The eXchange Operation and Tetradic Voice-Leading in the Music of Jobim

8/18/2019 The eXchange Operation and Tetradic Voice-Leading in the Music of Jobim

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The eXchange Operation and Tetradic Voice-Leading in the Music of Jobim

Brian Bartling

05/05/0!5



The eXchange Operation and Tetradic Voice-Leading in the Music of JobimBrian Bartling

"no#n #orld#ide as a progenitor of the bossa no$a st%le& as #ell as the composer of a number

of inimitable 'a(( standards& )nt*nio +arlos Jobim holds a special place in the histor% of music,

)lthough the bossa no$a st%le is no#n mostl% for its samba-based rh%thms& Jobim instills in his music

a distincti$e melodic and harmonic character, One could point to a number of influences that

contribute to this . contemporar% )merican 'a(( and sho# tunes& popular Bra(ilian songs& 0 th centur%

rench composers& etc, . but ne$ertheless& Jobims harmonic st%le is no$el and original, 1is harmonies

deal #ith 2th chords as its basic unit& and are characteri(ed b% little $oice-leading #or bet#een

subse3uent chords4 #idespread use of common tones4 and a relati$el% high spread,! all situated #ithin a

tonal frame#or, ssentiall%& Jobims harmonies acts lie triads& but #ithin a tetradic uni$erse, 6t

#ould seem that this music is ripe for 7eo-8iemannian anal%sis& but alas& tetrachords are infamousl%

obstinate to these theories& #hich are mostl% built on triads, Much insight is %et to be gained b%

anal%(ing Jobims $oice-leading& so it is imperati$e to in$ent ne# s%stems& or e9pand older ones& to

co$er a broader harmonic language,

7th Chords in Neo-Riemannian Theory

:erhaps the most potent e9ample of an attempt to e9pand 7eo-8iemannian triadic theories to

the four-note uni$erse #as in )drian +hilds !;;< article, +hilds addresses the problem b% sho#ing a

passage from a +hopin caden(a& built entirel% from se$enth chords& and tries to reduce the tetrads into

triads in order to gi$e them a familiar :& L& or 8 label, 6t 3uicl% becomes apparent that this cannot be

! The term spread #as used in =+allendar& >uinn& ? T%moc(o 00<@, 6n this conte9t& it is a term used to describe $oice-leadings bet#een t#o chords in terms of their distance from the perfectl% clustered chord t%pe A0& & 0C, 6t can also beused to describe the 3ualit% of a single chord& in terms of its distance from the perfectl% clustered chord, ) chord #ithlo# spread #ill be clustered& #hereas a chord #ith high spread #ill be closer to the perfectl% e$en chord . in this case& afull%-diminished chord,

=+hilds !;;<@



done effecti$el%& since the higher dimensional tetrads e9hibit smooth $oice-leading that cannot be

accuratel% captured b% triadic labels, 6nterestingl%& one theorist has anal%(ed t#o brief passages from

different Jobim songs b% this $er% method, That is& reducing them to a triadic subset,D or the $er%

same reasons elucidated in +hilds article& this anal%sis is ineffecti$e& as it fail to account for ho# the

e9tra note& the 2th& beha$es,

+hilds attempted to sol$e this problem b% in$enting a s%stem that is closel% related to the spirit

of 7eo-8iemannian theor%, The :& L& and 8 transforms are& according to +hilds& are P!

related since

the% all describe mo$ement b% a semitone in one $oice #hile the other t#o $oices remain constant, E

Thus& +hilds limited himself to tetradic transformations that e9hibit a P

relation . t#o $oices differ

b% a semitone #hile the other t#o $oices remain constant, This #a%& the harmonic language

encompassed b% 7eo-8iemannian theor% is e9panded but still tractable& since +hilds can no# sho#

#hich t#o $oices mo$e and #hich t#o $oices sta% constant, 6n his notation& similar mo$ement is

denoted b% F #hile contrar% mo$ement is denoted b% +4 the inter$al-class that remains constant is

sho#n in the subscript4 and the inter$al-class that changes is sho#n in parentheses& also in the

subscript, or e9ample& a SD() transformation implies that there is similar mo$ement& the constant

inter$al-class is separated b% D semitones& and the changing inter$al-class is separated b% semitones,

The $oice-leading in +hopin then becomes clear #ith this s%stem& #hich is sho#n in Figure 1,

This is an effecti$e #a% to model $oice-leadings among P−related chords& or =E-2@ chords

that share an octatonic relationship& but is much too restricti$e to anal%(e music as tetrad-rich as

Jobims, This is sho#n in Figure 2& #hich displa%s a Jobim-lie chord progression, )lthough

mo$ement bet#een chords & D& and E in this figure isObject5

there are man% instances of tetradic

D =Briginsha# 0!@, Briginsha# omitted the se$enths in her anal%sis to sho#& #hat she percei$ed to be& Jobims use ofFlide . a triadic transformation, 6n m% anal%sis& 6 #ill sho# ho# one passage that she anal%(ed . the opening to theOne-Note Samba . uses some#hat of a parallel to this transformation& but in this four-note uni$erse,

E :csets X and Y are :-related = X : Y @ if G X G H GY G and X IY K0!& #here X IY is the s%mmetric difference of X and Y ,=+ohn 00D@ This phenomenon is #idel% studied in the literature& so it goes b% man% different names, : relations arealso no#n as ma9imal smoothnessN in =+ohn !;;@& and are a crucial element for situating relationships bet#eenpitch class sets #ithin a geometr%,



progressions that arent& such as the one bet#een chords ! and , 7ot onl% do three $oices mo$e in this

progression& but one $oice mo$es b% a #hole toneP There are man% cases of tetrads that dont e9hibit

P

relations& e$en in those among the =E-2@ chords that are under +hilds pur$ie#, The progression

bet#een Qdom2 and )R2& for e9ample& necessitates that each of the $oices mo$e b% semitone& sho#n

in Figure 3, This is among t#o chords that are& as #ill be sho#n& closel% related, Fo #hile +hilds

method is fruitful for anal%(ing progressions #ithin the =E-2@ octatonic frame#or& it breas do#n for

music that $entures outside of this net#or& and so it needs to be e9panded to anal%(e all of the

possibilities that are presented b% Jobims harmonies, This must not come at the cost of the t%pe of

clarit% and tractabilit% that is sho#n in +hilds s%stem,

Triadic Sum-Class Diagrams

One #a% to situate Jobims chords inside of a diagram is b% using sum-class diagrams, The

e9tent to #hich these illustrate $oice-leadings among =D-!!@ triads has been e9plored in =+ohn !;;<@&

and operate b% grouping together the summation of each of a chords $oices& mod !, The sum class

diagram for triads& sho#n in Figure 4& can be easil% graspedS the directed voice-leading

5

bet#een t#o

chords related b% a :L8-transform is !& and an% chord that is related in this #a% forms a hexatonic

system, There are four he9atonic s%stems that are formed among the triads that are oriented in Figure 4

b% cardinal points, Fum-classes 10 and 11 are north& 1 and 2 are east& 4 and are south& and 7 and ! are

#est, 6t taes t#o units of $oice leading to tra$el to an% he9atonic s%stems t#o neighboring s%stems&

as it is necessar% to tra$el through the t#o s%stems coupling chord . an augmented triad, B% perfectl%

di$iding the octa$e& an augmented triad can be transformed into si9 different triads b% raising or

lo#ering a $oice b% a semitone, There are four augmented triads& and thus four regions& or Weitzmann

Regions& in #hich this mo$ement is emplo%ed, 6t taes at least fi$e units of semitonal #or to mo$e

5 The directed $oice leading bet#een bet#een chords X and is gi$en b%S UVLF=X&@ H FM=@ . FM=X@& #herethe operation FM is the summation of a chords $oices, That is& gi$en a trichord X #ith $oices A9!& 9& 9DC& FM=X@H 9! W 9 W 9D mod !, =+ohn !;;<@



bet#een t#o complementar% =sum-class groups across Figure 4@ he9atonic s%stems,

The #a% to mo$e bet#een sum classes on this diagram is b% eXchange operations, These

operations are similar to directed $oice-leadings& but tae into account the direction in #hich $oices are

mo$ing, or e9ample& sum-classes 11 and 1 ha$e a directed $oice-leading of !& so the% eXchangeN !

semitone& #hich can be #ritten as X ! The directed $oice-leading bet#een and !0& on the other

hand& is commutati$e& but mo$ement is sho#n in the eXchange notation b% X !0 . t#o $oices

descend b% semitone, 6n +ohns diagram& operations that e9change a semitone in one $oice& notated

X ! , are sho#n to mo$e cloc#ise& #hereas operations that e9change a semitone in t#o $oices&

notated X !0, mo$e countercloc#ise, 6t should be noted that X !! and X , respecti$el%& are also

possible mo$ements on the diagram,

The Tetradic "nalogue

The no$elt% of the triadic representation lies in the interpla% bet#een sum-class mo$ements that

e9change ! semitone and mo$ements that e9change semitonesS function and regional interchanges

are represented b% directed $oice-leadings, The tetrachordal analogue to the =D-!!@ triadic sum-class

diagram of Figure 4 is the =E-2@ sum-class diagram of Figure & #hich sho#s the relationship

bet#een in$ersionall% related dominant-se$enth and half-diminished chords, ach chord on this figure

is related b% units of directed $oice-leading #or& so regional geograph% is unclear from this =E-2@

set, 7e$ertheless& there are some relationships that appear from Figure . there are three perfectl%-

s%mmetrical tetrachords& or half-diminished chords& #hich suggests that there are three eit(mann

regions and three octatonic regions . the tetradic analogue to he9atonic regions, This suggests that

there is a northern set& a south#estern set& and a southeastern set4 each region contains both a dominant

se$enth and a half-diminished se$enth chord, 6t is no# clear from this representation that the

Fince eXchange operations arent set classes& s#itching tN for !0N and eN for !!N isnt necessar%, or the rest of m%anal%sis& 6 #ill eep consistent #ith +ohns notation,



progression in Figure 3 is as closel% related as t#o tetrachords of differing sum-classes can beS the

directed $oice-leading from # to 7 is related b% a X !0

operation that mo$es countercloc#ise on the

figure, The% are also in the same octatonic region on the northern section of the figure, This e9ample

points to an interesting discrepanc% bet#een the t#o dimensions of sum-class group diagramsS the

directed $oice-leading bet#een t#o sum-class groups ma% occur in & D& or E $oices in the tetrachordal

figure& #hereas there is much less $ariet% in the trichordal figure, Mo$ement among the tetrachordal

sum-class groups operate in a more general manner . something that #ill be addressed in the anal%ses,

$Ne%t Ste&' Tetradic Diagrams

6t is crucial to the stud% of Jobims chords that this tetrachordal s%stem is e9panded to include

chords outside of set-class =E-2@, ) ne9t stepN tetrachordal figure could include all of the perfectl%-

e$en full%-diminished se$enth chords& #hich are the three chords #ith sums 2& (& and 10, These chords

fill inN the gaps b% placing some e$en numbers into Figure & a diagram of all odd sum-classes& thus

thro#ing X !

and X !! operations into the tetrachordal mi9, These full%-diminished chords are also

coupling chords bet#een the octatonic regions, The rest of the sum-class groups #ithin the octatonic

regions themsel$es can be filled in b% introducing minor se$enth chords . often used in 'a(( and the

music of Jobim, These chords are situated #ithin the same octatonic region as its dominant se$enth

and half-diminished se$enth neighbors since the% simpl% borro#N notes& acting as a single semitone

barrier bet#een them, The filled-inN $ersion of Figure & the full mod ! cloc face of sum-class

groups& #as sho#n in =Uouthett ? Fteinbach !;;<@ as their Po!er "o!ers diagram& #hich is sho#n in

Figure (, Their figure clearl% sho#s the orientation of the octatonic regions . a northern& southeastern&

and south#estern region . #ith the coupling full%-diminished chords, ) flo# chart analogue of the

Po!er "o!ers sum-classes is displa%ed in Figure 7& #hich sho#s the do#n#ard =cloc#ise@

mo$ement of X !

transformations,



Figure 7 sho#s the cru9 of sum-class representations of tetrachords, $er% node =sum-class

group@ holds a place in the figure& and mo$ement to each subse3uent node entails a single semitone

shift, urthermore& mo$ement to other nodes can be described b% eXchange operations in either

direction, This is sho#n in the generali(ed node graph of Figure #& #hich sho#s some basic

s%mmetries among the $oice-leadings of sum-class groups,

6n order to encompass the #hole of tetradic material that is used b% Jobim& it is necessar% to add

more possibilities to the nodes of Figure 7& resulting in an e$en higher le$el of diagrammatic order,

Three such chord t%pes are the ma'or se$enth chord& the minor =Q2@ chord& and the dominant se$enth

=b5@ chord& or the french augmented si9th chord, ith the addition of these three chord t%pes& the bul

of Jobims harmonic material is accounted for . the most commonl% used chords b% Jobim are =0D;@4

=0D<@4 =0D5<@4 =0<@4 =05<@4 =0E<@4 =0E2@4 =0!5<@4 =0!E<@, The set =0E<@ accounts for the

dominant se$enth =Q5@ chord that is used sporadicall% in Jobims music, )n augmented triad #ith an

added se$enth& this chord has a s%mmetrical nature that occupies the coupling nodes in the tetrachordal

sum-class diagrams, Figure ! adds the three ne# Jobim chords to this diagram . =0E<@ is omitted for

no#, ith the e9ception of the half-diminished nodes& e$er% node no# has t#o possibilities to choose

from& and a composer =Jobim@ has the abilit% to tra$el multiple different path#a%s . to use different

eXchange operations . across the same nodes, +hords of different 3ualities that occup% the same node

can be transformed into one another b% raising some $oice and lo#ering another $oice, This poses an

interesting problem for anal%(ing $oice-leading $ia directed $oice-leading sums and eXchange

operationsS tri$ial transformations and non-tri$ial transformations sometimes ha$e the same eXchange

$alue, hile these tools are effecti$e for situating chords #ithin a geometr%& the same o$ergenerali(ing

problem pre$iousl% addressed comes forth, That is& directed $oice-leading sums sometimes label the

same $alue to chord progressions that e9hibit multiple different t%pes of $oice-leading #or, 6t is

necessar%& then& to couple these diagrams #ith a similar t%pe of $oice-leading anal%sis that #as done b%



+hilds in his article,2

)dding more chord t%pes to the tetradic sum-class diagram ma% spa#n multiple different

path#a%s& but the higher orders of tetradic chord t%pes do not necessaril% function the same #a% as

their lo#er ordered counterparts, This is especiall% true for ma'or se$enth chords4 these chords ha$e

the same sum class as full% diminished chords because the% can be deri$ed from these chords b%

lo#ering the prime and raising the se$enth a single semitone, This does not in turn gi$e the ma'or

se$enth the same po#er as full%-diminished se$enth chords, hile the directed $oice-leading remains

the same& the amount of $oice-leading #or is se#ed a#a% from the complete e$enness that

characteri(es the full%-diminished se$enth eit(mann regions, 6t is crucial to note that chord t%pes that

occup% the same node are related onl% in that the% share common ground, ) suitable anal%sis of the

directed $oice-leading among tetrads #ould focus most of its energ% on the description of path#a%s

among the nodes& not the actual nodes themsel$es, Thus& an anal%sis of Jobims tetrachords using

Figure ! #ould need more specificit%4 the t%pe of $oice-leading motion and the total number of

mo$ing $oices should be distinguished,

S&eci)ic *oice-+eading among Directed *oice-+eadings

6n his notation& +hilds specified the t%pe of $oice-leading motion that e9ists bet#een t#o

chords, That is& there #ere al#a%s t#o $oices mo$ing& but #hether the% mo$ed in similar or contrar%

motion #as something that must be labeled, This distinction must also be labeled in this anal%sisS a

directed $oice-leading sum of 0 in similar motion is fundamentall% different than a directed $oice-

leading sum of 0 in contrar% motion, )s #as pointed out& the former transformation describes t#o

chords of a similar t%pe& #hile the latter describes t#o chords of different t%pes, )lso in +hilds

notation& he #as careful to point out the inter$al-class that remains constant and the inter$al-class that

mo$es, This is an artifact of triadic 7eo-8iemannian theor% that doesnt necessar% appl% as easil% in

2 =+hilds !;;<@



this case, Mo$ement among chord t%pes& number of $oices mo$ing& and distance in #hich indi$idual

$oices mo$e are all much freer in a tetradic Jobim anal%sis, Therefore& a more crucial distinction to be

made is the number of indi$idual $oices that are mo$ing, or e9ample& the progression in Figure 3 is

labeled as a X !0 progression& but #ithout no#ing ho# man% $oices are mo$ing& the X !0 label

doesnt full% elucidate the nature or function of the progression . a theorist ma% assume that $oices

are mo$ing instead of E& #hich #ould negati$el% affect the anal%sis, These t#o bits of information .

the t%pe of $oice-leading motion and the number of mo$ing $oices . gi$e the anal%tical s%stem the

crucial missing piece of specificit% that isnt manifest in eXchange operations alone,

The complete notation for labeling eXchange operations among Jobim chords is as follo#sS

X y( z)!

, #here ! refers to the t%pe of $oice-leading motion4 z refers to the number of mo$ing $oices&

and X y refers to the familiar eXchange operations among Figure !, The progression in Figure 3&

then& #ould be labeled X !0(E)#

This sho#s that the $oices progress in contrar% motion& all of the

$oices are mo$ing& and the motion describes a X !0

operation, This notation also implies that there

are no common tones bet#een the t#o chords& all notes progress b% semitone& there is a single

ascending semitone that cancels out one of the three descending semitones& and that the progression

generall% descends, 6t is apparent that the t#o e9tra pieces of specificit% in the eXchange notation

impl% a number of features #ithin a gi$en progression that arent specificall% stated, This e$en e9tends

to cases #hen a $oice mo$es b% more than a semitone& such as the progression bet#een chords ! and

in Figure 2, )n eXchange anal%sis using this notation of the Jobim-lie progression of Figure 2 is

sho#n in Figure 10,

Situating ,oim Chords .ithin /eometry

)s stated earlier& the prime forms of the chords that Jobim uses most often are =0D;@4 =0D<@4

=0D5<@4 =0<@4 =05<@4 =0E<@4 =0E2@4 =0!5<@4 =0!E<@, These are characteri(ed b% a minimal state



change& in #hich one such Jobim chord can be related to another b% a P! relationship, This& coupled

#ith the fact that each of these chords are characteri(ed b% a high spread& as mentioned in the first

paragraph& means that a geometrical representation of Jobims chord #ould be clustered around the

middle of a gi$en graph, That is& the nine Jobim chords listed abo$e #ould be closel% related to a

highl% s%mmetrical chord . be it the full%-diminished chord& a ma'or se$enth or another t%pe of chord

that shares a node #ith the full%-diminished chord& or the in$ersionall% robust dominant se$enth chord,

Borro#ing the color scheme from =+allendar& >uinn& ? T%moc(o 00<@& these chords #ould range

from red =a perfectl% s%mmetrical full%-diminished chord@ to orange and %ello#& and finall% a faded

%ello# that characteri(es the most clustered chord in the set . =0!E<@, There are no instances #here

these chords $enture into the #armer colors that describe the clustered chords& such as green or

tur3uoise& much less the deep blue chords that characteri(e the highest clusters =0000@ and their :

relations,

Borro#ing the tetrahedral graph that #as sub'ect to in$estigation in =+ohn 00D@& it can be

sho#n that the Jobim chords ha$e the properties 'ust described, This tetrahedral graph models $oice-

leadings among the ; T/6-t%pe tetrachordal classes& but our interest onl% lies in the ; tetrachordal t%pes

described abo$e, This representation includes a number of duplicates . si9 tetrachordal classes are

dra#n t#ice . #hich is an intrinsic feature of the figure, or the purposes of modeling these chords& all

the $erte9es are omitted and edges )+& )U& B+& and +U are omitted, This 'ust lea$es the middle of the

figure . those chords mared b% a high spread, 6n this figure& the 3ueen beeN of =0E2@ is situated in

the middle& and mo$ement among each of the chords occurs in triangle )UB and triangle BU+, ach

of these mo$ements are a minimal state change& mared b% a small :-relation among chords, Being in

a higher dimension& the geometr% that describes $oice-leading mo$ement among a tetrahedron =a three-

dimensional representation of a tetrahedron@ is a much better model than its lo#er-dimensional

analogue . the t%pe of model that is sho#n in Figure #, This is because the e9tra dimension=s@ clearl%

define the mo$ements of the indi$idual $oices& something that is of great concern to eXchange



operations, The full tetrahedral graph& along #ith the )BU and BU+ triangles that contain the

mo$ements of the indi$idual Jobim chords& is sho#n in Figures 11a-c,

"nalyses

)n ad$antage to using this s%stem to model paths among Jobims harmonies is that both general

=eXchange operations@ and specific =$oice-leading t%pes@ aspects of $oice-leadings are illustrated

ad'acentl%, This offers a number of insights to the #a%s in #hich Jobims parsimonious $oice-leading

operates, 6t #ill be sho#n that Jobim tends to fa$or a t%pe of $oice-leading in specific conte9ts& that

the degree to #hich a piece ho$ersN around a certain transformation directl% informs the harmonic

character of the piece as a #hole& and that similar transformations can operate on different chords that

%ield different results& %et tra$el the same distance,

One-Note Samba

The first piece that 6 #ill consider is the song Samba de $ma Nota S%& or the One-Note Samba

Bearing true to its title& the song repeats a single note& beginning #ith a U5 in the e% of Y& that is

repeated for the first eight measures coinciding #ith a familiar bossa no$a rh%thm, The same spirit is

repeated again for the ne9t four measures& but the melod% ascends up to its tonic Y5& #hich is coupled

#ith a transposition in the harmon%, The opening structure then returns for the ne9t four measures& and

the first section closes on a tonic #ith an added . Y or m2, The ne9t section departs completel%

from the static melodic figure b% a series of eighth-note runs in the )eolian mode, 6t appears that the

harmonic progressions change as #ell& but there is no sense of a Y minor sonorit% that is heard #ithin

this middle section, The song is then repeated Ual Fegno& and then concludes #ith a coda that

essentiall% repeats the last four measures of the first section, < ) reducti$e setch of the t#o sections&

complete #ith an eXchange anal%sis of the harmonic material& is sho#n in Figure 12,

The transformation c%cle that is presented bet#een the first fi$e chords& X ; (D)S

Z X 2 (E )S

Z

< =+hedia@



X !0()S

Z X !!(!) , appears a number of times throughout the first section, 6t is e9pected for this

section to e9hibit some semblance of similarit%& as the descending harmonic progression of the first fi$e

chords are repeated in some form four times in the section& but it is surprising to see 'ust ho# similar

the transformational structure is bet#een each subse3uent chord in the first section, The most often

used chord is the first one . X ; (D)S

. #hich describes mo$ement bet#een a minor se$enth chord to a

dominant se$enth chord on a root a semitone lo#er, This mo$ement is done times in the first section&

but the transformation is performed another time . bet#een a Ub2 chord and a +M2 chord, hereas

the root mo$ement is similar& the transformation bet#een chord t%pes is different, 6nstead of arri$ing at

a dominant se$enth chord& a dominant se$enth chord goes to a ma'or se$enth chord& #hich suggests

that this is the ne9t transformation in a chain of X ; (D)S

)lso& #hile the normal transformation eeps the

third held constant #hile the rest of the $oices descend a semitone& this anomalous transformation

eeps the se$enth held constant, This transformation can be imagined as a tetradic analogue to the

triadic Flide transformation& #hich is the label that Briginsha# ga$e to these transformations in this

same passage b% omitting the se$enth,; )lthough Flide and X ; (D)S

beha$e similarl% . a single $oice is

held constant #hile the rest of the $oices descend a semitone& thro#ing out the se$enths #ill also thro#

out this second X ; (D)S

transformation& #hich is a crucial ingredient to the $oice-leading nature of this

#or,

6t is possible to find other instances of the same transformation used among differing chord

t%pes in this section, X !!(!) , for e9ample& is used to connect )bM2 and )b2 in the first phrase& and to

ser$e as its cadence, X !!(!) is used again in the transposed phrase& but this time bet#een +M2 and

+m=Q2@, This latter chord is an unstable sonorit% and is relati$el% brief in the music& because it acts as

a connecting chord to the cadence on the subse3uent )R2, This also mo$es b% a single semitone& but it

; =Briginsha# 0!@



descends a #hole step, Thus& the larger mo$ement bet#een the +M2 and the )R2 in this phrase is

actuall% another t%pe of X ; Fpecificall%& it is X ;()

S, #hich is the most often used transformation in

the middle& or B section,

X ; ()S is the first transformation in this section& and it occurs three times, 6t is interesting to

note that different t%pes of the X ;

transform are used in both t%pes of the sections, This suggests that

the t#o sections arent as different as the% seem . the% ma% be melodicall% and harmonicall% dissimilar&

but the path#a% that the $oice-leading follo#s is similar, This also sho#s that b% mo$ing multiple

$oices& fundamentall% different chord progressions can beha$e remarabl% similarl%, The ) section& for

e9ample& most often emplo%s the minor se$enth to dominant se$enth #ith a root lo#ered a semitone& as

pre$iousl% mentioned, This B section& on the other hand& is mared b% root motion that ascends a

perfect fourth& but that mo$es bet#een a $ariet% of different chord t%pes, There is also a general

preference for mo$ing t#o $oices in this section& #hile eeping the other t#o $oices constant, Thus&

#hile the melodic character and the harmonic chord t%pes are fundamentall% different& the holdo$er of

a t%pe of X ;

transform operates as a lining bridge bet#een the t#o sections, 6n fact& the X ;(D)S

transform from the ) section occurs once in this sectionS bet#een Bbm2 and b2, This progression

e9hibits root motion that progresses up a perfect fourth& and this specific t%pe of transformation is not

seen in the first section of the song, The X ;

transform& and especiall% the X ;(D)S

from the first

section& tends to sho# up in snea% #a%s throughout the One-Note Samba,

6n fact& the harmonic path#a% of this song tends to be relati$el% similar, That is& the eXchange

operations that operate in the One Note Samba all tend to ho$er around the X ; transform, This isnt

surprising& since this song is aboutN a static melodic note& in a bossa no$a rh%thm& that is held o$er a

smoothl% changing chord structure, One #ould e9pect& then& that the chordal structure operates in a

smooth& uniform #a%& especiall% since the song itself sounds lie it describes a uniform descending





note held in the first t#o phrases& preparing for a recapitulation of the opening melodic material, !0

6n Figure 13& the first three phrases encompass all of the chords up to the first YQ2 in the

second line . the subse3uent chords are all in the last phrase, Much lie the One Note Samba, &oo's

&i'e (ecember e9hibits a general descending motion throughout the entire period, But unlie the

pre$ious song& this song doesnt ho$er around the X ; (D)S

transformation& at least not in the first three

phrases, 6nstead& $arious forms of X !0

seem to be the main focus in the first three phrases, ) single

semitone that descends a #hole step #hile the rest are held constant& or the X !0(!) transformation& is

the first eXchange $alue, This form of $oice-leading doesnt seem to return in the follo#ing music&

#ith the e9ception of the repeated progression in the second phrase, 6nstead& $ariations of the X !0

transformation are usedS the ne9t eXchange $alue holds t#o $oices constant #hile the other t#o $oices

descend a semitone& or the X !0()S

transform, This mo$ement describes motion bet#een another

progression later is the period . that bet#een YQR2 and +Q2 . #hich actuall% happens in the same

metrical place as its original form in the third phrase, 1ere is another e9ample of prima acie unrelated

progressions that e9hibit the same $oice-leading& and metrical& beha$ior, )nother instance of a X !0

transform& a rare contrar% $oice-leading transform X !0(D )#

in #hich t#o semitones cancelN each other&

happens bet#een +Qm2 and Q2=Q5@, 1ere is an instance of the& also rare& Q2=Q5@ chord that contains a

s%mmetrical& augmented triad that inhabits a coupling node, This is& in fact& the same node that is held

b% the opening )M2 sonorit% . node !0,

There are man% instances of the X !0

transform in the first three phrases of the period& and the

other eXchange operations in these three phrases tend to ho$er around this transform, That is& there

seems to be a preference for someho# descending t#o semitones bet#een each chord, The last phrase&

though& mo$es a#a% from this X ;

preference in fa$or of X ; ()S

This is reflected in the song itselfS a

!0 =+hedia@



still& slo#l% changing melodic and harmonic idea that reflects the title is temporaril% morphed into a

#armer phrase& complete #ith a faster changing melodic and harmonic structure, The harmonic

structure in the last phrase is mared b% root motion that mo$es b% a perfect fourth, Thus& the

samet%pe of motion that Jobim used in the B section of One-Note Samba is used in this last phrase of

&oo's &i'e (ecemberS t#o notes descend a total of three semitones to mo$e bet#een chords that are

displaced b% a fourth in the root . the X ;(D)S

transform, Jobim tends to use progressions that e9hibit

some t%pe of X !0 transform to describe stillness and coldness& and the X ;(D)S

transform to act as a

#armer& more fluid counterpart to X !0

Waters of March

The song Waters o .arch tends to be a bit more sporadic in its use of eXchange

transformations, ) harmonic reduction of the first eight measures of this song& coupled #ith its

eXchange notations& is sho#n in Figure 1, ) notable aspect of this song is the descending chromatic

bass line in its head moti$e, This suggests that Jobim& once again& tends to fa$or progressions that

descend& or mo$e do#n#ards on Figure !, Melodicall%& a gesture that outlines a descending ma'or

third . UQ5 to BE . is repeated a number of times o$er a smoothl%-changing chordal structure, This is

especiall% true in the first eight measures in Figure 1, )fter this first phrase& a similar rh%thmic idea

is maintained #hile a perfect fourth is outlined& but the bass sips b% a larger inter$al, Then the

melodic line returns at a 3uicer pace,!!

Much lie the One Note Samba& the first phrase of the Waters o .arch also tends to ho$er

around a t%pe of X ; transform& but this isnt ob$ious in the first four transformations, 6n fact& all of

the chords that are under the pur$ie# of 62 tend to beha$e sporadicall% in terms of eXchange $alues,

This progression& X !0()S

Z X < (D)S

Z X 2 (D)S

Z X !!(D)#

, doesnt repeat or center around an%

eXchange $alue& but the last three transforms sho# that three $oices mo$e, This& although not

!! =+hedia@



necessaril% transformationall% related& sets a pattern that settles onto the X ; (D)S

operation seen in the

penultimate transformation, 6nterestingl%& these t#o transformations also mae appearances in m%

One-Note Samba anal%sis, The X ; (D)S

transform bet#een 2 and M2 #as also used in the One-Note

Samba bet#een a dominant se$enth and ma'or se$enth chord that are related b% a semitone motion in

the root . bet#een Ub2 and +M2, This #asnt the most fre3uentl% used t%pe of X ; (D)S

in the song . it

#as onl% the secondar%N t%pe of this transformation used onl% once in the ) section of One-Note

Samba, This is also the case for the X ; ()S

used bet#een M2 and +QR2 in Figure 1, This $ersion of

the transformation #as not the t%pe that #as used as a main feature in the B section of One-Note

Samba& but rather it #as used bet#een +M2 and )R2 in Fection ) . the transformation denoted b% an

arro# in Figure 12,

The most remarable transformation sho#n in Figure 1 is that bet#een B2M/Q and 2&

labeled as X !!(D)#

in m% anal%sis, 2 doesnt belong in the e% of B ma'or4 instead& it acts as a bV2

that resol$es into 6V =or 6V2M@ b% mo$ing three $oices do#n a semitone =all four #ould resol$e do#n

a semitone if the ne9t chord #as a dominant se$enth@, 6n order to mo$e to this outside sonorit%& the

B2M/Q chord mo$es do#n a node =or up a node if referring to Figure !@ b% aObject67

transformation&

but does so b% mo$ing three $oices in contrar% motion, T#o $oices cancel each other out& resulting in a

transformation that e9erts more $oice-leading #or relati$e to distance tra$eled,

aters of March mo$es more sporadicall% around a X ;

transform . so much so that the first

four progressions dont seem to focus around an% sonorit% . resulting in a song that is less concentrated

on a $oice-leading path and seems to be less harmonicall% uniform than One 7ote Famba or Loos

Lie Uecember, The song 1o# 6nsensiti$e goes e$en further in this direction b% #aling more

sporadicall% around a X !0 transform,



How Insensitive

Much lie Waters o .arch& /o! 0nsensitive e9hibits a descending bass line among the first

eight chords, This bass line is much closer to a true lament bass . a perfect fourth is spanned from

tonic to dominant& and each indi$idual step is harmoni(ed, The song itself repeats a similar melodic

line four times in a slo# bossa no$a rh%thm, The first iteration of the melod% starts on Q5 . the fifth

of the B minor e% . and the second iteration of the melod% is the original transposed do#n a semitone,

The third and fourth melodic lines are also similar to the first melod%& but the% both are transposed

do#n another #hole step . both start on a U5 . and the last melod% ends on the tonic note BE,!

The eXchange $alues of the first t#o melodies of /o! 0nsensitive& sho#n in Figure 1(& are

noticeabl% sporadic, There are occasional instances of ascending transformations =up#ard motion in

igure <@ bet#een Bm2 and Q2/)Q =the 2 th in the first chord is imagined as being an octa$e lo#er@ and

+M2 and +QR2& and there is an instance of a X O

transform bet#een QR2 and 2/YQ, $en though

harmonic mo$ement is sporadic& the eXchange operations tend to ho$er around t%pes of X !0 .

especiall% the X !0()S

transform that occurs t#ice in the passage, )s #as stated pre$iousl%& ho$ering

around t%pes of the X !0

transform suggests a darer mood& #hich is manifest in this song, The

relati$e harmonic comple9it% that results from sporadic eXchange mo$ements is another e9pressi$e

feature that further contributes to the songs character,

The songs of )nt*nio +arlos Jobim ought to be of great significance to the neo-8iemannian

literature due to their close& parsimonious $oice-leadings that operate #ithin a tonal frame#or,

nfortunatel%& the songs ha$ent been sub'ect to much anal%sis because the tools that are needed .

tetradic parallels to triadic frame#ors . arent %et trul% de$eloped to full% anal%(e this t%pe of music,

6n this paper& steps ha$e been taen to ameliorate the situation b% suggesting that tetrads& more

specificall% the t%pe of tetrads that Jobim uses most often& can be situated #ithin sum-class diagrams,

! =+hedia@



Mo$ements bet#een chords in these sum-class diagrams can be described $ia eXchange operations&

#hich are directed $oice-leadings that denote direction, 6t #as then found that an e9tra degree of

specificit% #as needed& so the concepts of number of $oices mo$ing and $oice-leading t%pe #ere added

to the notation, 6t is m% hope that this form of anal%sis offers a satisfactor% frame#or for

understanding Jobims $oice-leadings& thus situating it #ithin a neo-8iemannian conte9t,



iliogra&hy

Briginsha#& Fara B,:, ) 7eo-8iemannian )pproach to Ja(( )nal%sis,N 7ota BeneS ) +anadiannderground Journal of Musicolog%& Vol, 5& 7o, =0!@S )rticle 5,

+allendar& +lifton& 6an >uinn& and Umitri T%moc(o, [Yenerali(ed Voice-Leading Fpaces,[Science =00<@S D0-E,

+hedia& )lmir, FongbooS Tom Jobim,N Lumiar ditora Vol, !-D =!;;0@,

+hilds& )drian, Mo$ing be%ond 7eo-8iemannian TriadsS 9ploring a Transformational Modelfor Fe$enth +hordsN Journal of Music Theor%& Vol, E& 7o, & 7eo-8iemannian Theor% =)utumn&!;;<@S !<!-!;D,

+ohn& 8ichard, ) Tetrahedral Yraph of Tetrachordal Voice-Leading Fpace,N Music Theor%Online& Vol, ;& 7o, E =October 00D@,

+ohn& 8ichard, F3uare Uances #ith +ubes,N Journal of Music Theor%& Vol, E& 7o, & 7eo-8iemannian Theor% =)utumn& !;;<@& <D-;,

+ohn& 8ichard, Ma9imall% Fmooth +%cles& 1e9atonic F%stems& and the )nal%sis of Late-8omantic Triadic :rogressions,N Music )nal%sis Vol, !5& 7o, ! =!;;@& ;-E0,

Uouthett& Jac& and :eter Fteinbach, :arsimonious YraphsS ) Ftud% in :arsimon%& +onte9tualTransformations& and Modes of Limited Transposition,N Journal of Music Theor%& Vol, E& 7o, & 7eo-8iemannian Theor% =)utumn& !;;<@& E!-D,



Figure 1 )drian +hilds notation for $oice-leadings among dominant se$enth and half-diminished

se$enth chords =+hilds !;;<@

Figure 2, ) sample Jobim-lie progression sho#s that +hilds notation doesnt full% e9tend to all 2 th

chords,

Figure 3, This progression necessitates that e$er% note mo$e b% a semitone, This is a t%pe of an X !0 progression,



Figure 4 The sum class diagram for =D-!!@ triads& from =+ohn !;;<@

Figure The sum class diagram for =E-2@ tetrads,



Figure ( The :o#er To#ersN diagram from =Uouthett ? Fteinbach !;;<@,



Figure 7 ) flo# chart analogue to the pre$ious figure,



Figure ! ) tetradic sum-class group diagram #ith added Jobim chords,



Figure # ) generali(ed node graph of tetradic eXchange operations



Figure 10 )n eXchange anal%sis of Figure 2,

Figures 11a-c Jobims chords can be situated #ithin the tetrahedron of ; T/6-t%pe tetrachord classesdescribed in =+ohn 00D@, Most of the chords are in the middle of the figure . in triangle )UB and

triangle BU+,



Figure 12 )n eXchange anal%sis of the One Note Samba =+hedia@,



Figure 13 )n eXchange anal%sis of &oo's &i'e (ecember =+hedia@,

Figure 14 )n eXchange anal%sis of Waters o .arch =+hedia@,



Figure 1 )n eXchange anal%sis of /o! 0nsensitive =+hedia@,

Documents

The eXchange Operation and Tetradic Voice-Leading in the Music of Jobim