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 ==
=== Basics ===
* [[Monzos and interval space]]
* [[Vals and tuning space]]
* [[Temperament mapping matrices]]
* [[Dual list]]
* [[Mathematical theory of regular temperaments]]
* [[Mathematical theory of saturation]]


=== Interval Metrics ===
== Intervals ==
=== 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]]
* [[Abc, high quality commas, and epimericity]]


=== 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]]
* [[Transversal generators]]
* [[Fokker blocks]]
* [[Weak Fokker blocks]]
* [[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]]
* [[Consonant class scale]]
=== Other ===
* [[The Riemann zeta function and tuning]]
* [[Modulatory space]]
* [[Chordal space]]
* [[Logarithmic approximants]]
* [[Don Page comma]]
* [[Abc, high quality commas, and epimericity|''abc'', high quality commas, and epimericity]]
* [[Planar temperament]]
* [[Projection pair]]
* [[Temperament addition]]
* [[Cluster temperament]]
* [[Delta-N ratio]]
* [[Lumma stability]]
* [[Distributional evenness]]
* [[Distributional evenness]]
* [[Maximum variety]]
* [[Maximum variety]]
* [[Modal UDP Notation]]
* [[Lumma stability]]
* [[Consonant class scale]]
* [[Structure metric]]
* [[Structure metric]]
* [[Permutation product set]]
* [[Hahn distance]]
* [[Non radical intervals with musical significance]]
* [[Maximal dissonance tuning]]
* [[:Category:Theory|Category:Theory]]
* [[:Category:Math|Category:Math]]


== Musical Information Theory ==
=== Lattices and scale shapes ===
* [[Harmonic entropy]]
* [[Projection pair]]
* [[Categorical Entropy, Mutual Information, and Channel Capacity]]
* [[Modulatory space]]
* [[Detempering]]
* [[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]]


== Lists of factorizations ==
== More pages ==
* [[List of 3-limit factorizations]]
* [[:Category: Theory]]
* [[List of 5-limit factorizations]]
* [[:Category: Math]]
* [[List of 7-limit factorizations]]
* [[List of 11-limit factorizations]]
* [[List of 13-limit factorizations]]


[[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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