winders dissertation thesis
DESCRIPTION
Scholarly dissertation on musicology and serialismTRANSCRIPT
Cycles of Cycles: Ordering Principles Suggested
by George Perle’s Twelve-Tone Tonality
by
Christopher Clay Winders
Submitted in Partial Fulfillment
of the
Requirements for the Degree
Doctor of Philosophy
Supervised by
Professor Dave Headlam
Department of Music TheoryEastman School of Music
University of RochesterRochester, New York
2008
ii
from the fall of man
iii
Curriculum Vitae
The author was born in Raleigh, North Carolina on July 23, 1976. He attended the
North Carolina School of the Arts from 1997 to 2003, and graduated with a Bachelor of
Music degree in 2001, and with a Master of Music degree in 2003. He came to the Eastman
School of Music of the University of Rochester in the Fall of 2003 and began graduate
studies in Composition. He received the McCurdy Prize in 2004. He pursued his research
in Composition under the direction of Professor Ricardo Zohn-Muldoon and his research
in Music Theory under the direction of Professor Dave Headlam.
iv
Acknowledgements
The author wishes to thank his primary advisor, Professor Dave Headlam, for his
assistance in the development of this thesis. In particular, the importance of his extensive
knowledge of George Perle’s Twelve-Tone Tonality cannot be overstated. Many thanks
also go to the reading committee, Professors Jeannie Guerrero, Ricardo Zohn-Muldoon
and Carlos Sanchez-Gutierrez for their time and patience in the editing and revision of this
thesis.
v
Abstract
This thesis investigates correspondences between transformational networks
developed by Henry Klumpenhouwer and David Lewin and the compositional system,
Twelve-Tone Tonality, of George Perle. These correspondences are then utilized to
generate an ordering system which utilizes certain aspects of Perle’s cyclic sets. The first
portion of the paper traces the origin of cyclic sets from the usage of interval cycles in the
late 19th and early 20th Centuries, followed by an introduction to Perle’s cyclic sets and
how they intersect with Klumpenhouwer networks. After this introduction the relationship
between different cyclic set array alignments and the representative K-net transformations
is explored more fully. Ordering principles are then discussed as well as a demonstration
of how one can generate ordered chord progressions drawn from Perle sets and the relevant
K-net analyses. This section is followed by a similar demonstration using triadic arrays.
Finally two compositional designs illustrate possible foreground projections of the ordered
chord progressions.
vi
Table of ContentsIntroduction 1
Chapter 1 From Interval Cycle Through The Cyclic Set 2 To Klumpehnouwer Networks
1.1 The Interval Cycle 6
1.2 Klumpenhouwer Networks And Perle’s Cyclic Sets 8
Chapter 2 Toward New Ordering Principles 16
Chapter 3 Triadic Arrays 29
Chapter 4 From Compositional Spaces To Compositional Designs 38
Conclusion 41
Compositional Designs 42
Bibliography 44
vii
List of Tables
Table 1 .1 Cyclic Set Array in Sum and Difference Alignments 10
Table 2.1 Sum 5,0; 6,1 Array 18
Table 2.2 Axis-dyad Chord Progression (Single Axis-dyad Sum 3) 18
Table 2.3 Axis-dyad Chord Progressions (Sum 5,0; 6,1) 20
Table 2.4 Sum 0,1; 5,6 Array (With Resultant Sum Cycle) 25
Table 2.5 Axis-dyad Chord Progression (Single Axis-dyad Sum 3) 26
Table 2.6 Axis-dyad Chord Progressions (Sum 0, 1; 5, 6) 27
Table 3.1 Sum 5, 0; 6, 1 Array (With Resultant Sum Cycle) 29
Table 3.2 Split of Cyclic Set into Dyadic Series 30
Table 3.3 Triadic Divisions of Resultant Sum Cycle (e, 1) and a Possible 31 Triadic Array
Table 3.4 Additional Possible Triadic Array (e, 1 Resultant Cycle) 32
Table 3.5 Additional Alignments of Triadic Array (e, 1 Resultant Cycle) 32
Table 3.6 Initial Axis-dyad Chord Progression from the Triadic Array 35
Table 3.7 Axis-dyad Chord Progressions from a Triadic Array 36
Table 4.1 Design 1 - Generating Array 38
Table 4.2 Design 1 - First Two Axis-dyad Chord Progressions 39
Table 4.3 Design 2 - Generating Array 41
Table 4.4 Design 2 - First Two Axis-dyad Chord Progressions 41
viii
List of Figures
Figure 1.1 Trichords from Berg’s Lyric Suite Row in K-nets 9
Figure 1.2 Trichords Exhibiting Negative Isography 9
Figure 1.3 Axis-dyad Chord and Its Constituent Members 11
Figure 1.4 Hyper Relation Between Trichordal K-nets 12
Figure 1.5 Hexachordal K-net from a Difference Alignment 12
Figure 1.6 Hexachordal K-net from a Sum Alignment 13
Figure 1.7 Strongly Isographic K-nets Related by a Parallel Shift 14 of the Array
Figure 1.8 Strongly Isographic K-nets Related by a Symmetrical Shift 14 of the Array
Figure 1.9 <W>-related K-nets Drawn from an Asymmetrical Shift of 16 the Array
Figure 2.1.1 Generic Cyclic Set Array Algorithm 17
Figure 2.1.2 Cyclic Set Array Algorithm Example 18 Figure 2.2 Strongly Isographic Hexachordal K-nets 21 (Sum 3 Axis-dyad Chords)
Figure 2.3 Strongly Isographic Hexachordal K-nets 21 (Sum e Axis-dyad Chords)
Figure 2.4 <W>-related Axis-dyad Chords from Adjacent Sum Progressions 22
Figure 2.5 <T5>-related Cyclic Set Segments 23
Figure 2.6 Hexachordal K-net from a Difference Alignment 23
Figure 2.7 Hexachordal K-net from a Sum Alignment 24
ix
List of Figures (cont’d)
Figure 2.8 Strongly Isographic K-nets Related by a Parallel Shift 24 of the Array
Figure 2.9 Strongly Isographic K-nets Related by a Symmetrical Shift 25 of the Array
Figure 2.10 Strongly Isographic K-nets Related by a Asymmetrical Shift 25 of the Array
Figure 2.11 Strongly Isographic Hexachordal K-nets 28 (Sum 3 Axis-dyad Chords)
Figure 2.12 <W>-related Axis-dyad Chords from Adjacent Sum Progressions 28
Figure 3.1 Nine-node K-graph 33
Figure 3.2 Initial Axis-dyad Chord from the Triadic Array 33
Figure 3.3 Triadic Axis Chords Related by a Parallel Shift of the Array 34
Figure 3.5 Initial Triadic Axis Chords from the Sum 3 Progression 37
Figure 3.6 Triadic Axis Chords from Adjacent Sum Progressions 37
x
List of Examples
Example 1 Excerpt from Wozzeck 4
Example 2 Berg’s Master Array 5
Example 3 Compositional Design 1 42
Example 4 Compositional Design 2 43
1
INTRODUCTION
Aspects of large-scale structuring in George Perle’s system of Twelve-Tone Tonality,
including ordering aspects, first led me to some of the investigations in this study. Over the
course of this essay I will trace some parallels between Perle’s cyclic set structures and the
relevant Klumpenhouwer network (K-net) relationships which have been shown to be very
helpful in analyzing pitch-class sets extracted from Perle sets.1 I will then discuss a new
approach to the ordering of axis-dyad chords (i.e. hexachordal segments of arrays) drawn
from multiple alignments of a Perle array. This ordering method relies on the construction
of a cyclic set array such that in a specific alignment the vertical sums result in another
cyclic set of twelve distinct, non-repeating pitch-classes, similar to a twelve-tone row.
The pitch classes of this series then function as both local rotational determinates and
structurally as inversional sum orderings. Each sum is associated with twelve unique axis-
dyad chords, yielding a total compositional space of 144 axis-dyad chords generated from
a single cyclic set array.
David Lewin mentions possible correlations between Klumpenhouwer networks
and aspects of George Perle’s twelve-tone tonality, stating “I suspect that isographies of
the Networks, and networks-of-Networks, could reveal aspects of large-scale structuring
in Perle’s music beyond the extent of the formal analyses which Perle himself presents.”2
It is precisely these aspects of large-scale structuring (including ordering aspects) that led
me to some of the later investigations of this essay. In a response to Lewin’s footnote,
Perle wrote a letter to the editor of Music Theory Spectrum (published in the Autumn,
1993 issue) in which he states that “Any one of Klumpenhouwer’s triadic networks may
thus be understood as a segment of a cyclic set, and the interpretations of these and of
1 Headlam, Dave. Perle/Lewin Seminar. Department of Music Theory. Eastman School of Music. Spring 2005. This seminar traced the parallels between Lewin’s transformational networks, Klumpenhouwer networks and aspects of Perle’s twelve-tone tonality. I found interesting alignment properties in certain types of cycle set arrays which have led me to ordering principles and their application discussed in this essay.2 Lewin, David. “Klumpenhouwer Networks and Some Isographies that Involve Them.” Music Theory Spectrum. Vol. 12, No. 1. (Spring 1990), pp. 83-120.
2
the ‘networks-of-networks’ discussed in Lewin’s article plus consequent and concomitant
relations may be defined and efficiently and economically represented in this way.”3 A
later set of articles in the same issue expounds upon the K-Net nature of cyclic sets and
the cyclic set nature of K-Nets, including in the case of Stoecker, an additional isographic
relation (“axial isography”) among K-Nets. Concurrently Gretchen Foley also published
an article exploring the utility of K-nets in analyzing arrays (pairs of cyclic sets) in Perle’s
music. Foley’s discussion of cyclic set arrays will provide a foundation for the analysis
of the relationships of axis-dyad chords within this generated compositional space as well
as with those spaces generated from other arrays and array alignments.4 A further section
will begin to explore other types of arrays, specifically triadic arrays (those built upon
three cyclic sets), in order to establish a generalized relationship between these arrays
and possible resultant sum cycles (either twelve-note sets or otherwise). I will conclude
with a brief discussion of foreground applications of the ordered axis-dyad progressions,
including projection of the transformational relationships as well as possible voice-leading
guidelines.
CHAPTER 1: FROM INTERVAL CYCLE THROUGH THE CYCLIC SET TO
KLUMPENHOUWER NETWORKS
Early uses of the interval cycle in a scalar form are found in the “exotic” or magical
musics of the Russian school associated with Nicolai Rimsky-Korsakov. Here and in later
music by Igor Stravinsky, whole-tone scales (int. 2-cycles) and octatonic scales (derived
from 3-cycles) mix with tonal music in juxtapositions of diatonic and “other” scales used
to differentiate the worldly versus the supernatural. Elliott Antokoletz writes of Stravinsky’s The
3 Perle, George. “[Letter from George Perle].” Music Theory Spectrum. Vol. 15, No. 2. (Autumn, 1993), p. 300.4 I use the term “compositional space” with the particular connotation of Robert Morris in his article “Compositional Spaces and Other Territories,” Perspective of New Music, Vol. 33, No. 1/2, (Winter/Summer, 1995), pp. 328-358.
3
Firebird, “The Firebird’s array [of interlocking interval cycles] belongs to one of two musico-
dramatic spheres of the ballet - the magical realm of the Firebird is generally based on symmetrical
pitch collections derived from the interval cycles, whereas the human realm of the Prince and
Princess is generally set within folklike musical contexts primarily based on the asymmetrical
properties of the diatonic modes.”5 The example given by Antokoletz from the suite illustrates
Stravinsky’s use of three interlocking 5-cycles doubled at the third to generate a series of augmented
triads; interval cycles are used to create both diatonic (asymmetrical) and symmetrical collections
on the surface of the music. A later example found in Stravinsky’s music (and also explored in
the same article by Antokoletz6) is the opening of Le Sacre du Printemps. The bassoon’s opening
melody consists of the “white-note” fifth cycle emphasizing the pitch class C and secondarily A
(arising from the cycle C - G - D - A - E - B). A C#ppp introduced by the horn in measure 2 is then
used to initiate a new 5-cycle, F# - C# - G# - D#, the “black note” set from which the pitches of
the english horn melody at rehearsal 2 are drawn. Antokoletz points out the symmetry exhibited
by these collections in that the boundary pitches of the second set are in a tritone relation to the
emphasized pitches of the first.7 These boundary pitches are filled in with yet another fifth cycle
which creates the secondary set.
An even more explicit example of a composer’s thought and implementation of interval
cycles is found in a letter written by Alban Berg to his teacher and mentor Arnold Schoenberg on
July 27, 1920.8 The musical portion of the letter consists of twelve staves, one for each interval
cycle 1-12. Beginning on C2 the bottom staff is a chromatic scale ascending until the cycle
reaches C(3). The next stave above begins on the same pitch following the whole-tone scale
ending on C4. The next follows with an interval 3, then 4, etc. so that all 12 interval cycles
are represented horizontally. What Berg found interesting (which he describes as a “theoretical
trifle”) is the resultant vertical cycles found in this particular formation of horizontal cycles. The
5 Antokoletz, Elliott. “Interval Cycles in Stravinsky’s Early Ballets.” Journal of the American Musicological Society. Vol. 39, No. 3. (Autumn, 1986), p. 580.6 Antokoletz, pp. 600-608.7 Ibid., p. 602.8 This letter is reproduced in George Perle’s “Berg’s Master Array of the Interval Cycles,” The Musical Quarterly, Vol. 63, No. 1. (Jan., 1977), pp. 1-30.
4
left-most column is a “unison” cycle consisting of the same pitch class while each subsequent
column begins a half-step up (as per the bottom horizontal int. 1) and follows int. 2, then
int. 3 and so on. Berg’s array is reproduced on the following page. The compositional
significance of the “theoretical trifle” is found in an example that Perle provides from Act
II, scene 3, of Wozzeck composed around the same time as the letter described above.9
The excerpt includes a four-voice ascending gesture in which the lowest voice is an
int. 1, the next above, an int. 2, then an int. 3 and finally an int. 4 (a portion of the excerpt is
provided on p. 6). According to Perle, “Berg’s array of the interval cycles, far from being a
mere “trifle,” reflects a significant and persistent feature of his musical language, from the
second song of Opus 2, Schlafend trägt man mich in mein Heimatland, which still employs
a key signature, through his last work, the twelve-tone opera Lulu.”
Example 1: Excerpt from Wozzeck
9 Perle. “Master Array,” p. 2.
Wozzeck, Act II, scene 3, mm. 380f
(Cl.)
p
#
mf # #
(Solo Va.)
p
b
mf b # b #
(Ob.)
p
mf # # b
(Solo Vc.)
p
mf b b #
5
Example 2: Berg’s Master Array
IC 12
etc.
IC 11
b
b b
etc.
IC 10
b
b #
b b
etc.
IC 9
# b
b b
etc.
IC 8
#
b
b
betc.
IC 7
b b b b
b
etc.
IC 6
b # # b # #
IC 5
b b b # b b
IC 4
b # b b
IC 3
# # # b b #
IC 2
# b # # b b
IC 1
IC 0
IC 1
#IC 2
IC 3
b
IC 4
IC 5
#
IC 6
b#
IC 7
IC 8
b
IC 9
IC 10
b
IC 11
IC 12
Berg’s Master Array
6
CHAPTER 1.1: THE INTERVAL CYCLE
We will define a generic interval cycle as a series of pitch classes generated by
a single repeated interval. It is convenient to describe interval cycles as pitch-class sets
generated by a directed pitch-class interval, with the caveat that the completion of the
cycle, circles back to the original note. We may group complementary directed pitch-class
intervals into interval classes in the description of interval cycles; thus an IC 1/11 cycle is
either an “ascending” or “descending” chromatic scale.
Int. 1/11 (chromatic scale):
A A# B C C# D D# E F F# G G# (A
Int. 2/10 (whole-tone scale):
A B C# D# F G (A
Int. 3/9 (diminished 7th chord):
A C Eb Gb (A
Int. 4/8 (augmented triad):
A C# F (A
Int. 5/7 (circle of 5ths):
A E B F# C# G# D# A#/Bb F C G D (A
Only ints. 1/11 and 5/7 generate all twelve pitch classes within a single cycle - a property
very important to the construction of cyclic sets and resultant sum cycles. In order to
arrive at the missing pitch classes one must combine transpositions of these scales, such
as the two whole-tone scales where any odd T-level completes the aggregate. There is a
specific int. 2 which will yield all twelve pitch classes that will be discussed later. Another
scale which came into prominent use during the same period as the whole-tone scale (also
widely used by the Russian school of the late 19th and early 20th Centuries) is the octatonic
scale. This scale is built on the alternation of whole and half steps:
A Bb C C# D# E F# G (A
7
Another way of looking at this scale is as two interlocking 3/9 cycles beginning a half-step
apart:
A C D# F# (A
Bb C# E G (Bb
The concept of interlocking cycles is of utmost importance to the construction of cyclic
sets, as will be demonstrated below.
George Perle points out a special type or extension of the concept of an interval
cycle, related to the interlocking cycles of the octatonic collection, in his analysis of
the Lyric Suite by Alban Berg.10 Perle notes that the basic row for this piece is actually
constructed of two interlocking 5/7 cycles, one ascending and the other descending:
F E C A G D Ab Db Eb Gb B B
The cycles are clear if the row is written in integer notation in a retrograde rotation:
2 9 4 e 6 1 (8 7 0 5 t 3 8 (1
This type of interlocking of inversionally-related interval cycles is what Perle calls a cyclic
set. A cyclic set is then defined as two interlocking interval cycles in which the generating
cyclic interval of one cycle is paired with its complement (or descending form) in the other
cycle. Cyclic sets exhibit some extremely interesting features. The paired diagonal dyads
exhibit a constant sum - the southeast-oriented dyads share the inversional sum of 9 while
the northeast-oriented dyads share the inversional sum of 4. These sums between adjacent
pitch classes remain constant throughout a cyclic set and will be called “adjacency sums.”11
The difference between these two sums is that of the generating cyclic interval. Thus any
10 Perle, George. “Berg’s Master Array of the Interval Cycles.” The Musical Quarterly, Vol. 63, No. 1, (Jan., 1977), pp. 21-23.11 Perle calls these sums “tonic sums” in order to differentiate their inportance in his system from other sums, such as vertical sums within an axis-dyad chord. I have chosen the more generic term “adjacency sum” to remove some of the tonal connotations of the terminology. In fact my ordering priciples rely more heavily on the vertical axis dyad sum than the constant adjacency sums of a particular cyclic set.
8
cyclic set can be identified by its adjacency sums and may also be represented by any
trichord that exhibits those same sums.
CHAPTER 1.2: KLUMPENHOUWER NETWORKS AND PERLE’S CYCLIC SETS
Trichords which within themselves exhibit the sum and difference properties of
an entire cyclic set are called cyclic segments; they are the smallest cardinality pitch class
set which can represent a cyclic set. Because each of these cyclic segments share the
same inversional sum pair and cyclic interval, their transformational structure will also
be consistent. This transformational structure is well represented by the transformational
networks of David Lewin, specifically those investigated by Lewin and Henry
Klumpenhouwer. Lewin defines a Klumpenhouwer network as “any network that uses T
and/or I operations to interpret relations between pcs.”12 The basic network involves three
pitch classes in a transformational network of two I operations and a single T operation,
allowing both inversional and transpositional relationships to be represented in a visual
graph. Pitch classes are contained within nodes connected by arrows that identify the
transformational operations between the nodes. One of the great strengths of transformational
networks is the capability of representing the transformations between nodes, regardless
of what is contained within them. At this more abstract level in which empty nodes are
connected by operational arrows the network is referred to as a graph, which can represent
not only transformational relationships between pitch classes, but between pitch-class sets,
other transformational graphs, or any object which meets the operational criteria for the
connecting arrows. The following example places several cyclic segments from the Berg
Lyric Suite row into Klumpenhouwer networks:
12 Lewin, David. “Klumpenhouwer Networks and Some Isographies that Involve Them.” Music Theory Spectrum. Vol. 12, No. 1. (Spring 1990), p. 84.
9
Figure 1.1: Trichords from Berg’s Lyric Suite Row in K-nets
The first type of network isomorphism that Lewin introduces involves networks with
identical node/arrow relations as in the examples above. This isomorphism Lewin calls (from
Klumpenhouwer) strong isography, which represents the closest possible transformational
relationship between two K-nets (I will abbreviate the Klumpenhouwer nomenclature from
now on).
Another type of isography exists in the Berg cyclic set (twelve-tone row). If one
forms triangular shaped networks between the pitch-class sets <7, 9, 0> or <5, e, t> one
finds that the triangle must be flipped. The nodes and arrows of the flipped networks are
no longer strongly isographic with the networks above.
Figure 1.2: Trichords Exhibiting Negative Isography
The I-values in these networks have been reversed from the networks above and now sum
to a constant (4 + 9 = 1 = 9 + 4), while the T-value is the complement (MOD 12) to the
T-values in the other networks (5 + 7 = 12 = 0). Lewin and Klumpenhouwer define this
relationship as negative isography (one thinks of a photographic negative as an analogy).
In general orderings of K-nets can follow the musical surface or be arranged according to
some “underlying” transformational relationship. For instance either of the above groups
of networks could be revised to be strongly isographic with the other group by reversing
the content of the nodes connected by the T-arrow. Thus the T-value would become not the
T5
I4 I9 9
7 0T5
I4 I9 e
5 t
2 9
7
T7
I9 I4
9 4
0
T7
I9 I4
4 e
5
T7
I9 I4
10
complement, but the same value and the I-values would also reverse to match those of the
other networks. In this study however I will maintain the orderings found in the original
cyclic sets in order to maintain the highest amount of connectedness between constructions
and analyses of constructions.
Perle’s compositional method consists of combining cyclic sets into arrays: “paired
set forms, in all of their alignments.”13 These alignments fall into two categories based upon
the alignment of the generating cyclic interval of each cyclic set. If the cyclic intervals are
aligned in a complementary manner, i.e. the top is an ascending IC 7 and the bottom is a
descending IC 7/ascending IC 5, then the array is in a sum alignment. On the other hand
if the two generating intervals are equivalent and in the same direction, the array is in a
difference alignment. The directions are calculated by directed pitch-class intervals: pcs
<2 9 4 e 6 . . .> is an ascending 7-cycle or a descending 5 -cycle (One will also note that
this is a IS 2/t cyclic set). The nomenclature of sums and difference alignments arises from
the method for measuring the distance between the vertical dyads between sets as either a
transpositional or inversional relationship.
Difference alignment:
5,0: 3 2 t 7 5 0 0 5 7 t 2 3 9 8 4 1 e 6 6 e 1 4 8 9 6,1: 5 1 0 6 7 e 2 4 9 9 4 2 e 7 6 0 1 5 8 t 3 3 t 8 Difference cycle: 2 1 2 1 2 1 2 1 2 1 . . .
Sum alignment:
5,0: 3 2 t 7 5 0 0 5 7 t 2 3 9 8 4 1 e 6 6 e 1 4 8 9 1,6: 1 0 6 7 e 2 4 9 9 4 2 e 7 6 0 1 5 8 t 3 3 t 8 5 Sum cycle: 4 2 4 2 4 2 4 2 4 2 . . .
Table 1.1: Cyclic Set Array in Sum and Difference Alignments
13 Perle, George. Twelve-Tone Tonality, p. 21.
11
In the above arrays the transpositional differences are measured in alternating directions
as the cyclic set is comprised of two IC 7s, one ascending and one descending. In the
sum alignment directed measurement is unnecessary as the inversion operation is non-
directional.
The basic compositional cell which Perle extracts from his arrays are axis-dyad
chords, hexachords consisting of a cyclic segment drawn from each cyclic set. The
term axis-dyad refers to the two central, vertically-oriented pitch classes from which the
adjacency sums of a cyclic set are measured; the other four pitch classes of the axis-dyad
chord Perle calls a cyclic chord (from which one measures the cyclic intervals for the
sets).
Axis-dyad chord: 3 2 t Axis-dyad: 2 Cyclic chord: 3 t 5 1 0 1 5 0
Figure 1.3: Axis-dyad Chord and Its Constituent Members
For now we are dealing only with cyclic set arrays generated by the same cyclic interval,
thus the two cyclic segments of an axis-dyad chord will either share or have complementary
T-values in their respective K-nets. Unless the adjacency sums for the two cyclic sets
are identical or sum to a constant (which can be represented by either strong or negative
isography), one needs an additional Klumpenhouwer transformation to account for the
relationship between the K-nets of the two cyclic segments. In both sum and difference
alignments, the adjacency sums have the same sums and differences, just aligned differently:
5 - 6 = 0 - 1 = 11 (or 1 if reversed) and 5 + 1 = 6 + 0 = 6. Thus a cyclic segment drawn
from one cyclic set is related by <T1> or <I6> (Lewin calls these “hyper” relationships
denoted by angle brackets) to any cyclic segment from the other cyclic set of the array.
This relationship is called positive isography which results when T-values are maintained
12
between graphs and a constant is added to the I-values - the <Tn> relation, where n is the
constant added to the I-values; the relationship is called negative isography, <In`>, as noted
earlier, when complementary T-values are used and the inversional sums add to the constant
n. In the above case the <T1> relationship exists between all cyclic segments drawn from
a difference alignment, while the <I6> exists between segments in a sum alignment. This
is apparent in the relationship between the adjacency sums in each alignment: Difference:
5 - 6 = 0 - 1 = 11 = 1; Sum: 5 + 1 = 0 + 6 = 6.
Figure 1.4: Hyper Relation Between Trichordal K-nets
This hyper-relation between cyclic sets is not the only relationship to be found internally
within the transformational structure of an axis-dyad chord. Gretchen Foley illustrates
hexachordal K-nets for axis-dyad chords drawn from both types of array alignments which
allow comparison between chords, instead of just between the two constituent cyclic
segments of the chords.14 In a difference alignment the K-net of an axis-dyad chord will
have the T-values of each cyclic segment, four adjacency sums (two from each cyclic
segment) and then the secondary T-values measured vertically from each member of the
two cyclic segments (for these K-nets, T-values are measured in the same direction).
Figure 1.5: Hexachordal K-net from a Difference Alignment
14 Foley, Gretchen C. “Arrays and K-Nets: Transformational Relationships Within Perle’s Twelve-Tone Tonality.” Indiana Theory Review, Vol. 23 (Spring/Fall 2002), pp. 69-97.
3 t
2
1
5 0
T7
I5 I0
T1T2 T2
T7
I6 I1
3 t
2
T7
I5 I0
5 0
1
T7
I6 I1
<T1>
13
In a sum alignment the K-net will also have the T-values of each cyclic segment, the four
adjacency sums and then secondary sums measured vertically from each member of the
two cyclic segments.
Figure 1.6: Hexachordal K-net from a Sum Alignment
While the type of alignment affects how the vertical dyads of an axis-dyad chord
are measured, analytical necessity may override the measurement rules set out originally
by Perle. If a particular type of isography is desired to make an analytical distinction one
may use vertical sums in a difference alignment or transpositional differences in a sum
alignment; it is the relation of the networks themselves which is most important.
Several types of isography arise depending on the shifting of one cyclic set against
the other. Foley defines the following three types of shifting: parallel shifting, symmetrical
shifting and asymmetrical shifting.15 A parallel shift occurs when both the upper and lower
cyclic sets slide by an equal number of positions in the same direction.
5,0: 3 2 t 7 5 0 0 5 7 t 2 3 9 8 4 1 e 6 6 e 1 4 8 9 6,1: 5 1 0 6 7 e 2 4 9 9 4 2 e 7 6 0 1 5 8 t 3 3 t 8
The two underlined axis-dyad chords are separated by a parallel shift of ten positions; the
even number of positions will maintain the alignment type, ensuring the networks will be
strongly isographic:
15 Foley, p. 77.
3 t
2
0
1 6
T7
I5 I0
I2I4 I4
T5
I1 I6
14
Figure 1.7: Strongly Isographic K-nets Related by a Parallel Shift of the Cyclic Set
Parallel shifts by an even number of positions will maintain the type of array alignment,
while a shift by an odd number of positions will result in a flip or reversal of the array
alignment, yielding axis-dyad chords which will be negatively isographic.
A symmetrical shift results when both the upper and lower cyclic sets slide by an
equal number of positions in opposite directions. As in the above example a slide by an
even number of positions will maintain the type of alignment.
5,0: 3 2 t 7 5 0 0 5 7 t 2 3 9 8 4 1 e 6 6 e 1 4 8 9 6,1: 5 1 0 6 7 e 2 4 9 9 4 2 e 7 6 0 1 5 8 t 3 3 t 8
The underlined axis-dyad chords are generated by a symmetrical shift; the upper cyclic
segment of the second axis-dyad chord (5 0 0/3 3 t) comes from shifting the upper cyclic
set four positions to the right, while the lower cyclic segment comes from shifting the
lower cyclic set four positions to the left. These chords are also strongly isographic:
Figure 1.8: Strongly Isographic K-nets Related by a Symmetrical Shift of the Cyclic Set
T7
I5 I0
I3I8 It
T7
I6 I1
3 t
2
1
5 0
T7
I5 I0
I3I8 It
T7
I6 I1
5 0
0
3
3 t
<T0>
3 t
2
1
5 0
T7
I5 I0
T1T2 T2
T7
I6 I1
2 9
3
2
4 e
T7
I5 I0
T1T2 T2
T7
I6 I1
15
In this case while the array is still in a difference alignment, I have chosen to examine the
vertical sums between the cyclic segments in order to illustrate the strongly isographic
relationship, i.e. the I-values between the vertical dyads are maintained between the K-nets
of these particular axis-dyad chords. If one had chosen to measure the vertical differences
between the segments, as is normal in a difference alignment, one would then need a
transformation that maintains I-values and measures the difference between T-values - a
K-net relationship described below.
The final type of shift that Foley presents is asymmetrical shifting, resulting from
both the upper and lower cyclic sets sliding by an unequal number of positions in opposite
directions.
5,0: 3 2 t 7 5 0 0 5 7 t 2 3 9 8 4 1 e 6 6 e 1 4 8 9 6,1: 5 1 0 6 7 e 2 4 9 9 4 2 e 7 6 0 1 5 8 t 3 3 t 8
The underlined axis-dyad chords are related by an asymmetrical shift; the upper cyclic
segment of the second axis-dyad chord comes from shifting the upper cyclic set four
positions to the right, while the lower segment comes from a shift of the lower cyclic set
by six positions to the left. We are no longer dealing with strong or negative isography
between the K-nets of these axis-dyad chords. In positive isography the T-values remain
the same while a constant is added to the I-operators - the inverse is the case here. It is the
cyclic intervals and adjacency sums which remain the same, while the internal T-values
change by a constant. None of the isographies of Lewin or Klumpenhouwer deal with
this contingency and it is Foley who sites the <W> transformation proposed by Shaugn
O’Donnell as an appropriate solution: “In this type of transformation, the I-values are
identical while the T-values show a fixed transpositional relationship.”16
16 Foley, p. 77.
16
Figure 1.9: <W>-related K-nets Drawn from an Asymmetrical Shift of the Cyclic Set
CHAPTER 2: TOWARD NEW ORDERING PRINCIPLES
With these transformational relations between Klumpenhouwer networks in place
as tools for analyzing the relationships between axis-dyad chords I will now return to
cyclic set array construction and a particular intersection between the ordering of axis-dyad
chords drawn from these arrays and twelve-note series that are themselves cyclic sets.
I will denote an interval cycle generated by the vertical sums of a cyclic set array
alignment a resultant sum cycle.17 Every array can generate a resultant sum cycle, however
they are not all of equal compositional value, in that some resultant cycles will not include
all twelve pitch classes or may only be an alternation of two sums (these resultant cycles
may be useful in other contexts, however not in the ordering paradigm under discussion).
This being established, I will limit the immediate discussion to the combination of 5/7
cycles and the resultant 2/t cycle. (Because of the nature of the whole-tone scale - its
composition of only even-numbered interval classes - only 2/t cycles with odd sums will
generate all twelve pitch classes). Once the proper array is constructed in an appropriate
alignment that generates a useful resultant sum cycle, this resultant cycle is then used to
govern the foreground presentation of axis-dyad chords drawn from the cyclic set array.
Each pitch class of the resultant sum cycle is treated as a local harmonic determinant,
17 In Chapter 13 of Twelve-Tone Tonality, Perle uses the term “resultant set forms” to distinguish “tonic alignments” of arrays - those in which some axis dyads are equivalent to a segment of one of the cyclic sets - from the other remaining (i.e. resultant) alignments of a given cyclic set array (T-TT, p. 51).
3 t
2
1
5 0
T7
I5 I0
T1T2 T2
T7
I6 I1
5 0
0
t
8 3
T7
I5 I0
T2T3 T3
T7
I6 I1
<W1>
17
grouping the twelve axis-dyad chords whose axis-dyads sum to that pitch-class number.
Within each of these groups, the resultant sum cycle also determines the upper note of the
axis-dyad (beginning with the current local sum being used in that group). The lower pitch
class of the axis-dyad follows the TnI of the resultant sum cycle where n is the sum of the
twelve-chord progression.
The following is a generic algorithm for generating the cyclic set arrays just
described:
Interval System (j, k); Adjacency Sums (m, n; o, p)
Limitations: • m + o AND n + p must be ODD (ensures that the adjacency sums of the resultant sum cycle are odd) • j + k = 2 (MOD 12) (ensures the resultant sum cyclic is of interval system 2/10) • m + j = n (interval between adjacency sums is the interval system of the cyclic set) • o + k = p • The choice of starting pitch classes (u and x) is arbitrary (or based upon purely musical or auditory considerations).
m n m n m (u) (v) (u + j) (v - j) (u + 2j) (u - 2j) . . . o p o p o (x) (y) (x + k) (y - k) (x + 2k) (y - 2k) . . .
(m + o) (n + p) (m + o) (n + p) (m + o) (u + x) (v + y) (u + j + x + k) ((v - j) + (y - k)) (u + 2j + x + 2k) ((u - 2j) + (y - 2k))
Figure 2.1.1: Generic Cyclic Set Array Algorithm
Example: IS (7, 7) Sums (5, 0; 6, 1) j = 7, k = 7, m = 5, n = 0, o = 6, p = 1Limitations:
• m + o must be odd: 5 + 6 = 11 AND n + p must be odd: 0 + 1 = 1 (thus the adjacency sums of the resultant sum cycle will be (11, 1))• j + k = 2 (MOD 12): 7 + 7 = 14 = 2 (thus the interval system of the resultant sum cycle will be 2) • m + j = n: 5 + 7 = 12 = 0 (the interval system is represented in the difference between the adjacency sums)
18
• o + k = p: 6 + 7 = 13 = 1
5 0 (t + 7 = 17 = 5) (7 + 5 = 12 = 0)
(3) (2) (3 + 7 = t) (2 - 7 = -5 = 7) (3 + 2(7) = 17 = 5) . . .
6 1 (0 + 6 = 6) (6 + 7 = 13 = 1)
(5) (1) (5 + 7 = 12 = 0) (1 - 7 = -6 = 6) (5 + 2(7) = 19 = 7) . . .
(5 + 6 = e) (0 + 1 = 1) . . .
(3 + 5 = 8) (2 + 1 = 3) (t + 0 = t) (7 + 6 = 13 = 1) (5 + 7 = 12 = 0) . .
(8 + 3 = e) (3 + t = 13 = 1) (t + 1 = e) (1 + 0 = 1)
Figure 2.1.2: Cyclic Set Array Algorithm Example
Full array in this initial alignment:
T0 T0R T6 T6R
3 2 t 7 5 0 0 5 7 t 2 3 9 8 4 1 e 6 6 e 1 4 8 9 IS (7) Sums (5,0)
5 1 0 6 7 e 2 4 9 9 4 2 e 7 6 0 1 5 8 t 3 3 t 8 IS (7) Sums (6,1)
8 3 t 1 0 e 2 9 4 7 6 5 8 3 t 1 0 e 2 9 4 7 6 5 IS (2) Sums (e,1)
Table 2: Sum 5,0; 6,1 Array
The transpositional relationships between hexachords is T6, similar to the T6 relationships
inherent within the array. In addition all of the alignments which generate this particular
resultant sum cycle are sum alignments.
From this alignment one can draw the first ordered progression based upon the sums
of the resultant sum cycle. The initial axis-dyad chord 3 2 t / 5 1 0 begins the following
group:
Sum 3 (of the axis dyad):
3 2 t / 5 1 0 9 8 4 / e 7 6 8 9 3 / 0 6 7 2 3 9 / 6 0 1 1 4 8 / 7 e 2 7 t 2 / 1 5 8 t 7 5 / t 8 5 4 1 e / 4 2 e e 6 6 / 9 9 4 5 0 0 / 3 3 t 0 5 7 / 8 t 3 6 e 1 / 2 4 9 Table 2.1: Axis-dyad Chord Progression (Single Axis-dyad Sum 3)
19
A closer look at the generated columns reveals interesting transformations and
relationships created by grouping these particular alignments of the array in this particular
order:
• The first column exchanges pairs of adjacent sums from the resultant cycle (8-3, t-1, e-0,
etc.).
3 8 1 t e 0 9 2 7 4 5 6
• The second column is a rotation of the resultant sum cycle beginning with order position
7 (exchanging the hexachords).
2 9 4 7 6 5 8 3 t 1 0 e
• The third column is a rotation and retrograde of the resultant sum cycle beginning with
order position 3 (pitch class t).
t 3 8 5 6 7 4 9 2 e 0 1
• Columns four through six are operations upon an inversion of the resultant sum cycle:
T0: 3 8 1 t e 0 9 2 7 4 5 6
T8I: 5 0 7 t 9 8 e 6 1 4 3 2
• The fourth column is the inversion noted above.
• The fifth column is a rotation and retrograde of the inverted resultant sum cycle beginning
with order position 9 (pitch class 1).
1 6 e 8 9 t 7 0 5 2 3 4
• The sixth column is a rotation of the inverted resultant sum cycle beginning with order
position 2 (pitch class 0).
0 7 t 9 8 e 6 1 4 3 2 5
Note still that while the resultant sum cycle (rotations, rotated retrogrades and inversions
- both with rotations and retrogrades) is represented in the columns in the above example,
cyclic segments of the generating cyclic set array are represented in the rows. The individual
cyclic set segments govern the rotations themselves (with literal pitch classes). In addition,
20
because the adjacency sums of the two cyclic sets are constant throughout the progression,
each member of an axis-dyad will always have the same neighboring pitch-classes yielding
invariant trichords from each cyclic set. In essence the compositional space created by
generating progressions of axis-dyad chords in the manner just described consists of all the
distinct cyclic segments of a particular generating array alignment sliding independently of
one another, ordered by the resultant sum cycle.
Because all of the axis-dyad chords are generated by the independent sliding of the
array’s cyclic sets against one another, all of the chords will exhibit some sort of strong
isography or the <W> relationship, depending upon the type of shift from one chord to
another. As one will recall, parallel and symmetrical shifting results in strong isography,
while asymmetrical shifting is represented by the <W> transformation. I will continue the
progression of axis-dyad chords begun earlier in order to illustrate the types of isography
which are to be found at different points in the series of progressions (once again the
ordering of sums follows the resultant sum cycle).
Sum 3 Sum t Sum 1 Sum 0 Sum e Sum 2 3 2 t / 5 1 0 8 9 3 / 5 1 0 1 4 8 / 9 9 4 t 7 5 / 1 5 8 e 6 6 / 1 5 8 0 5 7 / 9 9 4 8 9 3 / 0 6 7 1 4 8 / 0 6 7 t 7 5 / 0 6 7 e 6 6 / 0 6 7 0 5 7 / 0 6 7 9 8 4 / 0 6 71 4 8 / 7 e 2 t 7 5 / 3 3 t e 6 6 / e 7 6 0 5 7 / e 7 6 9 8 4 / 3 3 t 2 3 9 / 7 e 2t 7 5 / t 8 5 e 6 6 / 2 4 9 0 5 7 / t 8 5 9 8 4 / 2 4 9 2 3 9 / t 8 5 7 t 2 / 2 4 9e 6 6 / 9 9 4 0 5 7 / 1 5 8 9 8 4 / 1 5 8 2 3 9 / 9 9 4 7 t 2 / 5 1 0 4 1 e / 5 1 00 5 7 / 8 t 3 9 8 4 / 4 2 e 2 3 9 / 8 t 3 7 t 2 / 4 2 e 4 1 e / 8 t 3 5 0 0 / 4 2 e9 8 4 / e 7 6 2 3 9 / e 7 6 7 t 2 / 3 3 t 4 1 e / 7 e 2 5 0 0 / 7 e 2 6 e 1 / 3 3 t2 3 9 / 6 0 1 7 t 2 / 6 0 1 4 1 e / 6 0 1 5 0 0 / 6 0 1 6 e 1 / 6 0 1 3 2 t / 6 0 17 t 2 / 1 5 8 4 1 e / 9 9 4 5 0 0 / 5 1 0 6 e 1 / 5 1 0 3 2 t / 9 9 4 8 9 3 / 1 5 84 1 e / 4 2 e 5 0 0 / 8 t 3 6 e 1 / 4 2 e 3 2 t / 8 t 3 8 9 3 / 4 2 e 1 4 8 / 8 t 35 0 0 / 3 3 t 6 e 1 / 7 e 2 3 2 t / 7 e 2 8 9 3 / 3 3 t 1 4 8 / e 7 6 t 7 5 / e 7 66 e 1 / 2 4 9 3 2 t / t 8 5 8 9 3 / 2 4 9 1 4 8 / t 8 5 t 7 5 / 2 4 9 e 6 6 / t 8 5
Table 2.2: Axis-dyad Chord Progressions (Sum 5,0; 6,1)
21
The other six progressions of the chart are in a strict T6 relation to this first half (including
both the sums of the resultant sum cycle and the actual pitch classes of the individual
axis-dyad chords). Within this chart one finds only the two types of transformational
relationships mentioned above (either strong isography or the <W> relation). In the first
type of relationship found in the set of progressions, the axis-dyad chords which share a
common sum (in the following example the chords are the first two of the sum 3 group) are
strongly isographic because they are related by a symmetrical shift:
Figure 2.2: Strongly Isographic Hexachordal K-nets (Sum 3 Axis-dyad Chords)
In order to arrive at the second chord from the first, one shifts the top cyclic set two positions
to the left and the bottom cyclic set 2 positions to the right (always referring to the initial
array alignment). An example from another sum group (the first two chords of sum e):
Figure 2.3: Strongly Isographic Hexachordal K-nets (Sum e Axis-dyad Chords)
The internal sum values between cyclic segments of the above strongly isographic K-nets
are those of the resultant sum cycle and the sum of each adjacent group of axis-dyad chords.
T7
I5 I0
I3I8 It
T7
I6 I1
3 t
2
1
5 0
T7
I5 I0
I3I8 It
T7
I6 I1
8 3
9
6
0 7
<T0>
e 6
6
5
1 8
T7
I5 I0
IeI0 I2
T7
I6 I1
0 7
5
6
0 7
T7
I5 I0
IeI0 I2
T7
I6 I1
<T0>
22
For instance all of the axis-dyad chords in the sum e group have the internal I-values <0, e,
2>, which is both a cyclic segment of the resultant sum cycle and the sums of the current
group and the adjacent groups.
The second transformational relationship is found between sum groups of axis-dyad
chords. While the cyclic segments from each cyclic set are still invariant, the alignments
are governed purely by the axis-dyad sums as they follow the resultant sum cycle. From
one sum group to another one finds only the <W> relation to be applicable. The following
series of chords are the first from the initial four sums of the resultant sum cycle:
Sum 3 Sum t Sum 1 Sum 0
Figure 2.4: <W>-related Axis-dyad Chords from Adjacent Sum Progressions
If one follows the remainder of the series with the <Wn> relation, n will continue to alternate
between 7 and e.
The preceding discussion has dealt only with arrays of IC 5/7, however the other
logical IC for the current ordering paradigm is IC 1/e. It shares the two most important
array properties with IC 5/7: each individual cyclic set will contain all twelve pitch classes,
and when the two cyclic sets are combined into an array the resultant sum cycle will also
be an IC 2/t. I will demonstrate that similar ordered progressions and K-net relationships
are availabe in both IC 5/7 and IC 1/e. In order to show these parallels I will take the same
initial axis-dyad chord from the preceding examples, 3 2 t / 5 1 0, and reorder the pitch
classes into an IC 1/1 array; this particular set class allows for it to generate both IC 7/7 and
3 t
2
1
5 0
T7
I5 I0
T1T2 T2
T7
I6 I1
8 3
9
1
5 0
T7
I5 I0
T8T9 T9
T7
I6 I1
<W7>
1 8
4
9
9 4
T7
I5 I0
T7T8 T8
T7
I6 I1
<We>
t 5
7
5
1 8
T7
I5 I0
T2T3 T3
T7
I6 I1
<W7>
23
IC 1/1:18
0,1: 2 t 3 9 4 8 5 7 6 6 7 5 8 4 9 3 t 2 e 1 0 0 1 e 5,6: 0 5 1 4 2 3 3 2 4 1 5 0 6 e 7 t 8 9 9 8 t 7 e 6
Table 2.3: Sum 0,1; 5,6 Array
The difference between the adjacency sums of the two cyclic set is 5, resulting in a <T5>
relationship between the K-nets of cyclic segments drawn from each set in this difference
alignment.
Figure 2.5: <T5>-related Cyclic Set Segments
Similar to the hexachordal K-nets in the sum alignment of the IC 7/7 example, each network
will have the T-values of the cyclic interval, four adjacency sums (two from each cyclic
segment) and the secondary T-values measured vertically from each member of the two
cyclic segments:
Figure 2.6: Hexachordal K-net from a Difference Alignment
One can still note the <T5> relation between the two cyclic segments within the overall
structure of the hexachordal K-net. In the sum alignment of this array the axis-dyad chord
18 This particular set-class has 3 instances of both int. 1 and int. 5 (the cyclic interval) so that it can generate both int. 1 and int. 5 cyclic sets.
2 3
t
T1
I0 I1
0 1
5
T1
I5 I6
<T5>
2 3
t
5
0 1
T1
I0 I1
T5Tt Tt
T1
I5 I6
24
K-net will also have the T-values of each cyclic segment, the four adjacency sums and then
secondary sums measured vertically from each member of the two cyclic segments:
Figure 2.7: Hexachordal K-net from a Sum Alignment
As in the above network one can now see the negative isography (resulting from the change
in alignment) between the two cyclic segments (complementary T-values and I-values
which sum to 6, i.e. <I6>. As before when relating several K-nets from the same array,
one looks to the type of shift involved for the type of isography that will arise; parallel and
symmetrical shifts will yield strong isography, while an asymmetrical shift will require the
<W> relation. The following two underlined axis-dyad chords are related by a parallel
shift, both segments of the second chord are drawn from a shift of twelve positions to the
right, and will be strongly isographic:
0,1: 2 t 3 9 4 8 5 7 6 6 7 5 8 4 9 3 t 2 e 1 0 0 1 e 5,6: 0 5 1 4 2 3 3 2 4 1 5 0 6 e 7 t 8 9 9 8 t 7 e 6
Figure 2.8: Strongly Isographic K-nets Related by a Parallel Shift of the Cyclic Set
2 3
t
1
5 4
T1
I0 I1
IeI7 I7
Te
I6 I5
8 9
4
e
6 7
2 3
t
5
0 1
T1
I0 I1
I3It I4
T1
I5 I6
T1
I0 I1
I3I2 I4
T1
I5 I6
<T0>
25
The following two underlined chords are related by a symmetrical shift, the upper segment
from four positions to the right, while the lower segment is drawn from a shift of four
positions to the left; these chords will also be strongly isographic.
0,1: 2 t 3 9 4 8 5 7 6 6 7 5 8 4 9 3 t 2 e 1 0 0 1 e 5,6: 0 5 1 4 2 3 3 2 4 1 5 0 6 e 7 t 8 9 9 8 t 7 e 6
Figure 2.9: Strongly Isographic K-nets Related by a Symmetrical Shift of the Array
As noted earlier an asymmetrical shift results not in strong isography, but requires the <W>
relation. The upper segment of the second chord results from a shift of four positions to the
right, while the lower segment comes from a shift of six positions to the left.
0,1: 2 t 3 9 4 8 5 7 6 6 7 5 8 4 9 3 t 2 e 1 0 0 1 e 5,6: 0 5 1 4 2 3 3 2 4 1 5 0 6 e 7 t 8 9 9 8 t 7 e 6
Figure 2.10: Strongly Isographic K-nets Related by an Asymmetrical Shift of the Array
Having illustrated the K-net relations between cyclic segments and axis-dyad chords drawn
from an IC 1/1, the next topic involves the resultant sum cycle and ordered axis-dyad chord
progressions generated by this array alignment.
4 5
8
7
t e
2 3
t
5
0 1
T1
I0 I1
I3It I4
T1
I5 I6
T1
I0 I1
I3I2 I4
T1
I5 I6
<T0>
4 5
8
8
9 t
2 3
t
5
0 1
T1
I0 I1
T5Tt Tt
T1
I5 I6
T1
I0 I1
T0T5 T5
T1
I5 I6
<W7>
26
0,1: 2 t 3 9 4 8 5 7 6 6 7 5 8 4 9 3 t 2 e 1 0 0 1 e (2 t 3 9 . . .
5,6: 0 5 1 4 2 3 3 2 4 1 5 0 6 e 7 t 8 9 9 8 t 7 e 6 (0 5 1 4 . . .
5,7: 2 3 4 1 6 e 8 9 t 7 0 5 2 3 4 1 6 e 8 9 t 7 0 5 (2
Table 2.4: Sum 0,1; 5,6 Array (With Resultant Sum Cycle)
The configuration of this array is subtly different than the earlier explored IC 7/7, in that
the point of the T6 relation inherent in any cyclic set occurs nine positions into the upper
set and seven positions into the lower set., i.e. the transposition is between the segment
beginning <6, 7, 5> and <0, 1, e> in the upper and <3, 2, 4> and <9, 8 t> in the lower. This
feature has no real effect on the ordering, except that it clearly illustrates the cyclic nature
of the compositional space.
From this alignment one can draw the first ordered progression based upon the
sums of the resultant sum cycle. The initial axis-dyad chord begins the following group:
Sum 3 (of the axis dyad):
2 t 3 / 0 5 1 8 4 9 / 6 e 7 5 7 6 / 9 8 t e 1 0 / 3 2 4 0 0 1 / 2 3 3 6 6 7 / 8 9 9 7 5 8 / 7 t 8 1 e 2 / 1 4 2 t 2 e / 4 1 5 4 8 5 / t 7 e 9 3 t / 5 0 6 3 9 4 / e 6 0
Table 2.5: Axis-dyad Chord Progression (Single Axis-dyad Sum 3)
• The first column is a retrograde of the resultant sum cycle beginning with order position
1 (pitch class 2) and moving to the left as the cycle wraps around.
2 5 0 7 t 9 8 e 6 1 4 3
• The second column is a rotation of the resultant sum cycle beginning with order
position 9 (pitch class t).
t 7 0 5 2 3 4 1 6 e 8 9
27
• The third, fourth and sixth columns are operations upon an inversion of the first
column:
T0: 2 5 0 7 t 9 8 e 6 1 4 3
T2I: 0 9 2 7 4 5 6 3 8 1 t e
• The third column exchanges pairs of adjacent sums beginning with order position 8
(pitch class 3) and moving to the right.
3 6 1 8 e t 9 0 7 2 5 4
• The fourth column is the T2I inversion of the first column as noted above.
• The fifth column is T3 of the first column (the retrograde of the resultant sum cycle).
5 8 3 t 1 0 e 2 9 4 7 6
• The sixth column maintains the order of adjacent sums from the inverted resultant
cycle, but is a retrograde of the pairs themselves.
1 t 3 8 5 6 7 4 9 2 e 0
Having shown the transformational relationships within a single axis-dyad
progression, one must again turn to the relationships between progressions. The following
is a continuation of the previous progression:
Sum 3 Sum 4 Sum 1 Sum 6 Sum e Sum 82 t 3 / 0 5 1 5 7 6 / 8 9 9 0 0 1 / 4 1 5 7 5 8 / 4 1 5 t 0 e / 8 9 9 9 3 t / 0 5 15 7 6 / 9 8 t 0 0 1 / 1 4 2 7 5 8 / 9 8 t t 2 e / 1 4 2 9 3 t / 9 8 t 8 4 9 / 1 4 20 0 1 / 2 3 3 7 5 8 / 6 e 7 t 2 e / 6 e 7 9 3 t / 2 3 3 8 4 9 / t 7 e e 1 0 / t 7 e7 5 8 / 7 t 8 t 2 e / 3 2 4 9 3 t / 7 t 8 8 4 9 / 3 2 4 e 1 0 / 7 t 8 6 6 7 / 3 2 4t 2 e / 4 1 5 9 3 t / 4 1 5 8 4 9 / 8 9 9 e 1 0 / 0 5 1 6 6 7 / 0 5 1 1 e 2 / 8 9 99 3 t / 5 0 6 8 4 9 / 5 0 6 e 1 0 / 5 0 6 6 6 7 / 5 0 6 1 e 2 / 5 0 6 4 8 5 / 5 0 68 4 9 / 6 e 7 e 1 0 / 2 3 3 6 6 7 / t 7 e 1 e 2 / t 7 e 4 8 5 / 2 3 3 3 9 4 / 6 e 7e 1 0 / 3 2 4 6 6 7 / 7 t 8 1 e 2 / 3 2 4 4 8 5 / 7 t 8 3 9 4 / 3 2 4 2 t 3 / 7 t 86 6 7 / 8 9 9 1 e 2 / 0 5 1 4 8 5 / 0 5 1 3 9 4 / 8 9 9 2 t 3 / 4 1 5 5 7 4 / 4 1 51 e 2 / 1 4 2 4 8 5 / 9 8 t 3 9 4 / 1 4 2 2 t 9 / 9 8 t 5 7 6 / 1 4 2 0 0 1 / 9 8 t4 8 5 / t 7 e 3 9 4 / t 7 e 2 t 3 / 2 3 3 5 7 6 / 6 e 7 0 0 1 / 6 e 7 7 5 8 / 2 3 33 9 4 / e 6 0 2 t 3 / e 6 0 5 7 6 / e 6 0 0 0 1 / e 6 0 7 5 8 / e 6 0 t 2 e / e 6 0
Table 2.6: Axis-dyad Chord Progressions (Sum 0, 1; 5, 6)
28
As before the remainder of the progression is in a strict T6 relation to the first half
(including both the sums of the resultant sums cycle and the actual pitch classes of the
individual axis-dyad chords). Chords drawn from a single progression (those possessing
a single axis-dyad sum) will be strongly isographic, as the are all related by a symmetrical
shift of the array. In the following example the chords are the first three from the sum 3
progression.
Figure 2.11: Strongly Isographic Hexachordal K-nets (Sum 3 Axis-dyad Chords)
As stated earlier, the relationship between sum groups requires the <W> relation, in that
the transpositional relation between the cyclic segments of the axis-dyad chords proves
more analytically fruitful than the inversional relations. The following example extracts
the initial axis-dyad chord from the first four sum progressions of the above chart:
Figure 2.12: <W>-related Axis-dyad Chords from Adjacent Sum Progressions
Again if one follows the remainder of the series with the <Wn> relation, n will continue
to alternate between 5 and 1. There are of course many other types of interval cycles
2 3
t
5
0 1
T1
I0 I1
I3I2 I4
T1
I5 I6
5 6
7
8
9 t
T1
I0 I1
I3I2 I4
T1
I5 I6
0 1
0
3
2 3
T1
I0 I1
I3I2 I4
T1
I5 I6
<T0> <T0>
2 3
t
5
0 1
T1
I0 I1
T5Tt Tt
T1
I5 I6
5 6
7
9
8 9
T1
I0 I1
TtT3 T3
T1
I5 I6
0 1
0
1
4 5
7 8
5
1
4 5
T1 T1
I0 I0I1 I1
Te T4T4 T9T4 T9
T1 T1
I5 I5I6 I6
<W5> <W1> <W5>
29
which can form cyclic set arrays, however not all contain all twelve pitch classes within
the individual cyclic sets, nor do they necessarily form a resultant sum cycle that is also a
twelve-tone series. The following discussion examines another type of array which does
satisfy the aforementioned requirements.
CHAPTER 3: TRIADIC ARRAYS
Chapter 29 of Perle’s Twelve-Tone Tonality introduces the concept of triadic arrays
in which three cyclic sets are combined. I would like to apply the resultant sum scale
orderings to these arrays as well, but first I will introduce the concept within the context
of Perle’s discussion. Dyadic collections, i.e. pairs of cyclic sets “can be converted into
triadic collections of a single sum by the addition of a third pitch-class number: n to each
dyad of sum 0, n-2 to each dyad of sum 2, n-4 to each dyad of sum 4, and so on.”19
Basically the constant subtracted from the added pitch class number is equal to the sum of
the dyad. The operation splits one of the dyads of the original pair into its own dyadic sum.
Because I want to maintain the resultant sum scale after the process of converting a dyadic
array into a triadic array, a single sum is not desirable. However the process of generating
all the possible triadic groups for a single sum is useful, and is employed individually for
each sum of the resultant sum cycle. I will illustrate how one converts a dyadic cyclic set
array into a triadic array, while maintaining a twelve-note series for the resultant sum cycle,
utilizing the same array as the past few examples.
5,0: 3 2 t 7 5 0 0 5 7 t 2 3 9 8 4 1 e 6 6 e 1 4 8 9 6,1: 5 1 0 6 7 e 2 4 9 9 4 2 e 7 6 0 1 5 8 t 3 3 t 8
e, 1: 8 3 t 1 0 e 2 9 4 7 6 5 8 3 t 1 0 e 2 9 4 7 6 5
Table 3.1: Sum 5, 0; 6, 1 Array (With Resultant Sum Cycle)
In converting this dyadic array into a triadic array, one must decide which cyclic set
to maintain and which set to divide into the appropriate additional cyclic set pair. Once this
19 Perle, George.. Twelve-Tone Tonality, p. 146.
30
decision is made a series of cyclic sets may be employed while still maintaining the desired
resultant sum cycle. All that is required is that the adjacency sums are maintained from the
dyadic array into the triadic array. The 6,1 set in the above example can be split into the
following pairs of pairs and still result in the triadic sum groups (which include sums 5,0
of the static cycle) with sums e, 1:
Split Resultant Adjacency Sums Sum 6 Sum 1 Sum e Sum 1 0, 6 0, 1 5, 0, 6 0, 0, 1 1, 5 1, 0 5, 1, 5 0, 1, 0 2, 4 2, e 5, 2, 4 0, 2, e 3, 3 3, t 5, 3, 3 0, 3, t 4, 2 4, 9 5, 4, 2 0, 4, 9 5, 1 5, 8 5, 5, 1 0, 5, 8 6, 0 6, 7 5, 6, 0 0, 6, 7 7, e 7, 6 5, 7, e 0, 7, 6 8, t 8, 5 5, 8, t 0, 8, 5 9, 9 9, 4 5, 9, 9 0, 9, 4 t, 8 t, 3 5, t, 8 0, t, 3 e, 7 e, 2 5, e, 7 0, e, 2
Table 3.2: Split of Cyclic Set into Dyadic Series
Any of the triadic groups of resultant adjacency sums from the sum e column may be paired
with any group from the sum 1 column, however the difference between each respective
member of the group will be the generating interval for the corresponding cycle. For
example if one pairs the 5, 3, 3 group with the 0, 8, 5 group the corresponding cycles will
be IS 5 (5 - 0), IS 7 (3 - 8), and IS t (3 - 5). The adjacency sums of the three cyclic sets is
only one consideration; the other and perhaps more important consideration is the resultant
vertical sum of each member of the two newly generated sets, which must still sum to the
appropriate resultant sum cycle. One must then generate a triadic sum grouping for each
of the members of the resultant sum cycle (or at least the first three to identify the actual
alignments of the 3, 8 and 3,5 cyclic sets), which must match the horizontal adjacency
sums and the vertical resultant sums.
31
Sum 8 Sum 3 Sum t Sum 1 Sum 0 Sum e Sum 2 Sum 9
3, 5, 0 2, 1, 0 t, 0, 0 7, 6, 0 5, 7, 0 0, e, 0 0, 2, 0 5, 4, 0 3, 6, e 2, 2, e t, 1, e 7, 7, e 5, 8, e 0, 0, e 0, 3, e 5, 5, e 3, 7, t 2, 3, t t, 2, t 7, 8, t 5, 9, t 0, 1, t 0, 4, t 5, 6, t 3, 8, 9 2, 4, 9 t, 3, 9 7, 9, 9 5, t, 9 0, 2, 9 0, 5, 9 5, 7, 9 3, 9, 8 2, 5, 8 t, 4, 8 7, t, 8 5, e, 8 0, 3, 8 0, 6, 8 5, 8, 8 3, t, 7 2, 6, 7 t, 5, 7 7, e, 7 5, 0, 7 0, 4, 7 0, 7, 7 5, 9, 7 3, e, 6 2, 7, 6 t, 6, 6 7, 0, 6 5, 1, 6 0, 5, 6 0, 8, 6 5, t, 6 3, 0, 5 2, 8, 5 t, 7, 5 7, 1, 5 5, 2, 5 0, 6, 5 0, 9, 5 5, e, 5 3, 1, 4 2, 9, 4 t, 8, 4 7, 2, 4 5, 3, 4 0, 7, 4 0, t, 4 5, 0, 4 3, 2, 3 2, t, 3 t, 9, 3 7, 3, 3 5, 4, 3 0, 8, 3 0, e, 3 5, 1, 3 3, 3, 2 2, e, 2 t, t, 2 7, 4, 2 5, 3, 4 0, 9, 2 0, 0, 2 5, 2, 2 3, 4, 1 2, 0, 1 t, e, 1 7, 5, 1 5, 4, 3 0, t, 1 0, 1, 1 5, 3, 1
5, 0: 3 2 t 7 5 0 0 5 7 t 2 3 9 8 4 1 e 6 6 e 1 4 8 9 3, 8: 5 t t 5 3 0 8 7 1 2 6 9 e 4 4 e 9 6 2 1 7 8 0 3 3, 5: 0 3 2 1 4 e 6 9 8 7 t 5 0 3 2 1 4 e 6 9 8 7 t 5
e, 1: 8 3 t 1 0 e 2 9 4 7 6 5 8 3 t 1 0 e 2 9 4 7 6 5
Table 3.3: Triadic Divisions of Resultant Sum cycle and a Possible Triadic Array
The most important part of the conversion process is the checking of either of the changing
sums (either the second or third of each group) against the corresponding sum of the
previous group in order to maintain the adjacency sums of the two new cyclic sets. The
other pitch class of the grouping is then automatic, based upon the desired sum in the
resultant sum scale. A brief walkthrough will illustrate:
The initial group <3, 5, 0> is chosen for its closeness to the initial vertical dyad
<3, 5> of the dyadic array. One then looks to either the 5 or 0 of this group and to the sum
3 column in order to match the two 3 sums of the new cyclic sets; in this case the pitch
classes t and 3 generate the needed adjacency sums. This procedure is then repeated using
the <2, t, 3> group and then looking to the sum t column for the pitch classes required
for the 8 and 5 adjacency sums: <t, t, 2>. This process continues for the rest of the array,
alternating between the two adjacency sum pairs as one chooses among the sum groups
32
for each required resultant sum. One final note before another example, the interval cycles
represented are 5, 7 and 2, which sum to the desired 2 cycle for the resultant sum cycle
(which also fulfills the need for odd resultant adjacency sums). The following example
might prove more musically relevant as the triadic sum retains the 5, 0 cyclic set, the 6, 1
set in an alternate alignment (1, 6) as well as a new 2 cycle:
5, 0: 3 2 t 7 5 0 0 5 7 t 2 3 9 8 4 1 e 6 6 e 1 4 8 9 1, 6: 4 9 9 4 2 e 7 6 0 1 5 8 t 3 3 t 8 5 1 0 6 7 e 2 5, 7: 1 4 3 2 5 0 7 t 9 8 e 6 1 4 3 2 5 0 7 t 9 8 e 6
e, 1: 8 3 t 1 0 e 2 9 4 7 6 5 8 3 t 1 0 e 2 9 4 7 6 5
Table 3.4: Additional Possible Triadic Array (e, 1 Resultant Cycle)
The same array in multiple alignments will generate the same resultant sum cycle, as long
as the initial vertical group is a member of the sum 8 column and the adjacency sums of the
new cyclic sets remain the same.
5, 0: 3 2 t 7 5 0 0 5 7 t 2 3 9 8 4 1 e 6 6 e 1 4 8 9 1, 6: 1 0 6 7 e 2 4 9 9 4 2 e 7 6 0 1 5 8 t 3 3 t 8 5 5, 7: 4 1 6 e 8 9 t 7 0 5 2 3 4 1 6 e 8 9 t 7 0 5 2 3
e, 1: 8 3 t 1 0 e 2 9 4 7 6 5 8 3 t 1 0 e 2 9 4 7 6 5
5, 0: 3 2 t 7 5 0 0 5 7 t 2 3 9 8 4 1 e 6 6 e 1 4 8 9 1, 6: e 2 4 9 9 4 2 e 7 6 0 1 5 8 t 3 3 t 8 5 1 0 6 7 5, 7: 6 e 8 9 t 7 0 5 2 3 4 1 6 e 8 9 t 7 0 5 2 3 4 1
e, 1: 8 3 t 1 0 e 2 9 4 7 6 5 8 3 t 1 0 e 2 9 4 7 6 5
Table 3.5: Initial Triadic Axis Chords from the Sum 3 Progression
Triadic arrays pose some difficult problems in the generation of analytically useful
K-net relationships. However there are certain aspects that are inherent in the arrays
above which may prove helpful. The array contains a pair of IC 5/7 which will have the
same K-net relations as the dyadic IC 5/7 array discussed earlier. In a single alignment
parallel and symmetrical shifts will result in strong isography while an asymmetrical shift
33
is represented by the <W> relation. Since this array has an additional cyclic set of IC 2/2
one must explore both inversional and transpositional relationships between the dyadic
array and the third cyclic set. It turns out that it is a combination of relationships that is
required in the comparison of K-nets drawn from triadic arrays. The following examples
will maintain the isographies that have been discussed regarding dyadic arrays and focus
upon the relationship between those and cyclic segments of the IC 2/2 set. Returning to
how the shifting of arrays affects isography, I will stipulate that the shifting will only occur
between the dyadic array and the third set, ensuring that the K-nets of the dyadic array will
remain intact.
The following is a generic graph for an individual axis-dyad chord drawn from a
triadic array:
Dyadic array:
Figure 3.1: Nine-node K-graph
This graph focuses only on the internal I-values within the dyadic array and the T-values
between the third cyclic set and the lower cyclic set of the dyadic array. The other
relationships are the adjacency sums of the individual cyclic sets and the cyclic interval
which will remain constant as long as the type of alignment does not change. If one applies
this graph to the initial axis-dyad chord of the above array (the last listed on the previous
page) the following results:
Figure 3.2: Initial Axis-dyad Chord from the Triadic Array
T TT
I I I
3 2 t
e 2 4
6 e 8
T7 T3 T4
I2 I4 I2
34
Once again a parallel shift from the above axis-dyad chord to a second will yield strong
isography between the two segments of the dyadic array, however the comparison of the
T-values between the third set and the lower of the dyadic set will require the <W> relation.
This example illustrates the isographic relationship between two axis-dyad chords drawn
from a shift of six positions to the right.
Figure 3.3: Triadic Axis Chords Related by a Parallel Shift of the Array
The axis-dyad chords are related by a shift of 4 positions to the right; recall that a shift of
an even number of positions maintains the type of alignment:
Figure 3.4: Triadic Axis Chords Related by a Parallel Shift of the Array
The strongly isographic relationship between paired segments drawn from the dyadic array
is expected, however the <W> relation shows an unexpected pattern if one examines all
of the parallel shifts by an even number of positions. If one begins with a shift by zero
positions (in this case an identity operation) one finds that the strong isography results:
<T0> between paired segments of the dyadic array and <W0> between segments of the IC
2. From this identity operator every shift by two positions, to either the left or the right
results in a <Wn-3> relationship (while n is the initial <W> value). This pattern arises from
the T+-2 operation inherent in the IC 2 and a special relation between the adjacency sums of
3 2 t
e 2 4
6 e 8
T7 T3 T4
I2 I4 I2
0 5 7
2 e 7
0 5 2
Tt T6 T7
I2 I4 I2
<W3>
<T0>
3 2 t
e 2 4
6 e 8
T7 T3 T4
I2 I4 I2
5 0 0
9 4 2
t 7 0
T1 T9 Tt
I2 I4 I2
<W6>
<T0>
35
the lower dyadic set and the IC 2 set. If one looks to the diagonal sums of the adjacency
sums between the lower dyadic cycle and the third set one arrives at the “key” of the array
of these two cycles.20 In this case the key is 8, e (1+7; 5+6). In the majority of Perle’s
examples, the key is measured between arrays of the same cyclic interval, however this
is not the case in this array and it is the difference between the two “key” sums, 3, which
yields the 3-cycle in the <W> relation.
From this triadic array one can draw the first ordered progression, again based upon
the sums of the resultant sum cycle.
Sum 3
3 2 t / e 2 4 / 6 e 8 8 9 3 / 6 7 e / 6 e 8 1 4 8 / 1 0 6 / 6 e 8 t 7 5 / 4 9 9 / 6 e 8 e 6 6 / 3 t 8 / 6 e 8 0 5 7 / 2 e 7 / 6 e 8 9 8 4 / 5 8 t / 6 e 8 2 3 9 / 0 1 5 / 6 e 8 7 t 2 / 7 6 0 / 6 e 8 4 1 e / t 3 3 / 6 e 8 5 0 0 / 9 4 2 / 6 e 8 6 e 1 / 8 5 1 / 6 e 8
Table 3.6: Initial Axis-dyad Chord Progression from the Triadic Array
One will imediately notice the last three columns of the progression are invariant; referring
to the resultant adjacency sum chart on page 27, one pitch class of the three is static, while
the others shift against one another for the remainder of the axis-dyad sum. The initial
three columns share the same relation to the resultant sum cycle as the first example of
ordered progressions (cf. p. 17).
• The first column exchanges pairs of adjacent sums from the resultant cycle (8-3, t-1,
e-0, etc.).
3 8 1 t e 0 9 2 7 4 5 6
20 Perle, George, Twelve-Tone Tonality, p. 47.
36
• The second column is a rotation of the resultant sum cycle beginning with order
position 7 (exchanging the hexachords).
2 9 4 7 6 5 8 3 t 1 0 e
• The third column is a rotation and retrograde of the resultant sum cycle beginning with
order position 3 (pitch class t).
t 3 8 5 6 7 4 9 2 e 0 1
• Column four is T2I of column one.
e 6 1 4 3 2 5 0 7 t 9 8
• Column five is T5I of column two.
2 7 0 9 t e 8 1 6 3 4 5
• Column six is T2I of column three.
4 e 6 9 8 7 t 5 0 3 2 1
• As noted above columns seven through nine consist of an invariant trichord completing
the axis-dyad sum 3.
Continuing the ordered progressions of axis-dyad chords (on the following page):
Sum 3 Sum t Sum 1 Sum 03 2 t / e 2 4 / 6 e 8 8 9 3 / 9 4 2 / 8 9 t 1 4 8 / e 2 4 / t 7 0 t 7 5 / 1 0 6 / 0 5 2 8 9 3 / 6 7 e / 6 e 8 1 4 8 / 4 9 9 / 8 9 t t 7 5 / 2 e 7 / t 7 0 e 6 6 / 0 1 5 / 0 5 21 4 8 / 1 0 6 / 6 e 8 t 7 5 / 7 6 0 / 8 9 t e 6 6 / 1 0 6 / t 7 0 0 5 7 / e 2 4 / 0 5 2t 7 5 / 4 9 9 / 6 e 8 e 6 6 / 6 7 e / 8 9 t 0 5 7 / 0 1 5 / t 7 0 9 8 4 / 2 e 7 / 0 5 2e 6 6 / 3 t 8 / 6 e 8 0 5 7 / 5 8 t / 8 9 t 9 8 4 / 3 t 8 / t 7 0 2 3 9 / 9 4 2 / 0 5 20 5 7 / 2 e 7 / 6 e 8 9 8 4 / 8 5 1 / 8 9 t 2 3 9 / t 3 3 / t 7 0 7 t 2 / 4 9 9 / 0 5 29 8 4 / 5 8 t / 6 e 8 2 3 9 / 3 t 8 / 8 9 t 7 t 2 / 5 8 t / t 7 0 4 1 e / 7 6 0 / 0 5 22 3 9 / 0 1 5 / 6 e 8 7 t 2 / t 3 3 / 8 9 t 4 1 e / 8 5 1 / t 7 0 5 0 0 / 6 7 e / 0 5 27 t 2 / 7 6 0 / 6 e 8 4 1 e / 1 0 6 / 8 9 t 5 0 0 / 7 6 0 / t 7 0 6 e 1 / 5 8 t / 0 5 24 1 e / t 3 3 / 6 e 8 5 0 0 / 0 1 5 / 8 9 t 6 e 1 / 6 7 e / t 7 0 3 2 t / 8 5 1 / 0 5 25 0 0 / 9 4 2 / 6 e 8 6 e 1 / e 2 4 / 8 9 t 3 2 t / 9 4 2 / t 7 0 8 9 3 / 3 t 8 / 0 5 26 e 1 / 8 5 1 / 6 e 8 3 2 t / 2 e 7 / 8 9 t 8 9 3 / 4 9 9 / t 7 0 1 4 8 / t 3 3 / 0 5 2
Table 3.7: Axis-dyad Chord Progressions from a Triadic Array
37
As in the earlier nine-node K-net example, the network compares only the inversional
relationship between the cyclic segments of the dyadic array and the transpositional
relationship between the third set and the lower of the dyadic array. The following figure
relates the first three axis-dyad chords of the sum 3 progression; each is related to one
another via the same set of transformations.
Figure 3.5: Initial Triadic Axis Chords from the Sum 3 Progression
The following K-nets of the initial axis-dyad chords are drawn from the series of resultant
sum progressions:
Figure 3.6: Triadic Axis Chords from Adjacent Sum Progressions
The <W> relationships of the dyadic array and the third set consistently sum to 9 (interval-
class 3) recalling the special key relationship within the adjacency sums of the generating
triadic array.
3 2 t
e 2 4
6 e 8
T7 T3 T4
I2 I4 I2
8 9 3
6 7 e
6 e 8
1 4 8
1 0 6
6 e 8
T0 T5T8 T1T9 T2
I2 I2I4 I4I2 I2
<W5> <W5>
<T0> <T0>
3 2 t
e 2 4
6 e 8
T7 T3 T4
T8 T0 T6
8 9 3
9 4 2
8 9 t
1 4 8
e 2 4
t 7 0
Te TeT7 T7T8 T8
T1 TtT5 T2Te T8
<W4> <W0>
<W5> <W9>
38
CHAPTER 4: FROM COMPOSITIONAL SPACES TO COMPOSITIONAL DESIGNS
Once the construction of a compositional space is complete there are numerous
methods for translating the interrelationships found in the pre-compositional charts into
musical structures of pitch, rhythm and timbre. This “composing out” of the compositional
space can be direct into a finished manuscript or it can generate an interim compositional
design consisting of varying levels of specificity. One must also consider what relationships
are important regarding the transformational networks of the preceding discussion. I
will present two compositional designs which will demonstrate different methodological
approaches to the translation of space into design.
Design 1:
The first design treats the ordered progressions as a six-voice chorale. Each
voice follows the columns of the ordered axis-dyad progression exposing the rotational
relationships between the pitch-class series. As the chorale continues each resultant sum
progression is rendered as a single harmonic area governed by a single inversional sum.
Rhythm in this design is related directly to the pitch class number, but in three different
strands. The first and “top” strand treats C (pitch-class 0) as a dotted-eighth note; C# (pitch-
class 1) adds another basic dotted-eighth unit for a dotted-quarter. Each subsequent pitch
class adds another basic unit so that for example B (pitch-class 11) is the longest note value
of eleven units of dotted-eighth length. The entrance point of each voice is determined by
the length of the first pitch in each series. The generating array:
2,3: 2 0 3 e 4 t 5 9 6 8 7 7 8 6 9 5 t 4 e 3 0 2 1 1 e,0: 4 7 5 6 6 5 7 4 8 3 9 2 t 1 e 0 0 e 1 t 2 9 3 8 1,3: 6 7 8 5 t 3 0 1 2 e 4 9 6 7 8 5 t 3 0 1 2 e 4 9
Table 4.1: Design 1 - Generating Array
39
The first two ordered progressions: Sum 7 Sum 8 2 0 3 / 4 7 5 1 1 2 / 4 7 5 1 1 2 / 5 6 6 0 2 1 / 5 6 6 0 2 1 / 6 5 7 3 e 4 / 2 9 3 3 e 4 / 3 8 4 t 4 e / 7 4 8 t 4 e / 8 9 3 5 9 6 / 0 e 1 5 9 6 / 1 t 2 8 6 9 / 9 2 t 8 6 9 / t 1 e 7 7 8 / t 1 e 7 7 8 / e 0 0 6 8 7 / e 0 0 6 8 7 / 0 e 1 9 5 t / 8 3 9 9 5 t / 9 2 t 4 t 5 / 1 t 2 4 t 5 / 2 9 3 e 3 0 / 6 5 7 e 3 0 / 7 4 8 2 0 3 / 3 8 4
Table 4.2: Design 1 - First Two Axis-dyad Chord Progressions
The column rotations and transformations are the focus of this design, which will be briefly
described below:
• The first column is a rotation and retrograde of the resultant sum cycle beginning with
order position 9 (pitch class 2) and moving left.
2 1 0 3 t 5 8 7 6 9 4 e
• The second column is a rotation of the resultant sum cycle beginning with order
position 7 (pitch-class 0).
0 1 2 e 4 9 6 7 8 5 t 3
• The third column exchanges adjacency sums of the resultant sum cycle beginning with
order position 6 (pitch class 3); the other of the 3-0 pair is found at the end of the series.
3 2 1 4 e 6 9 8 7 t 5 0 (3)
• The fourth through sixth columns are operations upon an inversion of columns one
through three.
Column one: T0: 2 1 0 3 t 5 8 7 6 9 4 e
Column four: T6I: 4 5 6 3 8 1 t e 0 9 2 7
40
Column two: T0: 0 1 2 e 4 9 6 7 8 5 t 3
Column five: T8I: 7 6 9 4 e 2 1 0 3 t 5 8
Column three: T0: 3 2 1 4 e 6 9 8 7 t 5 0
Column six: TtI: 5 6 3 8 1 t e 0 9 2 7 4
The second progression, sum 8, exhibits very similar rotational and inversional relationships,
as well as the same type of K-net relationships as described earlier in this paper within and
between the ordered progressions.
Design 2:
The second design treats the constituent elements of axis-dyad chords individually
in order to showcase the symmetry of the cyclic set array paradigm. The axis dyad is
projected followed by the remainder of either the remaining cyclic chord or the full axis-
dyad chord. In a similar manner to Berg’s master array this design consists of all twelve
progressions presented simultaneously in twelve “voices;” in this case the term voice refers
to a line of music for each of the twelve progressions alternating between an axis dyad and
its full axis-dyad chord.
Treatment of rhythm is similar to the first design in that the length of notes is related
to pitch via a numerical parallel to pitch-class numbers. However each voice is governed
by a single sum from the resultant sum cycle and it is this number which determines the
length of each note-event throughout the design. In this case the basic unit is the sixteenth
note so that all the note events governed by sum 0 will be a dotted-half note apart. While
the sums from the resultant sum cycle guide the length of note events (or the amount of
time between them), it is the actual pitches of the axis dyad which dictate the number and
type of note event. Specifically the top pitch of an axis dyad yields the number of note-
events consisting of the axis dyad only, while the bottom pitch of the axis dyad yields the
number of note events consisting of the related axis-dyad chord. Each voice then continues
through the remainder of the progression. In order to create a building saturation of the
41
axis-dyad chord progressions, the end of all the series are aligned and the beginnings are
staggered. The following is the generating array for the design:
5,0: t 7 5 0 0 5 7 t 2 3 9 8 4 1 e 6 6 e 1 4 8 9 3 2 6,1: 1 5 8 t 3 3 t 8 5 1 0 6 7 e 2 4 9 9 4 2 e 7 6 0 e,1: e 0 1 t 3 8 5 6 7 4 9 2 e 0 1 t 3 8 5 6 7 4 9 2
Table 4.3: Design 2 - Generating Array
The following are the first six ordered progressions from this alignment (the others are a T6
transposition); the resultant sum series follows the cycle in retrograde:
Sum 0 Sum e Sum 2 Sum 9 Sum 4 Sum 7t 7 5 / 1 5 8 e 6 6 / 1 5 8 0 5 7 / 9 9 4 9 8 4 / 5 1 0 2 3 9 / 5 1 0 7 t 2 / 9 9 4e 6 6 / 0 6 7 0 5 7 / 0 6 7 9 8 4 / 0 6 7 2 3 9 / 0 6 7 7 t 2 / 0 6 7 4 1 e / 0 6 70 5 7 / e 7 6 9 8 4 / 3 3 t 2 3 9 / 7 e 2 7 t 2 / 7 e 2 4 1 e / 3 3 t 5 0 0 / e 7 69 8 4 / 2 4 9 2 3 9 / t 8 5 7 t 2 / 2 4 9 4 1 e / t 8 5 5 0 0 / e 7 6 6 e 1 / t 8 52 3 9 / 9 9 4 7 t 2 / 5 1 0 4 1 e / 5 1 0 5 0 0 / 9 9 4 6 e 1 / 1 5 8 3 2 t / 1 5 87 t 2 / 4 2 e 4 1 e / 8 t 3 5 0 0 / 4 2 e 6 e 1 / 8 t 3 3 2 t / 4 2 e 8 9 3 / 8 t 34 1 e / 7 e 2 5 0 0 / 7 e 2 6 e 1 / 3 3 t 3 2 t / e 7 6 8 9 3 / e 7 6 1 4 8 / 3 3 t5 0 0 / 6 0 1 6 e 1 / 6 0 1 3 2 t / 6 0 1 8 9 3 / 6 0 1 1 4 8 / 6 0 1 t 7 5 / 6 0 16 e 1 / 5 1 0 3 2 t / 9 9 4 8 9 3 / 1 5 8 1 4 8 / 1 5 8 t 7 5 / 9 9 4 e 6 6 / 5 1 03 2 t / 8 t 3 8 9 3 / 4 2 e 1 4 8 / 8 t 3 t 7 5 / 4 2 e e 6 6 / 8 t 3 0 5 7 / 4 2 e8 9 3 / 3 3 t 1 4 8 / e 7 6 t 7 5 / e 7 6 e 6 6 / 3 3 t 0 5 7 / 7 e 2 9 8 4 / 7 e 21 4 8 / t 8 5 t 7 5 / 2 4 9 e 6 6 / t 8 5 0 5 7 / 2 4 9 9 8 4 / t 8 5 2 3 9 / 2 4 9
Table 4.4: Design 2 - Axis-dyad Chord Progressions
CONCLUSION
The ordering principles outlined in this paper aim to take advantage of the depth
of George Perle’s twelve-tone tonality while also exploring an inherent and specific
relationship with serial treatment of a twelve-note set. One can enjoy the freedom of
unordered axis-dyad chords which can be ordered at a further background level by a series
drawn from the original array. This ordering paradigm attempts to capture a structured
freedom from the wealth of compositional possibilities described by Perle in his many
analyses and theoretical writings.
42
Exam
ple 3: Com
positional Design 1
#
b
b
#
b
#
#
b
b
3
# 3
3
b3
b
3
b
3
#
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10
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b
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3
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# 3
#
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3
3
COMPOSITIONAL DESIGNS
43
Exam
ple 4: Com
positional Design 2
150
sum
0
sum
e
sum
2b
##
##
sum
9
sum
4
## ##
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sum 7
#
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44
BIBLIOGRAPHY
Antokoletz, Elliott. “Interval Cycles in Stravinsky’s Early Ballets.” Jounal of the American Musicological Society, Vol. 39, No. 3, (Autumn, 1986), pp. 578-614.
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Headlam, Dave, “Perle’s Cyclic Sets and Klumpenhouwer Networks: A Response,” Music Theory Spectrum, Vol. 24, No. 2, (Autumn, 2002), pp. 246-258.
-------------------, Perle/Lewin Seminar, Department of Music Theory, Eastman School of Music, Spring, 2005.
Klumpenhouwer, Henry, “Aspects of Row Structure and Harmony in Martino’s Impromptu Number 6,” Perspectives of New Music, Vol. 29, No. 2. (Summer, 1991), pp. 318-354.
Lambert, Phillip, “Isographies and Some Klumpenhouwer Networks They Involve,” Music Theory Spectrum, Vol. 24, No. 2. (Autumn, 2002), pp. 165-195.
Lewin, David, “Klumpenhouwer Networks and Some Isographies that Involve Them,” Music Theory Spectrum, Vol. 12, No. 1. (Spring, 1990), pp. 83-120.
-----------------, “Thoughts on Klumpenhouwer Networks and Perle-Lansky Cycles,” Music Theory Spectrum, Vol. 24, No. 2. (Autumn, 2002), pp. 196-230.
Morris, Robert D., “Voice-Leading Spaces.” Music Theory Spectrum, Vol. 20, No. 2, (Autumn, 1998), pp. 175-208.
O’Donnell, Shaughn J., “Transformational Voice Leading in Atonal Music,” Ph.D. diss., City University of New York (1997).
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