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{{interwiki | |||
| 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. | |||
[[ | It is recommended to read the pages listed on [[general theory]] first before exploring this page, as they lay the groundwork. | ||
[[ | |||
[[ | == 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]] | ||
[[ | |||
[[ | === 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]] | ||
[[ | |||
[[ | === 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]] | ||
[[k*N subgroups]] | |||
[[ | == 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]] | |||
=== 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]] | |||
=== Exterior algebra === | |||
* [[Exterior algebra]] | |||
* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]] | |||
* [[Wedgie]] and [[Plücker coordinates]] | |||
* [[Recoverability]] | |||
* [[Hodge dual]] | |||
* [[Interior product]] | |||
=== 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]] | |||
=== 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]] | |||
== Scales == | |||
=== MOS scales === | |||
* [[Periodic scale]] | |||
* [[MOS scale]] | |||
* [[Mathematics of MOS]] | |||
* [[MODMOS scale]] | |||
* [[Modal UDP Notation]] | |||
* [[Graph-theoretic properties of scales]] | |||
=== Scale measurements === | |||
* [[Consonant class scale]] | |||
* [[Distributional evenness]] | |||
* [[Maximum variety]] | |||
* [[Lumma stability]] | |||
* [[Structure metric]] | |||
* [[Hahn distance]] | |||
=== Lattices and scale shapes === | |||
* [[Projection pair]] | |||
* [[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]] | |||
== More pages == | |||
* [[:Category: Theory]] | |||
* [[:Category: Math]] | |||
[[Category:Math]] | |||
[[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
- Categorical entropy, mutual information, and channel capacity
- Harmonic entropy
- The Riemann zeta function and tuning
- Maximal dissonance tuning
Interval classification
- Delta-N ratio
- Non radical intervals with musical significance
- abc, high quality commas, and epimericity
- Don Page comma
Regular temperaments
Foundational theory
- Mathematical theory of regular temperaments
- Monzos and interval space
- 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
- k*N subgroups
Exterior algebra
- Exterior algebra
- Dave Keenan & Douglas Blumeyer's guide to EA for RTT
- Wedgie and Plücker coordinates
- Recoverability
- Hodge dual
- Interior product
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 dual norms and Tp tuning space
- Weil norm, Tenney–Weil norm, and TWp interval and tuning space
- Tp tuning
- Constrained tuning
- Logarithmic approximants
Temperament classification
- Rank-2 temperament (include linear temperaments)
- Rank-3 temperament (include planar temperaments)
(Rank-1 temperaments are equal-step tunings.)
- 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
Scale measurements
- Consonant class scale
- Distributional evenness
- Maximum variety
- Lumma stability
- Structure metric
- Hahn distance
Lattices and scale shapes
- Projection pair
- 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