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
• Regular Temperaments
• Saturation

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

The Exterior algebra

• Wedgies and Multivals
• Recoverability
• The dual
• Interior product

Temperament representations

• Abstract regular temperament
• Normal lists
• Gencom
• Dual list

Temperament Measures and Optimization

• TOP 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
• 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

Musical Information Theory

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