society for music theory pitch-class transformation in free jazz - steven block

Society for Music Theory

Pitch-Class Transformation in Free JazzAuthor(s): Steven BlockReviewed work(s):Source: Music Theory Spectrum, Vol. 12, No. 2 (Autumn, 1990), pp. 181-202Published by: University of California Press on behalf of the Society for Music TheoryStable URL: http://www.jstor.org/stable/746167 .Accessed: 02/07/2012 12:00

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Pitch-Class Transformation in Free Jazz

Steven Block


Steven Block


Steven Block


Steven Block

Free jazz originated with performances at the Five Spot Cafe by Cecil Taylor in 1957 and by Ornette Coleman in 1959, both of whom were then in the early stages of their careers.1 Taylor, Coleman, and John Coltrane helped to forge a new era in jazz, characterized by the introduction of harmonic, rhythmic, and timbral freedoms that affected the nature of jazz in a manner analogous to the great changes in Western concert music at the turn of the twentieth century. But while concert music of the formative first quarter of the twentieth century has re- ceived considerable analytical attention, particularly with respect to pitch and compositional design, hardly any analyses of free jazz compositions have attempted to go beyond merely de- scriptive commentary.2

The term "free jazz" comes from Coleman's landmark album of the same title, issued in 1960. In this work, which features the realization of Coleman's ideas about freedom in music to their fullest extent, two quartets perform a 36-minute improvisation using nontonal material without the benefit of a fixed meter or fixed entries of the ensemble. One can hear this

'Ronald M. Radano, "The Jazz Avant-Garde and the Jazz Community: Action and Reaction," Annual Review of Jazz Studies 3 (1985), 72.

2Gunther Schuller, who sponsored Coleman's work for a time, is the author of A Collection of the Compositions of Ornette Coleman (New York: MJQ Mu- sic, 1961), which contains transcriptions of and commentaries on Coleman's music. Ekkehard Jost's Free Jazz (Vienna: Universal, 1974) must be regarded as the primary work on the subject; it consists mostly of commentary on the music of several artists but also includes a few transcriptions.













music as both a reaction to and an extension of be-bop. The

principal differences between free jazz and be-bop are: (1) the elevation of the rhythm section from its formerly subsidiary role to a status equal to that of the melodic instruments; (2) improvisations that are no longer based upon chord changes; and (3) a harmonic language that does not necessarily incorporate "jazz tonality." The idea of "playing outside"-that is, of not basing an improvisation upon chord changes-is the primary distinction between free jazz and tonal jazz.3

Analysis of free jazz compositions by Coleman, Coltrane, Taylor, and the younger composer Anthony Braxton- compositions that are wholly different in style and tonal emphases-shows that pitch organization in free jazz can be very sophisticated; even ostensibly tonal compositions in this style cannot be adequately understood simply by reference to the harmonic underpinnings of the more tonal sections or to some surface-generative process of motivic invention. As in the analysis of early twentieth-century music, set-theoretical tools prove to be very useful in the analysis of free jazz.

Taylor's music has been labeled atonal by many of his peers. This label may be appropriate to Taylor's compositional de- signs, which are often at least superficially similar to those of Bartok and Stravinsky, but it is misleading in its implication

3It should be understood that functional tonality used in jazz by the be-bop era had evolved to incorporate implications (both in voice-leading and in root progression) that were different from those of the common-practice period in Western music.
















182 Music Theory Spectrum 182 Music Theory Spectrum 182 Music Theory Spectrum 182 Music Theory Spectrum

that Taylor's music is not really jazz and in its implication that Taylor's music has no tonal references. In an article published in 1965, Taylor describes his music as "constructionistic," meaning "based on the conscious working-out of a given material."4 The working-out of material is an additive or subtractive process in which motives or pitch material are not only reinter- preted and reworked but also altered slightly from phrase to phrase in a chain of progression that may span a long period of time. The final material may not, therefore, necessarily be understood as related to the original except in the sense that it lies at the opposite end of a musical process.

An excerpt from Taylor's Air above Mountains (1976) illus- trates the additive surface transformation that is typical of Tay- lor's improvisations (see Ex. la).5 The passage has been divided into the three primary gestures which alternate in the music, each of which represents a different musical lexicon. The first gesture is chromatic, the second whole tone, and the third diatonic in that it is the 3-9[0,2,7] trichord arranged as ascending fifths.

For the most part, each gesture is distinct and each one, as it is sounded, is permuted or reiterated in different ways. The first chromatic gesture remains consistent throughout the music and almost always occurs as the tetrachord C#-D-D#-E. The C~ which is added briefly in the opening statement and more prominently in the third statement (second system) expands this tetrachord to a chromatic pentachord. The third, diatonic gesture is always heard as ascending fifths from Ct. The second gesture is the most flexible in that it is first heard as a 3-8[0,2,6] trichord, D-E-Bb, and is always scored as a whole tone plus a tritone, even when it appears as a subset of later gestures. In the second statement of this motive (labeled 2a) this becomes a

4Tom Darter, "Piano Giants of Jazz: Cecil Taylor," Contemporary Key- board (May 1981), 56.

5The transcription in Example la appeared originally in Darter, "Piano Gi- ants of Jazz," 57. The transcribed passage begins about 4'30" into Side 1 of Air above Mountains (Buildings Within) (Inner City IC-3021).
















four-note whole-tone collection, 4-25[0,2,6,8], arranged as two tritones surrounding a major second. In m. 3, gesture 2a is repeated and further extended into the whole-tone pentachord (2b). Thus, both whole-tone and chromatic gestures are addi- tively developed as pc collections.

The relationship between these gestures, however, is more complex than a threefold alternation and variation scheme. There is a strong sense of a tonic D since it is the one pitch, in the first gesture, to which the music constantly returns and since in gesture 2, which is heard in several registers, D is always in the lowest octave. The third gesture features the lowest note of the passage, C,, scored in three octaves and always resolved to D in each of these registers by the music that follows. The final notes of the passage are clearly cadential; each gesture has a cadential formula built into it as well (Ex. lb).

One hears an interpenetration of these collections most clearly in m. 6, where the gesture of ascending perfect fifths is briefly transformed (GO mapped into A) into a transposition of gesture 2. This first appearance of the diatonic gesture con- cludes with an enunciation of the interval C#-D# (leading tone and enharmonically respelled ,2, respectively) in three registers. When this gesture is repeated in m. 6 and Cf and DO are paired once more, an A joins them to form a 3-8 whole-tone trichord one semitone lower than the 3-8 of gesture 2. The diatonic gesture has therefore been transformed into a whole-tone gesture. One can thus make a case for the most important mappings of this passage being the chromatic trichord CSt-D-D-, the diatonic trichord C#-D(-G#, and the whole-tone trichord C# -Dt-A (see Ex. Ic). The relationship between the chromatic and diatonic trichord here is the T6MI relationship.6 The rela-

6This has already been shown to have importance in tonal jazz since it maps a chromatic circle to the circle of fifths and underlies a particularly important relationship for chord progression in tonal jazz, the tritone substitution. See Robert Morris, review of John Rahn, Basic Atonal Theory, Music Theory Spectrum 4 (1982), 152-154; Henry J. Martin, "Jazz Harmony" (Ph.D. dissertation, Princeton University, 1980).













Transformation in Free Jazz 183 Transformation in Free Jazz 183 Transformation in Free Jazz 183 Transformation in Free Jazz 183

Example la. Gestural analysis of an excerpt from Cecil Taylor's Air above Mountains (Buildings Within). Transcription by Jim Aikin




Accidentals apply only to the note they precede. A




(2) 3-8 (1) (2) 3-8 (1) (2) 3-8 (1) (2) 3-8 (1)

) (1) 4-1 ( a) 5-1 (1) 3 '

B

(2a) 4-25

(2b) 5-33 (3) 3

; j'jjj^ja^ ^ ?H' j .r? Tf

(la) 3

9f: Li,[--:' L 3

(2) (2b) (2) i I (1)

|.. (2a) (3) i , # m

. P,;:J. J.

::r^^S^ffl nf 17 r h

) (1) 4-1 ( a) 5-1 (1) 3 '

B

(2a) 4-25

(2b) 5-33 (3) 3


(la) 3

9f: Li,[--:' L 3

(2) (2b) (2) i I (1)

|.. (2a) (3) i , # m

. P,;:J. J.


) (1) 4-1 ( a) 5-1 (1) 3 '

B

(2a) 4-25

(2b) 5-33 (3) 3


(la) 3

9f: Li,[--:' L 3

(2) (2b) (2) i I (1)

|.. (2a) (3) i , # m

. P,;:J. J.


) (1) 4-1 ( a) 5-1 (1) 3 '

B

(2a) 4-25

(2b) 5-33 (3) 3


(la) 3

9f: Li,[--:' L 3

(2) (2b) (2) i I (1)

|.. (2a) (3) i , # m

. P,;:J. J.

::r^^S^ffl nf 17 r h I j

I 1

I j

I 1

I j

I 1

I j

I 1

16 V t Tt I I

16 V t Tt I I

16 V t Tt I I

16 V t Tt I I

I? I? I? I?

#:t :; #; $_

#:t :; #; $_

#:t :; #; $_

#:t :; #; $_


Example lb. Cadential patterns within the gestures of Example la Example lb. Cadential patterns within the gestures of Example la Example lb. Cadential patterns within the gestures of Example la Example lb. Cadential patterns within the gestures of Example la

4-1

4-T eit

4-25

AM ,t.

4-1

4-T eit

4-25

AM ,t.

4-1

4-T eit

4-25

AM ,t.

4-1

4-T eit

4-25

AM ,t. -..---B U

3-9

Example lc. Mapping of the important trichords from Example la

-..---B U 3-9


-..---B U 3-9


-..---B U 3-9


3-1

X #~ 1

3-7 i :

(m. 6)

3-8

3-1

X #~ 1

3-7 i :

(m. 6)

3-8

3-1

X #~ 1

3-7 i :

(m. 6)

3-8

3-1

X #~ 1

3-7 i :

(m. 6)

3-8

tionship between the other two trichords, though, is undefined. From trichord to trichord, the leading tone and 2 remain invariant. It is D[ , the tonic, which can be perceived as mapping into 4 in moving to the whole-tone trichord and mapping into #4 when the circle of fifths dyad is enunciated. The transformation is still tonally suggestive, especially in its relation to the blues.

The passage taken from Air above Mountains shows imme- diate relationships between differing chord forms in different languages. Taylor's composition "Tales (8 Whisps)" (1966),







provides an opportunity to study a more sophisticated transformation over a longer span of time. Here the transformation is similar to a type that often occurs in Bartok, in which a passage previously octatonic is reshaped in the diatonic yet remains rec- ognizable. In "Tales," an eight-movement work for seven players, the outer sections are clearly related on the surface by in- strumentation, texture, and gesture.

The opening measures of "Tales" are shown in Example 2a. On one level, this passage is a florid elaboration of an El blues scale with the important notes clearly emphasized. First, the 3 and 4 3 are emphasized in mm. 1-2, followed by the dominant which then twice arpeggiates up to the tonic in m. 3 of the example. The next pitches to occur are 4 and 04 in mm. 4-5; this phrase ends with the leading tone. The motion from leading tone to tonic is picked up again in mm. 8-9. As in the previous Taylor excerpt, the development of material here follows an additive course.

Despite these tonal emphases, the basis of "chord" changes to come are evident from a tetrachordal segmentation of the primary gestures of the opening (Ex. 2b). In this mostly chromatic environment, the segmentation of these tetrachords follows a general pattern whereby semitone dyads and, more rarely, whole-tone dyads are arranged around some inner interval to form a larger pitch grouping.

At the end of the composition, the gestures are not distinctly or overtly related to those of the opening because the connections can only be understood as parts of a developmental process. Example 3a shows the conclusion of "Tales" with some pitch-class and gestural analysis. The first gesture, in all its vari- ations, is a diatonic 5-20[0,1,5,6,8] pentachord; gesture 2 is a more general category used for extended "chromatic" gestures; gesture 3 is octatonic; gesture 4 is made up of the Z- related 6-Z6[0,1,2,5,6,7] and 6-Z38[0,1,2,3,7,8] hexachords; and gesture 5 represents the cluster passages, both whole-tone and chromatic, at the conclusion. Using this labeling, one can follow some of the simultaneously unfolding processes. The













Transformation in Free Jazz 185

Example 2a. Opening measures of Cecil Taylor's "Tales (8 Whisps)." Transcription by the author from Unit Structures (Blue Note BST-84237)







Piano Piano Piano Piano

J =176 (tempo somewhat free)

1 , .

~drum v I v (D rr s r s

~ !

i7 I rq rrrirrrq . r f h-i--I--i 1 LI f' ~


1 , .


~ !



1 , .


~ !



1 , .


~ !

i7 I rq rrrirrrq . r f h-i--I--i 1 LI f' ~ l.U!1 3~1 ! i !

|I _ ! 1 .

Piano A: bbbbb

3 14 15

Drum Set7 -

poco accel. > (may be a fist cluster) =34 p , Poco accel. > > >' >,-=304

P,iano

bbbbbb

3 3 1 7 3 8

)\' b bbbb ' f 17 ?rf 8

:bb;bbbb u r 1' r Drum Set:

drummer only: IP =264

Piano 9 >

Drum Set -- -

l.U!1 3~1 ! i ! |I

_ ! 1 .

Piano A: bbbbb

3 14 15

Drum Set7 -


P,iano

bbbbbb

3 3 1 7 3 8

)\' b bbbb ' f 17 ?rf 8



Piano 9 >

Drum Set -- -

l.U!1 3~1 ! i ! |I

_ ! 1 .

Piano A: bbbbb

3 14 15

Drum Set7 -


P,iano

bbbbbb

3 3 1 7 3 8

)\' b bbbb ' f 17 ?rf 8



Piano 9 >

Drum Set -- -

l.U!1 3~1 ! i ! |I

_ ! 1 .

Piano A: bbbbb

3 14 15

Drum Set7 -


P,iano

bbbbbb

3 3 1 7 3 8

)\' b bbbb ' f 17 ?rf 8



Piano 9 >

Drum Set -- - Y. p?a f *

^1)0

Y. p?a f *

^1)0

Y. p?a f *

^1)0

Y. p?a f *

^1)0


Example 2b. Tetrachordal segmentation of opening measures of "Tales" Example 2b. Tetrachordal segmentation of opening measures of "Tales" Example 2b. Tetrachordal segmentation of opening measures of "Tales" Example 2b. Tetrachordal segmentation of opening measures of "Tales"

m.2 4-7 m.2

I I I I

m.2 4-7 m.2

I I I I

m.2 4-7 m.2

I I I I

m.2 4-7 m.2

I I I I

4-9 m.4 I I

I _

4-9 m.4 I I

I _

4-9 m.4 I I

I _

4-9 m.4 I I

I _

I I I - _L 4-7 4-1 4-3

ideas of additive and subtractive variation are present, but they are not always on the level of the musical surface. Here it is pitch-class metamorphoses which control the structure. Several pairs of sets, for instance, are related by T2. These sets are outlined in gestures 2 and 2a (7-5[0,1,2,3,5,6,7] chromatic sets), both in m. 2; in gesture 2b (the abstract complement of 7-5) in m. 7 and in the included pentachord in 2a; in gesture 4 (the 6-Z6 hexachord) in m. 4 and in gesture 4' in m. 16; and ;n gesture 4a (6-Z38) in m. 5 and in 4b in m. 8. Another T2 relation, not shown in the score, exists between the 6-Z38 hexachord in 4c and the 6-Z38 hexachord outlined in the upper register notes of the first measure of this transcription.

There are more subtle ways in which these gestures are related as well. The diatonic 5-20 pentachord is a subset of a hexachord, 6-Z38, a hexachord which one cannot abstractly hear as diatonic. Thus, the addition of one pitch class can transform a smaller octatonic or whole-tone collection into a chromatic one. In general, a set associated with one superset can be chro- matically inflected to connect it with a different superset. In this light, one might conclude that the clusters which are so preva- lent in Taylor's music, yet are so often dismissed as mere textural devices, in fact play a real structural role, since these are the elements from which any of the gestures in the final passage (or, as shown previously, the openings blues passage) can be derived.

Example 3b is a reduction of mm. 9-11, which can be thought of as a gestural summary of this movement. A subtler transformation emerges from an interpretation of the tetra-

I I I - _L 4-7 4-1 4-3




I I I - _L 4-7 4-1 4-3




I I I - _L 4-7 4-1 4-3




4-3

chords within the segmented phrases as divided into semitone dyads. From this vantage point, one sees a type of pitch-class condensation taking place in this passage: a 4-9[0,1,6,7] tetrachord, or two semitone dyads surrounding a perfect fourth, is reduced to a 4-8[0,1,5,6] tetrachord (two semitone dyads surrounding a major third), which in turn is transposed up a semitone, and finally is reduced to a 4-3[0,1,3,4] tetrachord (two semitone dyads surrounding a whole tone). This contraction may not seem as "natural" as the kinds of extensions or reduc- tions that occur on the surface pitch level from gesture to gesture; but, since within the process itself two or three pitch classes are invariant from gesture to gesture, it is quite conceiv- able that the artist consciously composed these structural connections.

The concluding cluster passages in Example 3a may be heard as moving from an overall whole-tone sonority to a more chromatic gesture centering on chromatic trichords to a mix- ture of the two. The registral separation in m. 17 segments gesture 5b' into a 4-10[0,2,3,5] tetrachord (two whole-tone dyads surrounding a semitone), and a 4-3[0,1,3,4] tetrachord which is the same collection of pitches that concluded the summary passage just discussed (Ex. 3b). As this cluster gesture becomes further condensed, the final sound can be heard as a V7 of G. This seems puzzling, since if one were to hear this final passage as tonal, Dl, would be the tonic since that pitch is segmented out by register, often in the bass. Moreover, gesture 3 returns twice in the guise of a dominant ninth of Db . It does seem pref- erable, on the whole, to hear Taylor's music as nontonal even

4-3



4-3



4-3



m.1 rn m.l I m.1 rn m.l I m.1 rn m.l I m.1 rn m.l I

nm.1-2 I_

nm.1-2 I_

nm.1-2 I_

nm.1-2 I_ b .- E b .- E b .- E b .- E

-4 : _ V F: 'O- 1 - 11 dph

I 1-I ?

I 4

.-e I ML NL I I I - -4 : _ V F:

'O- 1 - 11 dph I

1-I ? I

4 .-e I ML NL I I I -

-4 : _ V F: 'O- 1 - 11 dph

I 1-I ?

I 4

.-e I ML NL I I I - -4 : _ V F:

'O- 1 - 11 dph I

1-I ? I

4 .-e I ML NL I I I -


Example 3a. Final measures of "Tales" segmented into related gestures and pc sets. Transcription by the author







accel ------ tempo accel ------ tempo accel ------ tempo accel ------ tempo

=240 accel. - - - tempo

3 ^ . L_


3 ^ . L_


3 ^ . L_


3 ^ . L_

( 7 / 1, 1 12 1 '

I I /

L; 12b subset of gestures 2 4

b

79:2 n ..... ~ ! , ,ue ,,t^ k V (1) 5-20, diatonic i i (la) literal subset i i

(2) 7-5, chromatic of gesture 1 (2a) 75, related by (3) ctatonic I/Adomiant inth 1 s u beT2 to gesture 2

(4) 6-Z6 (4a) 6-Z38 T

~jÎ . ~: : . b?~~~~~~~~~~~

( 7 / 1, 1 12 1 '

I I /


b



(4) 6-Z6 (4a) 6-Z38 T

~jÎ . ~: : . b?~~~~~~~~~~~

( 7 / 1, 1 12 1 '

I I /


b



(4) 6-Z6 (4a) 6-Z38 T

~jÎ . ~: : . b?~~~~~~~~~~~

( 7 / 1, 1 12 1 '

I I /


b



(4) 6-Z6 (4a) 6-Z38 T

~jÎ . ~: : . b?~~~~~~~~~~~

~]-eL-r r- L, ' M

T " 1 m ~]-eL-r r- L, ' M

T " 1 m ~]-eL-r r- L, ' M

T " 1 m ~]-eL-r r- L, ' M

T " 1 m

I I I (not clear) I_ (3a) Octatonic I (4b) 6-Z38, related by T2 (3) rearrangement, upper register preserved

to gesture 4a (2c') extension of 2c Summary Passage

^=200 1 E .I )I .I I I= 336 k - ~ ~ ~ ~ ~ ~ ~ ~ ~~~e = 200 , -' C' a~-33



^=200 1 E .I )I .I I I= 336 k - ~ ~ ~ ~ ~ ~ ~ ~ ~~~e = 200 , -' C' a~-33



^=200 1 E .I )I .I I I= 336 k - ~ ~ ~ ~ ~ ~ ~ ~ ~~~e = 200 , -' C' a~-33



^=200 1 E .I )I .I I I= 336 k - ~ ~ ~ ~ ~ ~ ~ ~ ~~~e = 200 , -' C' a~-33

a =336 A

a =336 A

a =336 A

a =336 A 3 3 3 3 I 0-t1 I 0-t1 I 0-t1 I 0-t1

1 (inclusion of previous 5-20 is literal)




-2 -2 -2 -2

188 Music Theory Spectrum

3a (cont.)

= 240


3a (cont.)

= 240


3a (cont.)

= 240


3a (cont.)

= 240

175 ,1,1 ~ --- ! ', (5) Clusters (whole tone) 6f --

'1' ' e Tr r r :=r===


'1' ' e Tr r r :=r===


'1' ' e Tr r r :=r===


'1' ' e Tr r r :=r=== (5a) Clusters (chromatic) (5a) Clusters (chromatic) (5a) Clusters (chromatic) (5a) Clusters (chromatic)

(5b) (5b) (5b) (5b) (5b') (5b') (5b') (5b') (5b) condensation (5b) condensation (5b) condensation (5b) condensation

Example 3b. Reduction of mm. 9-11 from Example 3a

4-9 4-8 4-8 1 I 11 4.-8 1 1 1 1 11 1 A I I I 4-RIf I I I I I I4 -


4-9 4-8 4-8 1 I 11 4.-8 1 1 1 1 11 1 A I I I 4-RIf I I I I I I4 -


4-9 4-8 4-8 1 I 11 4.-8 1 1 1 1 11 1 A I I I 4-RIf I I I I I I4 -


4-9 4-8 4-8 1 I 11 4.-8 1 1 1 1 11 1 A I I I 4-RIf I I I I I I4 -

4-3 !

4-3 !

4-3 !

4-3 !

I I

"^y '~.w-4 - 1 #0- f f "- - I i I

I /

i T 5-20 6-Z38 5-20

I l I I l I l l l I

Q__b- b^ __ b,^ bw bb, _ b. b _,,

I I

"^y '~.w-4 - 1 #0- f f "- - I i I

I /

i T 5-20 6-Z38 5-20

I l I I l I l l l I

Q__b- b^ __ b,^ bw bb, _ b. b _,,

I I

"^y '~.w-4 - 1 #0- f f "- - I i I

I /

i T 5-20 6-Z38 5-20

I l I I l I l l l I

Q__b- b^ __ b,^ bw bb, _ b. b _,,

I I

"^y '~.w-4 - 1 #0- f f "- - I i I

I /

i T 5-20 6-Z38 5-20

I l I I l I l l l I

Q__b- b^ __ b,^ bw bb, _ b. b _,, r r r r * - * - * - * - r r r r

though it often contains tonal jazz references. Nevertheless, taking the cue from the final five notes of the piece being M- related to the important 5-20 pentachord in this passage, one can hear the closing notes as a T6MI chord substitution for the dominant of D,. Therefore, these final cluster passages can be understood as summaries of the measures in the entire excerpt, both in terms of derivational structure and tonal orientation.




Though Coltrane emerged as an important jazz artist before Taylor and Coleman, he turned to his own free jazz style only during his last period (1964-67), after Taylor's and Coleman's music had become widely known. Coltrane's first solo in Ascen- sion is a good example of his use of two primary generative devices within a quasi-modal framework: interval cycles and tri- chordal source sets. Coltrane's improvisations have




accel.------- nt.------------- accel.------ rit.------- accel.------- nt.------------- accel.------ rit.------- accel.------- nt.------------- accel.------ rit.------- accel.------- nt.------------- accel.------ rit.-------

I I I I I I I I


traditionally been described in terms of modal shifts as the primary indicators of harmonic movement and the modes themselves as the primary generators of pitch material. However, this kind of description fails to clarify whether Coltrane heard the modes simply as source collections or whether he exploited the structural properties of the mode to emphasize, for instance, the modal final and the dominant.7

At the opening of Ascension, Coltrane's tenor sax line is simultaneously imitated in the same register by the other players (see Ex. 4a for a transcription of Coltrane's opening line). Overall, the rhythmic emphasis marks the opening as a "Bb blues" statement with the scale enunciated from i to 5 and including both l,3 and | 3. The ensemble, as a whole, ignores t 3; thus one primarily hears Bl, D,, and Eb, which constitute a 3-7[0,2,5] trichord. The added Dt, which occurs only in Col- trane's part, forms a 3-2[0,1,3] trichord with the upper portion of the scale and a 3-3[0,1,4] moving down to the Bb tonic.

When Coltrane later begins his first solo (see Ex. 4b), these same trichords are fundamental to the pitch progression within the mode.8 The solo is launched from the 3-7 trichord now formed by the seventh, tonic, and third of the mode and an alternation between 3-2 and 3-3 trichords is the subject of the next 15 bars. The pitches explored here are not those from the opening but, instead, the lowest third of the B b scale with the leading tone.

The second part of the B, Aeolian solo features a longer play on the first trichord of the solo, 3-7; the entire passage, in fact, can be segmented into 3-7 trichords. One interesting relationship is that of the tetrachord formed in the first part by the

7Such a problem arises, for example, when Jost (Free Jazz, 92-93) catego- rizes the first part of the solo as being in B Aeolian with an added A as a

leading tone. Considering the generality of this description, Bb minor seems

equally suitable as a classification; but in either case, no real insight is gained into Coltrane's choice of specific pitch structures.

8The transcription in Example 4b is taken from Jost, Free Jazz, 92-93; the analytic summary alongside is my own.

























alternation of 3-2 and 3-3 trichords (4-3) and its T8M-related set (4-26[0,3,5,8] formed by the overlapping of two 3-7 trichords in mm. 16-20. This is a result of the transformation that maps the chromatic semitone of leading tone-to-tonic in the first tetrachord into the perfect fifth dominant-to-tonic in the second tetrachord.

In the final run of the first section, the first whole-tone structure is enunciated in mm. 23-24 when the 3-6[0,2,4] trichord, F-E ,-Dl, is sounded, a harbinger of the subsequent structural importance of the whole-tone cycle. The accumulation of whole-tone structures in the second section proceeds in a way that is analogous to the use of the 3-7 trichord (as major second followed by minor third) at the end of the first section. At the opening of the second section, Coltrane sounds a 3-8[0,2,6] whole-tone trichord and follows in upward scalar motion with a 5-30[0,1,4,6,8] chord (which embeds two 3-8 trichords). This 5-30 is then altered to the whole-tone pentachord 5-33 [0,2,4,6,8], which appears in ascending scalar form in m. 27.

Measures 27-30 mark the beginning of a section which parallels the opening of the solo. Here, instead of alternating between trichords, Coltrane primarily reiterates the pitch classes of one whole-tone tetrachord while varying them rhythmically. The tetrachord is scored as a descending whole-tone trichord followed by a downward tritone leap. The repetition and re- working of the chord clearly connect to the opening passage of the solo in section 1. The aural association of the two passages is made obvious through the use of the leap, which preserves contour. Coltrane's scoring ensures that one hears the last three notes in the second motive, which form the interval succession of whole tone and tritone, as a 2:1 expansion of the 3-3 trichord at the opening (semitone and minor third). Proceeding from section 1 to section 2, then, the listener notices: (1) the increase in cardinality of the chordal generator-or, at least, of the emphasized chord; (2) the contour expansion; (3) the intervallic expansion, with the chromatic generators 3-2 and 3-3 being re- placed by whole-tone generators.











Example 4a. Opening of John Coltrane, Ascension (Impulse version). Transcription by the author from Impulse A-95 Example 4a. Opening of John Coltrane, Ascension (Impulse version). Transcription by the author from Impulse A-95 Example 4a. Opening of John Coltrane, Ascension (Impulse version). Transcription by the author from Impulse A-95 Example 4a. Opening of John Coltrane, Ascension (Impulse version). Transcription by the author from Impulse A-95

(s) (s) (s) (s)

e_- I -

LI -.- I_ .

e_- I -

LI -.- I_ .

e_- I -

LI -.- I_ .

e_- I -

LI -.- I_ . _- 3W- 3 v_

I l I I 1 3-2 3-3 3-3

_- 3W- 3 v_ I l I I 1

3-2 3-3 3-3

_- 3W- 3 v_ I l I I 1

3-2 3-3 3-3

_- 3W- 3 v_ I l I I 1

3-2 3-3 3-3

In section 2, the nature of the whole-tone trichord and tetrachord as generators of pitch structure becomes quite evident. A brief allusion to chromaticism is heard in mm. 31-32 (see analysis in Ex. 4b), after which the same whole-tone tetrachord as scored previously is repeated twice. Following is a five-note ascending pattern from F4 to Bb5 in the highest register. This is 5-13[0,1,2,4,8], which is T8M-related to the second 5-30 rising pentachord of this second section (see Ex. 5c), in this case, D #4 in the first chord mapped onto B 4 in the second.

From the point where two whole-tone trichords a semitone apart are enunciated in m. 31, the primary sonority for some ten measures is a 6-22[0,1,2,4,6,8] hexachord; most of the emphasis is on the whole-tone pentachord embedded within it. B appears twice in this passage (once as a chromatic passing tone), but it is the overlapping whole-tone structures that form the primary sonic component, further emphasizing the rework- ing of the A-B-C#-DO whole-tone tetrachord.

When the improvisation starts moving away from this tetrachord at the end of this transcribed portion, several allied sonorities are formed, the last of which is 6-34[0,1,3,5,7,9]. This hexachord is T8M-related to the 6-2[0,1,2,3,4,6] formed by the upbeat C~ to m. 32 and the whole-tone pentachord in the next three measures (see Ex. 4c). Thus, in each section of the solo, some of the larger pc sets are related by T8M. Further-










more, both the literally included whole-tone pentachord and the 6-22 hexachord which comprise much of the pitch material of section 2 are invariant under that operation. Thus the T8M operation is a clue to the generative nature of Coltrane's improvisation. The idea of a structure based upon related pitch- class sets already goes considerably beyond any account based upon modes. From the foregoing analysis, it is clear that a specific series of operations can describe the relations between the pitch-class sets in an improvisation. Like composers of Western concert music in the twentieth century, jazz figures such as Col- trane had begun to discover the transformations that connect different kinds of pitch-class materials.9

As a composer, Anthony Braxton has transcended the com- monly opposed categories of composed music and improvised music; many of his works fit into both. His education and his association with other new music performers, notably Frederic Rzewski and the ensemble Musica Elettronica Viva, also have

90ne further relationship in the Coltrane unifies the two sections of this improvisation: in the endings of both, at the smallest level, the whole-tone and 3-2 trichords can be interpreted as generators of the improvisation. In addition, both sections embed 4-22, a set which can be segmented into overlapping 3-6 and 3-7 trichords, C# (Db)-D? (E!)-F-G (Al). However, since the operation T6MI is so endemic to jazz, the discovery that M relations are part of Coltrane's improvisational method should perhaps not be considered surprising.










(Coltrane playing) A I,




3-7 3-7 3-7 3-7


Example 4b. First solo by Coltrane in Ascension (HMV version). Transcription by Ekkehard Jost. Starred sets are T8-M related







Section 1 transcription

ibbbb


ibbbb


ibbbb


ibbbb

I> : > >

Cr J>! ^ LLd L.jr L-L_F .!r Lj. f ri inc.

I> : > >


I> : > >


I> : > >


J=215

)>s ! ctL:rLm't r J=215



)>s ! ctL:rLm't r analysis 3.7 3-3 3-2 inc. *4-3

vb, b ' . .. ' :"" , , 1

6vt 7-r rr^. ^ 16'

-

L"b;9,b etc. - r of r

'b '~~~ rf~~19 P f

T-i-r?r r ?24 3

3-7 3-7 3-7 3: 3-7 3-7 3-7 3-7 iA 3-7 3-7

*I I.I ? a~ I

analysis 3.7 3-3 3-2 inc. *4-3

vb, b ' . .. ' :"" , , 1

6vt 7-r rr^. ^ 16'

-


'b '~~~ rf~~19 P f

T-i-r?r r ?24 3

3-7 3-7 3-7 3: 3-7 3-7 3-7 3-7 iA 3-7 3-7

*I I.I ? a~ I

analysis 3.7 3-3 3-2 inc. *4-3

vb, b ' . .. ' :"" , , 1

6vt 7-r rr^. ^ 16'

-


'b '~~~ rf~~19 P f

T-i-r?r r ?24 3

3-7 3-7 3-7 3: 3-7 3-7 3-7 3-7 iA 3-7 3-7

*I I.I ? a~ I

analysis 3.7 3-3 3-2 inc. *4-3

vb, b ' . .. ' :"" , , 1

6vt 7-r rr^. ^ 16'

-


'b '~~~ rf~~19 P f

T-i-r?r r ?24 3

3-7 3-7 3-7 3: 3-7 3-7 3-7 3-7 iA 3-7 3-7

*I I.I ? a~ I

4-22 I

4-22 I

4-22 I

4-22 I

5-23

3-6 1 1

I I I 3-2

4-11

5-23

3-6 1 1

I I I 3-2

4-11

5-23

3-6 1 1

I I I 3-2

4-11

5-23

3-6 1 1

I I I 3-2

4-11

6-32 6-32 6-32 6-32

I I I I

- Lu - Lu - Lu - Lu


4b (cont.)

Section 2

q

25 3,7 rr6 7

3-8 (3,5,9) *530(1,3,5,9,10) 5-33 (1,3,5,9,11) 4-21

3-8 3-8

30 3 3 3 r3 -

3 ~3 3 ~ (mm. 31-34) ***6-21 4-22 3-6 3-6 I 4-21

I I I I Il

6 6' (mm. 26-30, 3242) 6-22

** 5-13 4-22 1 \

1 1

,.---'---- ---^' ^ I~ _ ___ ^ r _ J


4b (cont.)

Section 2

q

25 3,7 rr6 7

3-8 (3,5,9) *530(1,3,5,9,10) 5-33 (1,3,5,9,11) 4-21

3-8 3-8

30 3 3 3 r3 -

3 ~3 3 ~ (mm. 31-34) ***6-21 4-22 3-6 3-6 I 4-21

I I I I Il

6 6' (mm. 26-30, 3242) 6-22

** 5-13 4-22 1 \

1 1

,.---'---- ---^' ^ I~ _ ___ ^ r _ J


4b (cont.)

Section 2

q

25 3,7 rr6 7

3-8 (3,5,9) *530(1,3,5,9,10) 5-33 (1,3,5,9,11) 4-21

3-8 3-8

30 3 3 3 r3 -

3 ~3 3 ~ (mm. 31-34) ***6-21 4-22 3-6 3-6 I 4-21

I I I I Il

6 6' (mm. 26-30, 3242) 6-22

** 5-13 4-22 1 \

1 1

,.---'---- ---^' ^ I~ _ ___ ^ r _ J


4b (cont.)

Section 2

q

25 3,7 rr6 7

3-8 (3,5,9) *530(1,3,5,9,10) 5-33 (1,3,5,9,11) 4-21

3-8 3-8

30 3 3 3 r3 -

3 ~3 3 ~ (mm. 31-34) ***6-21 4-22 3-6 3-6 I 4-21

I I I I Il

6 6' (mm. 26-30, 3242) 6-22

** 5-13 4-22 1 \

1 1

,.---'---- ---^' ^ I~ _ ___ ^ r _ J 5-33 5-33 5-33 5-33


4b (cont.) 4b (cont.) 4b (cont.) 4b (cont.)

tfl .' -

-A

*** 6-34

I I I I

tfl .' -

-A

*** 6-34

I I I I

tfl .' -

-A

*** 6-34

I I I I

tfl .' -

-A

*** 6-34

I I I I I 1 4-22

6-33

6-33

\1 I 4-22

I 1 4-22 6-33

6-33

\1 I 4-22

I 1 4-22 6-33

6-33

\1 I 4-22

I 1 4-22 6-33

6-33

\1 I 4-22

had some influence on his work. Example 5a displays the head of a 1975 composition, "489M..."10 It is not a head tune in the usual sense, however, since it is repeated twice and makes up the first half of the composition, the second half being an improvisation freely based upon this material. A certain tonal character, including an emphasis on the tonic B, is evident; so, however, is aggregate completion, a feature often exhibited in




lOThe full title is: 489 M

70-2--(TH-B) M.

The transcription in Example 5a is modeled upon Braxton's own manuscript, which is reproduced in Ronald Radano, "Anthony Braxton and His Two Musi- cal Traditions" (Ph.D. dissertation, University of Michigan, 1985), 244. The rhythmic notation is not always exact but is generally proportional.


70-2--(TH-B) M.



70-2--(TH-B) M.



70-2--(TH-B) M.


Braxton's nontonal works. The last note, E, is saved for the very end of the composition, essentially signaling the return to B by way of the lower fifth.

A number of significant relations can be found within this short head tune; especially interesting is the pattern of related sets revealed by an examination of the hexachords formed at the opening and closing of phrases. Nine hexachords identified in Example 5a are strongly related. 6-Z44[0,1,2,5,6,9] is the opening hexachord of the piece and is included in the phrase (upper register pitches) in the fifth system. One could think of this piece as divided on the basis of texture into sections in which motion is varied as opposed to those in which the motion occurs in ascending or descending scalar runs. The fact that the second 6-Z44 occurs after three successive scalar runs thus links it by way of textural considerations to the opening. Note, in








Example 5a. Head tune for Anthony Braxton's "489M...," from Five Pieces 1975 (Arista AL-4064) Example 5a. Head tune for Anthony Braxton's "489M...," from Five Pieces 1975 (Arista AL-4064) Example 5a. Head tune for Anthony Braxton's "489M...," from Five Pieces 1975 (Arista AL-4064) Example 5a. Head tune for Anthony Braxton's "489M...," from Five Pieces 1975 (Arista AL-4064)

**5-13 1 1~~~~~~~~~~~~~~~ **5-13 1 1~~~~~~~~~~~~~~~ **5-13 1 1~~~~~~~~~~~~~~~ **5-13 1 1~~~~~~~~~~~~~~~

7__, - _ -- _

M T8 (1,3,5,9, 10)-- (5,3, 1,9,2) - (1, 11,9,5, 10)

^ I I I i

(9, 11,0, 1,3,5) - (9, 7, 0,5, 3, 1)T--85,3,8, 1, 11,9)

fact, that the two 6-Z44's are different by only one pc, Bb sub- stituting for Al in the second iteration. And the Z-related set 6-Z19[0,1,3,4,7,8] is the closing hexachord of the third system (including the drone B).

Example 5b shows the progression of the all-interval tetrachords included in each hexachord. These sets are most often transformed by T3 (in the case of 4-Z29[0,1,3,7] or T5. This pattern suggests that here the all-interval tetrachords may be the core of an improvisation built around the use of these chords under relatively few operations.

Ornette Coleman's classic composition, "Lonely Woman," from his album The Shape of Jazz to Come (1959), can be sec- tioned into the head tune (Ex. 6) and two different types of choruses. The first chorus is eight measures long and is accompa- nied by a chromatic ascent from D to F in the bass each time it is heard; the second chorus is approximately 22 measures long, the same as the length of the head tune with introductory measures. Since the head tune and first chorus remain in the minor mode, and since there is a constant return of the D-minor pedal, one might simply call this piece tonal. However, such a view, though it does express the relationship of "Lonely Woman" to the harmonic past of jazz and its roots in the blues, does not contribute much to an understanding of what

7__, - _ -- _

M T8 (1,3,5,9, 10)-- (5,3, 1,9,2) - (1, 11,9,5, 10)

^ I I I i

(9, 11,0, 1,3,5) - (9, 7, 0,5, 3, 1)T--85,3,8, 1, 11,9)




7__, - _ -- _

M T8 (1,3,5,9, 10)-- (5,3, 1,9,2) - (1, 11,9,5, 10)

^ I I I i

(9, 11,0, 1,3,5) - (9, 7, 0,5, 3, 1)T--85,3,8, 1, 11,9)




7__, - _ -- _

M T8 (1,3,5,9, 10)-- (5,3, 1,9,2) - (1, 11,9,5, 10)

^ I I I i

(9, 11,0, 1,3,5) - (9, 7, 0,5, 3, 1)T--85,3,8, 1, 11,9)




=96 1 n =96 1

n =96 1

n =96 1

n 6-Z44 6-Z44 6-Z44 6-Z44

I *.PNg I I

I *.PNg I I

I *.PNg I I

I *.PNg I I

! I 6-18

6-30

6-Z25 6- Z25 1 6-Z38

6-Z24 I 1 I I 6-Z24 t 6-Z19 t

9:

6-Z25

6.2 b3^bc^.^ 6

t 6-Z23

6-Z24

Ii 19 . - b, , b? b <y 1 ' i l l i 96-, v

6-Z44 b I

^JJ| I r ? 0o o? o

! I 6-18

6-30

6-Z25 6- Z25 1 6-Z38

6-Z24 I 1 I I 6-Z24 t 6-Z19 t

9:

6-Z25

6.2 b3^bc^.^ 6

t 6-Z23

6-Z24

Ii 19 . - b, , b? b <y 1 ' i l l i 96-, v

6-Z44 b I

^JJ| I r ? 0o o? o

! I 6-18

6-30

6-Z25 6- Z25 1 6-Z38

6-Z24 I 1 I I 6-Z24 t 6-Z19 t

9:

6-Z25

6.2 b3^bc^.^ 6

t 6-Z23

6-Z24

Ii 19 . - b, , b? b <y 1 ' i l l i 96-, v

6-Z44 b I

^JJ| I r ? 0o o? o

! I 6-18

6-30

6-Z25 6- Z25 1 6-Z38

6-Z24 I 1 I I 6-Z24 t 6-Z19 t

9:

6-Z25

6.2 b3^bc^.^ 6

t 6-Z23

6-Z24

Ii 19 . - b, , b? b <y 1 ' i l l i 96-, v

6-Z44 b I

^JJ| I r ? 0o o? o V V V V

Example 4c. Example 4c. Example 4c. Example 4c.

**5-30 A i I

**5-30 A i I

**5-30 A i I

**5-30 A i I


Example 5b. Progressions of all-interval tetrachords included as subsets in the hexachords of Braxton, "489M..." Example 5b. Progressions of all-interval tetrachords included as subsets in the hexachords of Braxton, "489M..." Example 5b. Progressions of all-interval tetrachords included as subsets in the hexachords of Braxton, "489M..." Example 5b. Progressions of all-interval tetrachords included as subsets in the hexachords of Braxton, "489M..."

4-Z29: 4-Z29: 4-Z29: 4-Z29:

(2,3,5,9)

TiI T7I

(11, 10,8,4) (5,4,2, 10)

\ / T3 T3

(8,7,5, 1)

i I3 T3

(5,4,2, 10)

T5

(10,9,7,3)

T3

(1,2,4,8)

(2,3,5,9)

TiI T7I

(11, 10,8,4) (5,4,2, 10)

\ / T3 T3

(8,7,5, 1)

i I3 T3

(5,4,2, 10)

T5

(10,9,7,3)

T3

(1,2,4,8)

(2,3,5,9)

TiI T7I

(11, 10,8,4) (5,4,2, 10)

\ / T3 T3

(8,7,5, 1)

i I3 T3

(5,4,2, 10)

T5

(10,9,7,3)

T3

(1,2,4,8)

(2,3,5,9)

TiI T7I

(11, 10,8,4) (5,4,2, 10)

\ / T3 T3

(8,7,5, 1)

i I3 T3

(5,4,2, 10)

T5

(10,9,7,3)

T3

(1,2,4,8)

4-Z15:

(2, 1, 10,8)

TS

(9,8,5,3) I

T1I

(4, 58, 10)

T5

(9, 10, 1,3)

T2I

(5,4, 1, 11)

4-Z15:

(2, 1, 10,8)

TS

(9,8,5,3) I

T1I

(4, 58, 10)

T5

(9, 10, 1,3)

T2I

(5,4, 1, 11)

4-Z15:

(2, 1, 10,8)

TS

(9,8,5,3) I

T1I

(4, 58, 10)

T5

(9, 10, 1,3)

T2I

(5,4, 1, 11)

4-Z15:

(2, 1, 10,8)

TS

(9,8,5,3) I

T1I

(4, 58, 10)

T5

(9, 10, 1,3)

T2I

(5,4, 1, 11)

Coleman achieves here. Much more informative with respect to compositional design are certain remarkable pitch-class relationships in the piece. Example 6 includes a tetrachordal segmentation of the opening head tune. The evidence of a tightly thought-out melodic line is compelling: only a few tetrachord types account for the construction.l That the bassist, Charlie Haden, has worked closely with the ensemble both in the head and throughout the entire work is evident from the fact that his pitches are almost all shared with the melody. In the last two phrases of the head, however, the contrapuntal relationships are less dependent on imitation. The repeated fifth A-E which

"The D-minor scale (including the raised sixth and seventh along with their natural correspondents), considered as a set of nine notes, includes abstractly all 29 of the possible tetrachords.







opens the penultimate phrase forms 4-11[0,1,3,5] with the horns; the same pitch classes occur at the opening of the second melodic phrase (m. 10) as well. Above the pedal, Haden chro- matically fills in the minor third, D-F, which both anticipates the chorus and parallels the upper voices. In the final phrase, if G# is interpreted as a passing tone (Haden actually slides up to A), the notes formed above the pedal form 4-22[0,2,4,7]; the same pitch classes, D-F-G-A, are the opening melodic notes of "Lonely Woman."

There is some significance even in the abstract relationships that exist between several pairs of sets in the head tune- specifically, 4-Z15[0,1,4,6] and 4-Z29, 4-4[0,1,2,5] and 4-14[0,2,3,7], and 4-22 and 4-2[0,1,2,4], which are all M- related. Even more interesting here, however, is the fact that the links between these sets are all by way of the same specific pitch-class operation, T3M (see Ex. 7).12 (1) The opening four notes of the melody are transformed through T3M into C-Ct- D-E, a 4-2 tetrachord, in the third phrase. (2) In the fourth phrase, which has the quality of a quick but pointed interjection that interrupts the melodic flow in a beautiful way, one important set formed is D-E-F-A, 4-14. This is a T3M transform of B- C-CS-E, 4-4, which is part of the harmonic support in the last notes of the previous phrase. This set is formed (by Coleman's own part) in a rather curious way: its penultimate note, C", du- plicates an octave lower one of the pitches in the melody; but its last note, B t, is not a melodic doubling. (3) The ending of the third phrase has B-C#-D-E, 4-10[0,2,3,5], as a harmonic component. Set 4-10 maps into itself under multiplication and can thus be interpreted as a T3M transform of the 4-10 formed melodically at the end of the second phrase. (4) The first four notes of the fourth phrase, 4-Z29, are related by T3MI to the 4-Z15 tetrachord formed in the opening phrase. (5) The most extraor- dinary relationship of all is the one that emerges between the general diatonic nature of the entire passage and the melodic







'2This would include the TgM operation, since it is the inverse of T3M. '2This would include the TgM operation, since it is the inverse of T3M. '2This would include the TgM operation, since it is the inverse of T3M. '2This would include the TgM operation, since it is the inverse of T3M.


Example 6. Segmentation of head, Orette Coleman, "Lonely Woman." Transcription by the author from The Shape of Jazz to Come (Atlantic SD-1317). (Parentheses indicate M-related pairs; asterisks indicate T3-M-related sets)







A trumpet and alto sax A trumpet and alto sax A trumpet and alto sax A trumpet and alto sax

^ v ~ - I - I _ I I

4-14 (44) I . 1 4-3

double bass (at pitch)

4-Z15 (4-Z29) *4-22 (4-2) I

I+ I Phrase 1] P --I---

*4-2 (4-22) *4-10 4-11 4-22 (4-2) I /

1*4-10 * 4 1Phrase (4-4) Phrase 2] I I Phrase 3 4]

i > r rr |f r I? 1i^ 61 ,) 1 1 Xi 10 *44 (4-14) 4-Z29 (4-Z15)

4-1 Phrase I 4-3 Phrase 61 419 [r ,1 Phrase 6] 4-19 \

16 4-9 f 4f 15) 4 4-19 16 4-Z29 (4-Z15) I I 4-19

2 > ^^ \- 1' |] H 1 i m iH 7

^ v ~ - I - I _ I I

4-14 (44) I . 1 4-3


4-Z15 (4-Z29) *4-22 (4-2) I


*4-2 (4-22) *4-10 4-11 4-22 (4-2) I /


i > r rr |f r I? 1i^ 61 ,) 1 1 Xi 10 *44 (4-14) 4-Z29 (4-Z15)


16 4-9 f 4f 15) 4 4-19 16 4-Z29 (4-Z15) I I 4-19

2 > ^^ \- 1' |] H 1 i m iH 7

^ v ~ - I - I _ I I

4-14 (44) I . 1 4-3


4-Z15 (4-Z29) *4-22 (4-2) I


*4-2 (4-22) *4-10 4-11 4-22 (4-2) I /


i > r rr |f r I? 1i^ 61 ,) 1 1 Xi 10 *44 (4-14) 4-Z29 (4-Z15)


16 4-9 f 4f 15) 4 4-19 16 4-Z29 (4-Z15) I I 4-19

2 > ^^ \- 1' |] H 1 i m iH 7

^ v ~ - I - I _ I I

4-14 (44) I . 1 4-3


4-Z15 (4-Z29) *4-22 (4-2) I


*4-2 (4-22) *4-10 4-11 4-22 (4-2) I /


i > r rr |f r I? 1i^ 61 ,) 1 1 Xi 10 *44 (4-14) 4-Z29 (4-Z15)


16 4-9 f 4f 15) 4 4-19 16 4-Z29 (4-Z15) I I 4-19

2 > ^^ \- 1' |] H 1 i m iH 7 4-11 4-11 4-11 4-11

4-22 (4-2) 4-22 (4-2) 4-22 (4-2) 4-22 (4-2)

Vr w w w -w Vr w w w -w Vr w w w -w Vr w w w -w


Example 7. T3M and T3MI relations in the head of "LonelyWoman"




T3M

4-22 Y 4-2

1 a . ._

T3M

4-22 Y 4-2

1 a . ._

T3M

4-22 Y 4-2

1 a . ._

T3M

4-22 Y 4-2

1 a . ._

T3M

4-4 4-14

r-I I -

T3M

4-4 4-14

r-I I -

T3M

4-4 4-14

r-I I -

T3M

4-4 4-14

r-I I -

Fr I rF 1 Fr I rF 1 Fr I rF 1 Fr I rF 1

T3MI

4-Z15 A 4-Z29

r-3,--j I I ,~~~~~~~ I

T3MI

4-Z15 A 4-Z29

r-3,--j I I ,~~~~~~~ I

T3MI

4-Z15 A 4-Z29

r-3,--j I I ,~~~~~~~ I

T3MI

4-Z15 A 4-Z29

r-3,--j I I ,~~~~~~~ I I F I FV . I F I FV . I F I FV . I F I FV .

T3M

4-10 4-10

,? ~II

T3M

4-10 4-10

,? ~II

T3M

4-10 4-10

,? ~II

T3M

4-10 4-10

,? ~II

T3M T3M T3M T3M

4-22 (2,5,7,9)

I M

(10, 11, 1,9)

T3

4-2 (0,1,2, 4)

4-14 (2,4,5,9)

MI

1 T3

(1, 11,4,0) 4-4 (11,0,1,4)

4-Z15 (1,2, 5, 7)

MI

(7, 2,11, 1)

T3

(10, 5,2,4) 4-Z29 (5,4,2,11)

4-10 (2,4,5,7)

M

(10, 8, 1, 11)

T3

(1, 11,4,2) 4-10 (11,1,2,4)

4-22 (2,5,7,9)

I M

(10, 11, 1,9)

T3

4-2 (0,1,2, 4)

4-14 (2,4,5,9)

MI

1 T3

(1, 11,4,0) 4-4 (11,0,1,4)

4-Z15 (1,2, 5, 7)

MI

(7, 2,11, 1)

T3

(10, 5,2,4) 4-Z29 (5,4,2,11)

4-10 (2,4,5,7)

M

(10, 8, 1, 11)

T3

(1, 11,4,2) 4-10 (11,1,2,4)

4-22 (2,5,7,9)

I M

(10, 11, 1,9)

T3

4-2 (0,1,2, 4)

4-14 (2,4,5,9)

MI

1 T3

(1, 11,4,0) 4-4 (11,0,1,4)

4-Z15 (1,2, 5, 7)

MI

(7, 2,11, 1)

T3

(10, 5,2,4) 4-Z29 (5,4,2,11)

4-10 (2,4,5,7)

M

(10, 8, 1, 11)

T3

(1, 11,4,2) 4-10 (11,1,2,4)

4-22 (2,5,7,9)

I M

(10, 11, 1,9)

T3

4-2 (0,1,2, 4)

4-14 (2,4,5,9)

MI

1 T3

(1, 11,4,0) 4-4 (11,0,1,4)

4-Z15 (1,2, 5, 7)

MI

(7, 2,11, 1)

T3

(10, 5,2,4) 4-Z29 (5,4,2,11)

4-10 (2,4,5,7)

M

(10, 8, 1, 11)

T3

(1, 11,4,2) 4-10 (11,1,2,4)

8-1 8-1 8-1 8-1

, 17jb p , 17jb p , 17jb p , 17jb p A A A A

I I I I M

M T3 8-1 (4, 5, 6, 7, 8, 9, 10, 11)-- (8, 1, 6, 11, 4, 9, 2, 7) - (11, 4, 9, 2, 7, 0, 5, 10) or 8-23 [2, 4, 5, 7, 9, 10, 0] + [11]

(D minor with added raised sixth)

I I I I M

M T3 8-1 (4, 5, 6, 7, 8, 9, 10, 11)-- (8, 1, 6, 11, 4, 9, 2, 7) - (11, 4, 9, 2, 7, 0, 5, 10) or 8-23 [2, 4, 5, 7, 9, 10, 0] + [11]


I I I I M

M T3 8-1 (4, 5, 6, 7, 8, 9, 10, 11)-- (8, 1, 6, 11, 4, 9, 2, 7) - (11, 4, 9, 2, 7, 0, 5, 10) or 8-23 [2, 4, 5, 7, 9, 10, 0] + [11]


I I I I M

M T3 8-1 (4, 5, 6, 7, 8, 9, 10, 11)-- (8, 1, 6, 11, 4, 9, 2, 7) - (11, 4, 9, 2, 7, 0, 5, 10) or 8-23 [2, 4, 5, 7, 9, 10, 0] + [11]


=-r I 1u1 1 =-r I 1u1 1 =-r I 1u1 1 =-r I 1u1 1

I _ r I 1b b I +i r-1 I _ r I 1b b I +i r-1 I _ r I 1b b I +i r-1 I _ r I 1b b I +i r-1 I I I I


chromaticism which enters in the penultimate phrase. The circle-of-fifths mapping, applied to a diatonic passage, generally transforms it into a chromatic one. The penultimate melodic phrase (winds only) forms a chromatic octochord which spans the fifth E-B. This set, subjected to the T3M transform, yields the natural minor scale of D with a raised sixth (set 8-23).

In examining the tetrachordal sets that fall out of the D- minor tonality, one should consider whether the frequency of certain included sets might explain pitch choices in a specific context. The subset of 8-23 with the greatest such frequency (8) is 4-22. Sets 4-Z29 and 4-11 are also frequently represented as subsets of 8-23 (they occur 4 and 6 times respectively) and are also prominent in this passage. Set 4-2(2), however, is not as frequently represented as a subset yet occurs at an important melodic point (see above, point (1)). Furthermore, 4-13(4), 4-16(4), and 4-27(4) are of comparable frequency yet are not profiled in the head. Set 4-13 can be formed as a subset of seventh and ninth chords often encountered in jazz, and 4-27 is the dominant seventh or half-diminished seventh; these chords are thus more likely to be outlined in a tonally oriented work. That the head tune does not include such sets indicates that the sets which are formed are part of a chosen pitch structure. This analysis is certainly more informative than the all-too-often repeated account of Coleman's improvisations as non-functional chord changes within a tonally centered framework.

Example 8a shows the melodic line of both first and second choruses of "Lonely Woman" with a segmentation by tetrachord.13 Once more, the small number of sets helps project the taut structure of the passage. In addition, the restricted field of pitch-class operations (To, T3, T5, ToM, T0lMI) accounts for many of the phrase-to-phrase transformations which lend such great aural cohesion to the work. For example, the two in-

13In most circumstances, the segmentation follows Coleman's phrasing. The only exceptions occur where a fourth pitch is appropriated from the beginning of a phrase to complete a tetrachord at the end of the previous phrase.













stances of 4-12[0,2,3,6] formed successively in two phrases of the second chorus have the musical sense of a fragment which is announced and then completed in the following phrase as the melodic line moves to the highest point of the chorus. More specifically, however, the second set is related to the first by T3, which same operation links at least two other phrases in the chorus (see Ex. 8b). Another example is the relation involving 4-24 and 4-2. The two sets occur successively in the music (sepa- rated by phrasing); later in the same passage, the T0lMI transforms of both sets appear-in reverse order (see Ex. 8b). The aural relationship is particularly striking in this passage, since the clearly whole-tone character of 4-24 is preserved (4-24 maps into itself under M).

It is also useful to examine some of the larger pc collections in this piece. The third chorus can be segmented into eight melodic phrases on the basis of the soloist's pauses. Five of the phrases contain one of the two Z-related sets, 6-Z3[0,1,2,3,5,6] and 6-Z36[0,1,2,3,4,7]. None of the five instances of these is a pitch-class duplicate of any other, a clear indication that Cole- man must be thinking intervallically as he improvises the line. Some of the specific transformations of this passage are outlined in Example 9. The two five-note sets formed here are included in larger sets that have some prominence in this chorus. The first 5-29[0,1,3,6,8], for instance, is abstractly included in 6-Z47[0,1,2,4,7,9], a set which is related by the M transform to 6-Z36. If F, is added to the 5-29 (it occurs in the next phrase) at the opening of the section, the 6-Z47 formed would be related by ToMI to 6-Z36 in the following phrase and by T10MI to the 6-Z36 included in the last phrase (see Ex. 9). The second pentachord, 5-24[0,1,3,5,7], is included in 6-Z26[0,1,3,5,7,8], a set formed by the six notes which precede the final cadence on A. Specifically, if the G that enters as the first note of the next phrase is added to the 5-24 shown, the resulting set contains the same pitch classes as are sounded at the end of the chorus (see Ex. 9). Among the 6-Z3/Z36 pairs themselves, all 6-Z36 sets are related by T2 and the 6-Z3 pair is also related by T2. The first








Example 8a. Segmentation of the melodic line of the first two choruses in "Lonely Woman." Transcription by the author.

(Parentheses indicate M-related sets)










1st Chorus (A) 1st Chorus (A) 1st Chorus (A) 1st Chorus (A) 4-Z29 (4-Z15) 4-Z29 (4-Z15) 4-Z29 (4-Z15) 4-Z29 (4-Z15)

[-3----1 5 r-3 - Pt J *w;-~- 1 _ I

4-22 (4-2) 4-13 4-8 2nd Chorus (B) 4-14(44) 4-14 (44) I I 4-22 (4-2) 4-11

| --

j |*-'to I ( I n

] J , 3 "l , ]. J

4-11 4-Z29 (4-Z15) Head-- 4.Z29

4-24 4-2 (4-22)

-- i -3-I I r---- I

4-11 4-12 (4-27)

, J - I 1- ' - r I - ' J I! 4-12 (4-27)

4-22 (4-2) 4-24 4-3 (4-26) 424

4-10 I 4-27 (4-12)

4-27 (4-12) 4-13 1

. . a - I, I Gi I J i -- Ii I K

[-3----1 5 r-3 - Pt J *w;-~- 1 _ I

4-22 (4-2) 4-13 4-8 2nd Chorus (B) 4-14(44) 4-14 (44) I I 4-22 (4-2) 4-11

| --

j |*-'to I ( I n

] J , 3 "l , ]. J

4-11 4-Z29 (4-Z15) Head-- 4.Z29

4-24 4-2 (4-22)

-- i -3-I I r---- I

4-11 4-12 (4-27)

, J - I 1- ' - r I - ' J I! 4-12 (4-27)

4-22 (4-2) 4-24 4-3 (4-26) 424

4-10 I 4-27 (4-12)

4-27 (4-12) 4-13 1

. . a - I, I Gi I J i -- Ii I K

[-3----1 5 r-3 - Pt J *w;-~- 1 _ I

4-22 (4-2) 4-13 4-8 2nd Chorus (B) 4-14(44) 4-14 (44) I I 4-22 (4-2) 4-11

| --

j |*-'to I ( I n

] J , 3 "l , ]. J

4-11 4-Z29 (4-Z15) Head-- 4.Z29

4-24 4-2 (4-22)

-- i -3-I I r---- I

4-11 4-12 (4-27)

, J - I 1- ' - r I - ' J I! 4-12 (4-27)

4-22 (4-2) 4-24 4-3 (4-26) 424

4-10 I 4-27 (4-12)

4-27 (4-12) 4-13 1

. . a - I, I Gi I J i -- Ii I K

[-3----1 5 r-3 - Pt J *w;-~- 1 _ I

4-22 (4-2) 4-13 4-8 2nd Chorus (B) 4-14(44) 4-14 (44) I I 4-22 (4-2) 4-11

| --

j |*-'to I ( I n

] J , 3 "l , ]. J

4-11 4-Z29 (4-Z15) Head-- 4.Z29

4-24 4-2 (4-22)

-- i -3-I I r---- I

4-11 4-12 (4-27)

, J - I 1- ' - r I - ' J I! 4-12 (4-27)

4-22 (4-2) 4-24 4-3 (4-26) 424

4-10 I 4-27 (4-12)

4-27 (4-12) 4-13 1

. . a - I, I Gi I J i -- Ii I K

4-14 (4-4) 4-14 (4-4) 4-14 (4-4) 4-14 (4-4)

A .A ,' A% A .A ,' A% A .A ,' A% A .A ,' A%

I I I I V _ V _ V _ V _

I I I I I I I I


Example 8b. Two pitch-class phrase connections in the opening chorus of "Lonely Woman"







4-24 4-24 4-24 4-24

^ I^j r^m ^ '

r-'-J3 '

IJ I x.. IA J IIJ I I

Lrt' M -N 1 J

1 1 1 1~~~~~~~~~~~~~~~~42

^ I^j r^m ^ '

r-'-J3 '


Lrt' M -N 1 J

1 1 1 1~~~~~~~~~~~~~~~~42

^ I^j r^m ^ '

r-'-J3 '


Lrt' M -N 1 J

1 1 1 1~~~~~~~~~~~~~~~~42

^ I^j r^m ^ '

r-'-J3 '


Lrt' M -N 1 J

1 1 1 1~~~~~~~~~~~~~~~~42 4-22 4-22 4-22 4-22

4-24 (9, 7, 5, 1) - (3, 1, 11, 7) T

4-24 (1, 11, 9, 5)

4-2 (5, 6, 7, 9) MI (11,6,1,3) T

(9,4, 11, 1) 4-22 (9,11,1,4)

4-24 (9, 7, 5, 1) - (3, 1, 11, 7) T

4-24 (1, 11, 9, 5)

4-2 (5, 6, 7, 9) MI (11,6,1,3) T

(9,4, 11, 1) 4-22 (9,11,1,4)

4-24 (9, 7, 5, 1) - (3, 1, 11, 7) T

4-24 (1, 11, 9, 5)

4-2 (5, 6, 7, 9) MI (11,6,1,3) T

(9,4, 11, 1) 4-22 (9,11,1,4)

4-24 (9, 7, 5, 1) - (3, 1, 11, 7) T

4-24 (1, 11, 9, 5)

4-2 (5, 6, 7, 9) MI (11,6,1,3) T

(9,4, 11, 1) 4-22 (9,11,1,4)

T3

I T3

I T3

I T3

I

9- 9- 9- 9-

4-12 4-12 4-12 4-12

XbI I IJp iLtr - I .r. r i-- 1r a

:6; XbI I IJp iLtr - I .r. r i-- 1r a



:6;

4-12 (7,9,10,1)T3412 (10,1, 4-12 (7, 9, 10, 1) . . 4-12 (10,0, 1,4) 4-12 (7,9,10,1)T3412 (10,1, 4-12 (7, 9, 10, 1) . . 4-12 (10,0, 1,4) 4-12 (7,9,10,1)T3412 (10,1, 4-12 (7, 9, 10, 1) . . 4-12 (10,0, 1,4) 4-12 (7,9,10,1)T3412 (10,1, 4-12 (7, 9, 10, 1) . . 4-12 (10,0, 1,4)

6-Z3 formed in the chorus is related by ToM to the

6-Z25[0,1,3,5,6,8] shown in the segmentation in the last

phrase-and, in fact, 6-Z25 can also be considered a superset of the initial five notes. In this section, then, a pattern of relation-

ships based upon whole-tone transpositions begins to emerge. The harmonic consistency of "Lonely Woman" is just as

noteworthy as the linear consistency detailed above. Example 10 displays a harmonic segmentation of the latter half of the second chorus-that is, the portion following the eight- measure sequence which is included in all the choruses.14 Here

14Here again the segmentation generally follows the phrasing, although in this revolutionary section, where bassist and soloist are playing in two different

tempi, the phrasing is determined by the pauses of the more prominent voice.






















too, the majority of the sets have clear interrelations, especially the instances of 6-Z26 and 6-Z48[0,1,2,5,7,9]. The three 6-Z48's are actually two Ts-related sets scored so that there is a

departure from and return to the first form presented in the music. The 6-Z26 is formed at the end of the chorus in the measures preceding the final measure and, as so often happens in this piece, the same set occurs horizontally as well as vertically. While the 6-Z26/Z48 pairs cannot map into each other under the M operation, 6-Z26 maps under M into 6-Z4[0,1,2,4,5,6], the very first set formed in this part of the chorus. The specific relation between the two sets here is T2M; this lends further

emphasis to the sense that whole-tone transposition controls the pitch structure. In addition, another five-note set formed










4-2 4-2 4-2 4-2

4-12 4-12 4-12 4-12

4-24 4-24 4-24 4-24

I I I I I I I I


Example 9. Some relationships between sets and implied sets in the third chorus of "Lonely Woman." Transcription by the author







To M

I I

To M

I I

To M

I I

To M

I I

6-Z47 (phrase 1) I 5-29

6-Z47 (phrase 1) I 5-29

6-Z47 (phrase 1) I 5-29

6-Z47 (phrase 1) I 5-29

6-Z36 (phrase 2) 6-Z36 (phrase 2) 6-Z36 (phrase 2) 6-Z36 (phrase 2) TI1MI

6-Z47 (phrase 1) 6-Z36 (phrase 6) TI1MI



6-Z47 (phrase 1) 6-Z36 (phrase 6) I I I I I I I I

_^ 7- -^ -tt ^ ^-^# , 1,, (,5 2,.4. 11 ( -)

). I . 2. 4. 5) 6) .5. 2.4. 11, 6) (9,11, 2, 4, 5, 6)- (3, 5, 2, 4, 11,6) (1,3, 0, 2, 9, 4)

5-24 (phrase 3) included in 6-Z26 (phrase 8) 6-Z36 (phrase 2) T10 6-Z36 (phrase 6) T0 6-Z36 (phrase 4)

ToM

6-Z3 (phrase 5) 4 6-Z25 (phrase 8)

* I i l I

P----^+---^ 'r^ ' ^ II./ b-, . . _- AP -_

T2 M 6-Z3 (phrase 5) T 6-Z3 (phrase 7) (6, 7, 9, 10, 11, 0) 6, 11, 9, 2, 7, 0)

Example 10. Harmonic segmentation of second chorus, second section of "Lonely Woman." (Slashes separate Z-related sets; sets in parentheses are M-related)

6-Z4/6-Z37(6-Z26) (0, 1, 2, 9, 8, 10) 6-Z48/6-Z26(6-Z37) (0, 2,4, 7, 8, 9) I /

nLt.r- . .

-... 1

e? i?' 1-

.ÂH 1i^l Li

I _J

6-Z48 (0, 1, 2, 5, 7. 9)

6-Z48 (0, 2,4, 7, 8, 9) 6-33 (1, 2,4,7,9,11) 6-Z26/6-Z48(6-Z4) (110, 2, 4. 6, 7) l (1I247911 I tempo

^ i r J r r I r'H "- H- " V-ii fit

- - - - - - - - - - - c I - - - - - - - - - - - - - - - - - -3

Ae | ~ 17 ' ' - '~

_^ 7- -^ -tt ^ ^-^# , 1,, (,5 2,.4. 11 ( -)

). I . 2. 4. 5) 6) .5. 2.4. 11, 6) (9,11, 2, 4, 5, 6)- (3, 5, 2, 4, 11,6) (1,3, 0, 2, 9, 4)


ToM


* I i l I

P----^+---^ 'r^ ' ^ II./ b-, . . _- AP -_



6-Z4/6-Z37(6-Z26) (0, 1, 2, 9, 8, 10) 6-Z48/6-Z26(6-Z37) (0, 2,4, 7, 8, 9) I /

nLt.r- . .

-... 1

e? i?' 1-

.ÂH 1i^l Li

I _J

6-Z48 (0, 1, 2, 5, 7. 9)

6-Z48 (0, 2,4, 7, 8, 9) 6-33 (1, 2,4,7,9,11) 6-Z26/6-Z48(6-Z4) (110, 2, 4. 6, 7) l (1I247911 I tempo


- - - - - - - - - - - c I - - - - - - - - - - - - - - - - - -3

Ae | ~ 17 ' ' - '~

_^ 7- -^ -tt ^ ^-^# , 1,, (,5 2,.4. 11 ( -)

). I . 2. 4. 5) 6) .5. 2.4. 11, 6) (9,11, 2, 4, 5, 6)- (3, 5, 2, 4, 11,6) (1,3, 0, 2, 9, 4)


ToM


* I i l I

P----^+---^ 'r^ ' ^ II./ b-, . . _- AP -_



6-Z4/6-Z37(6-Z26) (0, 1, 2, 9, 8, 10) 6-Z48/6-Z26(6-Z37) (0, 2,4, 7, 8, 9) I /

nLt.r- . .

-... 1

e? i?' 1-

.ÂH 1i^l Li

I _J

6-Z48 (0, 1, 2, 5, 7. 9)

6-Z48 (0, 2,4, 7, 8, 9) 6-33 (1, 2,4,7,9,11) 6-Z26/6-Z48(6-Z4) (110, 2, 4. 6, 7) l (1I247911 I tempo


- - - - - - - - - - - c I - - - - - - - - - - - - - - - - - -3

Ae | ~ 17 ' ' - '~

_^ 7- -^ -tt ^ ^-^# , 1,, (,5 2,.4. 11 ( -)

). I . 2. 4. 5) 6) .5. 2.4. 11, 6) (9,11, 2, 4, 5, 6)- (3, 5, 2, 4, 11,6) (1,3, 0, 2, 9, 4)


ToM


* I i l I

P----^+---^ 'r^ ' ^ II./ b-, . . _- AP -_



6-Z4/6-Z37(6-Z26) (0, 1, 2, 9, 8, 10) 6-Z48/6-Z26(6-Z37) (0, 2,4, 7, 8, 9) I /

nLt.r- . .

-... 1

e? i?' 1-

.ÂH 1i^l Li

I _J

6-Z48 (0, 1, 2, 5, 7. 9)

6-Z48 (0, 2,4, 7, 8, 9) 6-33 (1, 2,4,7,9,11) 6-Z26/6-Z48(6-Z4) (110, 2, 4. 6, 7) l (1I247911 I tempo


- - - - - - - - - - - c I - - - - - - - - - - - - - - - - - -3

Ae | ~ 17 ' ' - '~ t t t t rr r r rr r r rr r r rr r r


melodically in the measures previous to this section is a subset of 6-Z26. Thus, the chord 6-Z4 is immediately preceded by its T2M transform melodically and is followed by the same transform at the end of the chorus. It would seem that in this chorus, at least, networks of sets related to the 6-Z3/Z36 pair and the 6-Z26/Z48 pair can be interpreted as controlling elements of the improvisation. This is all the more impressive considering that the two networks result from a counterpoint in which the bass is clearly an independent voice.

It should be clear from the foregoing that the complexities of the musical fabric in free jazz deserve far more attention than analysts have thus far devoted to them. The traditional interpretation of the head tune as the composition itself, and the music which follows as some kind of variation upon it, clearly is not appropriate to the hybrid language of many free compositions. Even in examples with clearly diatonic elements, such as "Lonely Woman," traditional, tonal jazz analyses cannot eluci- date the real relationships between pitch structures. To do the job properly, the analyst needs recourse to certain constructs familiar from analysis of early twentieth-century concert music, such as pitch-class set saturation, tonal centers in a nontonal context, and aggregate completion. Other constructs, not much employed in analysis of earlier music, are also useful here, notably the multiplicative operation, the transformation of embedded chords, and the rescoring of "end-sets" within larger sonorities. The consistent use of certain operations such as T3 and the multiplicative operation in some of these examples suggests that there may be empirical evidence for use in jazz of some of the other pitch-class universes that have been asserted, particularly by Robert Morris.15

The art of free jazz seems to require that the improvisers themselves steer away from arpeggiation of common chord for-










mations and progressions and think more in terms of relationships defined by interval class; this is true in both tonal and nontonal contexts. For this reason, free jazz has an affinity to early twentieth-century concert literature, in which composers were thinking along similar intervallic and structural lines.16 While early twentieth-century composers constructed their pc- relations, jazz musicians heard them in improvisation-which suggests that pitch-class and nontonal relations can develop naturally out of musical practice in the same way that tonal music grew out of modal music and nineteenth-century tonality grew out of that of the eighteenth century.

15Robert Morris, "Set Groups, Complementation, and Mappings Among Pitch-Class Sets," Journal of Music Theory 26 (1982), 101-144.

'6Simple segmentations, suggested largely by the performers' pauses, seem appropriate to analysis of free jazz because these compositions are often cre- ated and/or elaborated spontaneously. Since the relationships that fall out of such segmentations seem clear, rich, and structurally important, perhaps it would be worth considering the utility of simple segmentations in the analysis of early twentieth-century concert music.

ABSTRACT Set-theoretic methodology is applied to the music of Ornette Cole- man, John Coltrane, Cecil Taylor, and Anthony Braxton in order to reveal the wide variety of pitch-class transformation present in free jazz. Each composer's music has been classified somewhat differently by other analysts-Coltrane's as modal, Coleman's as diatonic, and Taylor's as nontonal-yet all the improvisations examined here are shown to be based on tightly constructed conceptions which make use of such twentieth-century constructs as the multiplicative operation, transformation of embedded chords, and the use of a small number of transformational operations which control the course of the composition.













society for music theory pitch-class transformation in free jazz - steven block

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