Week 9 Flashcards
Form-bearing dimensions in music• The utility of a form-bearing dimension depends on the degree to which
▫
it affords a large number of perceivable
configurations, patterns,
▫ it affords the potential for transformation and
development
▫ configurations on the dimension can maintain
perceptual invariance in the presence of changes along other dimensions
Vibrato rate, poor potential for form, cannot distinguish easily.
Form-bearing dimensions in music• A dimension’s capacity to bear form
▫ A dimension can bear form if configurations of
values along the dimension can be
encoded by the auditory system
organized
recognized
compared with other configurations presented
earlier
Excellent form bearers are pitch and rhythm
Abstract knowledge structures
• The perceived qualities of musical events that
give rise to structure are anchored to a learned system of relations.
• The system of learned relations constitutes
abstract (implicit) knowledge about the
structure of music for a given culture acquired through experience.
• In music, there are two types of knowledge:
▫ systems of relations among musical categories (a temporal, whole scale as a package)
▫ a lexicon of abstract, archetypical patterns or
idioms (temporal, like cadences)
On a continuum:
Categorization of values along a dimension
▫ Discretization of values for discriminability andmemorability
▫ Relatively small number of categories
▫ Well-defined categories or relations among
categories
In the majority of cultures in the world, musical structure is based on dimensions thatare divided into categories (pitch, duration, dynamics, timbre). In many cultures, this
categorization is shown by coding in symbols in notation systems or in language
(solfège for pitch in Western and Indian music, timbral solfège for tabla strokes in
Indian music).
One of the major aspects of this categorization is the discreteness of the musical
dimensions. One cannot build a system of relations among values that cannot be
easily discriminated. The importance of discretization is that we remember discrete
entities easier than continuous ones. If we cannot easily remember the relations
among values, we cannot learn them.
It is also important that there is a relatively small number of categories, which
depends on memory limitations.
We can also store systems in long-term memory with smaller numbers of categories
that have fairly fixed relations among them. The degree of fixity varies across cultures as we will se
Ordered relations among categories
▫ Category values and relations should have a solidsensory foundation for their function in musical
patterns
▫ Important properties of a system of relations
among musical categories:
focal values (louder, longer, more frequent)
asymmetrical structure of intervals
distinctive intervals
predisposition to certain sequential relations
hierarchical ordering of stability or dominance
relations
Potential form bearing dimension should be closely correlated with sensory
dimensions that effect perceptual grouping (simultaneous, sequential or
segmentational). Research suggests the dimensions of timbral brightness, pitch,
duration, dynamics, and spatial location have this capacity.
Focal values are those that occur frequently, have longer durations, and tend to be
louder. Focal values help a listener establish a framework when faced with an
unfamiliar musical style.
Listeners must be able to find their position within a system of pitch relations due to
the asymmetric structure of intervals among the categories in a scale and the rare or distinctive intervals in a scale. Major semitone, whole tone all equal harder to situate oneself.
The criterion “predisposition to certain sequential relations” captures aspects of
functional relations that are related to the frequency of occurrence of pairs or sets ofvalues in a given order (such as the leading tone moves to the tonic more frequently
statistically than the reverse) and this gives a sense of directed motion.
Some sensory and some cognitive interaction of intervals understanding for ex.
4 levels of pitch analysis
Examination of scale systems from many cultures suggests that scales are constructedthrough several levels of processing:
1) underlying psychophysical pitch function (assigns pitches to frequencies) Some universal, some cultural for
2) discretization of pitch continuum (all possible scale degrees)
3) scale structure (subset of tonal material used as a basis for a variety of
modal scales; comprises an interval set)
4) modal hierarchy (hierarchical organization of pitches in scale structure;
establishes dynamic patterns of expectation, tension and release)
Pitch chroma and pitch height,
Constraints on scale construction
- Descrimintizinf of intervals: Pitches must be discriminable from one another. In Western music, the smallest
interval between notes represents a frequency difference of about 5.9%. Humans arecapable of discriminating on the order of 1%. - Octave equivalence: Tones whose fundamental frequencies stand in a 2:1 ratio are treated as similar to one another.
- Moderate number of pitches within the octave: (7+2) When the octave is filled with the intervals of the scale, there should be only a
moderate number of different pitches. Miller (1956) argues that the number of
stimuli along a given dimension that people can categorize consistently is typically
7±2 - Uniform modular pitch Ike semitone: Equal temperament operates in a few cultures of the world.
Properties of scale structure
• Modal hierarchies (like the major scale) are
established by:
▫ focal pitches
frequency of occurrence, duration, position in
phrase
▫ asymmetrical pattern of large and small intervals indicate position within scale, constant framework
for encoding sounded pitches
▫ distinctive intervals
2 minor seconds (2 semitones), 1 tritone (6
semitones) in major scale
• Pattern repeated at the octave throughout the
frequency range
These results suggest that people’s mental representation of the major scale
structure shifted rigidly to align itself with the notes of the stretched scale. As the
tones of the stretched scale deviated from the learned pattern, this pattern was
shifted to accommodate the deviations. By the end of the scale, it was positioned onC#, thereby influencing the subsequent judgments accordingly. Shepard and Jordan
conclude that tones are assimilated to an internalized musical scale
Do have mental representation of major scale and due to recency can shift it over.
Shepard & Jordan (1984), Science]
These bar graphs show the percentage of listeners that rated the final test tone as
much lower (-2), a bit lower (-1), identical to (0), a bit higher (1) or much higher (2)
than the first tone in the scale. The results of the study showed that listeners judged
pitches that fit with the ascending stretched scale as being related to a C# Major scalerather than to a C Major scale (the hashed bars in the upper panel are higher at 0,
whereas the white bars are higher at -2). This suggests that listeners accept C# as the original starting tone, even though that particular C# was never presented. Similarly,
the pitches that fit with the descending stretched scale (from C# to C) were judged to fit better with the C Major scale than with the C# Major scale (hashed bars are higher at 0 and white bars are higher at 1 and 2).
Diatonic scales give the maximal variety of interval sizes, also known as?
_______ is a property where any interval of n steps is larger than any interval of n-1steps (in semitones). In other words, all intervals of two scale steps (as measured in semitones) are larger than any interval of one scale step.
completeness
Coherence
Variable interval scale systems
Variable intervals in a pentatonic scale as used by the Bedzan pygmies of central
Cameroon. The variability of the intervals between notes (bottom 2 levels) are
constrained by the variability of the total size of two steps in the lower part of the
scale (Trichord) and three steps in the upper part of the scale (Tetrachord), as well asby the whole range of the scale (Octave). The double-headed arrow inside the white box indicates the minimum size of the interval and the arrow extending into the gray boxes indicates the maximum size of the interval. The overlapping gray boxes for
adjacent intervals and for higher level intervals shows how changing one interval has to result in changes in the other interval(s) within the maximum and minimum
constraints. So the actual intervals can be different, but the relative sizes of the
intervals must conform to the global constraints.
Ecological vs experimental issue I see.
Shepards pitch helix model
Chroma, height,
Shepard’s (1956) pitch helix model represents aspects of pitch perception involving aheight component and a circular chroma component, as we have seen in the lecture on pitch perception.
The probe tone technique,
Carol Krumhansl (and Roger Shepard) developed the probe-tone technique in order
to test the effect of musical context (an evoked scale) on the perception of relations
among tones. The first type of context was used to establish the relatedness of a
given tone to an evoked scale context. The second approach tests how an establishedcontext affects the perception of relations among tones. By assuming that these
ratings represent an evaluation of the relative stability or hierarchical importance of a tone within a given tonal context, this method allowed Krumhansl to describe a map
of tonal hierarchies for each major and minor key.
Psyco q: Do we store psychological representations, bunch of other rela.
Musical q: when do we know we heard a wrong note. Unfamiliar music play wrong note non musiNs know.
Krumhansl presented musician listeners a major triad or a scale (ascending or
descending) to establish a tonal framework. Then she played them a pair of notes
(probe tones) and participants judged the similarity of the two tones within the given framework. Multidimensional scaling revealed 2-D and 3-D solutions in which the
pitches are grouped into 3 rings. The inner ring contained pitches of the tonic triad, the middle ring contained the other diatonic pitches and the outer ring contained the non-diatonic pitches. The triad tones are closer to one another than are the other
diatonic tones and the chromatic tones are the farthest from one another. The non- diatonic notes are arranged closest to the nearest diatonic neighbour in the scale. Related to pitch height!
The perceptual ratings support the notion that a representation of pitch could includea circular arrangement of chroma. The results show that tonal context does have a strong role and that factors of pitch height, pitch chroma, and diatonicism enter into play in the perception of pitch relations in such a context. Tonal hierarchy exists.
The cone representation, mathematical model.
Shepards mods pitch space
4 dimensions this time, ugh.
Leading to a ____helix model
5d Taurus donut.
Krumhansl and Shepard (1979) presented listeners with a context of a major scale minus the final tonic. They asked the participants to judge how well the following probe tone completed the scale. Shepard (1982) performed multidimensional scaling on the dissimilarities and recovered a 4D solution.
The plane representing dimensions1 and 2 roughly corresponded to the chroma circle with a gap between the two C’s
separated by an octave, suggesting an added effect of pitch height. The plane representing dimensions 3 and 4 corresponded to the circle of fifths, with C and C’ nearly superimposed, which suggests octave equivalence. Listeners with the least extensive musical backgrounds had the heaviest weights on dimension 1, whereas listeners with the most musical training had the heaviest weights on dimensions 3 and 4 (circle of fifths and octave equivalence).
Based on these results, Shepard (1982) developed several idealized models of the psychological structure of musical pitch. In the double helix (A), involving two intertwined spirals, each link along each strand represents a whole tone (2 semitones) and each link across the strands represents a semitone. Tone height is expressed vertically. If one were to project a line down from each link to the floor, pitches separated by an octave would be connected and the pattern projected would encompass the circle of fifths (B). Shepard’s model includes several important aspects of pitch perception in one structure: pitch height, chroma, the circle of fifths, and octave equivalence.
Shepard (1982) also proposed more complex models. The 4D model wraps the double helix into a toroidal surface. Each of the internal rings includes a whole tone scale and includes the circle of fifths. He also added pitch height into the 4D model toobtain a 5D model, which is a toroidal spiral. No pitch height.
Amazing work for mathematics.
Tonal Hierarchy amazing study cognition,
Using the probe-tone technique, Krumhansl developed the notion of tonal hierarchy. Krumhansl and Kessler (1982) used chord cadences and tonic triads as musical
context. Based on careful experimentation with musician listeners, these are the
tonal hierarchies for major and minor scales. The hierarchy is clearly tonic (C – 1st
scale degree), dominant (G – 5th scale degree), mediant (E – 3rd scale degree),
subdominant (F – 4th scale degree), followed by the other diatonic notes (scale degrees 2, 6, 7) and then the chromatic notes (not part of the scale – the black noteson the piano with respect to the scale of C major on all the white notes). The structure is quite different for the minor scale, with the mediant being stronger than the dominant. These hierarchies seem to differ for different classes of listeners: This structure is much weaker for the nonmusicians, who tend to show an effect of pitch proximity with a slight octave equivalence effect.
Tonal hierarchy, mental representations
Using the probe-tone technique, Krumhansl developed the notion of tonal hierarchy. Krumhansl and Kessler (1982) used chord cadences and tonic triads as musical context. Based on careful experimentation with musician listeners, these are the tonal hierarchies for major and minor scales. The hierarchy is clearly tonic (C – 1st
scale degree), dominant (G – 5th scale degree), mediant (E – 3rd scale degree), subdominant (F – 4th scale degree), followed by the other diatonic notes (scale degrees 2, 6, 7) and then the chromatic notes (not part of the scale – the black noteson the piano with respect to the scale of C major on all the white notes). The structure is quite different for the minor scale, with the mediant being stronger than the dominant. These hierarchies seem to differ for different classes of listeners: This structure is much weaker for the nonmusicians, who tend to show an effect of pitch proximity with a slight octave equivalence effect.
Krumhansl showed that the tonal hierarchy can be partly explained by the relative frequency of occurrence of notes in real music (as measured by several scholars onsubsets of the Classical and Romantic repertoire). The frequency of occurrence of
notes could be a statistical learning rule by which the tonal hierarchy is acquired implicitly through passive exposure to music, and the brain keeps a running tally of how often different notes occur as a function of their relation to the tonic note.
Build mental representation of a key, activating schema, can take hours days. Exposure phase of learning this unclear.
Key profiling refining
Here is the frequency of occurrence of all pitch classes in Bach’s C Major Prelude. Wecan compare it to the tonal hierarchy of C Major as determined by Krumhansl and
Kessler.
Key finding:
One way to measure the fit between the two is to compute their correlation
coefficient. If we compute the correlation coefficients of the actual profile with all possible Major key profiles, we get positive correlations with
C major, R = .96; F major, R = .50; and G major, R = .77.
Key map;
Krumhansl and Kessler used probe-tone data of C major and A minor contexts (top graph) to obtain a geometric map of musical keys. They assumed that closely related keys would have similar tonal hierarchies. The profiles for all major and minor keys were correlated to give a quantitative measure of key distance. MDS was applied and a 4D solution was found that mapped all 24 major and minor keys on a surface of a torus (represented here as a rectangular map). The keys were related based on the circle of fifths, parallel major/minor keys (C major vs. c minor), and relative major/minor keys (same key signature). Shows that keys are related, theorists been saying for centuries.
