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.  


== Scales ==
It is recommended to read the pages listed on [[general theory]] first before exploring this page, as they lay the groundwork.
=== Basics ===
 
* [[Periodic scale]]
== Intervals ==
* [[MOS scales]]
=== Interval metrics ===
* [[Mathematics of MOS]]
* [[Height]]
* [[MODMOS scales]]
* [[Benedetti height]]
* [[Graph-theoretic properties of scales]]
* [[Tenney norm]]
* [[Kees seminorm]]
* [[Wilson norm]]
* [[Weil norm, Tenney–Weil norm, and TWp interval and tuning space|Weil norm and Tenney–Weil norm]]
 
=== Chords ===
* [[Chordal space]]
* [[Chord complexity]]
* [[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]]


=== Lattices and scale shapes ===
=== Interval classification ===
* [[Transversal]]
* [[Delta-N ratio|Delta-''N'' ratio]]
* [[Transversal generators]]
* [[Non radical intervals with musical significance]]
* [[Fokker blocks]]
* [[Abc, high quality commas, and epimericity|''abc'', high quality commas, and epimericity]]
* [[Weak Fokker blocks]]
* [[Don Page comma]]
* [[Minkowski reduced bases for Fokker groups of certain vals]]
* [[Minkowski blocks]]
* [[The Seven Limit Symmetrical Lattices]]
* [[Tablets]]
* [[Hahn distance]]
* [[Comma-based lattices]]
* [[Consonant class scale]]
* [[Distributional evenness]]
* [[Maximum variety]]


== Regular temperaments ==
== Regular temperaments ==
=== Basics ===
=== Foundational theory ===
* [[Mathematical theory of regular temperaments]]
* [[Monzos and interval space]]
* [[Monzos and interval space]]
** [[Radical interval]]
* [[Vals and tuning space]]
* [[Vals and tuning space]]
* [[Temperament mapping matrices]]
* [[Temperament mapping matrix]]
* [[Dual list]]
* [[Dual list]]
* [[Mathematical theory of regular temperaments]]
* [[Mathematical theory of saturation]]
* [[Mathematical theory of saturation]]
* [[Temperament merging]]
* [[Temperament addition]]
* [[Temperament addition]]


=== Subgroup Temperaments ===
=== Subgroup temperaments ===
* [[Subgroup Temperament Families, Relationships, and Genes]]
* [[Subgroup temperament families, relationships, and genes]]
* [[Smonzos and svals|Subgroup monzos ("smonzos") and subgroup vals ("svals")]]
* [[Subgroup monzos and vals]]
* [[Tmonzos and tvals|Tempered monzos ("tmonzos") and tempered vals ("tvals")]]
* [[Tempered monzos and vals]]
* [[Subgroup basis matrices]]
* [[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]]
* [[Generator complexity]]


=== Temperament classification ===
=== Temperament classification ===
* [[Planar temperament]]
* [[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]]
* [[Cluster temperament]]
* [[Associated temperament]]
* [[Microtemperament]] and [[exotemperament]]
* [[Trivial temperament]]
* [[Sane and insane temperaments]]


== Intervals ==
== Scales ==
=== Interval metrics ===
=== MOS scales ===
* [[Height]]
* [[Periodic scale]]
* [[Benedetti height]]
* [[MOS scale]]
* [[Tenney height]]
* [[Mathematics of MOS]]
* [[Kees height]]
* [[MODMOS scale]]
* [[Wilson height]]
* [[Modal UDP Notation]]
* [[Weil height and Tenney-Weil height]]
* [[Graph-theoretic properties of scales]]
* [[Abc, high quality commas, and epimericity]]


=== Chords ===
=== Scale measurements ===
* [[Chord complexity]]
* [[Consonant class scale]]
* [[Chord matrices]]
* [[Distributional evenness]]
* [[Chordal space]]
* [[Maximum variety]]
* [[Lumma stability]]
* [[Structure metric]]
* [[Hahn distance]]


=== Musical information theory ===
=== Lattices and scale shapes ===
* [[Harmonic entropy]]
* [[Projection pair]]
* [[Categorical Entropy, Mutual Information, and Channel Capacity]]
* [[Modulatory space]]
* [[The Riemann zeta function and tuning]]
* [[Detempering]]
* [[Maximal dissonance tuning]]
* [[Generator preimage]]
* [[Fokker block]]
* [[Minkowski block]]
* [[Minkowski reduced bases for Fokker groups of certain vals]]
* [[7-limit symmetrical lattices]]
* [[Comma-based lattices]]
=== Combinatorics on words ===
* [[Ternary scale theorems]]


=== Interval classification ===
== More pages ==
* [[Delta-N ratio]]
* [[:Category: Theory]]
* [[Non radical intervals with musical significance]]
* [[:Category: Math]]
* [[Abc, high quality commas, and epimericity|''abc'', high quality commas, and epimericity]]
* [[Don Page comma]]
 
=== Lists of factorizations ===
* [[List of 3-limit factorizations]]
* [[List of 5-limit factorizations]]
* [[List of 7-limit factorizations]]
* [[List of 11-limit factorizations]]
* [[List of 13-limit factorizations]]
 
== Other ==
* [[Modulatory space]]
* [[Logarithmic approximants]]
* [[Projection pair]]
* [[Lumma stability]]
* [[Modal UDP Notation]]
* [[Structure metric]]
* [[Permutation product set]]
* [[:Category:Theory|Category:Theory]]
* [[:Category:Math|Category:Math]]


[[Category:Math]]
[[Category:Math]]
[[Category:Theory]]
[[Category:Theory]]
{{todo|review|comment=ensure the list items are under the correct subheading, and in the right order (most foundational first, most advanced last)}}

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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