Chapter 9 – Other PC Set Similarity Relations

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This section covers other ways in which two PC Sets can be related. Again, this can be a useful compositional technique. For example, you could choose a PC Set and compose a work which is made up of just the original PC Set plus other, closely related sets. Such a composition should have a fairly consistent harmonic color throughout. Similarly, if you are looking for dramatic color contrasts, you will likely want to avoid similarly related PC Sets.

9.1          Special Purpose Relations:  Rp, R0, R1, R2

  • Rp  =>  When two PC Sets are the same except for one different pitch class, i.e. one note different
    • Very useful for composers, this is one way to "morph" PC sets. For example, you can go from PC Set 1 to PC Set 2 by changing a single note, as long as the two sets are related by Rp.
    • But not too useful for analysis, since this relates many PC sets to many many other PC sets
  • R0  =>  When two PC Sets have the same number of pitch classes, but no interval vector entries in common, for example:
    • 4-2:(0,1,2,4) has interval vector <221100>
    • 4-13:(0,1,3,6) has interval vector <112011>
    • There is no interval which has the same count in both interval vectors.
    • Not a very useful measure, since it has to do with the relative strengths of the intervals, rather than the presence or total absence of intervals.
  • R1  =>  When two PC Sets have the same number of pitch classes, and their interval vectors are as similar as they can be without being equal
    • This will be the case when the 4 of the 6 entries in the interval vector are the same, and the remaining two entries are simply exchanged, for example:
    • 4-2: (0,1,2,4) has interval vector <221100>
    • 4-3: (0,1,3,4) has interval vector <212100>
    • Note the highlighted entries in the interval vector are the only ones which are different, and the two entries are merely exchanged from one to the other.
  • R2  =>  Just like R1, except that the two different entries are not merely an exchange of numbers. For example:
    • 5-10: (0,1,3,4,6) has interval vector <223111>
    • 5-Z12: (0,1,3,5,6) has interval vector <222121>
  • Note that R1 and R2 are also Rp.

9.2          Other techniques for generating related PC Sets

  • Rotational arrays:  Used by Oliver Knussen and Igor Stravinsky
  • Intervallic projection to relate subsets and supersets:
    • Add notes to a PC Set by projecting up from the top note by a certain interval
    • For example:  Quartal / Quintal harmony is created by projecting by adding a note to a PC set which is a perfect 4th or 5th above the last note added
    • Or this can be done with alternating intervals (i.e. first add a 5th, then a tritone, etc)

Copyright © 2004-2017 by Paul Nelson, all rights reserved.

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