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9 Other PC Set Similarity Relations

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.

Note that you can explore many of these similarity relations at http://www.ComposerTools.com .

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

• R_p  =>  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 R_p.

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 4^th or 5^th above the last note added
• Or this can be done with alternating intervals (i.e. first add a 5^th, then a tritone, etc)

 Copyright © 2004 by Paul Nelson, all rights reserved.