The set 7131 describes a trapezoid on a clock diagram, and its inversional symmetry is geometrically evident. Perhaps the most useful feature of these set names is their ability to show subsets and supersets. Sets whose To hear your set, click on the "Play" button that appears below the keyboard of these sets are in normal form, but the first is "packed more tightly (See Example 12.). To invert the set in the applet, click the "Invert" button on the For students interested in the analysis of the music of the Second Viennese School, or who are considering music theory as a profession, the study of atonal set theory, at least at an introductory level, would seem to be a prudent part of the undergraduate curriculum. mirror image on the clock. The interval created by elements 2 fifths apart is the major second, and the rectangle can therefore accommodate 5 of these. example above, we could have clicked the "Rotate" 4 times to reveal this (See Example 3.). would spell it (9,10,2). Such projects would help the student develop fluency in set manipulation and to think of pitch collections in terms of their properties and limitations. In short, the number of times 6 occurs in a set name will be double the correct entry in the interval vector. Playing them as a simultaneity allows you The pattern continues smoothly through the keys until finally, at the most remote point, the tritone reappears as the only common interval [000001]. We have seen how clocks and set names create a medium through which pitch and set relations can be explored. Their names will be related, however, as will be seen later. Octatonic collection refers to the set 21212121. Finally, like any theoretical system, set theory simplifies the object in order to bring otherwise hidden properties into relief. The shape rotated 1 hour clockwise represents the set transposed up one semitone. This, too, is evident from the set name: 3333 may be generated by repeating the digit 3, indicating that there are only 3 transpositions. I would agree with this argument, were it not that set theory, even at the elementary level, ought to be a general tool for learning about pitch structures, and not just applications limited to a particular repertory. Although it precipitated several reviews that were critical of the system even at the outset, the system (or a close relative) is now standard. He introduced this system of numbering the prime forms in his 1977 book bcasaima says: April 30, 2020 at 10:47 am . Example 3: The set 5133 and its inversion, 5331. This is true because there is a 1 in the first column. By making the investigation of set structure practical, set theory encourages students to think of their musical language in a logical, demystified way. Question 15 : Let A = {a, b, c} find the power set of A. Notice that the clock-face graph of the Realizing that (0,3,6,7) is better represented by (0,1,4,7) is far more complicated. With a team of extremely dedicated and quality lecturers, set theory practice problems will not only be a place to share knowledge but also to help students get inspired to explore and discover many creative ideas from themselves. (In practice, these may be major or minor keys.). To transpose the set itself, use the "<" and ">" buttons below the you to set the Y-axis to be the same as the X-axis (Normal) or the The underground naming process is not wholly alien to students who have been taught to identify tonal chords. Example 5: The diminished seventh chord 3333. The set 5133 is therefore a quadrilateral, and rotating the figure is analogous to transposing the set. enter and displays it in the "Interval Class Vector:" field. To hear your set, click on the "Play" button that appears below the keyboard to hear all of the notes at once, like a chord. Example 4: The inversion symmetric set 7131. to as 'serialism.'. Enter key. One must treat interval class 6 with special care, just as in calculating an interval vector directly from a standard set of pitch classes. starting set. The dominant seventh chord, or "major-minor seventh" chord (4332) may be shown to appear once, while the inversion symmetric minor seventh chord (4323) appears three times. Students could be asked to prove the general case that if Set A is a subset of an inversion symmetric Set B, then the inversion of Set A is also a subset. collectively referred to as "The Viennese School" of composers. So to get T(3)I of (1,2,7) you would first The diminished triad appears 8 times, however, due to its inversional symmetry. 14One may observe this symmetry in the diatonic polygon inscribed either on a clock where the hours are ordered chromatically or one ordered by circle of fifths. There is one occurence of interval Example 14: Chart illustrating sets and interval classes retained as the melodic minor collection (2222121) is transposed by interval class 5 or by interval class 1. The concept of rotation is just as intuitive, but does not as easily tempt the student to think in terms of pitches and registers. Complement" button. This makes a total of 8. An interval that is "lost" when one modulates from the key of C major to D major will not return if one modulates to the more distant key of A major. However, there is another important lesson that set theory can make clear to students. On the contrary, while the properties evident in set structure certainly play a crucial role in music using these sets, they by no means completely predict or determine the musical results. (0,1,3,6) as the prime form. systems that composers like Schoenberg and his followers used to These groups of pitch classes are called pitch-class sets. Keep in mind that sets So 512121 will contain the tetrachord 8121 twice, since the larger set may accommodate 8121 in these two ways: beginning with the five ((5+1+2) 1 2 1) and beginning with the second two ((2+1+5) 1 2 1). It seems much easier to remember that, say, -B is in a sense equal to A flat-B, and that both bear a similarity (harmonic) to D-F than to conjure up what "(8, 11)" sounds like. Click on the "Define Set..." button, select a set from lists of A look ahead to the A-flat collection reveals that while we return to the smaller, common tetrachordal subset (4413), we reach the maximum number of common major thirds (3). The general case is shown in Example 11.13 The example uses interval vectors to tally the intervals shared between the original key (here, C major) and the other major keys. Your sets will be graphed on the clockface and on the keyboard automatically. Any set name that can be completely generated by repeating a segment of itself will bear this property: the sum of the digits in the smallest segment will give the number of transpositions. Example 9 shows rectangles representing the keys of C major, F major and the 6 pitch classes they have in common. This returns a new set consisting of any note that was Many academics have seen the value of annular (or "clock") diagrams in set theory, as mod 12 arithmetic represents the notions of octave and enharmonic equivalence. This gives students an additional, geometrical way of understanding a chord and its name. That is, by adding together the 5 and 1 in 5133 produces 633, which is the diminished triad. 49-92. enter and displays it in the "Prime Form:" field.