Mathematical theory: Difference between revisions
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== Regular temperaments == | == Regular temperaments == | ||
=== Basics === | === Basics === | ||
* [[Mathematical theory of regular temperaments]] | |||
* [[Monzos and interval space]] | * [[Monzos and interval space]] | ||
** [[Fractional monzos]] | |||
* [[Vals and tuning space]] | * [[Vals and tuning space]] | ||
* [[Temperament mapping matrices]] | * [[Temperament mapping matrices]] | ||
* [[Dual list]] | * [[Dual list]] | ||
* [[Mathematical theory of saturation]] | * [[Mathematical theory of saturation]] | ||
* [[Temperament addition]] | * [[Temperament addition]] | ||
| Line 58: | Line 59: | ||
=== Exterior algebra === | === Exterior algebra === | ||
* [[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.'' | |||
* [[Wedgies and multivals]] | * [[Wedgies and multivals]] | ||
* [[Recoverability]] | * [[Recoverability]] | ||
* [[The dual]] | * [[The dual]] | ||
* [[Interior product]] | * [[Interior product]] | ||
=== Temperament Measures and Optimization === | === Temperament Measures and Optimization === | ||
* [[Optimization]] | * [[Optimization]] | ||
* [[Target tuning]] | |||
* [[Cangwu badness]] | |||
* [[Generator complexity]] | |||
* [[TOP tuning]] | * [[TOP tuning]] | ||
* [[BOP tuning]] | |||
* [[Tenney-Euclidean tuning]] | * [[Tenney-Euclidean tuning]] | ||
* [[Tenney-Euclidean temperament measures]] | * [[Tenney-Euclidean temperament measures]] | ||
* [[Tenney-Euclidean metrics]] | * [[Tenney-Euclidean metrics]] | ||
* [[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]] | ||
| Line 77: | Line 81: | ||
* [[Tp tuning]] | * [[Tp tuning]] | ||
* [[Constrained tuning]] | * [[Constrained tuning]] | ||
* [[Logarithmic approximants]] | * [[Logarithmic approximants]] | ||
Revision as of 02:04, 2 December 2024
Español 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
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
Regular temperaments
Basics
- Mathematical theory of regular temperaments
- Monzos and interval space
- Vals and tuning space
- Temperament mapping matrices
- Dual list
- 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
- 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.
- Wedgies and multivals
- Recoverability
- The 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
Scales
Basics
MOS scales
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
- Comma-based lattices
- Consonant class scale
- Projection pair
Scale measurements
- Hahn distance
- Distributional evenness
- Maximum variety
- Modulatory space
- Lumma stability
- Structure metric