Carol Krumhansl. (1990). Cognitive Foundations of Musical Pitch. Oxford: Oxford University Press,. Chapter 7; pp. 165-187.

Quantifying harmonic hierarchies and key distances

• This book addresses the central problem of music cognition: how listeners' responses move beyond mere registration of auditory events to include the organization, interpretation, and remembrance of these events in terms of their function in a musical context of pitch and rhythm. Equally important, the work offers an analysis of the relationship between the psychological organization of music and its internal structure. Combining over a decade of original research on music cognition with an overview of the available literature, the work will be of interest to cognitive and physiological psychologists, psychobiologists, musicians, music researchers, and music educators. The author provides the necessary background in experimental methodology and music theory so that no specialized knowledge is required for following her major arguments.

• Chapter 7 presents two experiments directed at measuring the hierarchy of structural stability that a tonal center induces on the set of chords of different types built on all chromatic scale steps. The data is then analyzed to uncover characteristics that may contribute to percieved hierarchy. These include the chord type, tonal consonance, whether or not the chord is diatonic in the context key, and the frequency with which the chords are sounded tonal-harmonic music.

• Consolidation of a system of harmony. Sounding of tones in certain harmonic relations becomes (theoretically and compositionally) an organizational principle that was distinguishable from melodic considerations. XVIII and XIX century – common practice period. Rameau, Schoenberg, Piston codified it precisely since harmonic practice became very conventionalized.
• Roles of diatonic chords: I, IV, V; II, VI; VII, III
• Intervallic structure of major, minor, diminished, augmented chords
• Establishing the key – 3 main (central) chords, cadences
• Modulation as a main priciple governing form in western tonal music
• Hierarchy of harmonic stability – distribution of chords in tonal-harmonic compositions
• 2 EXPERIMENTS measured the percieved relative structural significance of chords in tonal contexts. C major & minor, F# major and minor scales/chord progressions (sine waves) were used (fig. 7.1). All major, minor and diminished chords in root position were played afterwards. Subjects had to rate how well the chord "fits" with the preceding key defining concept. 1 – fits poorly, 7 – fits well. Results in tables 7.3. and 7.4.
• Could tonal consonance per se account for the effect of chord type found in experiments? Using tabulated values from Hutchinson & Knopoff showed that those values do not conform well with Krumhansl’s results.
• Did chords that are in the basic set of harmonies tend to be given higher ratings than chords outside the set? Table of chords I-VII is corelated with table 7.3, and the correlation for minor-key contexts is greater (significant) than that for major key contexts (narrowly missed beign significant). Normalizing values and then corelating them leads us to the bottom of table 7.6. The results are now significant, and we can conclude that chords in the diatonic set receive higher ratings than other chords of the same type.

• The ratings of 36 chords can be predicted by the positions of the component tones in the tonal hierarchy of the context key (using probe tone values from chapter 2). Same thing happened using frequency distibution with which the chords occur in real music instead of probe tones.

• Harmonic hierarchy of diatonic triads. I, IV, V strongest ratings. II, VI following them, III and VII last (III & VI higher in minor key – why?). Finer discriminations in major keys showes that the listeners may have been less certain about the harmonic functions of chords in minor keys.
• How does this correspond with the frequency of use of diatonic chords in real music? Values are taken from Budge’s tables (1943.):

2. V

3. IV

4. II

5. VI

6. VII

7. III

This agrees with experimental results, except results for IV chord! Explanation: IV chord in the experiment sounds like a logical continuation of the sequence, and also forms strong candence with the last V-I progression.

• Deriving key distances from harmonic hierarchies. 2 steps: (1) generating the harmonic hierarchy for each of 24 major and minor keys, and (2) intercorrelating the normalized harmonic hierarchies for all major or minor keys (table 7.10). We also have the results plotted in Figure 7.2.
• Multidimensional scaling analysis. Similar harmonic hierarchies – proximal points, dissimilar harmonic hierarchies – distant points. First one – circle of 5ths. Second one – emphasize relative/parralel relationship. Those are 2-dimensional configurations (fig. 7.3.). The 4-dimensional configuration is represented as Figure 7.4.

• Harmony & Voice-leading Course Page