2. Multiple levels of implication-realisation at the authentic cadence Richard Parncutt and Annemarie Seither-Preisler University of Graz, Austria Presented at: Music & Emotion, Durham, England, 31 August - 4 September 2009 SysMus Graz

3. Why is musical syntax like it is? Can we predict probability distributions of pitch-time patterns in a well-defined style?...starting from a few “first principles”? perceptual, cognitive, social, historical...perceptual, cognitive, social, historical...

4. Whence the authentic cadence?  Clarification of tonic  cognitive efficiency  But why this harmony, this voice leading? Historical account 1. Medieval 2-part cadence M6-P8 (e.g. DB-CC) 2. add a third voice  double leading-tone cadence 14 th Century Ars Nova: Vitry, Machaut  authentic cadence 15 th Century: Dunstable, Dufay, Ockeghem Why this change? Why is authentic cadence so stable?

5. Avoid circular arguments Explain by non-musical phenomena  mathematics of frequency ratios (Pythagoras) psychophysics of pitch perception (Aristoxenus) psychophysics of pitch perception (Aristoxenus)

6. Implication-realisation theory fulfilment of expection = realisation of implication  emotion   Example: melodic gap-fill implication: rising leap realisation: stepwise descent Meyer, L. B. (1956). Emotion and meaning in music. Chicago: U Chicago Press.

7. Realised implications in tonal music  melodic gap-fill, rising leap  falling step  authentic cadence, chains of falling fifths  rising leading tones, falling appogiaturas  thematic repetition...and people like it! (Sloboda, 1991)

8. Thwarted expectations exceptional, but essential  manipulate attention and emotion  create conflict (social metaphor)  new expectation of “happy end” (the norm) Examples  delayed melodic gap fill (baa baa black sheep)  interrupted cadences (Mozart arias)  tonic avoidance (Wagner: Tristan)

9. Authentic cadence V-I implication-realisation  “ultimate” satisfaction? 1. Rising semitone (leading tone to tonic) 2. If seventh  triad: tension-relaxation 3. Entire passage  final triad (Schenker) 4. Falling fifth between roots 1. Why do leading tones tend to rise by m2? 2. Why Mm7? Why major or minor triad? 3. What aspect of passage? Of final triad? 4. Why P5? Why fall rather than rise?

10. 1. Origin of the leading tone Prevalence of scale steps in Gregorian chant (Parncutt & Prem, ICMPC 2008) Most prevalent: G and D. C>B, F>E (exception: E as final) Theory: tones are preferred if their harmonics are in diatonic scale

11. 2. Why major and minor triads? 19 Tn-types of cardinality 3 after Rahn (1980) prime form 012013014015016024025026027036037048 inversion 023034045056035046047 Most consonant Tn-types of cardinality 3 fourth/fifth (fusion) no major/minor second (roughness).

12. 3. Chroma prevalence anticipates chroma salience Aarden, B. (2003). Dynamic melodic expectancy. PhD dissertation, Ohio State University. major keyminor key

13. 4. Why falling fifth between roots? competing theories Common notes or pitches chords 1 and 2 have something in common Root newness root of chord 2 is not a note in chord 1 Implication-realisation implied pitches* in 1  real pitches in 2 *missing fundamentals

14. Prevalence of diatonic progressions rising P4 falling P4 rising 3rd falling 3rd rising M2 falling M2 total maj-maj 6419006291 maj-min 601295077 min-maj 5201155349 min-min 215001027 total15045324175244 Eberlein, R. (1994). Die Entstehung der tonalen Klangsyntax (pp. 422- 423). Frankfurt: Peter Lang. J. S. Bach7 chorales; kleine harmonische Labyrinth HändelTrio sonata Op. 5 No. 5 MozartMissa brevis KV 65 (Kyrie, Gloria, Agnus Dei) BeethovenMass in C (Kyrie, Gloria) MendelssohnMotets Op. 78, Nos. 1 & 2 Assumption: Asymmetry began in 15 th Century and grew

15. Why fourth/fifth intervals? Common notes? No. of common notes Non-directional interval between roots prevalence 0secondlow 1fourthhigh 2thirdmedium 3unisonhigh* *= sustained chord

16. Practical constraints on common notes Does a “progression” imply same rhythm in each voice?  zero common notes?  Just one common note is better: helps perceptual coherence helps perceptual coherence helps tuning in performance helps tuning in performance avoids parallel fifths avoids parallel fifths  Is that why one common note is preferred? Does that in turn explain why fourth/fifths preferred? But what about the cycle of fifths?

17. Neural net model (Bharucha) Spontaneous emergence of cycle of fifths from exposure to triads or tonal music? Psychological reality of cycle of fifths?

18. Interval asymmetry: Root newness e.g. dominant preparation: imply tonic without playing it  tension Diatonic interval between roots Preferred direction PredictedActual second-*rising thirdfallingrising fourthrising * BUT: 2 rising fourths + rising second = octave

19. Virtual triangle (Kanizsa, 1955) Reconstruction of foreground object from elements Virtual pitch (Terhardt, 1976) Reconstruction of a missing fundamental frequency (F0) from harmonics fundamental (F0) overtones frequency Virtual objects – Virtual pitch SPL

20. Basics of pitch perception Things that everyone agrees about  Pitches correspond either to individual spectral components (spectral) individual spectral components (spectral) harmonic patterns of components (virtual) harmonic patterns of components (virtual)  Pitches vary in salience Predictions of spectral and temporal models are about the same Predictions of spectral and temporal models are about the same

21. Missing fundamentals in major triads harmonics above pitch that are present in the chord P5M3m7M2 pitch relative to root M2--P1M3 P4P1--P5 M6M3-P5- m7---P1 Rank order of salience: M6, P4, M2, m7

22. Missing fundamentals in minor triads harmonics above pitch that are present in the chord P5M3m7M2 pitch relative to root M2--P1- P4P1-m3P5 m6m3P1-- m7---P1 Rank order of salience: P4/m6, M2, m7

23. Experimental data Parncutt, 1993 Stimuli in one trial: A chord of OCTs, then a single OCT Listeners rate how well tone follows chord Diamonds: Mean ratings Squares : Theoretical predictions

24. Physicalspectraand calculated experiential spectra “Pitch category”: 48 = C4, 60 = C5 etc. (Parncutt, 1989) Pure tone Harmonic complex tone Octave complex tone

25. Physical spectra and calculated experiential spectra “Pitch category”: 48 = C4, 60 = C5 etc. (Parncutt, 1989) Minor triad Tristan chord

26. Implication-realisation model of falling fifth progressions  CEG implies F and A in CEG-CFA, implications are realised in CEG-CFA, implications are realised  CEbG implies F and Ab in CEbG-CFAb, implications are realised in CEbG-CFAb, implications are realised  Also explains falling third progressions prevalent because of IR prevalent because of IR less prevalent than fifths because less IR less prevalent than fifths because less IR

27. Pitch salience and common notes Consider two chords: C and Am/C Prediction: Most salient pitch in both is C  Chords with 2 common notes are not different relative major-minor (Riemann: parallel)  Fourth progressions > third progressions

28. Overtone spectrum: Elementary physical dimension Virtual pitch: Musical gestalt dimension 2.- 4. 5.- 10. 1. Individual differences in pitch perception Auditory ambiguity test (Seither-Preisler)

29. 0 25 50 75 100 major second major third forthfifthmajor sixth Melodic interval of missing F0 F0-responses (%) amateurs Non-musicians Professionals Individual differences in pitch perception Seither-Preisler, A. et al. (2007). JEP: HPP, 33(3), 743–751

30. Pitch salience and music history Revised thesis: Missing fundamentals influence historical development of syntax because some (not all) listeners, performers, composers perceive them

31. Advantages of virtual pitch approach pitch commonality - implication-realisation  Bottom-up: underlying scale not assumed not circular not circular prevalence of any chromatic progression? prevalence of any chromatic progression?  Same model explains similarity of successive tones, chords, keys  Explain perceptual coherence of progression  IR explains why progressions are emotional

32. Why is authentic cadence based on a falling fifth between roots? Why fifth interval? pitch commonality  perceptual coherence one common note  feeling of progression two implied pitches are realised Why falling? Harmonic aspect: root newness or IR Melodic aspect: leading tone rises