Such is the case in the Prelude to "The Afternoon of a Faun," where, in measures 8-10, for example, a 624 may be heard as part of a dominant seventh chord based on B-flat or as part of a whole tone pentachord. For instance, consider the octatonic set 21212121, the polygon for which is shown in Example 8. Any number over 11 will For the collection defining the key of C major, this axis passes through pitch classes D and .14 Because C and E lie at analogous points on either side of the axis, some kind of equivalence between these two pitch classes (in general, the tonic and mediant), would seem to be predicted. 10The 1 in the last column indicates that 2 pitch classes are retained if the set is transposed a tritone (e.g. Using set names and diagrams in this way, many properties of the diatonic set that play crucial roles in tonal music become evident. T(n)I just means that first you invert your original set, and then Example 2: The set 5133 at two transposition levels. (0,1,3,6) as the prime form. Over the years, scholars have made improvements in the system. Hence the four digit set name 5133 represents a tetrachord. This, of course, is the kind of lesson that belongs in any academic curriculum. By imagining 3315 to be a polygon inscribed in a circle, beginning on the "5" will produce 5331. Freida made the following statements about the sets above. We see a startling difference in the properties of these sets on transposition. n, T(n)I returns the exact same pitches as the original set. This is true because there is a 1 in the first column. Numbers" as your criteria for obtaining a preset. One must treat interval class 6 with special care, just as in calculating an interval vector directly from a standard set of pitch classes. This is because the major triad (543) and the minor triad (534) each appear three times each, but the diminished triad (633) appears only once. Note that since consecutive elements create perfect fifths (or fourths), the rectangle (which contains 7 elements) can encompass 6 of these intervals. Since (4,5,7,10) is packed more tightly to the left than clock-face graph. The work needed to produce or even to verify that statement consists largely of consulting a table and a bit of trial and error—neither of which promotes thinking about music. All rights reserved. A next step in this study would be to examine how various transpositions of, say, the harmonic minor collection effect its relationship to the major scale, and so on. example above, we could have clicked the "Rotate" 4 times to reveal this matrix with 2's along the diagonal: Hint: If a set is combinatorial, you should be able to click the "Rotate" 5 times in the body of the matrix. it was flipped horizontally. Elements in a set do not "repeat". For example, the major second C-D exists in the keys of G, C, F, B-flat and E-flat major, which appear consecutively on the circle of fifths. on the music faculty at Yale University. between 10 and 2 (4). Playing them as a simultaneity allows you Example 13: Chart illustrating sets and interval classes retained as the harmonic minor collection (3121221) is transposed by interval class 5 or by interval class 1. google_ad_width = 728; Although it precipitated several reviews that were critical of the system even at the outset, the system (or a close relative) is now standard. Octatonic collection refers to the set 21212121. At the same time, one discovers that compositional invention (whether it be in a single work, or in the development of a style) finds fresh ways to use and transform these properties. Here is a brief list of just a few popular forte numbers: Allen Forte is starting set. As one expects, the melodic minor collection behaves in yet different ways, as a study of Example 14 would illustrate. For any two sets X and Y, either there is a one-to-one function from X into Y or a one-to-one function from Y into X. Notice that the clock-face graph of the Because each entry in the interval vector is unique, the diatonic collection enjoys the greatest variety in number of pitches retained upon transposition: from 2 to 6, as indicated by the interval vector.10 Hence, each modulation will have its own characteristic sound: some keys are closely related, and some are remote. Playing the set as a 49-92. Prime form representations are also referred to as "Set classes." As its set name illustrates, the diminished seventh chord (3333) inverts into itself. example, the pitch class sets (1,2,7), (8,2,3), and (0,11,6) all belong between 2 and 9 (7) is larger than the intervals between 9 and 10 (1) or right-hand side of the screen. In short, the number of times 6 occurs in a set name will be double the correct entry in the interval vector. 13I wish to thank Brian McCartin for this diagram. Within the octatonic set, the major triad appears 4 times, as does its intervallic inversion, the minor triad. Because the set A = {a, e, i, o, u} contains 5 elements. By solving a problem—such as finding the largest common subset between the diatonic and octatonic collections—students discover abstract properties as well as develop manipulative skills useful to analysis.9 For example, one could note that the dominant seventh chord (4332) and the whole tone chord (222222) share a trichordal subset 624. It also means that there are only 6 enharmonically distinct tritones, since there are only 12 major keys, and each major key shares its one tritone with one other key. This suggests that such a sonority could play a crucial role in music that explores the relationship between the diatonic and whole tone systems.