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Index
Title Page Dedication Acknowledgments Editor’s Preface Introduction
Two Reasons for Writing This Book How This Book Is Organized The Method: Art over Science Learning by Singing What Is Theory For? Who Is Theory For? For Whom This Book Is Written How to Use This Book Tools The Teacher Question
PART ONE HARMONIC PURITY: FEELING THE NUMBERS
Chapter 1: Seeing Air and Touching Sound
Seeing Touching Numbering
Chapter 2: Singing Unison with a Drone
Entrainment in Two Worlds Singing Sa Simplicity
Chapter 3: Singing Octaves
No Babies or Dogs Nomenclature: Octave versus 2:1
Chapter 4: Singing Perfect Fifths
Triple Nature Octave Reduction The Rhythmic 3:2 versus the Harmonic 3:2 Nomenclature: Reading versus Singing Naming Your Place: Solfege and Sargam Singing Pa
Chapter 5: Singing Major Thirds
Pentamerous Nature The Concurrent Study of Just Intonation and Equal Temperament Octave Reduction of the Fifth Partial; Rhythmic 5:4 versus Harmonic 5:4 The Three-and-Five Problem Summarized
Chapter 6: A Map of Harmonic Relationships
The Powers of Two as Sterile The Powers of Three as Generative: Compounding Singing a Compound Tone: Re A Map Emerges The Major Seventh The Harmonic Path Scale Practice with Five Notes: The Flying Swan The Augmented Fourth Scale Practice with Six Notes Where Does This Stuff Come From? What Are We Doing?
Chapter 7: The Perfect Fifth Below: Hello, Mother
Reciprocal Tones Dominant and Subdominant Compared Singing Practice with Ma Internalizing Ma
Chapter 8: The Major Third Below
Singing Reciprocal Thirds Scale Practice with Komal Dha A Map of Affects Komal Dha in a Seven-Tone Scale
Chapter 9: The Central Section of the Map Completed
The Major Sixth: Dha, 5:3 The Minor Third: Komal Ga, 6:5 The Real Secret of the Minor Third The Minor Seventh: Komal Ni, 9:5 The Minor Second: Komal Re, 16:15 Reciprocal Mystery The Center Is Complete
Chapter 10: The Five-Limit Lattice of Twelve Notes
Drawing the Map Review and Clarification of Terminology Harmonic Paths Lattice Practice
Chapter 11: Available Modes: Lydian through Phrygian
The Lydian Mode Digression on Terminology The Ionian Mode The Mixolydian Mode The Dorian Mode The Aeolian Mode The Phrygian Mode Is This a Rainbow?
Chapter 12: Mixed Modes
Naming the Tetrachords Modes with Four Consecutive Whole Steps Modes with Five Consecutive Whole Steps Modes Containing Augmented Seconds The Remaining Possibilities
Chapter 13: Internalizing the Lattice
Tell Me Again Why I Should Do All This Singing Vibrato Double Think Between the Notes The Territory Singing Practice
Chapter 14: Numbers and Mysteries
Ratios of the Twelve Lattice Tones to Tonic Ratios of the Tones to One Another Why Are There Seven Tones in a Scale? What Is a Degree? Terminology Check Treating Names as If They Were True
Chapter 15: Chromatic Pairs
Diatonic versus Chromatic Half Steps Major and Minor Thirds Major and Minor Sevenths Major and Minor Sixths Learning Major/Minor Pairs Perfect and Augmented Fourths Major and Minor Seconds Using All Twelve Tones
Chapter 16: Beyond Twelve Notes: The Extended Five-Limit Lattice
Cents Extending the Central Spine of Fifths Extending the Ga Spine of Fifths The Ga-Ga Spine of Fifths The Ga-Ga-Below Spine of Fifths: Fb The Extended Five-Limit Lattice
Chapter 17: The Seventh Partial and Beyond
Each Prime Its Own Realm The Seven Realm Prime Numbers beyond Seven
Chapter 18: The Practice of Pure Harmony
PART TWO THE SELECTIVE USE OF EQUAL TEMPERAMENT
Chapter 19: Leaping from Paradise
One Foot in Eden The Need for Simplification The Two Possibilities within Fixed Pitch Approximation versus the Real Thing Low-Prime Harmony as Norms Where Does the Ear Draw the Line? The Hunger for the Center Our Plan
Chapter 20: Six Unambiguous Chords
A First Glimpse at Equal Temperament: The Overtone Chord Drawing the Major Triad Three-Chord Music: The Ping-Pong Cadence The Whole Cadence Three-Chord Music Drawing the Minor Triad Cadences in Minor Harmonic Rhythm
Chapter 21: Twelve Unambiguous Chords
The Relative Minor Parallel versus Relative The Twelve Triads in the Lattice of Twelve Notes Matchstick Harmony Adding a Melody What Are We Doing? The Reason behind Unambiguous Harmony
Chapter 22: Unambiguous Triadic Harmony in the Extended Lattice
Harmonic Journeys Harmonic Balance Adding Melodies with Passing Tones Where Are We?
Chapter 23: Dronality in Equal Temperament: Lydian through Phrygian
The Overtone Chord versus the Chord of Fifths The Lydian Mode in Equal Temperament The Major (Ionian) Mode in Equal Temperament The Mixolydian Mode in Equal Temperament The Dorian Mode in Equal Temperament The Aeolian Mode in Equal Temperament The Phrygian Mode in Equal Temperament Lydian to Phrygian and Back Again in Equal Temperament
Chapter 24: Dronality in Equal Temperament: Mixed Modes
Gaps in Stacks Modes with Four Consecutive Whole Steps Modes with Five Consecutive Whole Steps Modes Containing Augmented Seconds
Chapter 25: Chromatic Pairs and the Magic Mode
Major and Minor Sevenths Combined Major and Minor Thirds Combined Major and Minor Sixths Combined Perfect and Augmented Fourths Combined Major and Minor Seconds Combined Two Simultaneous Chromatic Pairs Three and Four Simultaneous Chromatic Pairs Five Simultaneous Chromatic Pairs: The Magic Mode
Chapter 26: Pushing the Magic Mode Envelope
Testing the Rules The Augmented Tonic The Great Diesis in Equal-Tempered Dronality Super-Lydian and Sub-Phrygian Seventh-Partial Tones Things Not Considered The Harmonic Map of the Magic Mode Boundary Weather How to Proceed Magic Mode Compositions
PART THREE THE FUNCTIONAL COMMAS OF EQUAL-TEMPERED TONAL HARMONY
Chapter 27: Comma Phenomena
What Is a Comma? The Pythagorean Comma The Didymic Comma The Great Diesis The Diaschisma The Four Commas Compared Modulation Where Are We?
Chapter 28: Didymic Pairs of Tones
The Didymic Saga So Far The Didymic Comma of the Major Sixth Degree The Didymic Comma of the Second Degree The Didymic Comma of the Seventh Degree
Chapter 29: Didymic Pairs of Triads
The Usual Preliminary Advice Staff Orientation and Compass Orientation The Didymic Comma of the ii Triad Pathfinder versus Homebody Each Comma Works in Both Directions The Didymic Comma of the bVII Triad The Didymic Comma of the II Triad The Didymic Comma of the bvii Triad The Fifteen Pairs of Didymic Triads The Table of Didymic Triads More about Practice
Chapter 30: The Great Diesis: The Beauty of the Beast
The Minor Sixth and the Augmented Fifth Compared Comparing the Rising and Falling Harmony of the Great Diesis in Equal Temperament
Chapter 31: The Tonal Array of the Great Diesis
The Overtonal Diesis Family The Minor Second/Augmented Prime Diesis The Minor Third/Augmented Second Diesis The Minor Seventh/Augmented Sixth Diesis The Reciprocal Diesis Family The Major Sixth/Diminished Seventh Diesis The Major Third/Diminished Fourth Diesis The Major Seventh/Diminished Octave Diesis The Augmented Fourth/Diminished Fifth Diesis Pentamerous Harmony Triadic Harmony Associated with the Great Diesis The Modal Truth of the Great Diesis The Uses of the Lattice
Chapter 32: The Diaschismaa
Singing through the Db/C# Diaschisma in Just Intonation The Dominant Seventh Chord Augmented Sixth Chords The Ambiguity of Function between Dominant Seventh Chords and Augmented Sixth Chords The Ga-Balanced Family of Diaschismas The Ga-Ga Family of Diaschismas in C The Array of Diaschismic Pairs in C The Power of the Diaschisma
Chapter 33: Quasi-Functional and Nonfunctional Commas
Is There a Septimal Comma? The Septimal Seventh/Dominant Seventh Comma, 64:63 The Blu/Minor Comma, 36:35 The Blu/Ga-Ga Comma, 225:224 Other Commas
Chapter 34: The Shape of Tonality
The Folding Plane The View from the Center A Picture of Three Commas Position and Expressivity The Dark Side Ambiguity at the Center Virtual Return The Value of Ambiguity The Best Strategy A Time Line
PART FOUR HARMONIC PRACTICE, ANALYSIS, AND REVIEW OF THE THEORY
Chapter 35: Practical Matters
The Coexistence of Just Intonation and Equal Temperament Key, Degree, and Path: Positional Analysis Nomenclature When to Analyze and When Not
Chapter 36: A Small Syllabus of Diatonic Sequences
Descending Root Motion by Thirds Distribution of the Comma Ascending Root Motion by Thirds Descending Root Motion by Fifths Internalizing the Wisdom of Harmonic Sequences Root Motion by Fifths Ascending Extended Use of Descending Fifths: Inversions and Seventh Chords The Nomenclature of Seventh Chords The Harmonic Qualities of Seventh Chords The Ultimate Sequence of Descending Fifths Modes Other Than Major A Practice Involving Modulation Root Motion by Seconds Summary of Root Motion and Voice Leading Additional First-Inversion Sequences One More Sequence Practicing in Keys
Chapter 37: Cadential Practices: Parallel Borrowing
Scales against Cadential Chords Borrowing from Minor Borrowing from Major The “Altered Scale” What Can Be Added to a Dominant Seventh Chord?
Chapter 38: The Long Cadence: A Comprehensive View of Cadential Energy
The Long Cadence: General Characteristics The Long Cadence Chord by Chord Plagal Cadences Extending the Territory: Locrian Lives The Double Leading-Tone Penultimate Deceptive Cadences Half Cadences Commas in Cadences Overtonal/Reciprocal Balance Standing Dissonances Tonicization versus Modulation Dr. Overtone Keeps His Word How to Practice Cadential Energy
Chapter 39: Tonicization and Modulation
Common-Tone Modulation: The Pivot Tone Common-Chord Modulation: The Pivot Chord Secondary Dominants and the Diatonic Family of Keys Borrowing from Minor The Extended Family of Keys Where Are We? A Conspiracy of Deceptions A Map of the Domain of G7 How To Use the Conspiracy When the Pivot Involves a Comma Great Diesis Practices An Opposite Method Full Diesis Practice
Chapter 40: Modal Modulation
A Way of Composing in Modulating Modality Composing from a Given Key Chain Constructing Key Chains The Composition Hints for Composition The Value of the Method The Modal Limit in Equal Temperament: The Modulating Magic Mode
Chapter 41: A Syllabus of Cyclic Sequences
The Art of Virtual Return Directional Organization of the Sequences Movement by Fifths Movement by Major Thirds Movement by Minor Seconds Movement by Minor Whole Tones Movement by Minor Thirds A Graphic Summary of Cyclic Sequences Where Are We?
Chapter 42: Symmetrical Harmony and the Dissolution of Tonality
The Equal-Tempered Chromatic Scale The Equal-Tempered Whole-Tone Scale Diminished Scales Augmented Scales Equal Temperaments with Other Than Twelve Tones Beyond Tonality and Symmetry: Atonality
Chapter 43: The Notation of Positional Analysis
The Three Lattices Time-Related Notations for the Three Lattices Raga in Tone Lattice Notation Medieval Music in Tone Lattice Notation Unambiguous Modality in Equal Temperament: The Magic Mode Bartók in Tone Lattice Notation Fauré in Chord Lattice Notation Mozart in Positional and Traditional Analysis: Sonata in C, K.545 Beethoven in Key Lattice Notation: Sonata in F Minor, Op. 57 A Jazz Standard in Key Lattice Notation Harmony and Form
Chapter 44: Review of the Theory
The Value of the Theory Flaws and Limits of the Theory The Doctor’s Comfort Whose Theory Is It?
Afterword
The Limits of Twelve-Tone Equal Temperament The Limits of Just Intonation Why Is Just Intonation Not Taught? Future Music and the Collective Ear Brain Harmony Music as a Model of Knowing What Music Stands For Some Parting Advice
Glossary of Terms Glossary of Singable Tones in Just Intonation Footnotes Sources
Books Recordings Contemporary Composition
About the Author About Inner Traditions • Bear & Company Copyright & Permissions General Index Index of Ratios Index of Most-Referenced Examples, Figures, and Tables
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