp tuning space]]
* [[Tenney–Euclidean temperament measures]]
* [[Weil norm, Tenney-Weil norm, and TWp interval and tuning space]]
* [[Tenney–Euclidean metrics]]
* [[Tp tuning]]
* [[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]]
* [[Constrained tuning]]
* [[Target tuning]]
* [[Logarithmic approximants]]
* [[Cangwu badness]]
 
* [[Fractional monzos]]
=== Temperament classification ===
* [[Generator complexity]]
* [[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]]
 
== Scales ==
=== MOS scales ===
* [[Periodic scale]]
* [[MOS scale]]
* [[Mathematics of MOS]]
* [[MODMOS scale]]
* [[Modal UDP Notation]]
* [[Graph-theoretic properties of scales]]


=== Lattices and Scales ===
=== Scale measurements ===
* [[Transversal]]
* [[Consonant class scale]]
* [[Transversal generators]]
* [[Distributional evenness]]
* [[Fokker blocks]]
* [[Maximum variety]]
* [[Weak Fokker blocks]]
* [[Lumma stability]]
* [[Minkowski reduced bases for Fokker groups of certain vals]]
* [[Structure metric]]
* [[Minkowski blocks]]
* [[The Seven Limit Symmetrical Lattices]]
* [[Tablets]]
* [[Hahn distance]]
* [[Hahn distance]]
* [[Comma-based lattices]]


=== Other ===
=== Lattices and scale shapes ===
* [[The Riemann zeta function and tuning]]
* [[Projection pair]]
* [[Modulatory space]]
* [[Modulatory space]]
* [[Chordal space]]
* [[Detempering]]
* [[Logarithmic approximants]]
* [[Generator preimage]]
* [[Don Page comma]]
* [[Fokker block]]
* [[Abc, high quality commas, and epimericity|''abc'', high quality commas, and epimericity]]
* [[Minkowski block]]
* [[Planar temperament]]
* [[Minkowski reduced bases for Fokker groups of certain vals]]
* [[Projection pair]]
* [[7-limit symmetrical lattices]]
* [[Temperament addition]]
* [[Comma-based lattices]]
=== Combinatorics on words ===
* [[Ternary scale theorems]]


== Musical Information Theory ==
== More pages ==
* [[Harmonic entropy]]
* [[:Category: Theory]]
* [[Categorical Entropy, Mutual Information, and Channel Capacity]]
* [[:Category: Math]]


[[Category:Math]]
[[Category:Math]]
[[Category:Theory]]
[[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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