172A. Cognitive Psychology of Music
(Introduction) |
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172A. Cognitive Psychology of Music (Introduction)
Undergraduate, non-major course
Week 5
Summary of lectures
The Octave; Pitch
Chroma - Pitch Height.
C3In music, an example of data reduction in terms of categories is the interval of an octave.
Octave: Interval (: distance between two notes) with a frequency ratio 2/1. Notes separated by octave intervals are characterized (universally) by a high degree of sameness / smoothness. This 'sameness' is referred to as:
Pitch chroma: The distinctive quality of a specific tone, separating it from the rest of the tones within an octave.
Pitch chroma essentially describes perceptual 'differences'/'distances' of pitches within an octave and the perceptual sameness of pitches separated by one or more full octaves.
It is reflected in the fact that the different note names (i.e: C, D, E, F, G, A, B, C, D ...) repeat periodically for every 2/1 increase in frequency (every octave) with the addition of a subscript (i.e. C4) indicating how high or low this pitch is relative to some reference pitch; in other words indicating its
Pitch height: term describing the perceptual 'highness' or 'lowness' of a pitch; it is related to frequency.
Example 1: C3 and C4 are two notes with the same pitch chroma (C) but different pitch height (3 vs. 4); the frequency value of C4 is 2 times that of C3.
Example 2): The intervals A3 (220Hz) to A4 (440Hz) and A4 (440Hz) to A5 (880Hz) are both octaves with the tree notes (A3, A4, A5) having the same pitch chroma but different pitch heights. In terms of their pitch height, octaves are equidistant perceptually although they are not equidistant in terms of Hz.
Remember Fechner's psychophysical law stating that the perceptual magnitude of stimuli relates logarithmically to the physical magnitude of stimuli. (i.e. Hertz versus Pitch; Intensity in W/m2 versus Intensity in dB)Notes represented by different names have different pitch chromas and different pitch heights.
For example: For notes with the same subscript (in other words within the same octave) the pitch height increases as we move to a name at the right of the name scale (starting with C in our example), while the pitch chroma also changes.
i.e.
D3 E3 F3 G3 A3 B3 C4 D4 ...
Chromatic Scale; Intervals; Basic music notation.
With the Octave as the basic category, the relevance of Miller's rule to the way musical systems are constructed is reflected in the fact that, in practically all music cultures, all pitch systems used to construct melodies employ a maximum of 7±2 individual pitches. This is not to say that only 7±2 pitches will be available. The western musical system has 12 pitches available within an octave and some other cultures have musical systems with even more. No musical system however violates Miller's 7±2 rule in terms of the number of functional pitches used in melodic units.
Those functional pitches employed in melodies belong to pitch subsets called Scales:Scale: A system prescribing the frequency relationships between 7±2 pitches within an octave.
In the Western musical system the two most common and important scales are the major scale and the minor scale.Before we examine their differences we need to examine a very important aspect that both those scales (: systems of pitch relationships within an octave) share:
Both scales satisfy the earlier mentioned (end of week-4 notes) goal for:
maximum variety (maximum intervalic variety, defined operationally in terms of the maximum number of different intervals possible within a given scale).In Western music the Octave is divided in 12 equal intervals called
Semitones. Semitones are the modular interval units of the Western musical system. Semitone-intervals are also called 'minor seconds' or 'half steps'. Remember that, in terms of perception, equal interval does not mean equal distance in Hertz but equal ratios between Hertz values. So a note that is a semitone away from another note (semitone interval / minor second / half tone) differs in frequency not by a fixed amount in Hertz but by a fixed ratio: 21/12 (= twelfth root of 2 = 1.05946).
i.e. When we say that F4 is a semitone higher than E4 we mean that the frequency of F4 = frequency of E4 * 21/12
In the following table all 12 semitones within an octave are included, using the 7 names (C, D, E, F, G, A, B, C, D ....) and the symbols: # (= sharp = 1 semitone higher than the previous note) and b (= flat = 1 semitone lower than the next note).
| C3 | |
C#3 (Db3) |
|
D3 | |
D#3 (Eb3) |
|
E3 | |
F3 | |
F#3 (Gb3) |
|
G3 | |
G#3 (Ab3) |
|
A3 | |
A#3 (Bb3) |
|
B3 | |
C4 |
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) |
The set of those 12 notes makes up the
chromatic scale : a set of all 12 modular units/building blocks within an octave, in the Western musical system. A scale that contains all 12 semitones, outlining 12 successive minor second intervals.
The table below indicates all the possible intervals within the chromatic scale, starting (as an example) from C3.
# of Semitones note-pairs Interval name 0 C3 unison 1 C3 minor second 2 C3 major second 3 C3 minor third 4 C3 major third 5 C3 perfect fourth 6 C3 augmented fourth (tritone) 7 C3 perfect fifth 8 C3 minor sixth 9 C3 major sixth 10 C3 minor seventh 11 C3 major seventh 12 C3 octave
Each interval has a unique sonic character or 'signature sound'. Combinations of these signature sounds can create cognitively recognizable patterns.
Beating; Critical band; Consonance; Dissonance.
Some of the above intervals are generally considered consonant and some dissonant:
i.e. Consonant intervals (smooth, non-discordant)
a) Unison: 0 semitones (maximally consonant),
b) Octave: 12 semitones,
c) Perfect fifth: 7 semitones,
d) Perfect fourth: 5 semitones,
....i.e. Dissonant intervals (rough, discordant)
a) Minor second: 1 semitone (maximally dissonant),
b) Major second: 2 semitones,
c) Augmented fourth: 6 semitones,
d) Major seventh: 11 semitones,
....The concept of consonance and dissonance is of great importance, especially in Western music. One of the most basic principles in the development of melody has been contrasts in terms of a back and forth motion between consonance and dissonance. (i.e. The opening of the song "Maria" in Leonard Bernstein's "West Side Story". The initial dissonant interval of an augmented fourth is resolved into the consonant interval of the fifth.)
Click here for an audio example.
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Whether one combination [of tones] is rougher or smoother than another depends solely on the anatomical structure of the ear, and has nothing to do with psychological motives. But what degree of roughness a hearer is inclined to … as a means of musical expression depends on taste and habit; hence the boundary between consonances and dissonances has frequently changed … and will still further change… (Helmholtz, 1885.)
The concepts of consonance and dissonance are approached in our class based on the physical (sound wave properties) and physiological (ear properties) frames of reference. The approach applies to acoustic or sensory consonance/dissonance and is not addressing evaluative or general contextual musical issues.
In other words it will not address the fact that, what is considered musically consonant (smooth, right, acceptable, pleasing) or dissonant (rough, wrong, unacceptable, disturbing) changes:
a) with time (historical context),
b) with tradition (cultural context), or even
c) within a single tradition, style, or even piece of music (musical context).
As has been the case with all other aspects of musical communication, understanding music involves types of 'knowing' (: explicit/implicit rules, schemata, categories, etc.) that go beyond physics or physiology. The concept of consonance/dissonance is no exemption.
Superimposing (adding) two sine waves with similar frequency values
(sine wave 1: F1, sine wave 2: F2, F1<F2)
results in a wave with an amplitude that fluctuates at a rate: F2 - F1. This amplitude fluctuation is called beating or beats and the rate of this amplitude fluctuation is called beating rate or beats per second.
When the rate of amplitude fluctuation (F2 - F1) is small (1-10Hz) a single tone is heard with frequency: (F2+F1)/2 and loudness that fluctuates (tremolo) at a rate equal to the amplitude fluctuation rate (F2-F1).
If the beating rate increases but still remains smaller than the frequency width of a critical band (~ 1/3 octave), the resonance regions along the basilar membrane will be less than a 'critical band' apart. Their interaction is perceived as roughness.
As the interval between two tones decreases their respective disturbances on the basilar membrane (critical bands) overlap to an increasing extend.
(From Campbell & Greated, 1987: "The Musician's Guide to Acoustics". New York: Schirmer Books, p.58.)As it has already been noted ('week 3'; 'the ear'), the term critical band refers to the specific area on the basilar membrane that goes into vibration in resonance with an incoming simple tone. Its length is determined by the elastic properties of the basilar membrane and psychoacoustical studies indicate an average value of approx. 1.2mm., representing ~1/3 of an octave.
Since critical bandwidth is determined by perceptual experiments based on the presence or absence of beats/roughness we can define:
critical bandwidth: the frequency separation between two simultaneous sines necessary for beats/roughness to disappear and for the tones to sound clearly apart.
In other words critical band refers to a distance in millimeters on the basilar membrane while critical bandwidth refers to a frequency distance in Hz.
The degree of dissonance of intervals is therefore determined by the extend of roughness generated from the beating between all the different components of the complex tones involved.Amplitude fluctuation rates representing semitone intervals result in maximum roughness, a roughness that decreases as the rate (and therefore the separation between resonance regions on the basilar membrane) increases.
The relative consonance or dissonance (or better acoustic/sensory consonance or dissonance) of different intervals can then be found by comparing the number of semitone and tone separations between the components of the two notes (see figure 8 in the 'week 3 graphs' page.) So:
Consonance: Term referring to the perceptual 'smoothness' of an interval. The further apart on the basilar membrane the resonance regions for the components of the two notes, the less 'rough' the beating and the more consonant (smooth) the interval.
Dissonance: Term referring to the perceptual 'roughness' of an interval. It is the result of beating between interval-note components with resonance regions along the basilar membrane that are less than a critical band apart.
Global / Local musical features.
Two 'cognitive realities' are represented in what we call:
Global features in a piece of music: features that characterize/schematize a piece as a whole; features that evolve throughout a piece of music; i.e. major/minor tonalities, overall rhythmic features, overall mood etc.
and
Local features in a piece of music: germinal features; small-scale musical events that serve as building blocks / points of departure for the construction/emergence of global features.i.e. Beethoven's Fifth symphony uses the major and minor 3rd intervals as signature intervals, creating a motif (local feature) that becomes an important building block for the entire first movement.
A minor 3rd is also used as a signature interval in the song "By the waterfall" (from the film: Foot Light Parade, 1933).i.e. In the main theme from the film "Jurassic Park" (music by John Williams) we can identify the perfect 5th as a signature interval, as a local feature.
It is contrasted by a second theme that uses narrower (with less semitones) intervals.
The title theme from the film "2001, A Space Odyssey" ('Also Sprach Zarathustra' by Richard Strauss) uses the perfect 5th and its complement the perfect 4th (perfect 5th + perfect 4th = octave) followed by altering major and minor chords. It creates a local feature that essentially outlines the most important aspects of the Western tonal system. It is contrasted by a section with a different organization.
In the same film, Ligetti's Requiem accompanies the 'monolith' scene. This is one of the many possible examples that could illustrate that music can be created without necessarily following the rules of the Western tonal system. However, the idea of local and global features is in such cases still relevant. They may be outlined in terms of timbral, temporal and dynamic aspects, as well as in terms of the ways that the rules are being broken. (Art is a game of expectations).
Musical scales as psychological constructs.
Both major and minor scales contain 7 out of the 12 available pitches, distributed in such a way as to include five 2-semitone intervals (whole steps) and two 1-semitone intervals (half steps).
This distribution makes it possible for the major and minor scales (with only 7 pitches) to reproduce all the intervals present in the chromatic scale (with 12 pitches). In other words, this distribution satisfies the requirement for maximum variety.In the following table we see that the main difference between major and minor scales is in the way those intervals are distributed.
i.e. For the major scale, the first semitone interval (half step) is between the 3rd and 4th notes while for the minor scale the first semitone interval (half step) is between the 2nd and 3rd notes. (The starting note of the scale is called the Tonic and its name characterizes the scale: i.e. C major if the starting note of a major scale is C; A minor if the starting note of a minor scale is A etc.)
C major Scale |
C minor scale (natural minor) |
A major Scale |
A minor scale (natural minor) |
Balzano has shown mathematically that if we want to get maximum variety (maximum interval-combination possibilities) from only 7 pitches out of an octave, we must follow the exact configuration present in the major and minor scales. That is, we have to divide the octave in 12 log units (: 21/12 : twelfth root of 2) and select 7 in such a way as to include five 2-semitone intervals (whole steps/tones) and two 1-semitone intervals (half steps/tones). The graph below illustrates that all 12 intervals of a 12-tone scale are possible within 7-tone subsets configured similarly to the major or minor scales.
Notation of the major scale from C4 to C5 showing the intervals in semitones. Notice that all 12 intervals are possible, indicating that this specific scale (subset of the twelve tones in a semitone-divided octave) satisfies Miller's rule while, at the same time, it allows for maximal intervalic variety (From Kendall & Carterette, 1996: 93; Reader p. 108.)
The entire Western tuning (equal temperament: octave divided in 12 equal log units) and Western scale system (major / minor) is therefore based on this optimal combination of cognitive limits (Miller's rule), data reduction/pitch circularity (octave), and maximum variety (major/minor scales). The Octave and Miller's rule represent two musical universals.
The principle of coherence (: no distance/interval between any two successive scale notes/degrees should be larger than the addition of any two successive intervals of a scale) is closely related to maximum intervalic variety and the possibility of the development of full functional harmony within a scale.
Multidimensionality of Pitch (pitch spiral.)
Another important consequence of the above discussion is that pitch is multidimensional with one dimension being pitch height (frequency) and another being pitch chroma (separating pitches within an octave and linking pitches an octave apart).
A variable is uni-dimensional if all its values can fit on a single straight line. If for example A, B, & C represent 3 values of a variable as points in space then:
a) if AB + BC = AC then the variable is uni-dimensional and
b) if AB + BC ¹ AC then the variable is multidimensional.
Pitch is therefore multidimensional with the dimensions of pitch chroma (pitch chroma is represented by a circle and is therefore 2-dimensional) representing a circularity in pitch perception . The western chromatic scale breaks down the octave into 12 different pitch chromas. (right)
The result of this circularity is that, although the octave interval represents the largest physical distance (within a single octave) in terms of pitch height, it represents the smallest perceptual distance in terms of pitch chroma.
Pitch perception therefore wraps on the octave, with scales defining sets of different pitch chromas that repeat at different pitch heights for each new octave. This results in what may be called a pitch spiral (right).
The perceptual circularity of pitch is demonstrated by the so-called Shepard-tone scales (after Roger Shepard). Shepard scales present the paradox of a continuously ascending (or descending) pitch.
They are the auditory analog of the continuously ascending/descending staircases, first "discovered" by Penrose and later seen in many Escher paintings (left). The phenomenon of pitch circularity indicates that the octave represents perceptually a 'cognitive geometry' which does not exist in any form in the physical world. As far as 'music' is concerned, however, it is this cognitive geometry that constitutes 'reality' and not simple and 'real' frequency values. This 'cognitive reality' suggests that there are schemata (: implicit knowledge structures) that track both, pitch height and pitch chroma.
" The idea of computers talking to us, recognizing our speech, has been around for over 20 years. People at that time were overly optimistic about how quickly it would take place because they thought we could solve the problem simply by paying attention to the wave forms that were coming in. ..........What they didn't know then that we know now is that if you just look at things at that level, the sound level, speech is very, very ambiguous. It is only because of common sense and context that people are able to figure out what's being said..."
Quote, from Bill Gates, Microsoft, in a Reuters news
release.
Ethnomusicology Department - UCLA©
