Download Brief background The tonal hierarchy The first probe tone study

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The experiments provide a quantitative
measure of the hierarchical ordering imposed
on the individual tones in tonal contexts.
It will be argued that this hierarchy is, in some
sense, basic to the structuring of music itself
and also to the psychological response to music
Very general feature of music: one particular
pitch is established as a central reference pitch.
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The results obtained within the experiments discussed
in the chapter of reading parallel those found in other
areas of human cognition and perception, suggesting
that a general psychological principle is operating in
the particular musical case considered
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This pitch is called the tonic, or tonal center.
Adheres to the basic principle of tonality that,
defined in its most general sense, is the
centering of the pitch materials around one
particular tone
Brief background
The tonal hierarchy
The first probe tone study
Replication and extension to minor-key context
A derived measure of interkey distances
Spatial representation of interkey distances
Theoretical maps of key relationships
Principle: Particular perceptual and conceptual objects have
special psychological status
à Within categories certain members are normative, unique, self
consistent, simple, typical, or the best exemplars of the domain
à They are reference points to which other category members are
compared.
à colors are often described with respect to "focal" colors, such as
red, green, blue, and yellow
 A color may be described as off-red or brownish-red
à numbers are rounded off to other numbers with special
cognitive status, such as multiples of tens and hundreds.
 People say that 9 is almost 10, or that 95 is almost 100
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From a psychological point of view their exists a desire
to drive toward maximizing the efficiency of coding or
minimizing the complexity of cognitive objects
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The experiments discussed in the reading lead
to two different kinds of findings
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elements can be rated reliably in terms of "goodness”
or typicality
à This establishes a kind of hierarchical ordering on the
elements in the category
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the hierarchical ordering influences various
measures of perceptual or cognitive processing
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A tonal context designates one particular tone as most central. The
other tones all have functions specified with respect to this tone, in
terms of their relatedness to the tonic and secondary reference
points established by the tonic
One slight snag: Whereas other perceptual and cognitive reference
points are fixed, the tonic depends on the particular context. No
tone is inherently more "tonic" than others
Two types of hierarchy:
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Event hierarchies
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Tonal hierarchies
à describe the encoding of specific pieces of music;
à embody our tacit or implicit knowledge of the abstract musical structure of a
culture or genre
 All experiments on this reading for tonal hierarchy testing use this hierarchy
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Stability
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Tonality
the relationships existing between tones or tonal spheres
within the context of a particular style system
ƒ tonal systems are generally hierarchical
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à tones which are active tendency tones on one level may be
focal substantive tones on another level and vice versa
 in the major mode
‚ the tonic tone is the tone of ultimate rest toward which all
other ones tend to move.
 On the next higher level the third and fifth of the scale
‚ active melodic tones relative to the tonic, join the tonic as
structural tones; and all the other tones, whether diatonic or
chromatic, tend toward one of these.
 Going still further in the system:
‚ the full complement of diatonic tones are structural focal
points relative to the chromatic notes between them.
 Finally, any of these twelve chromatic notes may be taken as
substantive relative to slight expressive deviations from their
normal pitches
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It is taken to refer to the dimension along which musical tones differ, with
some tones producing an unstable effect and requiring resolution, and
other tones producing a stable effect and giving a sense of completion
This hierarchy has correlates in the names of the
notes in various theoretical systems for describing
music
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Basic names are given to the normative tones, and other
tones are described in relation to these
à less stable tones in the scale have names that reflect their
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relationship to the most stable tones, the tonic and the
dominant (a fifth above the tonic).
The third scale tone is called the mediant because of its
position between the tonic and the dominant.
The seventh degree of the scale (one scale step below the
tonic) is called the leading tone because it "leads to" the tonic.
The second degree of the scale (one scale step above the
tonic) is called the supertonic.
The fourth scale tone, which is a fifth below the tonic, is
called the subdominant
the sixth scale tone, midway between the subdominant and
the tonic, is called the submediant.
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The method used
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Two step process
Step 1:
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investigate the psychological ordering imposed on
the set of chromatic tones by contexts establishing
major and minor keys.
 The method suggests sounding incomplete scale contexts with all
possible tones of the chromatic scale and ask listeners to give a
numerical rating of the degree to which each of the tones completed
the scale
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Experiment 1:
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Step 2:
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The probe tone method
à when an "incomplete" scale is sounded, such as the successive
tones C, D, E, F, G, A, B, this creates strong expectations about
the tone that is to follow
Establish that the elements that dominate in the
quantified hierarchy have special perceptual and
cognitive status, with other elements heard in
relation to them
Used both ascending and descending incomplete C major
scales.
à The ascending scale was sounded in the octave below middle C;
it consisted of the sequence of notes C, D, E, F, G, A, B.
à The descending scale began two octaves above middle C; it
consisted of the notes, C, B, A, G, F, E, D.
à The probe tone came next in both
 equal-tempered semitones (the tones of the chromatic scale) in the
octave range from middle C to the C an octave above. Thus, the 13
probe tones were C, C#, D, D # , E, F, F # , G, G # , A, A # , B, and C’
à Tones were produced on an electronic organ, using the flute
stop
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Experiment 2:
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Participants
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Used the same incomplete scale contexts, but the
tones were produced by computer using digitized
sine waves converted to analog form
also included as probe tones the quarter tones
between the chromatic scale tones
University students of diverse musical backgrounds
Results
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Three distinct patterns
Pattern 1:
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Pattern 1:
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Pattern 2
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Conclusion
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Under the circumstances being tested, listeners are
unable to make these fine discriminations, and that
the quarter tones are assimilated to their chromatic
scale neighbors
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Pattern 3
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A subsequent probe tone experiment
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Attempting to replicate and extend the results of the first study.
Included a variety of different contexts to ensure that the basic
findings of the first experiment generalized to other contexts.
assumed that the scale context of the first study established the
expected key
à Would be supported if similar ratings were given when different
contexts thought to imply the same key were used
à Context used:
 Complete scales
 Tonic triads
 Three different chord cadences
 sounded in both major and minor keys
‚ to see whether analogous patterns would be found for the two modes
 a variety of different tonal centers or keys were used
‚ to make sure that our findings in the first experiment did not depend on
the choice of C as the tonic
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Major objective
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Obtain ratings of probe tones that were as stable and
reliable as possible, so that they could be used in
connection with subsequent experiments with more
complex musical contexts
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One final difference
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The task was to rate how well the final probe tone
"fit with" the context in a musical sense
Results
two things were done to minimize the effects of
influences that are not specifically musical
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decrease the chance that nonmusical response
strategies would be adopted, listeners had to have at
least 5 years of formal instruction in music
attempted to minimize the effect of pitch height
differences between the context and probe tones
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considers whether the experimentally measured
tonal hierarchies can be used to produce a measure
of interkey distance
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Interkey distance:
à keys are considered close if modulations between them are
relatively frequent
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Distance between major keys is represented by the "circle
of fifths”
à The name derives from the tonics of neighboring keys that
are separated by an interval of a fifth
à Around the circle, neighboring keys have scales that share all
but one pitch. For example, the scale of the key of C major (C,
D, E, F, G, A, B) has all but one tone in common with the
scale of the key of G major (G, A, B, C, D, E, F#)
When enharmonically equivalent tones are identified, the pattern of
interkey relatedness folds back on itself to form a closed circle
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When minor keys are introduced, the problem of
defining interkey distance is complicated considerably
Minor scales take a number of different forms
à The "natural" minor scale contains the same pitches as a major
scale
à these different forms of the minor scale make impossible any
simple definition of distances for minor keys based on scale
membership
à each minor key is considered closely related to two different
major keys that are not themselves closely related to each other
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 a minor key is considered close to the major key that has the same
scale pitches (but different tonic) as the natural minor scale, and the
two are called the "relative" major and minor of each other
 a minor key is considered closely related to the major key with which
it shares a tonic. That is, A major and A minor are closely related by
virtue of their shared tonic tone (A)
‚ This relationship between major and minor keys is called "parallel"
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it might be possible to obtain a quantitative measure of the
distances between keys
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two keys are close to the extent that they impose a similar pattern
of relative stability on the tones
If two keys have similar hierarchies, then modulations between
them should be able to be effected relatively easily
ways to measure the degree of similarity between rating profiles
à for each tone, to take the absolute value of the difference between the
two ratings for the two keys in question. These absolute values can be
added or averaged across the 12 probe tones to give a summary
measure of the difference between the two profiles
à take the sum of the squared differences between the corresponding
ratings for the two keys
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The correlation between profiles was computed for each
possible pair of major and minor keys (i.e., all major-major
key combinations, all minor-minor key combinations, and
all major-minor key combinations).
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The results are as follows
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It can be seen that the farther a major key is from C
major on the circle of fifths, the lower is the
correlation between its rating profile and that of C
major
Findings:
à For major keys
 the pattern of correlations corresponds to distances around
the circle of fifths.
à For major and minor keys
 we see the influence of both relative and parallel
relationships between major and minor keys.
à For minor keys
 we again see the influence of the circle of fifths, as well as
the relative and parallel major-minor relationships that
produce associations between minor keys mediated
through major keys
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Taking prior analysis one step further
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use the correlations to produce a spatial representation of the distances between keys
This representation simultaneously summarizes the relationships between all major and minor keys in a form that
is easily accessible visually
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The rule that governs the placement of the points is that the order of distances between points in the spatial
configuration should correspond as closely as possible to the order of similarity values
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If two objects have a high similarity value, then they should correspond to points close to each other in the coordinate
space
if two objects have a low similarity value, then they should correspond to distant points
we would expect the points for C major and A minor to be fairly close, but those for C major and F # major to be distant
Nonmetric multidimensional scaling
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transform a set of similarity values into a spatial representation of points
the algorithm uses only the relative magnitudes of the similarity values, not their absolute magnitudes, arithmetic
differences, or any other arithmetic combinations
Choosing the number of dimensions for the spatial configuration can be problematic.
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The points can be located in a one-dimensional space, two-dimensional space, as on a flat surface or plane, the
points can be located in three dimensional space, or indeed a space of higher dimensions
wants a spatial representation that is visually accessible and that substantially reduces the amount of information
contained in the original similarity values
As the number of dimensions increases, the stress value will necessarily decrease, but at the expense of losing visual
accessibility and having to estimate a larger number of parameters
Current testing used 4 dimemsions
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two dimensions to account for the circle of fifths that is reflected in the correlations among major keys and also in the
correlations among minor keys.
Two additional dimensions are required to account for the parallel and relative major-minor relationships
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two theoretical schemes will be described for
comparison with the multidimensional scaling
results