Abstract
Introduction
3-D Harmonic Space
Layout of 3-D Harmonic Space
Derivation of 3-D Harmonic Space from Harmonic Number
Transformational Theory - Lewin (1998)
3-D Harmonic Space Staff
Virtual Pitch and Harmony
Virtual Pitch Theory - Terhardt (1981)
Harmonic Distance
Distance Between Pitch Heights versus Distance from Pitch Height to Fundamental
Algorithm for extraction of pitch and pitch salience from complex tonal signals (1982)
Hierarchical Coding - Deutsch (2000), Lerdahl (2001)
Perceptual Distance
Interval, Pitch Difference, and Perceptual Distance
Harmonic Group
Representing Chords in Terms of Harmonic Groups
Representing Chord Pairs in Terms of Horizontal Harmonic Groups
Perceptual Grouping
Harmonic Position Assignment
Methods
Mapping
Aliasing
Harmonic Group Assignment (Optimization)
Logical Constraints
Alternate Organizations
Chord Harmonic Group Assignment
Chord Pair Harmonic Group Assignment
Transformation in 3-D Harmonic Space
Pitch to Pitch Mapping
Neo-Riemannian Theory - Cohn (2000)
Transformational Theory - Lewin (1998)
Introduction to Algebraic Groups
Voice Leading Algebraic Group
Harmonic Part Algebraic Group
Transformations Modelling Structure and Organization of Chords
Musical Concept (Meaning) of Pitch
A Harmonic Scale that Musicians Can Use
Pitch Height Scales
Harmonic Scales
How Musicians Conceive of Pitch
The Height of a Pitch on the Pitch Height Scale
The Position of a Pitch in 3-D Harmonic Space
Pitch Role
Justification for Separate Treatment of Pitch and Harmonic Aspects or Pitch
Nature of Independent Dimensions in Harmonic Space
3-D Harmonic Space Lacks Symmetry and Is Not a True Space
Color
Expression
Notation of 3-D Harmonic Space
Encoding Higher Dimensions in Lower Dimensions
Projections of Higher Dimensions onto Lower Dimensions
Harmonic Frame is Everywhere, and is Part of The Fabric of Harmonic Space
Harmonic Staves
Hierarchy of Harmonic Spaces
Mathematical Properties of Harmonic Space
Rational Numbers
Lattice and Greatest Lower Bounds
Symmetries
Tuning and Temperament
Tuning Systems and 3-D Harmonic Space