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Exploring Outcomes:

Thinking of Interval Direction As A Binary Operation

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.

When making a chart in this manner, utilizing every possible binary combination, the series should always end up on the pitch from which it began.  And regardless of the order in which the binary permutations are presented, as long as each permutation is used and is only used once, it will always end on the pitch from which it began.  The reason for this is simple - for every instance of an ascending interval there is a corresponding descending interval.

Chart 4 (below), for contrast, simply shows each individual directional permutation outcome starting from the same initial pitch.

In the first piece I composed using this system, Binary Systems, the order of binary operations is sequential. Chart 5 (below) shows the 5 interval set used to compose the first section of the piece. Excerpt 1 shows the first few measures of Binary Systems.

Excerpt 1 (from Binary Systems)

Additionally, in two of the early pieces I composed using this method, Binary Systems and The Enterprise, binary code was also used to generate the rhythmic figurations. Illustration 1 (below) shows specifically how this was realized in Binary Systems:

Illustration 1

It should be said that the use of binary rhythm is of secondary importance to the overall discussion in this document but is worth mentioning for a particular reason.  This use of binary rhythm in Binary Systems helped to resolve the issue concerning the distribution of pitches created by the series between the right and left hands.  In any given measure of the piece, the right hand's rhythmic figuration is the binary inversion of the left hand.  For example, if the right hand is playing 10011, the left hand is playing 01100.  Due to this configuration, the right and left hands never both play a new note simultaneously, thus allowing for the linear temporal distribution of the pitches created by the series in the piece.



The most direct way to examine this system in detail is to look at an entire piece created this way.  The Fourth Miniature for Clarinet  lends itself most readily to this in that the piece is short and is entirely monophonic, so any of the issues associated with harmony or pitch distribution do not come into consideration.

Chart 6 (below) shows the interval set.  Once again it is a dovetailing structure, with the last note from the previous permutation determining the first pitch of the subsequent permutation.

The first thing one may notice, and this is where the bracketed numbers are helpful, is the asymmetry of the distribution of permutations in this case.   So much like in the first half of the pitch sequence for Binary Systems, the first half of this system ploughs through all of the unique permutations in an orderly fashion, but the second half of this sequence deviates by jumping from inversion 6, labeled [-6], to inversion 2, labeled [-2].   An explanation for this behavior can be found in an interesting feature of this deviation:  the entire pitch sequence starting at  [-2] through the last pitch of [-5] is a continuous inversion of the pitch sequence beginning at the first note of [2] through the last note of [5], which is to say that a super-sequence occurs here.

Another aspect of this system, certainly not unknown to the composer at the time, is the distribution of pitches. In Chart 7 (below) the frequency of pitch occurrences is tabulated.

The primary piece of information that can be ascertained from this chart is that this system, at least in this case, does not inherently produce an even distribution of pitches. In this case, what this tabulation does reveal is a hierarchical distribution - with "G" being the main tone center and "C, F#, & A#" as secondary centers.  What is perhaps more unique is the singular appearance of "D".

As far as statistical properties, if all 12 pitches occurred with the same frequency, each pitch would occur 8.3% of the time or 6.75 times per 81 notes. In this particular distribution, the standard deviation is 2.17 times or 2.78%.  So in taking account of all of this, 5 pitches (A, F, C#, D#, E) occur within one standard deviation, 5 other pitches (G#, B, C, F#, A#)  lie between one and two standard deviations, and 2 pitches (G & D) lie between 2 or 3 standard deviations.  Although it isn't a "normal" distribution, it certainly reveals a defined bell-like structure, with G & C taking on primary dominance or centrality and A# & F# occupying a secondary position.

But if the duration of these pitches as found in the Fourth Miniature itself is accounted for, the weighting is something different.  Using EDUs  (Enigma Duration Units used by Finale®) to count the durations of each pitch occurrence the chart below reveals their temporal distribution [note: a quarter note = 1024 EDUs; an eighth note = 512 EDUs].

Merely looking at either duration or frequency of occurrences presents a incomplete or misleading picture of each pitch's role in the piece.  But the final column in the preceding chart shows how much duration was manipulated in order to alter the balance of pitches given purely by the system.


Another way to examine the implications of using this particular interval set can be seen in the first 17 measures of The Nine Members of the Asian Dawn (2014) .  The pitches used to compose this section are note-for-note identical to the pitches in The Fourth Miniature.   Given the completely different rhythmic treatment of the pitch material, on top of the obvious difference in instrumentation, the two passages sound worlds apart.

During measures 17 through 61 a completely different interval set is used which follows a completely different pattern of interval directionality.  In this set (chart 9 below) there is no avoidance of repetition regarding particular interval sequence permutations, but the overall goal of using the same underlying interval pattern to produce related material is still achieved.  In some ways, due to the use of repeated permutations, it could be argued that this material is more related than in the more pure interval sets shown previously.  Regardless, this set demonstrates yet another approach that can be taken using this system to generate intervallically related material.

Starting in the middle of measure 61 (G5) through the end of the piece the pitches are derived once again from the initial Fourth Miniature interval set but in retrograde.  


For the sake of comparison, and to quickly demonstrate how each interval sequence creates its own frequency profile, charts 10 an 11 (below) show the pitch frequencies found in the pitch structures examined previously.

If all pitches occurred with the same frequency, each pitch would occur 13.4 times.  In this case the standard deviation is 3.4 with 6 pitches having occurrences within 1 standard deviation of the mean and the other 6 pitches occur within 2 standard deviations.  Although this is also not a normal distribution, a hierarchy is still apparent.  


Chart 12 (below) shows only a portion of the pitch structure used to compose The Second Coming.  Underlined are only those pitches present in Excerpt 2. This work deserves noting if only because the order of permutations was not systematic or ordered.  In this case the order was intentionally chosen while the composition was being written.

Excerpt 2  (from The Second Coming)


Multiple instruments, chords and harmonies present specific issues when actually composing using the clearly linear outcomes produced by this method.  But these issues certainly aren't new and were dealt with in much the same manner as 12-tone composers approached this issue in the past.   Examples would include sharing the sequence of pitches among voices or having one voice do a portion of the sequence while another simultaneously occurring line uses the pitches from another portion of the sequence.  

In order to examine the harmonic implications of this system, returning to the pitches generated by Chart 3, the following is a list of all of the tetrachords produced by every 4-note sequence in that pitch series:

Each chord labeled with an asterisk (*) is a tetrachord created purely by the actual interval sequence.  All other tetrachords are the result of 4 note sequences arising from the order of permutations.  This is important to note for if the order of permutations is done differently, the 'other' tetrachords produced can and in most cases will be different.

If it can be assumed that inverted tetrachords are equivalent, or at least share a special relation to their prime relatives, then the tetrachords created purely by the interval sequence each must occur at least twice.   This is due to the fact that each permutation (e.g. 1001) has an inverted version (e.g. 0110) somewhere in the sequence and that these inversions end up creating the prime relative of each other.  In this case the tetrachords created purely by the interval sequence each occur only twice (4-14, 4-10, 4-4, & 4-6) and of the five other tetrachords found, 4-11 and 4-5 occur four times each, 4-16 occurs three times, 4-29 occurs twice, and 4-15 occurs only once.

But in order to determine if the distribution and frequency of tetrachords is unique for each interval sequence that is subjected to the same binary order, let's subject a different three interval sequence to the same treatment and analysis.

In this 8 different tetrachords occur (as opposed to the nine found in the previous exercise), with the four pure tetrachords occurring with greater frequency: 4-18 occurs four times, 4-13 & 4-14 occur three times each and 4-7 occurs twice.  Of the other tetrachords, 4-10 & 4-17 occur three times and 4-20 & 4-4 occur twice.

In so far as conclusions that can be drawn from this limited exercise it appears that even when subjected to the same order of directional permutation, different interval sets produce different sets of tetrachords (which should have been obvious even without this) and, of much greater significance, they produce different numbers of tetrachords at different rates of recurrence.  Interestingly enough though, it appears that each interval set possesses some internal logic, and in at least in these two examples, each has a tendency to create a hierarchy of chords, a feature which could certainly be compositionally utilized and exploited.

Further explorations of this method would certainly be aided by the use of software to generate charts such as these.  For obvious reasons this has yet to be developed.  But as this relatively new technique becomes more widely known and utilized, one can only assume that such software will inevitably be created.

NEWS 2023:  Software finally created !  Please visit:

The following is a list of the pieces composed so far utilizing this method:

The Phantom Zone (a revision of Broken with a new alternate 2nd movement) 2022

The Nine Members of the Asian Dawn (for xylophone) 2014
Broken (III) (for English horn, bassoon & cello) 2014

Spore (for Clarinet and Percussion) [non-quarter-tone edition] 2013
Not Yet Quite Dark (for cello) 2013
Spore (for clarinet and percussion) 2013

Broken (I) (for oboe, bassoon & cello) 2011
Premonitions of Death (for flute, viola and double bass) 2011
The Second Coming (for speaker, oboe and harpsichord) 2011
The Enterprise (for piano) 2011
Fourth Miniature for Clarinet  2009
Third Miniature for Clarinet   2009
Broken (II) (for soprano, bassoon, & cello)  2006
Binary Systems (original for piano)  2004


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©2023 Gregory Carl Pfeiffer.  All Rights Reserved.

Special Thanks to composer Alan Tormey for helping edit this !

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