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

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

3. 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

4. Musical interval

5. 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.
12/27/2018
Music Cognition - Jazwinski & Howard - Tulane University

6. 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

7. 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

8. 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

9. 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.
see also
12/27/2018
Music Cognition - Jazwinski & Howard - Tulane University

10. 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
12/27/2018
Music Cognition - Jazwinski & Howard - Tulane University

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

12. 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

13. 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

14. 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

15. 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

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

17. 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

18. 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

19. 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

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

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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

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

32. 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

33. 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

34. 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

35. Next Monday
Theories of rhythm