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.