Greg Pfeiffer

In an attempt to expand on the concept of using interval sets to generate pitch material, I discovered a novel approach of applying binary operations to alter interval direction.  In this system, every possible combination of ascending and descending is applied to an interval sequence to generate a list of pitch sequences which all share the same underlying interval pattern.  

The first aspect of this system which must be mentioned is that it only produces lists of pitch names, not notated music.  Secondly, within the confines of this system there are no intervals larger than a tri-tone, for instance an ascending Perfect 5th is interpreted as a descending Perfect 4th (e.g. C - G). On a superficial level one might intuit a relationship between this and interval class vector analysis, as both of these systems do not recognize any intervals larger than a tri-tone.  But since this system is purely linear there is no function within the system that looks at non-adjacent pitches for their intervallic relationship the way that vector analysis does.  And conversely, vector analysis does not concern itself with pitch (interval) order or linearity, which is the primary element of this system.

In this system, most intervals - minor 2nds, Major 2nds, minor 3rds, Major 3rds and Perfect 4ths - come in two varieties, ascending or descending, with the exception of the tri-tone which is symmetrical and produces the same pitch ascending or descending.

In order to illustrate this system, it is first necessary to list every possible combination of ascending and descending.  Chart 1 (below) is a list showing every binary permutation possible for a three interval set.

Chart 1

000        up, up, up                  [1]
100        down, up, up             [2]
010        up, down, up             [3]    
001        up, up, down             [4]
110        down, down, up        [-4]
101        down, up, down        [-3]
011        up, down, down        [-2]
111        down, down, down   [-1]

The order chosen in the above example follows a pattern -  after going through all combinations containing either all zeros or two zeros, the binary list proceeds as a mirror inversion. Each unique binary code is given a bracketed identification number and the negative id numbers indicate which codes are their inversions.

A product of this particular pattern of binary code ordering is that inversional relationships are kept separated in such a way as to avoid making larger inversional sequences or super-sequences.  In order to elaborate, if one were to order the binary code as follows, [1] [2] [3] [4] [-1] [-2] [-3] [-4], then the entire second half would be a continuous inversion of the first half.  However, in spite of all of that, there really is no necessity to utilize this or any particular ordering. Other ordering schemes, such as Gray code ordering (shown below)  also happen to avoid creating inversional super-sequences.   On top of all of this, one may not want to avoid larger inversional super-sequences anyway, so no particular ordering is paramount.

Chart 2 - Gray code

000 [1]
001 [2]
011 [3]
010 [4]
110 [-2]
111 [-1]
101 [-4]
100 [-3]

Chart 3 (below) demonstrates how the initial pattern from chart 1 applies to an actual 3 interval set. It should be noted that each subsequent permutation uses the last pitch produced from the one previous.  This dovetailing approach allows for every permutation of the interval set to be used in one continuous series.