Mathematical theory
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This is an easily accessed page that is intended to serve as a portal to the pedagogical enrichment of newcomers to the xenharmonic community as well as of more established members who are unaware of certain bodies of knowledge. As a result, it has been semi-protected to prevent vandalism.
Please be mindful of this when making edits to this page, or to the pages linked herefrom. |
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
Foundational theory
- 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
- 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
- Rank-2 temperaments (include linear temperaments)
- Rank-3 temperaments (include planar temperaments)
(Rank-1 temperaments are equal-step tunings.)
- Temperament families and clans
- Cluster temperaments
- Associated temperaments
- Microtemperaments
- Exotemperaments
- Trivial temperaments
- Sane and insane temperaments
Scales
MOS scales
- Periodic scale
- MOS scales
- Mathematics of MOS
- MODMOS scales
- 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
- Transversal
- Transversal generators
- Fokker blocks
- Weak Fokker blocks
- Minkowski blocks
- Minkowski reduced bases for Fokker groups of certain vals
- The Seven Limit Symmetrical Lattices
- Comma-based lattices