Mathematical theory: Difference between revisions
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Rearrange subheadings to be slightly more accurate (still a very long way to go though) |
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* [[Graph-theoretic properties of scales]] | * [[Graph-theoretic properties of scales]] | ||
=== Lattices and | === Lattices and scale shapes === | ||
* [[Transversal]] | * [[Transversal]] | ||
* [[Transversal generators]] | * [[Transversal generators]] | ||
| Line 22: | Line 22: | ||
* [[Comma-based lattices]] | * [[Comma-based lattices]] | ||
* [[Consonant class scale]] | * [[Consonant class scale]] | ||
* [[Distributional evenness]] | |||
* [[Maximum variety]] | |||
== Regular | == Regular temperaments == | ||
=== Basics === | === Basics === | ||
* [[Monzos and interval space]] | * [[Monzos and interval space]] | ||
| Line 31: | Line 33: | ||
* [[Mathematical theory of regular temperaments]] | * [[Mathematical theory of regular temperaments]] | ||
* [[Mathematical theory of saturation]] | * [[Mathematical theory of saturation]] | ||
* [[Temperament addition]] | |||
* [[ | |||
=== Subgroup Temperaments === | === Subgroup Temperaments === | ||
| Line 79: | Line 69: | ||
* [[Generator complexity]] | * [[Generator complexity]] | ||
== Musical | === Temperament classification === | ||
* [[Planar temperament]] | |||
* [[Cluster temperament]] | |||
== Intervals == | |||
=== Interval metrics === | |||
* [[Height]] | |||
* [[Benedetti height]] | |||
* [[Tenney height]] | |||
* [[Kees height]] | |||
* [[Wilson height]] | |||
* [[Weil height and Tenney-Weil height]] | |||
* [[Abc, high quality commas, and epimericity]] | |||
=== Chords === | |||
* [[Chord complexity]] | |||
* [[Chord matrices]] | |||
* [[Chordal space]] | |||
=== Musical information theory === | |||
* [[Harmonic entropy]] | * [[Harmonic entropy]] | ||
* [[Categorical Entropy, Mutual Information, and Channel Capacity]] | * [[Categorical Entropy, Mutual Information, and Channel Capacity]] | ||
* [[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|''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 == | == Other == | ||
* [[Modulatory space]] | * [[Modulatory space]] | ||
* [[Logarithmic approximants]] | * [[Logarithmic approximants]] | ||
* [[Projection pair]] | * [[Projection pair]] | ||
* [[Lumma stability]] | * [[Lumma stability]] | ||
* [[Modal UDP Notation]] | * [[Modal UDP Notation]] | ||
* [[Structure metric]] | * [[Structure metric]] | ||
* [[Permutation product set]] | * [[Permutation product set]] | ||
* [[:Category:Theory|Category:Theory]] | * [[:Category:Theory|Category:Theory]] | ||
* [[:Category:Math|Category:Math]] | * [[:Category:Math|Category:Math]] | ||
[[Category:Math]] | [[Category:Math]] | ||
Revision as of 01:34, 2 December 2024
Español Tuning theory can become heavily mathematical, and here are links to some of the pages which wade into deeper waters.
Scales
Basics
Lattices and scale shapes
- 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
- Distributional evenness
- Maximum variety
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
- Temperament addition
Subgroup Temperaments
- Subgroup Temperament Families, Relationships, and Genes
- Subgroup monzos ("smonzos") and subgroup vals ("svals")
- Tempered monzos ("tmonzos") and tempered vals ("tvals")
- Subgroup basis matrices
- Meet and join
- Sane and insane temperaments
- k*N subgroups
Exterior algebra
- Wedgies and multivals
- Recoverability
- The dual
- 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
- Optimization
- TOP tuning
- Tenney-Euclidean tuning
- Tenney-Euclidean temperament measures
- Tenney-Euclidean metrics
- BOP tuning
- 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
- Target tuning
- Cangwu badness
- Fractional monzos
- Generator complexity
Temperament classification
Intervals
Interval metrics
- Height
- Benedetti height
- Tenney height
- Kees height
- Wilson height
- Weil height and Tenney-Weil height
- Abc, high quality commas, and epimericity
Chords
Musical information theory
- Harmonic entropy
- Categorical Entropy, Mutual Information, and Channel Capacity
- 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
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