Mathematical theory

Español Tuning theory can become heavily mathematical, and here are links to some of the pages which wade into deeper waters.

Scale Basics

• Periodic scale
• MOS scales
• Mathematics of MOS
• MODMOS scales
• Graph-theoretic properties of scales

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

Interval Metrics

• Height
• Benedetti height
• Tenney height
• Kees height
• Wilson height
• Weil height and Tenney-Weil height

Chords

• Chord complexity
• Chord matrices

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
• Douglas Blumeyer and Dave Keenan's Intro to exterior algebra 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

• 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 Norms, Tenney-Weil Norms, and TWp Interval and Tuning Space
• Tp tuning
• Target tunings
• Cangwu badness
• Fractional monzos
• Generator complexity

Lattices and Scales

• 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

Other

• The Riemann zeta function and tuning
• Modulatory space
• Chordal space
• Logarithmic approximants
• Don Page comma
• abc, high quality commas, and epimericity
• Planar temperament
• Projection pair
• Temperament arithmetic

Musical Information Theory

• Harmonic entropy
• Categorical Entropy, Mutual Information, and Channel Capacity