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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. 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 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 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 The experiments discussed in the reading lead to two different kinds of findings 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 the hierarchical ordering influences various measures of perceptual or cognitive processing 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: Event hierarchies 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 Stability Tonality the relationships existing between tones or tonal spheres within the context of a particular style system tonal systems are generally hierarchical 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 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 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 à à à à à 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. The method used Two step process Step 1: 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 Experiment 1: Step 2: 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 Experiment 2: Participants 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 Three distinct patterns Pattern 1: Pattern 1: Pattern 2 Conclusion 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 Pattern 3 A subsequent probe tone experiment 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 Major objective 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 One final difference 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 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 considers whether the experimentally measured tonal hierarchies can be used to produce a measure of interkey distance Interkey distance: à keys are considered close if modulations between them are relatively frequent 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 • 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 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" it might be possible to obtain a quantitative measure of the distances between keys 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 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). The results are as follows 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 Taking prior analysis one step further 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 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 à à à à 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 à 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. 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 à à 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 two theoretical schemes will be described for comparison with the multidimensional scaling results