Chapter 3

Introduction to Pitch-Class
Set Theory
;'q t'^c{
il,t\,
In Chapters I and 2 we have introduced and r"rsed the concept of pitch-class set. We
have seen that this concept is particularly useful to label small pitch-class collections
and to compare and relate such collections among themselves. We have not only dem-
onstrated how the system can be used analytically to uncover relationships that have
immediate musical interest, but also that it would be much more difficult to discover
some of these relationships without recourse to pitch-class set theory. In the following
chapters we will study music in which pitch centricity is normally avoided, and which
is often based on collections of pitch classes, used both motivically or as harmonic
building blocks. A general term to refer to such music is "atonal." To deal with this
type of music analytically, however, we will need to have a better grasp of pitch-
class set theory. Although set theory has been used in recent years as a compositional
method, it was originally formulated (by Milton Babbitt and Allen Forte) in the 1960s
and 70s, as a theoretical and analytical system particularly applicable to the study of
atonal music. We will devote the present chapter to a more systematic and formal pre-
sentation of set theory, after which we will be better prepared to succeed in our studies
of atonal music.l

lTwo pioneering articles in set theory are Milton Babbitt, "Set Structnre as a Compositional Determi-
nanr,".lourncrl of MusicTheory 5 (1961):12-94: reprinted inThe Collected Essc,ys of Milton Babbitt,
cd. Stepl-ren Pclcs, Stcphen Dombski, Andrew Mead, and Joseph Straus (Princeton: Pt incetou University
Press,2003), pp.86-108;and Allen Forte,'A Theory of Set-Cornplexes forMusic," Journolctf Mlusic
Theor1,8(1964): 136-83.Extensivepresentationsof settl'reory,towhichthischapterisindebted,can
be found in Allen Forte,The Structure of Atonal Music (New Haven: Yale University Press, 1973): John
Rahn, Ba.ric Atontrl Theory (New York: Schirmer, 1980); and Robert Morris, Compositiort with Pitch-
Classes: ATheory of Compositionol Design (New Haven: Yale University Press, 1987), and Class Notes
For Atonal Music Theory (Hanover, NH: Frog Peak Music, l99l). See also Joseph Straus, IntrodLtctiort
to Post-Tonal'[heory. A very irnportant book that plovicles a diff'erent ancl,compelling perspective on the
matters and materials studied in our Chapters 3 and 4 is David Lewin's Generalized Musical Intervals
unclTransformatiorts (Nerv I{aven: Yale University Press, l987).

69

F
 70 CHAPTER 3

PREI-,IMINARY CONCEPTS
Wearealreadyfamiliarwithsomeofthebasicprinciplesofpitch-classsettheory.As
with the same name' in any octave'
we know, a pitch class (pc) is a group of pitches
of octave equivalence: in set theory we
The concept of pitch class implies the p.op.tty
pitches with the same name' For instance'
clo not distinguish between octave-relatei
Cfi in any octave is equivalent to uny
any
oth"' Cil in any other octave' and they are all
the property of enharmonic equivalence
members of the same pitch class. Moreover,
also applies to pitch-class set theory. Although
infunctional tonality a cil and a Dl have
enharmonically spelled pitches
different functions and cannot be used interchangeably,
That is, C$ and Db are equivalent for
in set theory are members of the same pitch clasi.
our purposes' 'ng
pitch classes
:-+-^r"^ol thp nnnr-ent inteper notation.
We have also introduced the concept of integer
n By notatt
set operations much. more efficiently'
as integers we will be able to realize pit"h-.lurs That
with pitches and pitch classes.
Always think, however, that you are still dealing number (in the
a letter (F, for example) or a
is, whether you represent a pitch class with
caseofF,5),wearestilltalkingofwaystorepresentorlabelanessentialmr-rsicalele-
there are two systems of using integers
ment (a pitch class). we have also learned that
we assign integer 0' by convention' to
,t
to cJenote pitch classes. In the "fixed do" system,
and D[b) and go up the chromatic set
pitch-ciass C (or its enharmonic eqr-rivalents, B{
the "moveable do" system' integer
of pitch classes, counting half steps up from C' In
i
ri
set, no matter what that may be' In this
:i
0 will be assigned to the first pitch class of the
chapter we will use each of these systems for
diftbrent purposes' For the time being'
we will begin with the fixed ,lo ,yrt.,-,t. Figure
3'l shows the integer and pitch-class
(as a reminder, let's note again that en-
equivalences that we are already familiar with
ancl Gb' are represented by the same
harmonically equivalent pitch classes, such as Ff
integer).
twelve different pitch classes, although
Becattse of octave equivalence we have only
solve the cliscrepancy by doing all
pitch takes place in many clifferent octaves. we will
(abbreviation for ntoclulo 12)'
of our pitch-class set operations tn mod 12 arithmetic
(frorn 0 to 1l), and any integerlarger
Trris means that we will'se onry twelve i'tegers
equivalent integer within these twelve'
than 1r (or smarler flran 0) will be reduced to its
a single octave, and any pitch outside
To put this in musical terms, we will operate in
our octave. Thinking of a clock lace
this octave will be recl'ce d to its eclr,ival-nt within
(seeFigure3.2)willhelpyouunclerstandntocl]2.Ifyorrthinkofmidnightasbeing
to 0' 13 is equivalent
l2hours you begin the cycle again: 12 is equivalent
0 hotrrs, after
to its rnocl 12 cquivalentby
to l, to 2, 1-5 to 3, ancl so on. you can reduce any integer
14
given integer' For instance' 16 - 12 - 4
subtracting 12 (or a mLrltiple of 12) from the is3in
(so 16is4in nt,cl 12), lg - 12=7(so 19is7 rnrnctcll2),anr127 -24=3(21
mod l2).

c-0 c{-1 D-2 Dil-3 E-4 F-5 F$-6 G-7 Gil-8 A-9 Af-10 B-11
pitch classes
Figtrrc 3.1 Integer notation fbr thc twelve
 Introduction to Pitch-Class Set Theory 1I

Figure 3.2 The twelve pitch classes on a clock face

U=0 m2-l M2-2 m3=3 M3=4 P4-5 +4-6

P5-7 m6=8 M6=9 m7=10 M7=11 P8=12
Figure 3.3 Integer representation of intervals (by semitone content)

Interval and Interval Class

Similar to the way we represent pitch classes by integers, intervals can also be repre-
sented by integers. An integer representing an interval refers to the number of semitones
in that interval. Enharmonically equivalent intervals (such as an augmented fourth or
a diminished fifth) will be represented by the same integer. You can easily determine
the number of semitones for each interval by looking at the clock face in Figure 3.2 and
thinking of all intervals ascending from C. Figure 3.3 shows the number of semitones
for each interval (without listing all enharmonic equivalences). You should memorize
the integer representation of intervals.
Intervals in the realm of pitch classes (which we will refer to as pitch-class space)
are of a different nature than intervals in the realm of pitches (or pitch space). In pitch
space, intervals are directional (ascending or clescencling), and an interval and its in-
version are uot the saure. We will represent ascending motion with a + sign before thc
interval integer, and descending rnotion by means of a - sign. Thus, we can speak of
an ascending P4 from C (+5) or a descending P4 from C (-5). Moreover, an ascending
P4 from C, (C-F, +5), is not the samc as a descending P5 frorn C, (C-F, -7). We will
r-rse pitch space and directional intervals to describe actual pitch motion on the musical
surface. For instance, we know that Debussy's "The Sunken Cathedral" begins with the
ascending motive D-E-B, or (+2, +7). Moreover, intervals in pitch space can be simple
or compound. Thus, an ascending major tenth (+16) is not the same as an ascending
major third (+4).
In pitch-class space, on the other hand, intervals may still be ordered (if we take
into accot"tnt the distance between two pitch classes considered in a particular order) or

!
 72 CHAPTER 3

B-E=5
- 1r(B)

4(E)

E-B=J
Figure 3.4 Ordered pitch-class intervals

but
shortest distance between two pitch
classes),
unordered (if we take into account the their "simple"
we will reduce alr intervals to
in alr cases they win be mod 12; thatis, pitch classes
equivalents. An ordered pitch-crass
interval is the distance between two
direction)'
(and, by convention' always in an ascending
considered in a particular order
two pcs' subtract the first pc from the
To determine the ordered pc interval between E and B
second in mod 12. lnthe Debussy
motive, the ordered pc interval between
in
context' the same i*o ptt could be
presented
(4 and 11) is lI - 4 -J.lna different
11=5(thatis,16-11=5
theorderB_E,anclthentheorderedpcintervalwouldbe4- the
pc interval on a clock face, always measure
tn mod t2). Todetermine un o.d.."i 3'4'
(that is, ascending), as shown in Figure
distance between pcs clockwise
Morerelevantinpitch-classsettheoryistheco.n::ptol""::d,.redpitch.class
in which octave equivalence applies'
interval (also called interval class). In a space of a
and we can represent both by means
interval E-B is equivalent to interval B-E, of pitch classes, but
not the direction or order
single integer. wrrat matters to us here is pitch classes E
classes. The distance between
the shortest span between the two pitch smallest of these'
and B tn mocr 12 canbe 11 - 4 -_7
0r 4 - 11 = 5. we will choose the
an
interval between E and B' To determine
5, to represent the unorderecl pitch-class two
unorclered pitch-class interval on a
clock face' we measure the distance between
pitchclassesfollowingtheshortestpossiblepath,beitclockwiseorcounterclockwise,
as shown in Figure 3'5'
Inotherwords,invertedintervalsareequivalentinunorcleredpitch_classSpace. the
Because of octave equivalence, the
major :ri C-f is equivalent to its inversion'
an!.its inversion in one singre category'
minor 6th E_C. we can then group un i"nt"ruaf
whichwecallintervalclass(abbreviatedasic).Therearesevenintervalclasses,and
is made up of intervals 0
3'6' Thus' ic 0
their intervallic content is presented in Figure
t'p of intervals I and 11 (m2 and M7)'
and so
and ]2(unison and octave), ic 1 is made th:t-^d1:t::
class are complementary (that is'
on. Because two intervals in an interval
each ic add up to l}'The only ic that contatns
an octave), the two integers that represent
 Introduction to Pitch-Class Set Theory 73

B-E=E-B=5
11(B)

4(E)

Figure 3.5 Unordered pitch-class interval

Interval class Intervals

0 0, 12

1 1, 11

2 2, lo
a
-) 3,9
4 4,8
5 5,J
6 6

Figure 3.6 The intervai classes

a single interval is ic 6, because the tritone inverts into itself. We can now also see that
the intervals from the preceding examples are grouped as ic 5 (intervals 5 and J,P4 and
P5) and ic 4 (intervals 4 and B, M3 and rn6).

PITCH.CI.ASS SETS; GRDERTNGS

AND BASIC RELATIONSHXPS
A pitch-class set (pc set) is an unordered collection of pitch classes. Although the
pitch-class members of a pitch-class set may appear in any order on the actual musi-
cal surface (and this is what we mean by "unordered collection"), we will have to find
a system of organizing them in a way that will allow us to compare different sets. We
have already seen this principle applied in Chapters 1 and 2, as when we realized that

l. 1:
l.
 74 CHAPTER 3

in different
in which pitch classes were ordered
various motives from Stravinsky's Rite'
set class, (0251).
ways, all were members of the same

NOTE
introducecr in this sectio,, we can thinkfor instance
To herp us uncrerstancr the concepts
we present them as C-E_G, E-G_C, G_C-g, or aS
of pitch classes G-E-C. Whether "t^!1
a variety of other possible orrteringi,'*'
in'*'diately recogniz'e this collection 11
root posb
orcler all these collections as
c mojor triad" (in other words, we mentally taken
tion c maior triacls). In this
*, noi,
proces.'s,' for granted that-the "'!': "f .l::::
crcLsses does ttot- matter, an',d
we have easity icrentified all these dffirent Srouplngs
pitch-cLass
Moreover' we can also think of the
witlt a single term, "the c ntajor triacJ." that they all belong to
B-D-G, ora i-r-A and ectsily recognize
couections c_E-G, ,,major re'cognize the col.
ancl
the same CateSory, triads.,, We Can 8o one step farther
.lriacrs," ctl'triottgh these couections inclucle
rections c_E_G, G_B_E, anct E-A-c
as
we can cro att of this becarrse ,ve have a werl-establishetl
botrt major attd minor triads. to this system' vve
uncJ tertictn sonorities- Thanks
system to icrentifi l"i Irirt triacrs
catliclentifyandcompareterttcrltstlnoritiesandSrouptltemintosimilctrcategories.Itt We neecl
with nontertictn' nontriaclic colle-ctions'
tnost at,ncrl mttsic, however, we cleal to contpare col'
label any collection ancl
rc rlefine a similar system that wi-ll allow r'ts
to
rs exactly ivhat pitch-class set theory does'
Thlis
lectiors for similority or di,ssimilrr;;";'
''t
i nS we
,,.s will le0fn
We WIIL in LrLe
leorn IrL the JULLrrvvL'16 pages.
following y-"6""'
:!,

as its cardinal number' A set

with a
The number of elements in a set is known with a czrrdinal
and it is called a trichord. A set
cardinal number 3 has three elements, pentachord' one with a
a carclinal number 5 is a
number 4 is a tctrachord. A set with
with cardinal numbers 7 (septachord),
8
cardinar number 6 is a hexachord, and so on (monad
have cardinal numbers I and 2
(octachord), and 9 (nonachord). Sets may also
numbers l0' I l' and these five cardinalities
l}'but
and dyad, respectively) and cardinal
sets'
hau" less musical significance as unordered
(

Normal Order them

collections of pitch classes is to atrange
The first thing we need to do to compare uormal order (N'o')
such a'ordlri'g the
in the salrre, standzrrd ordering. we will call
Anorm.alorderisanarrangittgofllitchclctssesitl,ttscentlingltttmericctlsuccesslrin, pitch
cutcr in such ctway trtut trtey cover.trrc
srrtrtest possibre s1tct,. Take, for instance,
form of this colleciic'n is t0' 5' 2' 1' Exalrple 3'1
classcs Bt, F, I), ancl Dt. The numerical musical and
the shortest possible span with both
illustrates the complete process to find ordel all within
the pitch classes in ascencli'g
numericar notations. we hrst orrong" example' l'2' 5' t0'
the pitch.classes' as in' for
the same octave , beginning with any"of circr-rlar permuta-
to achieve all the possible
Then we neecl to rotate this arrang"."n, per-
(flr a set of n elements' there are n such circular
tions of this ascending ordering To rotate a
there wi* be four orderings)'
mutations or orclerings, so for our tetrachorcl leave the remaining
place the first element last' ancl
set (an operation we cail rotation),
 Introduction to Pitch-Class Set Theory 75

Rotation of a set

r2510 2510 1 10r25

9 11

elements as they are. Keep repeating the same process until you end up with the origi-
nal ordering again. As shown in Example 3.1, the four orderings of our set are (1,2,5,
l0), (2,5, 10, l), (5, 10, 1, 2), and (10, 1, 2,5). One more permutation would produce
otrr original arrangement, (1,2,5, 10). After we have all the possible orderings, we can
determine which one covers the shortest possible span. We can do so by looking at the
intervals between outer pitches (in our example, M6, M7, M6, and P5), or by subtract-
ing the first pc integer from the last (intervals 9, 11,9, and 5). We see that the shortest
span between outer pitches in our example is provided by the fourth ordering, so this
arrangement will be our normal order, which we represent in brackets: [0, 1,2,5).
It is not necessary, however, to go though all the orderings one by one as we have
done for this example (a very cumbersome process when we are dealing with large sets).
A more efficient procedure to determine the normal order of a pc set is as follows:
1. Arrange the pitch classes in ascending order, with the octave on top.
2. Look for the largest interval (computed as an ordered pitch-class interval) between
two adjacent pitch classes and relist the set starting with the upper pitch class of the
largest interval.
3. If more than one possible ordering results from the two plevious steps, you should
choose the ordering that is most closely packed to the left. To do so, first check the
interval between the first and the next-to-last pcs, and choose the ordering with
the smallest such interval. If that still results in a tie, check the interval between the
first and third-to-last pcs, and so on.
4. If it is not possible
to determine a normal order from step 3 because all steps pro-
duce ties, choose the orclering that begins with the smallest pc number.

Let's go back to our collection from Example 3.[:B!, F', D, Dt or 10, 5,2,I. We f,rst
list it in ascending order with the octave on top: L,2,5,10, l. We check ordeled pitch-
class intervals between each of the adjacent pitch classes (by subtracting each pc from
the pc to the right of it). and we see that the largest interval js 5 between pcs 5 and 10.
We relist the set starting with the upper pc of the largest interval, tlrat is, starting with
10, and the resulting ordering is our normal order, [10, 1,2,5). This process is shown
in Figure 3.7a. You can also visualize the procedure on a clock face, following the same

I
t6 CHAPTER 3

b.
a.

Bb-F-D-Db
10 5

:
| 2 5 10 1

VVV
1 3c
V
3 Largest
Interval
[10, 1 ,2, 5f
Figure 3'7 Normal order

Then' find the

steps, and as shown in Figure 3.7b.
First, list the pcs on the clock face'
and read the set clock-
largest interval (in our "xlmpl", the
interval between 5 and 10)'
is the N'O' for
iargest interval' [10' 1' 2' sJ'which
wise beginning on the upper pc of the
tntt bel2'3' 6' 81) and
(the normal order shoulcl
ill* rry the process with sets 8,3,2,6
r0,j,z,g,r(normalorderu,g,r0,I,zl).Ifyoutrywithset0,4,8,9,11'however'you or 8'
po'sible normal orders' 4' 8' 9' 11' 0'
will discover that steps 1 and 2 resulf in t*o between two pcs' interval 4
laigest interval
9, 0, 4, because there are two cases of
ll,
in Figuie 3'8' We thus need to determine which
between both 0-4 and 4-8, as shown check the interval
packe"d to the left. To do so, we
of these two orderings is most crosely
of the two orderings, this interval
between the first and the next-to-larip"r. For the rirst
is11_4_T.Forthesecondordering,itis0-$=12-8=4.Thesecondordering,
the smallest interval between the
normal orcler because it has
[8, 9, 11, 0, 4], is thus the
first and the next-to.-last Pcs'
Try the process with set 1,4,J,10. You
will lind that all four orderings feature the
(alr orderings are
same outer interval, and moreover
that step 3 does not break the tie
that
io appty step 4 arncl choosc the ordering
eq,ally packecl to the left). We the. nec6
begins with the smallest pc number,
which in this case is [1, 4' 7, 10].

Transpositional Equivalence
compar-
into a standard ordering, we can start
Now that we know how to arrange sets To
First we will consicler transposition.
ing diff,erent sets for equivalence relationships. first be arranged in
(Tn), the sets must
compare sets for transpositional equivalence Qre trons-
the satne ntunber of pitch classes
normai order. Two pitch-clcLss sets vvitlt
onto one anotlter (that is' they can
be
rnapped
positionally equiva'lent if they catt be opera-
the same nutmber (or transpositional
transformed into one another) by aclcling both
for instance' sets l2'3'J '8'gland [0'1'5'6J]'
tor) to each pitch class in thesel. Take,
 Introduction to Pitch-Class Set Theory l7

11

0^ 4 8 9 11 0
VVVVV
@@ 1

:.:___-:/11
l
i 9 11
11 |

------t---/

[8, 9, 1r,0,4]
Figure 3.8 Normal order for a set with multiple shortest-span orderings

already in normal order. Is there any number we can add to each pc in the first set to
map the first set onto the second set? Notice that, by convention, we will transpose the
first set into the second (and not the other way around), and we will use a positive trans-
positional operator (and not a negative one). By adding 10 to each pc in the first set, we
come up exactly with the second set. The second set is thus a transposition of the first
set, and the transpositional operator is 10 (t = 10). We can say that the second set is T,o
of the first set. Note that, by convention, we say that t = l0 (a positive value) and not -2
(a negative value). Try now determining whether the following pair of sets (all of which
are already in normal order) are T eqr-rivalent, and what the transpositional operator is
in each case: [0,1,3,4,6f and [2,3,5,6,8]; andl].,3,6,8] and [0,1,4,6].
To transpose a set, add the transpositional operator to each pc. For instance, to
transpose 12,3,6,81by T z you will add 2 to each pc of the set to produce [4,5,8,10]. You
can also colrpare two sets lbr transpositional equivalence by comparing their adja-
cency interval series, that is, the set of ordered pitch-class intervals between adjacent
pitch classes. In transpositionally equivalent sets, the adjacency interval series will be
the same, as shown in Figure 3.9. We will represent the adjacency interval series (AIS)
as a series of integers inside angles, as <1,4,1,1> for the set in Figure 3.9.2
The nrusical significance of tr anspositional equivalence is illustrated by the open-
ing measures of Webern's Five Movements for String Quartet, op. 5, III (1909), a piece

2For an article that discusses the properties anrJ implications of intervallic successions such as the AIS,
see Richard Chrisman, "Dcscribing Structural Aspects of Pitch-Sets Using Successive-Interval Arrays,"
Journol of Music Theory 2lll (1917): l-28.

gl-r

D
78 CHAPTER 3

Tro

'VVVV' 'Vvvv
12,3,7 , q.,,?l [q, ] , 1; q,71
t4 it r411
Figure3.gAdjacencyintervalseriesintranspositionallyequivalentsets

op. 5, III, mm. 1_6

Anton Webern, Five Movements for String Quartet,

12.3.71
)ro
r r,o.+1 (
il 0,0,2,a.s.611 Trr J

z---> -
[3,6,7]\
A
lll,2,3l [7,10'11]
Is,7.9,I I.0.1]/
)r'
[4,7,8] [3,6,7]
6
lo,
,7
t0.21lr'o
t0,3,41{
)T"

16 t4,r,s1)r.r
Sehr bewegt I 2

of this piece shown in

that we will study in some detail in Chapter 4' The fragment
and r-rumberecl. Normal orclers for ail these
Exan1ple 3.2 incrudes twelve sets boxed
are the transpositional equivalences among
sets are provided on the example, and so
acljacent sets. we see that sets are groupecl
in pairs or in gror'tps ol three transposition-
by Tr' sets 3 by Tz'sets 5 and
ally equivalent sets. Thus, sets 1 andZare relatecl i"tll
chains of tirree trichords each' relatecl re-
6 by T,,. The example concludes with two
10-1 l-12)' considering nonadjacent
spectively by TnrT,n (sets 7-8-9) and Tn-To.(sets
ttuntpositionally equivalent'
trichords, sets 1 ,2,'\,6, 10, 11, and 12 aie uil

I ttversionlrl F,qtrivalence
(and hence musical) property of trans-
As we have just seen, an important perceptual
property applies also to inversional
position is that it preserves intervals. The salne
a:
:
:.

.::

lntroduction to Pitch-Class Set Theory 79

Transposition and inversion

[8,9,0] [0,1,4] [8,9,0] [0,3,4] [8,9,0]

\'\---,
12,5,61 1] il,4,51
[7,8,1 \\\---, 1]
[7,8,1\'\---J 12,5,6)
\----, \-_-J
T4 Tol TzI Tol Tr I

equivalence. Example 3.3 will help you understand the concept of set inversion. In 3.3a,
a trichord is transposed by To. Both forms of the trichord present the same intervals
fbllowing the same contours (that is, up or down direction): minor second up, minor
third up. In 3.3b, however, we still have the same intervals, but now the second trichord
featr,rres them with their contour (up or down) reversed with respect to the first trichord.
Thus, we can say that the first trichord is rnade up of a minor third followed by a minor
second down from C, and the second trichord is made up of a minor third fbllowed by
a minor second r-rp from C. We have inverted the first trichord around the note C, and
thLrs the two sets are mirror reflections of each other around C. By convention, our
basic inversion will be around C, or pitch class 0. In our example, because the inver-
sion is around C, no'transposition is involved (or, rather, there is a transposition at the
unison, or at the 0 level). Hence the ToI label. If you take the already inverted trichord,
C-Et-Eh, and you transpose it up a major second as in Example 3.3c, you end up with
the trichord D-F-Ffi, which is a transposed inversion of or-rr original trichord, and thus
has the label TrI. Similarly, the sets in Example 3.3d invert around C or pc 0 (think of
C as the axis of syrnnletry between the two sets), and thLrs the inversion is ToI. In 3.3e,
we transpose the second set, [,4,-5], up a minor second to 12,5,6f, so the inversion is
nor,v T,L
We can then define a new type of set eq,nivalence, inversional equivalence (T,,t).
Two pitch-clcLss sets nt'e irtversionally equivcLlent if they cutt be mappecl onto one on-
other by inversion Jbllctwecl by transposition. We know that in Example 3.3b, set 2 is
an inversion of set l. We can verify this inversion by means of a simple procedure. ln
inversionally eqLrivalent sets, the adjacency interval series will be mutLrally retrograd-
able in some of the set's ordering (usually the N.O., but not necessarily). FigLrre 3.10a
shows that if we put the two sets from Example 3.3b in normal order, the respective
adjacency interval series for these two sets are <1,3> and <3,1>. These two AIS are
mutLrally retrogradable (they are the retrograde of each other), so these two sets are
inversionally equivalent. Now let us compare the two larger sets in Figure 3.10b, al-
ready given in N.O., 12,4,5,7,8] and [2,3,5,6,8]. The adjacency interval series for these
sets are <2,1,2,1> and <1,2,1,2> respectively; that is, they are the retrograde of each
other, hence the two sets are inversionally equivalent. Now show that sets [11,0,3,5] and
14,6,9,I01 are also inversionally equivalent.
80 CHAPTER 3

b.

AbA C E EbC
8,9,0 4,3,0
12.3. 5, 6, 8]
No t*dvor rvvr .12. 4,5, 7, B]
VVVV 'VVVV
AIS I3 31 2121 r 212
sets
in inversionally equivalent
Figure 3.10 Adjacency interval series

14,5,8, 0l [9, 0,
.VVV 1, 5]
N.O.
VVV
AIS 134 314

Rotate 5,8,0,4 1,5,9,0

VVV VVV
AIS 3 44 443
Figure3.llAlsininversionallyrelatedsetsthatrequirerotation

most sets in
inversional equivalence worksfor
This procedure to compare sets for sets the form that shows
howevett.'n
normal order. There are Some exceptions' 'o*"
arrangements. These excep-
one of the other rotational
the inversion is not the N.O., but later in this
tional sets are some inversionalv
,v**etric_al sets.(which we will study
For an example of
chapter), or also some sets that
feature a tie for the largest interval ad-
that do not feiure mutually retrogradable
two inversionalry related normal orders in Fig-
orders [4'5'8'0] and [9'0'1'51' shown
jacency interval series, consider the normal the normal orders'
."ti,+rand <3,1,4>. If we rotate
ure 3.11. Their respective AIS are <3'4'4> and
we see that the respective
however, into 5,8,0,4 and 1,5,9,0, i: ,1t",^::w equiva-
that these two sets are inversionally
<4,4,3>,mutually retrograbable, proving tie for the largest interval' in this
the type that features a
rent. Note that these sets are of
case between pitch classes 8'0
and 0'4'
---^- :s ir'. ^nlrr r
Anyinversionwillalsoinvolveatransposition,evenifit,sonlythetransposltlon
that two sets are
at the 0 level (around pitch class
c, or 0), or ToI' Thus' after we know
is i'volved in the
to determini the transposition that.
inversionally equivalent, we need
inversion.Wewillrefertothetranspositional^operatorappliedtoaninversion(then
equivalent, the first
sets in N.O. are inversionally
in TnI) as the index number. If two element with
the last in the other set' the second
element in one set corresponds with into the last, the second
(that is, the first pitch class inverts
the second_to_last, and so on each of these pairs
into the second-to-last, and so on),
in such a way that the addition of
should note'
as illustrated by Figure 3'12'.(we
of elements equals the index nrr-i"r, ttot work with normal orders'
this property oo"t
however, that here again sometimes for the property to work') In
other form
and then the set needs to be rotated to some
 F
i:
!l
: Introduction to Pitch-Class Set Theory 8l
:
a

'v"_,0
-/\
12,3, 5, 6, 8]
Figure 3.12 Determining the index number for an inversion

10 10 10 10 10
12,4,5,J,8]
subtractfroml0 8 6 5 3 2
reverse 12, 3, 5, 6, B] : Tro I
Figure 3.13 Inverting a set

Figure 3.12 we see that the index number for the 1nI operation between 12,4,5,7,8] and
f2,3,5,6,8f is l0 (the operation is thus TroI). In other words, in TnI (inversion around C
or 0), 0 inverts into 0, I inverts into 11,2 into 10, 3 into 9, and so on. The sum of any of
these pairs is 12. But in Tt.I, 0 inverts into 10, I into 9, 2 into 8, and so on, and the sum
of any of these pairs is now 10. Can you determine in this same way the index number
for the equivalence between sets [11,0,3,51 and14,6,9,101? And for sets [5,9,10,0] and
w,9,5,2)?
To invert a set by T',I, simply subtract each pc from n and reverse the order of the
resulting set (the inversion of a set in N.O. will normally be the retrograde of the N.O. of
the inversion, and that is why we need to reverse the orcler after performing the opera-
tion). In Figure 3.13 we see that we derive T,oI of 12,4,5,7,8f by subtracting each pc fron't
10 and then reversing the result. Now determine TnI of [11,0,3,5] in this same way.

NOTE

(Jnlike fran.spositiott, ittver,siott urtdoe,.r itsclf whert repeatetl. Thal is, every T,,l is its
ovyn inverse, .so the ope ratiotl goes botlt ways. Tlte inversion of set [2,1,5,7,8] by T,ol i.s
I [2,3,5,6,8], and the inversion oJ [2,3,5,6,8] again by T,oI takes us back to [2,4,5,7,8]
t
l
I Here again, the initial measures from Webern's Five Movements for String Quar-
S
tet, op. 5,III, will help us understand the significance of invcrsional equivalence among
pc sets. In Example 3.2, some musical events in m. 3 were left out of our commentary
on transpositional equivalence because the sets formed by these events are not related
by transposition, but rather by inversion. Example3.4 shows four pc collections, boxed

D
q
I

I
82 CHAPTER 3

I
I

I
op' 5'III' mm' 1-4
Webern, Five Movements for String Quartet'

Tol

[8,9,0] [0,3,4]
[n,to,r]
)r.,
ll ,ll ,01/
Sehr bewegt

lr .
semPrepPP

sets are related

sets 1 and 2 show us that these two
and numbered. The normal orders of of fact' these two sets
related by ToI (as a matter
by TnI, and, similarlY, sets 3 and 4 are 3'3 and Figure 3'10'
previ,ousiy in Example
are identical to two tlichords we studied two or three adjacettt
words' all groups of
the trichords [8,9,0] and [0,3,4])' In other or inversional
sets showr-r in both ExamPles 3'2 ancl
3.4 are reiirted by transpositional
equivalence. .

Set Class and Prime Form

(or
sets to determine the existence
The normal orcler of a set allows tts [o compafe however' still
urnong them' The nonnal order'
nonexistence) of equivalence relation,
representsspecificpitchclasses.Thatis,thenormalorclerisequivaletrttoreducingall tl:
position' This allow',,u: :liij,:'
inversions of a c major triacl to its root 'o ::i1l,1*
that they are all transpositionally eqlrlva-
triad with all other major triads and realize
all rninor triads and realize that they are
lent, ancl to compare the C major triard with
triads are mtttnal inversions)' Now we
inversionally equivalent (the major and minor triads under
us to incrr.rcre all these equivalent
should be able to find a raber that allows and inverted/
set can be transposecl twelve
one single category. In other words, a lT],"t
transposedanothertwelvetimes'andalltheseformsofthesetareequivalent.Thecol-
and it is represented by
is called the set class'
lection of equivalent forms of a same set
a numerical arrangement we call
prime form'
 Introduction to Pitch-Class Set Theory 83

8,4,3,1
N.O. [1, 3, 4, B]
Oatleft (0 2 3 1)
VVV
214
Figure 3.14 A shortcut to determine the prime form of a set

In Chapter I we learned a simple and quick procedure to determine the prime form
of a set. That procedure works for most sets, but not for all. We will review it here as
a quick shortcut to figure out prime forms, but we will also learn the more systematic
procedure that works for any set. First, let's review the shortcut:

l. Begin lrom the normal order.

2. Assign integer 0 to the first pitch class and renumber the rest of the pitch classes
accordingly. You can do so by transposing the normal order to begin on 0, or sim-
ply by producing the same adjercency interval series found in the normal order, but
now beginnin-q from 0.
3. The smallest possible intervals must be to the left. If they are to the right, read
from the right beginning with 0 (you will have to read the set's AIS backward and
beginning from 0), and then reverse the order. You will obtain the same result by
snbtracting all the members of the set from the last number in the set and then
reversing the order (in other words, in this step we are actually inverting the set to
bring the smallest intervals to the left).

Let r-rs apply these steps to some First, we will determine the prime fbrm
"*ornpl"s.
for set 8,4,3,1 (see Figure3.l4, where the process is shown both numerically and on a
clock face). The N.O. is [,3,4,8]. We zrssign 0 to tl're first pc and then prodr-rce the sanle
adjacency interval series lbLrncl in tl-re N.O. (that is, <2,[,4>).Alternatively, we trans-
pose the N.O. to be_ein with 0. In either case, the result is [0,2,3,7]. The first interval is
2, the last one is 4, so the smallest interval is to the left (there is a smaller interval in the
center, l, br-rt rotating the set to place it to the left or to the right would undo our N.O.
because then the set would not be arranged in the shortest possible span, and that would
go against our step l). So this is already this set's prime form, which we will notate in
parentheses withor-rt com mas: (0231).
r
We can also follow the same steps using a clock tirce. First, we list the N.O. on the
circle, and then we assign 0 to the first pc of the N.O. ancl count half steps clockwise. If
the srnallest intervals erre to the left of the resr-rlting transposition, as is the case with our
example in FigLrre 3.14, that will be the prime form.

D
 84 CHAPTER 3

4,7, 6,2
N.O. 12,4,6,7)
0 at left 0,2, 4,5
VVV
22r
right 5:t0
o at

Reverse (0 135)
0.2, 4, 5
(v'vt'
VV tZ
2l
the prime form of a set
Figure 3'15 A shortcut to determine

the right is 1' so this

WewillnowtrywithSet4J,6,z.TheN.o.is|2,4,6J],whichtransposedtobegin
The interval to the left is 2, the Lne to
on 0 becomes [0,2,4,5]. you can reacl [0'2'4'5] back-
varid prime form. we need to invert this set'
is not a set's AIS backward
read the
0 to the rightmost pc and
ward from 0 (that is, you assign to 0,1,3,5, and this is your prime
R"u"",," this se"t
from 0), and you Come up with 5,3,t,0.
and then we reverse the order,
form,(0135).Thisprocedurei,,no*ninFigure3.15.Notethatifwesubtractallthe
from the last number in this Set' 5,
members of |0,2,4,5] the N'O' on the circle
same result, (0135)' on a clock face' we first list
we obtain the smallest intervals are
it to begin on 0 as we did in Figure 3.r4,rfthe
and then transpose po of the N'O' and
we will assign 0 to the last
to the right of the resulting transpo'i'lon'
as shown in Figure 3'15'
count half steps countercl'cckwise'

!,$r'\
'y- NoJE
ure lhe pritrc forms ctf [5'9''10'0]'
[] l'0'2'41'
Yotr can ttovv try soi''te on your own' What F'igttre 3'B' IB'9'01'
that the tltree
ancl [9,0,3'5]? Can)'ort clentonstra'te "i'iu'at 'fiout
sonle set clrtss'/
[0,3,4],and[2,5,6],areall.nlember,srftlteS(In.Lesetclass?wnn'istheprinte.ftlrntfor
Are they mentbers oJ',the
the maior triocl'l For tlte ntirtor triacl?

for all sets is as follows:

A longer procedure that works
on 0 as You dicl in the Prevt-
fronl tiie N'O', and transpose the N'O' to begin
t. Begin
ous set of instructions'
12), and repeat the oPera-
all members of the set front
2. Invert the set (by subtracting transPose it to begin
the inversion in N'O'' and then
tions from step t' ihat is' p-ut
on 0. That is, choose
is more packed to the left'
3. From stepsand 2, choose the result that
I the next-to-last Pcs' If
the form that has the smallest
interval between the first and
ti

!i
 Introduction to Pitch-Class Set Theory 85

Original Set Inverted Set

4,J, 6, 2

N.O. 12, 4, 6,71

0 at left 0,2,4,5
lnvert set 10, 9,6,5
N.O. 15, 6,9, 10]

0 at left (0r3 5)
P.F. (0 13 5)

0,2,4,5 (0r35)
4 -)

Figure 3.16 The complete procedure to determine the prime form of a set

that still results in a tie, check the interval between the first ancl third-to-last pcs,
and so on.

Figure 3.16 shows the application of these steps to set 4,J,6,2. After we figure out
the N.O. and transpose it to begin on 0, we invert the N.O. Subtractin g[2,4,6,71from 12
gives us 10,8,6,5, which, in N.o., is [5,6,8,101. Transposing this N.o. to begin on 0 gives
us [0,1,3,5J. we compare the previous i0,2,4,5] to [0,] ,3,5], and we see that the latter is
more packed to the left; hence it is the correct prime form, (0135). To follow this proce-
- dure on a clock face, we first need to list both the N.O. of the original set and the N.O.
of the inverted set on circles, read both clockwise, and choose the olle most packed to
the left.
From Figtrres 3.15 and 3.16, we might think that we can always achieve the same
result with both methods to determine the prjme form. Figure 3.17, however, shows a
cztse of a longer set where we need to perforrn the second list of steps to arr.ive at the
correct prime fbrm. In Five Movements for String Quartet, op. 5, IV, Webern uses an
ascending figLrre on three occasions. The first of them is shown in Figure 3.17, where
we see that the N.O. for this set is [0,11,0,1,4,6,J]. Transposing this N.O. to begin on 0,
we come trp with L0,1,2,3,6,8,9]. Becausc the intervals from the left, <1,1,1>, are snialler
than the intervals ft-orn the right (reaciing backward fi-om the right, <1,2,3>), we might
be tempted to settle on this arrangement as a prime form. As it turns out, this is not
the correct prirne fortn. If we invert the set, pr-rt the inversion in N.O., ancl transpose it
to begin on 0, we discover that the fbrm f0,1,2,3,6,7,91 is more packecl to the left than
[0,1,2,3,6,8,9], so the prime form is indeed (0123619). The. process is illustratcd both
numerically and on a clock face in Figure 3.17.
We can now go back to Example 3.2 and veril'y that, among all the sets related by
transpositional equivalence, the sets numbered as 7,2,5,6, 10, 11, and l2 belong to the
same set class. Sets 7, 8, and 9, on the other hand, are also members of the same set

$
 86 CHAPTER 3

Original Set Inverted Set

Set: 0,4,6,1r,1,J,10
N.O. [10,11,0,1 ,4,6,J)
0 at left 0,1,2,3,6,,8,9
Invert set: 2,r,0,11,8,6,5
N.O. [11,0,1,2,5,6,8]
0 at left 0,1,2,3,6,7 ,9
P.F. (0t2361e)

0,r,2,3,6,8,9
"? 0123619
-----r-_--l
g7
Figure 3.17 The complete procedure to determine the prime form of a set

class, and similarly with sets 3 and 4. Identify each of the three set classes present tn
this example. If you refer now to Example 3.4, you can verify that each of the two pairs
of inversionally related sets also represents one of the set classes you just identified in
Example 3.2. Which are these two set classes?

x'*,,'
NOTE

The normal order representts an ordering of actual, specific pitch classes. We wilt use it
when we neecJ to compare particular collections of literal pitch classes for rryterations
sttch as transposition, int,ersion, antl some other literal operations we will sttLdy in this
chctpter (for instance,'literal cornplementarity and literal subset structure)' The prime
forit,
-
on the other hcmcl, is an abstrctct label thctt represents a set class. That is, it repre-
sen1s a1 the clffirent sets (all the transltositionally and inversional\t equivalent normal
orders) that make up ct set clctss. We witt use the primeformwhenwe want to refer to the
set clctss as a whole, to a set as a representative of a set class, or for some operations
tltat cletil with cLbstrctct, as oppr,tsed to literal, representcttions of sets (such a'v ttbstract
complementarity and abstract subset structtrre, to be discussed later in this chapter).

Interval-Class Vector
It is particularly useful, both for the composer and for the analyst, to know the complete
intervai content of a pitch-class set. That is, to know the list of all possible intervals that
can result from combining all pitch classes in a set in pairs. Because we are dealing
with pitch classes, what really matters is not so much the exact intervals but the interval

a1;===:

'
_
"'.'.'
=.-:;-=;i{ .:=- T=_=-=a-l
.' ::'..,,'':::1..' .1.:,
:' 'l:::.-ir
:
', =i.,,==,-
":i,i..;!!l
 Introdr-rction to Pitch-Class Set Theory 8l

\
^<-=-\
9r\
(0) 1 3 4 6 l2 2 3 I 1 1l
235
13
2

Figure 3.18 Calculating the interval-class vector of a set

classes. The interval-class vector (ICV) is precisely this, a list of the complete interval-
class content of a pc set. An interval-class vector contains six integers (which we will
represent in brackets and without commas), and each integer indicates the number of
occurrences of each ic. For instance, the ICV for set (01346) isl223ll1l. This means
that in set (01346) there are two cases of ic I (intervals 1-11), two of ic 2 (intervals
2-10), three of ic 3 (intervals 3-9), and one each of interval classes 4 (intervals 4-8),
5 (intervals 5-7), and 6 (interval 6).
Tb determine the ICV of a set, you can build a "triangle of differences," as shown
in Figure 3.18. With the set in prime form, disregard the openingzero, subtract the first
integer after the zero from each of the remaining integers, and write the results on a
line under the set. Then follow the same process with each subsequent line: subtract
the first integer from each of the remaining integers until you get to a line with a single
integer. When you have the complete triangle, count the occurrences of integers 1 or 1 1.
This will give yor-r the entry for ic 1 in your vector. Occurrences of integers 2 or 10 will
give you the entry for ic 2; occurrences of integers 3 or 9 provide the entry fclr ic 3; oc-
cun'ences of 4 or 8 provide the entry for ic 4; occurrences of 5 or J provide the entry
for ic 5; and occurrences of 6 provide the entry for ic 6. These entries account for the
vector 12231111 in our example. Now determine the ICV for set (0148) using a triangle
of clifl'erences.
t
One of the most immediate applications of the ICV is that it allows us to cornpare
t
the interval-class content of sets. Webern's Five Movements for String Quartet, op. 5,
.t
IV, opens with twcl tremolo figures in the violins, as shown in Example 3.5. The respec-
e
tive prirne fcrn-ns for these figure s are (0156) and (0167). The ICVs for each of these sets
t.l are l200l2ll and 12000221. We can see from this information that each of these sets
e
contains two instances of rc 2, two more of ic 5, and one or two of ic 6. On the other
,t hand, both sets display zero cases of ic2 ztnd ic 3, and only one case in one set of ic4.
:t We thus lcarn from tlie ICVs tliat the interval-class contentof these twcl sets is sr-rbstan-
tially similar, a property that is interesting from both the compositional and analytical
perspectives. Note also that the exarnple closes with a przzicato chord in the upper reg-
ister. The set class tbr this chord is also (0156).3

jThere
is a voluminor,rs literature on various ways of measuring similarity relations among sets. Sce,
te fbr instance, Robert Morris, "A Similarity Index fbr Pitch-Class Sets," Perspectives of New Mttsic 18
at (1979-80); 445-60, Eric Isaacson, "Similarity of Interval-Class Content Between Pitch-Class Sets: The
IcVSIM Relation," Jcturnal ctf Music Theory 34 (1990): l-28; and Michael Buchle r, "Broken and Unbroke n
rg
Interval CyclesandTheilUseinDeterminingPitch-ClassSetResernblance," Perspeclivesof NewMusic
al 3B/2 (2000):52-81.
'E!r---*-

88 CHAPTER 3

Webern, Five Movements for String Quartet, op' 5'IV mm' 1-2

(01s6) (0167) (01s6)

rcv [200121] rcv 12000221
-
am Steg-

mit Diim f-er

hfhr ilfD^ pi
hht

FFP
pw
)
am Steg przt
n mit Diimpfer L ----\ L :---\

-J- '-t

mit DiimPf-cr lppp (01261) ,-"'f:_ W

pw I

rnit DAmplcr .^

pw

The List o[ Set Classes

and nine pitch classes
A list of set classes (or prime forms) containing between three
l.a Complementary set classes
(trichords to nonachorai) can be fbund in Appendix
below; for the
are listed across from each other (we will explain set complementarity
listed across from nonachords, tetrachords
time being, simply notice that trichords are
from septachords, and So on' and that in
ZrcroSS from octachorcls, pentachorcls across
up to 12, as in 3 + 9 = l2)' Note fhat in
all cases complementary tardinal numbers acld
(Some sources in the
prime torms, *" *ill T and E for ten ancl eleven, respectively.
'se
literature use A lor ten and B for eleven')
In The structttre 0.f At,noL Mttsic, Forte assigned a clouble uumbet to
each set
F<I'te names appear in the leftmost and
class, which we will call the set's Forte name.
first integer in a Forte name (as' fbr in-
rightrnost columns in our list of set classes. The
class (that is, 3-6 has three pitch
stance, in 3-6) refers to the cardinal number of the set

aEarly iorms of this list appeared in Howard Hanson's 7'he Harmoric Materictls of T-wentieth-Cettlury
"The Source:,t:
Music(New york: nppr"i,in c"ntury-Crofts, 1960) and Donaldl4artino's i:9..tj::9q:
used lists appear in
(le6l): 224-:73. rhe most commonlv
;ffi;;;;;i;;;':; ;;;'r,'at o.f Music'rheort-_5/2
BalicAtonal Theorr-'ThelistinAppenclixlismodeled
Forte's TlrcstructttreoJ'AtonalMusicanclRahnls
of set classcs in Straus's Introclttction trt Prtst-Tonctl Theory,3rd
ed'' pp' 261-64' itsell mod-
after the list
elecl after Rahn's. Because of slightly diff'erent methods of
computing normal orders and prime forms' our
and di fters from Forte's in six piime forms (for set classes 5-20 ' 6-229 '
list (as wel I as Rahn's Straus"is;
6*31, 7 -218, 7 -20, and 8-26).
 Introduction to Pitch-Class Set Theorv 89

classes). The second integer is an order number that indicates where the set appears
in Forte's list (3-6 appears in the sixth place in the list of trichords). The second and
second-to-last columns are a list of set classes, including all the possible prime forms
with a cardinal number of 3 to 9. The third and third-to-last columns show the interval-
class vectors for all set classes. Finally, the central column indicates the degrees of
transpositional and inversional symmetry, two concepts explained below.

NOTE

Although the list was sornewhat reordered by Rahn, the original Forte names were
preservecl, but now out of order in sonte cases. Our list .follows Rahn's reordering,
including the original Forte nan'Les in their reordered state.

As you can probably realize at a glance, the list of set classes is a highly useful source.
In the first place, we can qLrickly verify whether what we think is a prime form does
or does not exist. For instance, you might try to figure out mentally the prime form for
set E-F-B, and come up with (017). A look at the list will tell you, however, that (017) is
not a possible prime form (it does not exist on the list). Why not? What is the real prime
form for this set? Moreover, the list provides you with a set's interval-class vector and
other useful infbrmation that we will discuss in the following sections.

Z-Related Sets
Z-related. sets are nonequivalent sets with identical interval-class vectors.S For each
Z set in the list of set classes you will find another 7, set with the same cardinal nLrm-
ber that has the same ICV. Find, fbr instance, set 5-236, (01247), with ICV l222l2ll.
S
Furrther down in the pentachords you will find set 5-212, (01356), with the same ICV.
S
These two sets ate Z related. What set is Z-related to 1-238?
e

n
PITCH.CI,ASS StrTS: FURTF{ER.
n
PROFEITTIES AND R ELATIGNSHHPS
Invariant Tones under Transposition
It
,cl Cornmon tones (or invariant tones) among various pitch or pitch class collections have
1- been used for a verriety of musical purposes in different musical periods and styles. In
:h tonal voice leading, tbr instance, it is usr-ral to leave common tones between adjacent
chorcls in the same voice, thus minimizing the rnotion from one chord to the next.
Some post-tonal composers (particularly Anton Webern) have also fhvored the use of
common tones or common pitch classes between adjacent collections in their composi-
tions, whereas other composers have favored the arvoidance of comrlon tones. Refer,

sAlthough the Z relation was named by Forte, its earliest fbrmulations are found in David Lewin's "The
lntervallic Content of a Collection of Notes," Journal of Music Thectr1,4 (1960): 98-101, and Hanson's The
H u rnt on i c IVI at e r i n I s oJ' Tw en tiet lt - C e nt r t ry M u.s i c.

$
1!4

100 CHAPTER 3

be nl| and (nl}) + 6' Consider set

TnI, the two centers of inversional symmetry will
symmetrical at TrI and TrI' lot JtI'S."
(0268). As we saw earlier, it is inversionally
E) and also 4 + 6 = l0 (pitch class Bb)'
centers of symmetry will be 812= 4 (pitch
class

ForTrI,thecente^ of ,y-rnetry willte2l2- 1 and I +6-7, orpitchclassesC$andG'

in Figure 3.24b- The axis of symmetry
The two axes E-Bb and C$-G are indeed shown
(0167) is inversionally sym-
can aiso fall between rwo pitch classes. we know that
metrical at T,I and TrI. For T,I, the centers of symmetry are ll2 = '5 and '5 + 6 - 6'5
(axis 3'5 = 9'5)' These two
(axis .5-6.5), and fo. frt they are 712 = 3.5 and 3.5 + 6 = 9'5

axes are shown in Figure 3'24b'

Terms for Review

inversional equivalence

$- ,l( \u"41 ::,*:'fi:ivalence, index number

set class
"f::{{,:;H;""" prime form

;^,;i,.tf,..o*^',
,)
,
vg-hn,tI ?'al; pitch-class space
interval-class vector
list of set classes
.\h;f 5t^^^*- iit.hspace
Forte name
Z-related sets
ordered Pitch-class interval
unorderecl Pitch-class interval
invariant tones
invariant tones under transposition
tnterval class
invariant tones under inversion
Pitch-class set
inclex vector
cardinal number
complement
trichord
aggregate
tetrachord
literal comPlement
Pentachord
abstract comPlernenl
hexachord
hexachordal comPlementaritY
subset

#n"l";l superset
literal subset
mouad
abstract subsets
clyad
transpositionally symmetrical set
l;,T:1"*"' inversionallY sYm n-retrical set
transPositional equivalence degree of tr anspositional symmetry
clegree of inversional symmetry

:fi:::il':J,*ff:::?J'