Following graduate studies at the University of Chicago and the Technical University of Berlin, William Caplin joined the Faculty of Music at McGill University in 1978 and remained there until his retirement in 2022. Caplin specializes in the theory of musical form and the history of harmonic and metric theories. His ground-breaking treatise Classical Form (1998) won the Wallace Berry Book Award from the Society for Music Theory. He subsequently served that society as both Vice-President and President. In 2011, Caplin was awarded the prestigious Killam Research Fellowship and was elected a Fellow in the Royal Society of Canada in 2015.

• TABLE OF CONTENTSPrefaceGuide to the Analytical AnnotationsCHAPTER 1 IDEAS OF CLOSURE1.1 Closure in Literature1.2 Closure in Music1.2.1 Leonard Meyer1.2.2 Kofi Agawu1.2.3 Robert Hatten1.2.4 Patrick McCreless1.2.5 Mark Anson-CartwrightPART 1 THE CLASSICAL CADENCE CHAPTER 2 GENERAL CONCEPTS OF THE CLASSICAL CADENCE2.1 Traditional Notions of Cadence2.2 Cadence as Formal Closure2.2.1 Formal Units Closed by Cadence; Cadence and Phrase2.2.2 Cadence and Higher-Level Formal Units2.3 Cadence as Harmony; Harmony as Cadence2.3.1 Progression Types2.3.1.1 Prolongational Progressions2.3.1.2 Sequential Progressions2.3.1.3 Cadential Progressions2.3.2 Ambiguity of Progression Types2.4 Bass-Line Melody and Cadence2.5 Cadential Arrival versus Cadential Function2.6 Cadential Function versus Cadential Content2.7 Limited Cadential Scope2.8 Cadential Function versus Postcadential Function2.8.1 Theoretical Background2.8.2 Postcadential Function as "Confirmation"2.8.3 Hepokoski and Darcy's Critique2.8.4 Two "Registers" of Closure2.8.5 Consecutive (Repeated) PACs2.9 "End" versus "Stop"2.10 Cadence as Punctuation2.11 Cadential Strength-Syntactical versus Rhetorical CHAPTER 3 BASIC CADENCE TYPES: MORPHOLOGY AND FUNCTION3.1 Perfect Authentic Cadence3.1.1 Morphology-Harmonic Content3.1.1.1 Stages of the Cadence Schema3.1.1.2 Stage 1-Initial Tonic3.1.1.3 Stage 2-Pre-dominant3.1.1.4 Stage 3-Dominant3.1.1.5 Stage 4-Final Tonic3.1.1.6 Complete versus Incomplete Cadential Progressions3.1.1.7 Boundaries of the Cadential Progression3.1.2 Morphology-Melodic Content3.1.2.1 Basic Patterns3.1.2.2 Varied Patterns3.1.2.3 Combined Patterns3.1.2.4 Additional Patterns3.1.3 Function3.1.3.1 Standard Functions of the PAC3.1.3.2 Exceptional Situations3.2 Imperfect Authentic Cadence3.2.1 Morphology3.2.1.1 Basic Tenor Stream (8/6/5/3); Varied (5/6/5/3)3.2.1.2 Prinner Cadence3.2.1.3 Other Patterns; Melodic Diversion3.2.1.4 Combined Patterns3.2.1.5 Ending on 5?3.2.2 Function3.2.2.1 Independent IAC3.2.2.2 Way-Station IAC3.2.2.3 Additional Functions of the IAC3.3 Half Cadence3.3.1 General Conditions for Half Cadence3.3.2 Morphology-Harmonic Content3.3.2.1 Stage 1-Initial Tonic3.3.2.2 Stage 2-Pre-dominant3.3.2.3 Stage 3-Dominant3.3.3 Morphology-Melodic Content3.3.3.1 Simple (L-K) HC3.3.3.2 Converging HC3.3.3.3 Expanding HC3.3.3.4 Miscellaneous Issues3.3.4 Function3.3.4.1 Levels of Phrase Functionality3.3.4.2 Levels of Thematic Functionality3.3.4.3 Some Functional Generalizations3.3.5 Special Cases3.3.5.1 Reinterpreted HC3.3.5.2 Reopened HC3.4 Additional Aspects of the Basic Cadence Types3.4.1 Meter3.4.1.1 A Ratio Model of Metrical Strength3.4.1.2 Metrically Weaker Cadences3.4.1.3 Hypermetrical Considerations3.4.1.4 Metrical Weighting-Syntactical or Rhetorical?3.4.2 Texture3.4.2.1 Textural Types3.4.2.2 Texture and Cadence3.4.2.3 Covered cadence3.5 Other Cadence Types3.5.1 "Contrapuntal Cadence"; Prolongational Closure3.5.2 "Plagal Cadence" CHAPTER 4 CADENTIAL DEVIATIONS4.1 Deceptive Cadence4.1.1 Morphology4.1.2 Function4.1.2.1 Way-Station Deceptive Cadence, Replacing a PAC4.1.2.2 Way-Station Deceptive Cadence, Replacing an IAC4.1.2.3 Way-Station Deceptive Cadence, Denied4.1.2.4 Independent Deceptive Cadence4.1.2.5 Noncadential Uses of a Deceptive Cadential Progression4.1.2.6 Deceptive Resolutions of the Dominant in Noncadential Contexts4.1.2.7 Elided Deceptive Cadence4.2 Evaded Cadence4.2.1 Morphology4.2.1.1 Harmonic Content4.2.1.2 Melodic Content4.2.2 Function4.2.2.1 Way Station4.2.2.2 One-More-Time Technique4.2.2.3 New Material4.3 Abandoned cadence4.3.1 Cadential Dominant Undermined by Inversion4.3.2 Cadential Dominant Replaced by Inverted Dominant4.3.2.1 Replaced by Lt4.3.2.2 Replaced by Le4.3.3 Cadential Dominant Replaced by Nondominant Harmony4.3.4 Prolongational versus Cadential; Harmonic Expansion4.3.5 Cadential Abandonment and Harmonic Reduction4.4 Dominant Arrival4.4.1 Arrival on a Final Dominant4.4.2 Arrival on a Terminal Dominant4.4.3 Premature Dominant Arrival4.4.3.1 PDA on a Final Dominant4.4.3.2 PDA on a Terminal Dominant4.4.3.3 Hyperdominant Prolongations4.4.4 Additional Issues4.4.4.1 Dominant Arrival of Limited Scope4.4.4.2 Reinterpreted Dominant Arrival4.4.4.3 Reopened Dominant Arrival4.4.4.4 Doppia PDA versus doppia HC4.4.4.5 Burstein's Critique4.5 Combinations of Deviations4.6 Cadential Ambiguities4.6.1 Evaded versus Deceptive (Elided)4.6.2 Evaded versus Authentic (Elided)4.6.3 Evaded versus IAC (Elided)4.6.4 Evaded versus Covered PAC4.6.5 Evaded versus HC (Dominant Arrival)4.6.6 Abandoned versus Deceptive4.6.7 Multiple Ambiguities4.6.7.1 Mozart, Piano Sonata in A Minor, K. 310, i, mm. 1-104.6.7.2 Haydn, String Quartet in D, Op. 71, no 3, iii, 1-10 CHAPTER 5 CADENTIAL EXPANSION5.1 Perfect Authentic Cadence5.1.1 Harmonic Content5.1.1.1 Stage 1-Initial Tonic5.1.1.2 Stage 2-Pre-dominant5.1.1.3 Stage 3-Dominant5.1.1.4 Diversity of Harmonic Content; Bass-Line Complexities5.1.1.5 Ambiguous Onset of an ECP5.1.2 Melodic Content5.1.2.1 Basic Simple (3/2/2/1); with Cadential Six-Four (3/2/3-2/1)5.1.2.2 Basic Soprano (5/4/2/1); with Cadential Six-Four (5/4/3-2/1)5.1.2.3 Basic Alto (3/2/7/1); with Cadential Six-Four (3/2/1-7/1)5.1.2.4 Basic Tenor (8/6/5/1); with Cadential Six-Four (8/6/5-4/1)5.1.2.5 Varied Simple (1/2/2/1); with Cadential Six-Four (1/2/3-2/1)5.1.2.6 Varied Soprano (3/4/2/1); with Cadential Six-Four (3/4/3-2/1)5.1.2.7 Varied Alto (1/2/7/1); with Cadential Six-Four (1/2/1-7/1)5.1.2.8 Varied Tenor (5/6/5/1); with Cadential Six-Four (5/6/5-5 (4)/1)5.1.2.9 Successive Combinations5.1.2.10 Simultaneous Combinations5.1.2.11 Additional Patterns5.2 Imperfect Authentic Cadence5.3 Half Cadence5.3.1 Minimum Length5.3.2 Harmonic Content5.3.2.1 Stage 15.3.2.2 Stage 25.3.2.3 Stage 35.3.3 Melodic content5.3.3.1 Simple HC5.3.3.2 Converging HC5.3.3.3 Expanding HC5.3.3.4 Converging/Expanding; Expanding/Converging5.4 Deviations5.4.1 Minimum Size of Expansion5.4.2 One-More-Time and Related Repetitions5.4.3 Deceptive-Authentic Combinations (DCP-ACP); Melodic Content5.4.4 Abandoned ECP5.4.5 Dominant Arrival5.5 A Cadential Conundrum5.6 Expanded Cadential Progressions in Beethoven's Symphonies5.6.1 Symphony No. 1 in C, Op. 21, i, Subordinate-Theme Group5.6.2 Symphony No. 2 in D, Op. 36, iv, Coda5.6.3 Symphony No. 3 in E-flat ("Eroica"), Op. 55, i, Second Subordinate Theme5.6.4 Symphony No. 5 in C Minor, Op. 67, iv, Subordinate Theme, Coda Theme5.6.5 Symphony No. 6 in F ("Pastoral"), Op. 68, iv, Subordinate Theme5.6.6 Symphony No. 7 in A, Op. 92, i, Coda Theme5.6.7 Symphony No. 9 in D Minor ("Choral"), Op. 125, i, Main Theme5.6.8 Symphony No. 9 in D Minor ("Choral"), Op. 125, i, Coda PART 2 CADENCE IN OTHER TONAL STYLES CHAPTER 6 CADENCE IN THE HIGH BAROQUE6.1 Authentic Cadence6.1.1 Perfect Authentic Cadence6.1.2 Harmonic Content6.1.3 Imperfect Authentic Cadence6.1.4 Cadential Deviations (Authentic Cadence)6.1.4.1 Deceptive Cadence6.1.4.2 Evaded Cadence6.1.4.3 Abandoned Cadence6.1.5 Cadential Extension and Expansion6.1.5.1 Consecutive PACs6.1.5.2 Expansion via Slower Tempo; Hemiola; Change of Tempo Marking6.1.5.3 Expanded Cadential Progression6.1.5.4 Deceptive-Authentic Combinations (DCP-ACP); Pulcinella6.2 Half Cadence6.2.1 Harmonic, Tonal, and Formal Functions6.2.2 Dominant Arrival6.2.3 Precadential Dominant Expansion6.3 Miscellaneous Issues6.3.1 Cadential Content versus Cadential Function; Cadences of Limited Scope6.3.2 Prolongational Closure6.3.3 "Plagal" Cadence; Tonic Arrival6.4 Cadence in Fugue: J. S. Bach's Well-Tempered Clavier6.4.1 General Concepts and Terminology6.4.2 Tonal Functions of Cadence in Fugue6.4.3 Formal Functions of Cadence in Fugue6.4.4 Subject-Ending Cadences6.4.5 Expositions Concluding with an S-Ending Cadence6.4.6 Final Cadence as S-Ending6.4.7 Cadential Deviations and Cadential Blurring6.4.8 Three Case Studies6.4.8.1 Fugue in D, WTC 16.4.8.2 Fugue in E-flat, WTC 16.4.8.3 Fugue in G-sharp Minor, WTC 1 CHAPTER 7 CADENCE IN THE GALANT ERA7.1 Morphology of the Galant Cadence7.1.1 Gjerdingen's Galant Clausulae7.1.2 Prinner Cadence7.1.3 Miscellaneous Variants7.1.4 Melodic "Overhang"7.2 Cadential Deviations7.2.1 Evaded Cadence7.2.2 Deceptive Cadence7.2.3 Abandoned Cadence7.2.4 Dominant Arrival7.3 Extension and Expansion Techniques7.3.1 Consecutive Cadences and Codettas7.3.2 Expanded Cadential Progression7.3.3 Pulcinella7.3.4 Dominant Expansions-Penultimate, Ultimate, Precadential7.3.4.1 Expanded Penultimate Dominant7.3.4.2 Internal HC (or Dominant Arrival)7.3.4.3 Precadential Dominant Expansion7.4 Galant Cadence in Relation to Formal Functions7.4.1 Main Theme7.4.2 Transition; Fusion Processes7.4.3 Subordinate Theme; Cadential Play7.4.4 Development7.4.5 Recapitulation CHAPTER 8 CADENCE IN THE ROMANTIC ERA8.1 Chromaticism and Dissonance8.2 Root-Position Harmonies8.3 Uniform Harmonic Rhythm and Density8.4 Off-Tonic Openings; ECP Openings8.5 Formal Circularity8.6 Symmetrical Grouping Structures8.7 Sequence versus Cadence8.8 Prolongational Closure8.9 Lack of Formal Closure8.10 Cadential Deviations8.10.1 Deceptive Cadence8.10.1.1 "Mondnacht"8.10.1.2 "Waldesgespräch"8.10.2 Evaded Cadence8.10.3 Dominant Arrival8.11 Ultimate versus Penultimate Dominants; Dissipated Cadence8.12 Prinner Cadence8.13 Analytical Notes to Chopin's Preludes, Op. 288.13.1 Prelude No. 1 in C8.13.2 Prelude No. 2 in A Minor8.13.3 Prelude No. 3 in G8.13.4 Prelude No. 4 in E Minor8.13.5 Prelude No. 5 in D8.13.6 Prelude No. 6 in B Minor8.13.7 Prelude No. 7 in A8.13.8 Prelude No. 8 in F-sharp Minor8.13.9 Prelude No. 9 in E8.13.10 Prelude No. 10 in C-sharp Minor8.13.11 Prelude No. 11 in B8.13.12 Prelude No. 12 in G-sharp Minor8.13.13 Prelude No. 13 in F-sharp8.13.14 Prelude No. 14 in E-flat Minor8.13.15 Prelude No. 15 in D-flat8.13.16 Prelude No. 16 in B-flat Minor8.13.17 Prelude No. 17 in A-flat8.13.18 Prelude No. 18 in F Minor8.13.19 Prelude No. 19 in E-flat8.13.20 Prelude No. 20 in C Minor8.13.21 Prelude No. 21 in B-flat8.13.22 Prelude No. 22 in G Minor8.13.23 Prelude No. 23 in F8.13.24 Prelude No. 24 in D Minor CHAPTER 9 CADENCE IN THE MID TO LATE NINETEENTH CENTURY9.1 Tonic Arrival9.2 Pre-dominant Arrival; Subdominant Arrival9.3 Plagal Cadence/Plagal Closure9.3.1 Theoretical Speculations9.3.1.1 Plagal Cadential Progression (PCP)9.3.1.2 Plagal Cadential Progressions: Moment-to-Moment Listening Contexts9.3.1.3 Plagal Cadence; Prolongational Closure; Plagal Closure9.3.2 Analyses of Plagal Closure9.3.2.1 Dependent Plagal Closure9.3.2.2 Independent Plagal Closure9.3.3 Plagal Closure versus Plagal Codetta9.3.4 Additional Characteristics of Plagal Closure9.3.4.1 6-5 Melodic Closure9.3.4.2 Phrygian Modality9.3.4.3 Home-Key Plagal Closure versus KL: HC.9.3.4.4 Subdominant with Added Sixth; 1-2-39.3.4.5 "Diminished" Plagal Closure9.3.5 Feigned Plagal Closure9.4 Prolongational Closure9.5 Detour Cadence9.6 Iconic Cadence9.7 Cadential Deviations9.7.1 A Deceptive Half Cadence?9.7.2 Evaded Cadence9.8 Cadential Ambiguities9.9 Four Case Studies9.9.1 Grieg, String Quartet in G Minor, Op. 27, Finale9.9.2 Dvorak, Symphony No. 7 in D Minor, Second Movement9.9.3 Liszt, Sonata in B Minor9.9.4 Wagner, Prelude to Tristan und IsoldeGlossary of TermsBibliographyIndex of Musical CompositionsGeneral Index

Cadence is a monumental work that undeniably pushes the discourse on cadential function toward greater nuance and rigor. Caplin's dedication to a fulsome exploration of the classical cadence is laudable, and he makes many persuasive arguments toward his larger theory of formal functions.

Following graduate studies at the University of Chicago and the Technical University of Berlin, William Caplin joined the Faculty of Music at McGill University in 1978 and remained there until his retirement in 2022. Caplin specializes in the theory of musical form and the history of harmonic and metric theories. His ground-breaking treatise Classical Form (1998) won the Wallace Berry Book Award from the Society for Music Theory. He subsequently served that society as both Vice-President and President. In 2011, Caplin was awarded the prestigious Killam Research Fellowship and was elected a Fellow in the Royal Society of Canada in 2015.

• TABLE OF CONTENTSPrefaceGuide to the Analytical AnnotationsCHAPTER 1 IDEAS OF CLOSURE1.1 Closure in Literature1.2 Closure in Music1.2.1 Leonard Meyer1.2.2 Kofi Agawu1.2.3 Robert Hatten1.2.4 Patrick McCreless1.2.5 Mark Anson-CartwrightPART 1 THE CLASSICAL CADENCE CHAPTER 2 GENERAL CONCEPTS OF THE CLASSICAL CADENCE2.1 Traditional Notions of Cadence2.2 Cadence as Formal Closure2.2.1 Formal Units Closed by Cadence; Cadence and Phrase2.2.2 Cadence and Higher-Level Formal Units2.3 Cadence as Harmony; Harmony as Cadence2.3.1 Progression Types2.3.1.1 Prolongational Progressions2.3.1.2 Sequential Progressions2.3.1.3 Cadential Progressions2.3.2 Ambiguity of Progression Types2.4 Bass-Line Melody and Cadence2.5 Cadential Arrival versus Cadential Function2.6 Cadential Function versus Cadential Content2.7 Limited Cadential Scope2.8 Cadential Function versus Postcadential Function2.8.1 Theoretical Background2.8.2 Postcadential Function as "Confirmation"2.8.3 Hepokoski and Darcy's Critique2.8.4 Two "Registers" of Closure2.8.5 Consecutive (Repeated) PACs2.9 "End" versus "Stop"2.10 Cadence as Punctuation2.11 Cadential Strength-Syntactical versus Rhetorical CHAPTER 3 BASIC CADENCE TYPES: MORPHOLOGY AND FUNCTION3.1 Perfect Authentic Cadence3.1.1 Morphology-Harmonic Content3.1.1.1 Stages of the Cadence Schema3.1.1.2 Stage 1-Initial Tonic3.1.1.3 Stage 2-Pre-dominant3.1.1.4 Stage 3-Dominant3.1.1.5 Stage 4-Final Tonic3.1.1.6 Complete versus Incomplete Cadential Progressions3.1.1.7 Boundaries of the Cadential Progression3.1.2 Morphology-Melodic Content3.1.2.1 Basic Patterns3.1.2.2 Varied Patterns3.1.2.3 Combined Patterns3.1.2.4 Additional Patterns3.1.3 Function3.1.3.1 Standard Functions of the PAC3.1.3.2 Exceptional Situations3.2 Imperfect Authentic Cadence3.2.1 Morphology3.2.1.1 Basic Tenor Stream (8/6/5/3); Varied (5/6/5/3)3.2.1.2 Prinner Cadence3.2.1.3 Other Patterns; Melodic Diversion3.2.1.4 Combined Patterns3.2.1.5 Ending on 5?3.2.2 Function3.2.2.1 Independent IAC3.2.2.2 Way-Station IAC3.2.2.3 Additional Functions of the IAC3.3 Half Cadence3.3.1 General Conditions for Half Cadence3.3.2 Morphology-Harmonic Content3.3.2.1 Stage 1-Initial Tonic3.3.2.2 Stage 2-Pre-dominant3.3.2.3 Stage 3-Dominant3.3.3 Morphology-Melodic Content3.3.3.1 Simple (L-K) HC3.3.3.2 Converging HC3.3.3.3 Expanding HC3.3.3.4 Miscellaneous Issues3.3.4 Function3.3.4.1 Levels of Phrase Functionality3.3.4.2 Levels of Thematic Functionality3.3.4.3 Some Functional Generalizations3.3.5 Special Cases3.3.5.1 Reinterpreted HC3.3.5.2 Reopened HC3.4 Additional Aspects of the Basic Cadence Types3.4.1 Meter3.4.1.1 A Ratio Model of Metrical Strength3.4.1.2 Metrically Weaker Cadences3.4.1.3 Hypermetrical Considerations3.4.1.4 Metrical Weighting-Syntactical or Rhetorical?3.4.2 Texture3.4.2.1 Textural Types3.4.2.2 Texture and Cadence3.4.2.3 Covered cadence3.5 Other Cadence Types3.5.1 "Contrapuntal Cadence"; Prolongational Closure3.5.2 "Plagal Cadence" CHAPTER 4 CADENTIAL DEVIATIONS4.1 Deceptive Cadence4.1.1 Morphology4.1.2 Function4.1.2.1 Way-Station Deceptive Cadence, Replacing a PAC4.1.2.2 Way-Station Deceptive Cadence, Replacing an IAC4.1.2.3 Way-Station Deceptive Cadence, Denied4.1.2.4 Independent Deceptive Cadence4.1.2.5 Noncadential Uses of a Deceptive Cadential Progression4.1.2.6 Deceptive Resolutions of the Dominant in Noncadential Contexts4.1.2.7 Elided Deceptive Cadence4.2 Evaded Cadence4.2.1 Morphology4.2.1.1 Harmonic Content4.2.1.2 Melodic Content4.2.2 Function4.2.2.1 Way Station4.2.2.2 One-More-Time Technique4.2.2.3 New Material4.3 Abandoned cadence4.3.1 Cadential Dominant Undermined by Inversion4.3.2 Cadential Dominant Replaced by Inverted Dominant4.3.2.1 Replaced by Lt4.3.2.2 Replaced by Le4.3.3 Cadential Dominant Replaced by Nondominant Harmony4.3.4 Prolongational versus Cadential; Harmonic Expansion4.3.5 Cadential Abandonment and Harmonic Reduction4.4 Dominant Arrival4.4.1 Arrival on a Final Dominant4.4.2 Arrival on a Terminal Dominant4.4.3 Premature Dominant Arrival4.4.3.1 PDA on a Final Dominant4.4.3.2 PDA on a Terminal Dominant4.4.3.3 Hyperdominant Prolongations4.4.4 Additional Issues4.4.4.1 Dominant Arrival of Limited Scope4.4.4.2 Reinterpreted Dominant Arrival4.4.4.3 Reopened Dominant Arrival4.4.4.4 Doppia PDA versus doppia HC4.4.4.5 Burstein's Critique4.5 Combinations of Deviations4.6 Cadential Ambiguities4.6.1 Evaded versus Deceptive (Elided)4.6.2 Evaded versus Authentic (Elided)4.6.3 Evaded versus IAC (Elided)4.6.4 Evaded versus Covered PAC4.6.5 Evaded versus HC (Dominant Arrival)4.6.6 Abandoned versus Deceptive4.6.7 Multiple Ambiguities4.6.7.1 Mozart, Piano Sonata in A Minor, K. 310, i, mm. 1-104.6.7.2 Haydn, String Quartet in D, Op. 71, no 3, iii, 1-10 CHAPTER 5 CADENTIAL EXPANSION5.1 Perfect Authentic Cadence5.1.1 Harmonic Content5.1.1.1 Stage 1-Initial Tonic5.1.1.2 Stage 2-Pre-dominant5.1.1.3 Stage 3-Dominant5.1.1.4 Diversity of Harmonic Content; Bass-Line Complexities5.1.1.5 Ambiguous Onset of an ECP5.1.2 Melodic Content5.1.2.1 Basic Simple (3/2/2/1); with Cadential Six-Four (3/2/3-2/1)5.1.2.2 Basic Soprano (5/4/2/1); with Cadential Six-Four (5/4/3-2/1)5.1.2.3 Basic Alto (3/2/7/1); with Cadential Six-Four (3/2/1-7/1)5.1.2.4 Basic Tenor (8/6/5/1); with Cadential Six-Four (8/6/5-4/1)5.1.2.5 Varied Simple (1/2/2/1); with Cadential Six-Four (1/2/3-2/1)5.1.2.6 Varied Soprano (3/4/2/1); with Cadential Six-Four (3/4/3-2/1)5.1.2.7 Varied Alto (1/2/7/1); with Cadential Six-Four (1/2/1-7/1)5.1.2.8 Varied Tenor (5/6/5/1); with Cadential Six-Four (5/6/5-5 (4)/1)5.1.2.9 Successive Combinations5.1.2.10 Simultaneous Combinations5.1.2.11 Additional Patterns5.2 Imperfect Authentic Cadence5.3 Half Cadence5.3.1 Minimum Length5.3.2 Harmonic Content5.3.2.1 Stage 15.3.2.2 Stage 25.3.2.3 Stage 35.3.3 Melodic content5.3.3.1 Simple HC5.3.3.2 Converging HC5.3.3.3 Expanding HC5.3.3.4 Converging/Expanding; Expanding/Converging5.4 Deviations5.4.1 Minimum Size of Expansion5.4.2 One-More-Time and Related Repetitions5.4.3 Deceptive-Authentic Combinations (DCP-ACP); Melodic Content5.4.4 Abandoned ECP5.4.5 Dominant Arrival5.5 A Cadential Conundrum5.6 Expanded Cadential Progressions in Beethoven's Symphonies5.6.1 Symphony No. 1 in C, Op. 21, i, Subordinate-Theme Group5.6.2 Symphony No. 2 in D, Op. 36, iv, Coda5.6.3 Symphony No. 3 in E-flat ("Eroica"), Op. 55, i, Second Subordinate Theme5.6.4 Symphony No. 5 in C Minor, Op. 67, iv, Subordinate Theme, Coda Theme5.6.5 Symphony No. 6 in F ("Pastoral"), Op. 68, iv, Subordinate Theme5.6.6 Symphony No. 7 in A, Op. 92, i, Coda Theme5.6.7 Symphony No. 9 in D Minor ("Choral"), Op. 125, i, Main Theme5.6.8 Symphony No. 9 in D Minor ("Choral"), Op. 125, i, Coda PART 2 CADENCE IN OTHER TONAL STYLES CHAPTER 6 CADENCE IN THE HIGH BAROQUE6.1 Authentic Cadence6.1.1 Perfect Authentic Cadence6.1.2 Harmonic Content6.1.3 Imperfect Authentic Cadence6.1.4 Cadential Deviations (Authentic Cadence)6.1.4.1 Deceptive Cadence6.1.4.2 Evaded Cadence6.1.4.3 Abandoned Cadence6.1.5 Cadential Extension and Expansion6.1.5.1 Consecutive PACs6.1.5.2 Expansion via Slower Tempo; Hemiola; Change of Tempo Marking6.1.5.3 Expanded Cadential Progression6.1.5.4 Deceptive-Authentic Combinations (DCP-ACP); Pulcinella6.2 Half Cadence6.2.1 Harmonic, Tonal, and Formal Functions6.2.2 Dominant Arrival6.2.3 Precadential Dominant Expansion6.3 Miscellaneous Issues6.3.1 Cadential Content versus Cadential Function; Cadences of Limited Scope6.3.2 Prolongational Closure6.3.3 "Plagal" Cadence; Tonic Arrival6.4 Cadence in Fugue: J. S. Bach's Well-Tempered Clavier6.4.1 General Concepts and Terminology6.4.2 Tonal Functions of Cadence in Fugue6.4.3 Formal Functions of Cadence in Fugue6.4.4 Subject-Ending Cadences6.4.5 Expositions Concluding with an S-Ending Cadence6.4.6 Final Cadence as S-Ending6.4.7 Cadential Deviations and Cadential Blurring6.4.8 Three Case Studies6.4.8.1 Fugue in D, WTC 16.4.8.2 Fugue in E-flat, WTC 16.4.8.3 Fugue in G-sharp Minor, WTC 1 CHAPTER 7 CADENCE IN THE GALANT ERA7.1 Morphology of the Galant Cadence7.1.1 Gjerdingen's Galant Clausulae7.1.2 Prinner Cadence7.1.3 Miscellaneous Variants7.1.4 Melodic "Overhang"7.2 Cadential Deviations7.2.1 Evaded Cadence7.2.2 Deceptive Cadence7.2.3 Abandoned Cadence7.2.4 Dominant Arrival7.3 Extension and Expansion Techniques7.3.1 Consecutive Cadences and Codettas7.3.2 Expanded Cadential Progression7.3.3 Pulcinella7.3.4 Dominant Expansions-Penultimate, Ultimate, Precadential7.3.4.1 Expanded Penultimate Dominant7.3.4.2 Internal HC (or Dominant Arrival)7.3.4.3 Precadential Dominant Expansion7.4 Galant Cadence in Relation to Formal Functions7.4.1 Main Theme7.4.2 Transition; Fusion Processes7.4.3 Subordinate Theme; Cadential Play7.4.4 Development7.4.5 Recapitulation CHAPTER 8 CADENCE IN THE ROMANTIC ERA8.1 Chromaticism and Dissonance8.2 Root-Position Harmonies8.3 Uniform Harmonic Rhythm and Density8.4 Off-Tonic Openings; ECP Openings8.5 Formal Circularity8.6 Symmetrical Grouping Structures8.7 Sequence versus Cadence8.8 Prolongational Closure8.9 Lack of Formal Closure8.10 Cadential Deviations8.10.1 Deceptive Cadence8.10.1.1 "Mondnacht"8.10.1.2 "Waldesgespräch"8.10.2 Evaded Cadence8.10.3 Dominant Arrival8.11 Ultimate versus Penultimate Dominants; Dissipated Cadence8.12 Prinner Cadence8.13 Analytical Notes to Chopin's Preludes, Op. 288.13.1 Prelude No. 1 in C8.13.2 Prelude No. 2 in A Minor8.13.3 Prelude No. 3 in G8.13.4 Prelude No. 4 in E Minor8.13.5 Prelude No. 5 in D8.13.6 Prelude No. 6 in B Minor8.13.7 Prelude No. 7 in A8.13.8 Prelude No. 8 in F-sharp Minor8.13.9 Prelude No. 9 in E8.13.10 Prelude No. 10 in C-sharp Minor8.13.11 Prelude No. 11 in B8.13.12 Prelude No. 12 in G-sharp Minor8.13.13 Prelude No. 13 in F-sharp8.13.14 Prelude No. 14 in E-flat Minor8.13.15 Prelude No. 15 in D-flat8.13.16 Prelude No. 16 in B-flat Minor8.13.17 Prelude No. 17 in A-flat8.13.18 Prelude No. 18 in F Minor8.13.19 Prelude No. 19 in E-flat8.13.20 Prelude No. 20 in C Minor8.13.21 Prelude No. 21 in B-flat8.13.22 Prelude No. 22 in G Minor8.13.23 Prelude No. 23 in F8.13.24 Prelude No. 24 in D Minor CHAPTER 9 CADENCE IN THE MID TO LATE NINETEENTH CENTURY9.1 Tonic Arrival9.2 Pre-dominant Arrival; Subdominant Arrival9.3 Plagal Cadence/Plagal Closure9.3.1 Theoretical Speculations9.3.1.1 Plagal Cadential Progression (PCP)9.3.1.2 Plagal Cadential Progressions: Moment-to-Moment Listening Contexts9.3.1.3 Plagal Cadence; Prolongational Closure; Plagal Closure9.3.2 Analyses of Plagal Closure9.3.2.1 Dependent Plagal Closure9.3.2.2 Independent Plagal Closure9.3.3 Plagal Closure versus Plagal Codetta9.3.4 Additional Characteristics of Plagal Closure9.3.4.1 6-5 Melodic Closure9.3.4.2 Phrygian Modality9.3.4.3 Home-Key Plagal Closure versus KL: HC.9.3.4.4 Subdominant with Added Sixth; 1-2-39.3.4.5 "Diminished" Plagal Closure9.3.5 Feigned Plagal Closure9.4 Prolongational Closure9.5 Detour Cadence9.6 Iconic Cadence9.7 Cadential Deviations9.7.1 A Deceptive Half Cadence?9.7.2 Evaded Cadence9.8 Cadential Ambiguities9.9 Four Case Studies9.9.1 Grieg, String Quartet in G Minor, Op. 27, Finale9.9.2 Dvorak, Symphony No. 7 in D Minor, Second Movement9.9.3 Liszt, Sonata in B Minor9.9.4 Wagner, Prelude to Tristan und IsoldeGlossary of TermsBibliographyIndex of Musical CompositionsGeneral Index

Cadence is a monumental work that undeniably pushes the discourse on cadential function toward greater nuance and rigor. Caplin's dedication to a fulsome exploration of the classical cadence is laudable, and he makes many persuasive arguments toward his larger theory of formal functions.

Following graduate studies at the University of Chicago and the Technical University of Berlin, William Caplin joined the Faculty of Music at McGill University in 1978 and remained there until his retirement in 2022. Caplin specializes in the theory of musical form and the history of harmonic and metric theories. His ground-breaking treatise Classical Form (1998) won the Wallace Berry Book Award from the Society for Music Theory. He subsequently served that society as both Vice-President and President. In 2011, Caplin was awarded the prestigious Killam Research Fellowship and was elected a Fellow in the Royal Society of Canada in 2015.

• TABLE OF CONTENTSPrefaceGuide to the Analytical AnnotationsCHAPTER 1 IDEAS OF CLOSURE1.1 Closure in Literature1.2 Closure in Music1.2.1 Leonard Meyer1.2.2 Kofi Agawu1.2.3 Robert Hatten1.2.4 Patrick McCreless1.2.5 Mark Anson-CartwrightPART 1 THE CLASSICAL CADENCE CHAPTER 2 GENERAL CONCEPTS OF THE CLASSICAL CADENCE2.1 Traditional Notions of Cadence2.2 Cadence as Formal Closure2.2.1 Formal Units Closed by Cadence; Cadence and Phrase2.2.2 Cadence and Higher-Level Formal Units2.3 Cadence as Harmony; Harmony as Cadence2.3.1 Progression Types2.3.1.1 Prolongational Progressions2.3.1.2 Sequential Progressions2.3.1.3 Cadential Progressions2.3.2 Ambiguity of Progression Types2.4 Bass-Line Melody and Cadence2.5 Cadential Arrival versus Cadential Function2.6 Cadential Function versus Cadential Content2.7 Limited Cadential Scope2.8 Cadential Function versus Postcadential Function2.8.1 Theoretical Background2.8.2 Postcadential Function as "Confirmation"2.8.3 Hepokoski and Darcy's Critique2.8.4 Two "Registers" of Closure2.8.5 Consecutive (Repeated) PACs2.9 "End" versus "Stop"2.10 Cadence as Punctuation2.11 Cadential Strength-Syntactical versus Rhetorical CHAPTER 3 BASIC CADENCE TYPES: MORPHOLOGY AND FUNCTION3.1 Perfect Authentic Cadence3.1.1 Morphology-Harmonic Content3.1.1.1 Stages of the Cadence Schema3.1.1.2 Stage 1-Initial Tonic3.1.1.3 Stage 2-Pre-dominant3.1.1.4 Stage 3-Dominant3.1.1.5 Stage 4-Final Tonic3.1.1.6 Complete versus Incomplete Cadential Progressions3.1.1.7 Boundaries of the Cadential Progression3.1.2 Morphology-Melodic Content3.1.2.1 Basic Patterns3.1.2.2 Varied Patterns3.1.2.3 Combined Patterns3.1.2.4 Additional Patterns3.1.3 Function3.1.3.1 Standard Functions of the PAC3.1.3.2 Exceptional Situations3.2 Imperfect Authentic Cadence3.2.1 Morphology3.2.1.1 Basic Tenor Stream (8/6/5/3); Varied (5/6/5/3)3.2.1.2 Prinner Cadence3.2.1.3 Other Patterns; Melodic Diversion3.2.1.4 Combined Patterns3.2.1.5 Ending on 5?3.2.2 Function3.2.2.1 Independent IAC3.2.2.2 Way-Station IAC3.2.2.3 Additional Functions of the IAC3.3 Half Cadence3.3.1 General Conditions for Half Cadence3.3.2 Morphology-Harmonic Content3.3.2.1 Stage 1-Initial Tonic3.3.2.2 Stage 2-Pre-dominant3.3.2.3 Stage 3-Dominant3.3.3 Morphology-Melodic Content3.3.3.1 Simple (L-K) HC3.3.3.2 Converging HC3.3.3.3 Expanding HC3.3.3.4 Miscellaneous Issues3.3.4 Function3.3.4.1 Levels of Phrase Functionality3.3.4.2 Levels of Thematic Functionality3.3.4.3 Some Functional Generalizations3.3.5 Special Cases3.3.5.1 Reinterpreted HC3.3.5.2 Reopened HC3.4 Additional Aspects of the Basic Cadence Types3.4.1 Meter3.4.1.1 A Ratio Model of Metrical Strength3.4.1.2 Metrically Weaker Cadences3.4.1.3 Hypermetrical Considerations3.4.1.4 Metrical Weighting-Syntactical or Rhetorical?3.4.2 Texture3.4.2.1 Textural Types3.4.2.2 Texture and Cadence3.4.2.3 Covered cadence3.5 Other Cadence Types3.5.1 "Contrapuntal Cadence"; Prolongational Closure3.5.2 "Plagal Cadence" CHAPTER 4 CADENTIAL DEVIATIONS4.1 Deceptive Cadence4.1.1 Morphology4.1.2 Function4.1.2.1 Way-Station Deceptive Cadence, Replacing a PAC4.1.2.2 Way-Station Deceptive Cadence, Replacing an IAC4.1.2.3 Way-Station Deceptive Cadence, Denied4.1.2.4 Independent Deceptive Cadence4.1.2.5 Noncadential Uses of a Deceptive Cadential Progression4.1.2.6 Deceptive Resolutions of the Dominant in Noncadential Contexts4.1.2.7 Elided Deceptive Cadence4.2 Evaded Cadence4.2.1 Morphology4.2.1.1 Harmonic Content4.2.1.2 Melodic Content4.2.2 Function4.2.2.1 Way Station4.2.2.2 One-More-Time Technique4.2.2.3 New Material4.3 Abandoned cadence4.3.1 Cadential Dominant Undermined by Inversion4.3.2 Cadential Dominant Replaced by Inverted Dominant4.3.2.1 Replaced by Lt4.3.2.2 Replaced by Le4.3.3 Cadential Dominant Replaced by Nondominant Harmony4.3.4 Prolongational versus Cadential; Harmonic Expansion4.3.5 Cadential Abandonment and Harmonic Reduction4.4 Dominant Arrival4.4.1 Arrival on a Final Dominant4.4.2 Arrival on a Terminal Dominant4.4.3 Premature Dominant Arrival4.4.3.1 PDA on a Final Dominant4.4.3.2 PDA on a Terminal Dominant4.4.3.3 Hyperdominant Prolongations4.4.4 Additional Issues4.4.4.1 Dominant Arrival of Limited Scope4.4.4.2 Reinterpreted Dominant Arrival4.4.4.3 Reopened Dominant Arrival4.4.4.4 Doppia PDA versus doppia HC4.4.4.5 Burstein's Critique4.5 Combinations of Deviations4.6 Cadential Ambiguities4.6.1 Evaded versus Deceptive (Elided)4.6.2 Evaded versus Authentic (Elided)4.6.3 Evaded versus IAC (Elided)4.6.4 Evaded versus Covered PAC4.6.5 Evaded versus HC (Dominant Arrival)4.6.6 Abandoned versus Deceptive4.6.7 Multiple Ambiguities4.6.7.1 Mozart, Piano Sonata in A Minor, K. 310, i, mm. 1-104.6.7.2 Haydn, String Quartet in D, Op. 71, no 3, iii, 1-10 CHAPTER 5 CADENTIAL EXPANSION5.1 Perfect Authentic Cadence5.1.1 Harmonic Content5.1.1.1 Stage 1-Initial Tonic5.1.1.2 Stage 2-Pre-dominant5.1.1.3 Stage 3-Dominant5.1.1.4 Diversity of Harmonic Content; Bass-Line Complexities5.1.1.5 Ambiguous Onset of an ECP5.1.2 Melodic Content5.1.2.1 Basic Simple (3/2/2/1); with Cadential Six-Four (3/2/3-2/1)5.1.2.2 Basic Soprano (5/4/2/1); with Cadential Six-Four (5/4/3-2/1)5.1.2.3 Basic Alto (3/2/7/1); with Cadential Six-Four (3/2/1-7/1)5.1.2.4 Basic Tenor (8/6/5/1); with Cadential Six-Four (8/6/5-4/1)5.1.2.5 Varied Simple (1/2/2/1); with Cadential Six-Four (1/2/3-2/1)5.1.2.6 Varied Soprano (3/4/2/1); with Cadential Six-Four (3/4/3-2/1)5.1.2.7 Varied Alto (1/2/7/1); with Cadential Six-Four (1/2/1-7/1)5.1.2.8 Varied Tenor (5/6/5/1); with Cadential Six-Four (5/6/5-5 (4)/1)5.1.2.9 Successive Combinations5.1.2.10 Simultaneous Combinations5.1.2.11 Additional Patterns5.2 Imperfect Authentic Cadence5.3 Half Cadence5.3.1 Minimum Length5.3.2 Harmonic Content5.3.2.1 Stage 15.3.2.2 Stage 25.3.2.3 Stage 35.3.3 Melodic content5.3.3.1 Simple HC5.3.3.2 Converging HC5.3.3.3 Expanding HC5.3.3.4 Converging/Expanding; Expanding/Converging5.4 Deviations5.4.1 Minimum Size of Expansion5.4.2 One-More-Time and Related Repetitions5.4.3 Deceptive-Authentic Combinations (DCP-ACP); Melodic Content5.4.4 Abandoned ECP5.4.5 Dominant Arrival5.5 A Cadential Conundrum5.6 Expanded Cadential Progressions in Beethoven's Symphonies5.6.1 Symphony No. 1 in C, Op. 21, i, Subordinate-Theme Group5.6.2 Symphony No. 2 in D, Op. 36, iv, Coda5.6.3 Symphony No. 3 in E-flat ("Eroica"), Op. 55, i, Second Subordinate Theme5.6.4 Symphony No. 5 in C Minor, Op. 67, iv, Subordinate Theme, Coda Theme5.6.5 Symphony No. 6 in F ("Pastoral"), Op. 68, iv, Subordinate Theme5.6.6 Symphony No. 7 in A, Op. 92, i, Coda Theme5.6.7 Symphony No. 9 in D Minor ("Choral"), Op. 125, i, Main Theme5.6.8 Symphony No. 9 in D Minor ("Choral"), Op. 125, i, Coda PART 2 CADENCE IN OTHER TONAL STYLES CHAPTER 6 CADENCE IN THE HIGH BAROQUE6.1 Authentic Cadence6.1.1 Perfect Authentic Cadence6.1.2 Harmonic Content6.1.3 Imperfect Authentic Cadence6.1.4 Cadential Deviations (Authentic Cadence)6.1.4.1 Deceptive Cadence6.1.4.2 Evaded Cadence6.1.4.3 Abandoned Cadence6.1.5 Cadential Extension and Expansion6.1.5.1 Consecutive PACs6.1.5.2 Expansion via Slower Tempo; Hemiola; Change of Tempo Marking6.1.5.3 Expanded Cadential Progression6.1.5.4 Deceptive-Authentic Combinations (DCP-ACP); Pulcinella6.2 Half Cadence6.2.1 Harmonic, Tonal, and Formal Functions6.2.2 Dominant Arrival6.2.3 Precadential Dominant Expansion6.3 Miscellaneous Issues6.3.1 Cadential Content versus Cadential Function; Cadences of Limited Scope6.3.2 Prolongational Closure6.3.3 "Plagal" Cadence; Tonic Arrival6.4 Cadence in Fugue: J. S. Bach's Well-Tempered Clavier6.4.1 General Concepts and Terminology6.4.2 Tonal Functions of Cadence in Fugue6.4.3 Formal Functions of Cadence in Fugue6.4.4 Subject-Ending Cadences6.4.5 Expositions Concluding with an S-Ending Cadence6.4.6 Final Cadence as S-Ending6.4.7 Cadential Deviations and Cadential Blurring6.4.8 Three Case Studies6.4.8.1 Fugue in D, WTC 16.4.8.2 Fugue in E-flat, WTC 16.4.8.3 Fugue in G-sharp Minor, WTC 1 CHAPTER 7 CADENCE IN THE GALANT ERA7.1 Morphology of the Galant Cadence7.1.1 Gjerdingen's Galant Clausulae7.1.2 Prinner Cadence7.1.3 Miscellaneous Variants7.1.4 Melodic "Overhang"7.2 Cadential Deviations7.2.1 Evaded Cadence7.2.2 Deceptive Cadence7.2.3 Abandoned Cadence7.2.4 Dominant Arrival7.3 Extension and Expansion Techniques7.3.1 Consecutive Cadences and Codettas7.3.2 Expanded Cadential Progression7.3.3 Pulcinella7.3.4 Dominant Expansions-Penultimate, Ultimate, Precadential7.3.4.1 Expanded Penultimate Dominant7.3.4.2 Internal HC (or Dominant Arrival)7.3.4.3 Precadential Dominant Expansion7.4 Galant Cadence in Relation to Formal Functions7.4.1 Main Theme7.4.2 Transition; Fusion Processes7.4.3 Subordinate Theme; Cadential Play7.4.4 Development7.4.5 Recapitulation CHAPTER 8 CADENCE IN THE ROMANTIC ERA8.1 Chromaticism and Dissonance8.2 Root-Position Harmonies8.3 Uniform Harmonic Rhythm and Density8.4 Off-Tonic Openings; ECP Openings8.5 Formal Circularity8.6 Symmetrical Grouping Structures8.7 Sequence versus Cadence8.8 Prolongational Closure8.9 Lack of Formal Closure8.10 Cadential Deviations8.10.1 Deceptive Cadence8.10.1.1 "Mondnacht"8.10.1.2 "Waldesgespräch"8.10.2 Evaded Cadence8.10.3 Dominant Arrival8.11 Ultimate versus Penultimate Dominants; Dissipated Cadence8.12 Prinner Cadence8.13 Analytical Notes to Chopin's Preludes, Op. 288.13.1 Prelude No. 1 in C8.13.2 Prelude No. 2 in A Minor8.13.3 Prelude No. 3 in G8.13.4 Prelude No. 4 in E Minor8.13.5 Prelude No. 5 in D8.13.6 Prelude No. 6 in B Minor8.13.7 Prelude No. 7 in A8.13.8 Prelude No. 8 in F-sharp Minor8.13.9 Prelude No. 9 in E8.13.10 Prelude No. 10 in C-sharp Minor8.13.11 Prelude No. 11 in B8.13.12 Prelude No. 12 in G-sharp Minor8.13.13 Prelude No. 13 in F-sharp8.13.14 Prelude No. 14 in E-flat Minor8.13.15 Prelude No. 15 in D-flat8.13.16 Prelude No. 16 in B-flat Minor8.13.17 Prelude No. 17 in A-flat8.13.18 Prelude No. 18 in F Minor8.13.19 Prelude No. 19 in E-flat8.13.20 Prelude No. 20 in C Minor8.13.21 Prelude No. 21 in B-flat8.13.22 Prelude No. 22 in G Minor8.13.23 Prelude No. 23 in F8.13.24 Prelude No. 24 in D Minor CHAPTER 9 CADENCE IN THE MID TO LATE NINETEENTH CENTURY9.1 Tonic Arrival9.2 Pre-dominant Arrival; Subdominant Arrival9.3 Plagal Cadence/Plagal Closure9.3.1 Theoretical Speculations9.3.1.1 Plagal Cadential Progression (PCP)9.3.1.2 Plagal Cadential Progressions: Moment-to-Moment Listening Contexts9.3.1.3 Plagal Cadence; Prolongational Closure; Plagal Closure9.3.2 Analyses of Plagal Closure9.3.2.1 Dependent Plagal Closure9.3.2.2 Independent Plagal Closure9.3.3 Plagal Closure versus Plagal Codetta9.3.4 Additional Characteristics of Plagal Closure9.3.4.1 6-5 Melodic Closure9.3.4.2 Phrygian Modality9.3.4.3 Home-Key Plagal Closure versus KL: HC.9.3.4.4 Subdominant with Added Sixth; 1-2-39.3.4.5 "Diminished" Plagal Closure9.3.5 Feigned Plagal Closure9.4 Prolongational Closure9.5 Detour Cadence9.6 Iconic Cadence9.7 Cadential Deviations9.7.1 A Deceptive Half Cadence?9.7.2 Evaded Cadence9.8 Cadential Ambiguities9.9 Four Case Studies9.9.1 Grieg, String Quartet in G Minor, Op. 27, Finale9.9.2 Dvorak, Symphony No. 7 in D Minor, Second Movement9.9.3 Liszt, Sonata in B Minor9.9.4 Wagner, Prelude to Tristan und IsoldeGlossary of TermsBibliographyIndex of Musical CompositionsGeneral Index

Cadence is a monumental work that undeniably pushes the discourse on cadential function toward greater nuance and rigor. Caplin's dedication to a fulsome exploration of the classical cadence is laudable, and he makes many persuasive arguments toward his larger theory of formal functions.