Expectation in music 2 Day 19 Music Cognition MUSC 495.02, NSCI 466, NSCI 710.03 Harry Howard Barbara Jazwinski Tulane University

Course administration Spend provost's money 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Music Cognition - Jazwinski & Howard - Tulane University Goals for today Background on musical interval Narmour (1990) Margulis (2005) Pearce & Wiggins (2004) 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Musical interval

Music Cognition - Jazwinski & Howard - Tulane University Interval An interval in music means the distance between two notes. Intervals may be described as: vertical (or harmonic) if the two notes sound simultaneously, or linear (or melodic) if the two notes sound successively. http://en.wikipedia.org/wiki/Interval_(music) http://simple.wikipedia.org/wiki/Interval_(music) 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

How to conceptualize melody A musical melody is commonly conceived as a sequence of pitches. For musicians, however, this is not the best way to conceive of a melody. A melody can be transposed to different pitch levels without losing its perceived identity. This suggests that a melody can be adequately described in relative, rather than absolute terms. Li06 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Relative pitch in the West For Western musicians, two systems are used to characterize relative pitch. Pitch distances or intervals The ratio of the frequencies of two notes. Tones in a scale or “scale degree” Tonic solfa syllables (do, re, mi, fa, etc.), denoted using the numbers 1 through 7. Li06 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Music Cognition - Jazwinski & Howard - Tulane University Units of measurement A semitone (also called a half step or a half tone) is the smallest musical interval used in Western music. It is the smallest distance in pitch between two notes. A whole tone is the distance of two semitones i.e. the distance between two notes which are separated by one other note in pitch. An octave is divided into twelve semitones, which are equal in size. 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Piano keyboard example The distance between two white notes that are side by side may be a whole tone (if there is a black note in between them) or a semitone (if there is no black note between). To go from a C to a C sharp (or D flat) is a semitone. To go from a C sharp (or D flat) to a D is a semitone. To go from a C to a D is a tone. http://simple.wikipedia.org/wiki/Semitone see also http://www.phys.unsw.edu.au/jw/notes.html 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Comparison of different interval naming systems # semitones Interval class Generic interval Common diatonic name Just interval perfect unison 1:1 1 minor second 16:15 2 major second 9:8 3 minor third 6:5 4 major third 5:4 5 perfect fourth 4:3 6 augmented fourth diminished fifth 45:32 64:45 7 perfect fifth 3:2 8 minor sixth 8:5 9 major sixth 5:3 10 minor seventh 16:9 11 major seventh 15:8 12 perfect octave 2:1 http://en.wikipedia.org/wiki/Interval_(music) 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Narmour (1990) The Analysis and Cognition of Basic Melodic Structures: the Implication-realisation Model. Chicago: University of Chicago Press.

The Implication-realization Model It attempts to characterize the set of implied continuations to an incomplete melodic pattern. Expectancies are influenced by two independent cognitive systems: A small number of Gestalt-like principles which are held to be innate and universal, i.e. ‘hard-wired’ [bottom-up] An unknown number of expectations acquired through extensive exposure to music in a given style [top-down] extra-opus knowledge about style specific norms, such as diatonic interpretations, tonal and metrical hierarchies and harmonic progressions, intra-opus knowledge about aspects of a particular composition such as distinctive motivic and rhythmic patterns. Pearce 04 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Music Cognition - Jazwinski & Howard - Tulane University Combination Since the top-down system is largely independent from the bottom-up system, it may generate implications which conflict with those generated by the bottom-up system. Pearce 04 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Music Cognition - Jazwinski & Howard - Tulane University The bottom-up system Melodic pitch intervals vary in the degree of closure that they convey. An interval which is closed signifies the termination of ongoing melodic structure An interval which is unclosed is said to be an implicative interval and generates expectancies for the following interval which is termed the realized interval. Pearce 04 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Implicative intervals The expectancies generated by implicative intervals are described in terms of several principles of continuation which are inspired by the Gestalt principles of proximity similarity good continuation symmetry Pearce 04 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Parameters of the model Proximity Registral direction Intervallic distance Registral return Closure Pearce 04 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Intervallic difference Small intervals imply a subsequent interval that is similar in size, while large intervals imply a consequent interval that is smaller in size. This principle can be taken to be an application of the Gestalt principles of similarity and proximity for small and large intervals respectively. Pearce 04 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Music Cognition - Jazwinski & Howard - Tulane University Registral return Registral return is a general implication for a return to the pitch region of the first tone of an implicative interval in cases where the realized interval reverses the registral direction of the implicative interval. This principle can be viewed as an application of the Gestalt principle of proximity. Pearce 04 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Music Cognition - Jazwinski & Howard - Tulane University Closure Closure is determined two conditions: A change in registral direction related to the Gestalt principle of symmetry Movement to a smaller sized interval. Degrees of closure exist corresponding to the satisfaction of both, one or neither of the conditions. Pearce 04 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

A model of melodic expectation. Music Perception 22:663-714. Margulis (2005) A model of melodic expectation. Music Perception 22:663-714.

Parameters of the model Stability Proximity Direction Mobility Margulis (2005) 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Music Cognition - Jazwinski & Howard - Tulane University Stability Stability captures the intuition that, in general, listeners expect relatively stable melodic events. Margulis (2005) 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Music Cognition - Jazwinski & Howard - Tulane University Proximity (Narmour) The principle of proximity describes a general implication for small intervals between any two tones. The implication is graded according to the size of the interval and can again be viewed as an application of the Gestalt principle of proximity. 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Music Cognition - Jazwinski & Howard - Tulane University Proximity (Margulis) Proximity captures the intuition that listeners expect subsequent events to be relatively proximate to preceding ones. that is, listeners expect small intervals to continue to additional small intervals in the established direction and large intervals to reverse to small intervals in the opposite direction. This amounts in part to theorizing that listeners expect subsequent events to be proximate to preceding ones. Margulis (2005) 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Proximity ratings increase as semitone distance increases Pitch Distance in Semitones 1 2 3 4 5 6 7 8 9 10 11 12 13 >14 Proximity Rating 36 32 25 20 16 12 9 6 4 2 1 0.25 0.02 0.01 Margulis (2005) 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Registral direction (Narmour) The principle of registral direction states that small intervals imply continuations in the same registral direction whereas large intervals imply a change in registral direction. The application of the principle to small intervals is related to the Gestalt principle of good continuation while its application to large intervals is related to the principle of symmetry. 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Music Cognition - Jazwinski & Howard - Tulane University Direction (Margulis) The expectation for reversal after large intervals between pitches may be considerably stronger than the expectation for continuation after small ones. Margulis (2005) 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Music Cognition - Jazwinski & Howard - Tulane University Direction ratings The x axis represents pitch interval size, and the y axis represents strength of directional expectation. Dark shading represents an expectation for reversal, and light shading indicates an expectation for continuation. The influence of the direction parameter varies according to the size of the interval between the two preceding notes. Very large intervals should generate a strong expectation for reversal, but very small intervals should generate only a moderate expectation for continuation. Margulis (2005) 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Music Cognition - Jazwinski & Howard - Tulane University Mobility The secondary factor mobility captures the general intuition that a melody will move. the mobility rating of a pitch x which is expected to follow a pitch y is 2/3 if x repeats y and 1 in all other cases. Margulis (2005) 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Music Cognition - Jazwinski & Howard - Tulane University Factor combination Before hierarchic levels are considered, an event’s expectancy rating may be generated by using the following formula. A pitch x is expected to follow pitch y by amount z: z = (s  p  m) + d, where s = the stability rating of x, p = the proximity rating of x, d = the direction rating of x, and m = the mobility rating of x (2/3 if x repeats y and 1 in all other cases). Margulis (2005) 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Rethinking gestalt influences on melodic expectancy Pearce & Wiggins (2004) Rethinking gestalt influences on melodic expectancy

An alternate source for bottom-up regularities They may reflect universal physical constraints on performance The relative difficulty of singing large intervals accurately; The fact that large intervals will tend towards the limits of a singer’s vocal range. Pearce & Wiggins (2004) 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Music Cognition - Jazwinski & Howard - Tulane University Musical development Pitch proximity is a strong influence for listeners of all ages. The influence of more complex predictors such as pitch reversal and registral return only became apparent with the older listeners. These results suggest that mature patterns of expectancy in melody are acquired through exposure to music which results in the progressive induction of increasingly complex musical regularities. Pearce & Wiggins (2004) 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Induction of statistical regularities According to the theory developed here, observed patterns of melodic expectancy can be accounted for in terms of the induction of statistical regularities from a repertoire of music. In order to examine this claim, previous data on melodic expectancy are reanalyzed in terms of a computational model embodying the theory. Pearce & Wiggins (2004) 12/27/2018 Music Cognition - Jazwinski & Howard - Tulane University

Next Monday Theories of rhythm