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<span style="display: block; text-align: right;">[[Teoría_Profunda|Español]]</span>
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Tuning theory can become heavily mathematical, and here are links to some of the pages which wade into deeper waters.


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.


=Scale Basics=
== Intervals ==
[[Periodic_scale|Periodic scale]]
=== 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]]


[[Mathematics_of_MOS|MOS scales]]
=== Chords ===
* [[Chordal space]]
* [[Chord complexity]]
* [[Chord matrices]]
* [[Permutation product set]]
* [[Tablet]]


<span style="">[[Mathematics_of_MOS|Mathematics of MOS]]</span>
=== Musical information theory ===
* [[Categorical entropy, mutual information, and channel capacity]]
* [[Harmonic entropy]]
* [[The Riemann zeta function and tuning]]
* [[Maximal dissonance tuning]]


[[MODMOS_Scales|MODMOS scales]]
=== 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]]


[[Graph-theoretic_properties_of_scales|Graph-theoretic properties of scales]]
== 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]]


=Regular Temperaments=
=== Subgroup temperaments ===
'''Basics:'''
* [[Subgroup temperament families, relationships, and genes]]
* [[Subgroup monzos and vals]]
* [[Tempered monzos and vals]]
* [[Subgroup basis matrix]]
* [[Meet and join]]
* [[k*N subgroups]]


[[Monzos_and_Interval_Space|Monzos and Interval Space]]
=== Exterior algebra ===
* [[Exterior algebra]]
* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]]
* [[Wedgie]] and [[Plücker coordinates]]
* [[Recoverability]]
* [[Hodge dual]]
* [[Interior product]]


[[Vals_and_Tuning_Space|Vals and Tuning Space]]
=== 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]]


[[Regular_Temperaments|Regular Temperaments]]
=== 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]]


[[Saturation|Saturation]]
== Scales ==
=== MOS scales ===
* [[Periodic scale]]
* [[MOS scale]]
* [[Mathematics of MOS]]
* [[MODMOS scale]]
* [[Modal UDP Notation]]
* [[Graph-theoretic properties of scales]]


[[dual_list|Dual List]]
=== Scale measurements ===
* [[Consonant class scale]]
* [[Distributional evenness]]
* [[Maximum variety]]
* [[Lumma stability]]
* [[Structure metric]]
* [[Hahn distance]]


[[Meet_and_Join|Meet and Join]]
=== 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]]


'''Linear algebra:'''
[[Category:Math]]
 
[[Category:Theory]]
[[Temperament_Mapping_Matrices_(M-maps)|Temperament Mapping Matrices (M-maps)]]
 
[[Tmonzos_and_Tvals|Tmonzos and Tvals]]
 
[[Subgroup_Mapping_Matrices_(V-maps)|Subgroup Mapping Matrices (V-maps)]]
 
[[Smonzos_and_Svals|Smonzos and Svals]]
 
[[Generalized_Tenney_Norms_and_Tp_Interval_Space|Generalized Tenney Norms and Tp Interval Space]]
 
[[Generalized_Tenney_Dual_Norms_and_Tp_Tuning_Space|Generalized Tenney Dual Norms and Tp Tuning Space]]
 
 
'''Exterior algebra:'''
 
[[Wedgies_and_Multivals|Wedgies and Multivals]]
 
[[The_wedgie|The wedgie]]
 
[[Recoverability|Recoverability]]
 
[[The_dual|The dual]]
 
[[Interior_product|Interior product]]
 
 
'''Abstract algebra:'''
 
[[Groups,_Lattices,_Z-modules,_and_Vector_Spaces|Groups, Lattices, Z-modules and Vector spaces]]
 
 
'''Temperament representations:'''
 
[[Abstract_regular_temperament|Abstract regular temperament]]
 
[[Normal_lists|Normal lists]]
 
[[Gencom|Gencom]]
 
 
'''Other:'''
 
[[Planar_Temperament|Planar Temperament]]
 
[[Transversal_generators|Transversal generators]]
 
==Temperament Measures and Optimization==
[[Target_tunings|Target tunings]]
 
[[TOP_tuning|TOP tuning]]
 
[[Tp_tuning|Tp tuning]]
 
[[Tenney-Euclidean_Tuning|Tenney-Euclidean tuning]]
 
[[Tenney-Euclidean_temperament_measures|Tenney-Euclidean temperament measures]]
 
[[Tenney-Euclidean_metrics|Tenney-Euclidean metrics]]
 
[[Cangwu_badness|Cangwu badness]]
 
[[Fractional_monzos|Fractional monzos]]
 
[[Generator_complexity|Generator complexity]]
 
==Lattices and Scales==
[[Transversal|Transversal]]
 
[[Fokker_blocks|Fokker blocks]]
 
[[Weak_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|Minkowski blocks]]
 
[[The_Seven_Limit_Symmetrical_Lattices|The Seven Limit Symmetrical Lattices]]
 
[[Tablets|Tablets]]
 
[[Hahn_distance|Hahn distance]]
 
[[Comma-based_lattices|Comma-based lattices]]
 
==Other==
[[The_Riemann_Zeta_Function_and_Tuning|The Riemann Zeta Function and Tuning]]
 
[[Modulatory_space|Modulatory space]]
 
[[Chordal_space|Chordal space]]
 
[[k*N_subgroups|k*N subgroups]]
 
[[Logarithmic_approximants|Logarithmic approximants]]
 
[[Don_Page_comma|Don Page comma]]
 
[[Sane_and_Insane_Temperaments|Sane and Insane Temperaments]]
 
[[ABC,_High_Quality_Commas,_and_Epimericity|ABC, High Quality Commas, and Epimericity]]      [[Category:links]]
[[Category:math]]
[[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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