1. hobbies and interests
2. guitar

THREE
TheVerticalDimension:
Chordsand Simultaneities
IN T R O D U C T IO N
That is,
The musicof the tonalerais almostexclusivelytertianin its harmonicorientation.
generally
the
be thoughtof asbeingconstructed
of stacked3rds, only exits harmoniescan
chordsand the
ceptionsbeing "voice-leadingchords" suchas the fanrily of augrnented-6th
That tonal music usedtertianharmonywas
chordsproducedby the omnibusprogression.
not the resultof a consciousdecisionon anyone'spart but insteadwas the resultofclassificationsof consonanceand dissonancearrd the developmentover centuriesof various
procedures.
The fact that the underlyin-e
voice-leacling
harmoniesin the tonal style are
known to be terlian makestlre labeling of chords and the identificationof nonchordtones
in tonalmusica relativelysimpletask.
Much of the musicof the twentiethcenturyis alsobasicallytertian,but thereis in addition a good deal of music usingchordsbuilt fiom 2nds,frorn 4ths,and tiorn combinations of various intervals.Even the tertian music tiequently usesnew kinds of teilian
sonorities.as we shall see.One resultof this unlimitedarrayof harmonicmaterialis that
the clistinction
betweenchordtonesand nonchordtonesis often difficult or impossibleto
make.Also, chordssometimesseemto result more or less accidentallyfrorn the combination of harmonically inc'lependent
lines. For thesereasons,many writers pret-erat times
"simultaneity,"
"sonority,"or "note complex"insteadof
"verticality,"
to useterrnssuchas
"chord." In this text. howeveq "chord" will be useclfreely along with the other terms to
referto any verticalcollectionof pitches,no matterhow it originates.
The presentchaptersurveysin an organizedway the chordsfound in twentiethcenturymusic.The contextsin which thesechordsareusedis a subjectthat involvesboth
Thesetopicswill be takenup in Chapters4 and-5.
voice leadingandharmonicprogression.
46
The VerticalDimension:Chords and Simultaneities 47
CONVENT I O NA L T E R T IAN SON O R IT IES
Tertiantriads and 7th chords are an important,if less preponderant,part of the harmonrc
vocabularyof twentieth-centurymusic. Certaincomposersmake more use of thesesounds
thanothersdo. Someof the works by composerssuchas SergeiRachmaninofT,
Gian Carlo
Menotti, and Aaron Copland,for example,might be expectedto containa high proportion
of triadsand 7th chords.whereasother composers,suchas Paul Hindemith,tend to reserve
the pure soundof a triad for importantcadencesor even for the end of a movement.Still
othercomposersrarely make use of thesemore traditionalsounds.Examplesof the use of
triads and 7th chords will be found in later chaDters.where voice leadine and harmonic
progressionare discussed.
Tertian sonorities"taller" than the 7th chord-9th chords. 1lth chords. and 13th
chords-are not an inrportantpart of tlre harmonicvocabularybefbrethe late nineteenthor
earlytwentiethcentury.In theory,any diatonic triad can be extendedto a I 3th chord befbre
its root is duplicated(seeExample 3-1). In practice,however,it is the dominant and secondarydominantchords,and to a lesserextentthe supertonicand submediantchords,that
tendto be singledout for this treatment.Chromaticalterations,especiallyof chordswith a
dominantfunction, are often used.Example3-2 illustratessomeof the possibilities.
EX A M P LE3- l
D i a to n i cl 3 th c h o rd s
#=#+
C:
I13
EXAMPLE3-2
11it3(etc.)
ii13
Altered Dominants
{s
t
===
d
C:
v?
vq b
V ttn (erc.)
Chordstaller than a 7th irrefrequentlyincomplete,posing cerlainproblemsin analysis. [n Example 3 3, for instance,the first chord rvould probably best be analyzedas an
incompletesupertonicI lth chord becauseofthe 2-5-l bassline, but one could also argue
thatit is a V]. And is the secondchordan incompletedorninant9th or an incompletedominant 13th'/The answerdependson whetherone hearsthe C5 in the melody as a chord tone
or as an appoggiatura,and either readingis defensible.The final chord is a tonic triad, the
F3 in the tenorbeingan ornamented
suspension.
48
The VerticalDimension:Chords and Simultaneities
EXAMPLE3-3
mm. l-5
Book ll ( l9 | 3),"Heaths"("Bruydres"),
Debussy:Preludes,
expressif
Calme-Doucement
?/
ii ll
Ab :
V 9?
I
Ninths and other tall chords can be inverted,of course,but inversionscan be problematical,
as we just saw with the iirI chord in Example 3-3. Refening back to Example 3-l , noticethat
the pitch-classcontent ofevery diatonic l3th chord is identical, meaning that every diatonic
13th chord could be analyzedas an inversion of every other diatonic 13th chord. The situation is only slightly less ambiguouswith 9th and l lth chords,as Example 3-4 illustrates.
EXAMPLE3-4
913?
F9?
lnverted 9th and I lth chords
g l 3?
p l 1?
A typically ambiguousexampleis seenin Example 3-5. The excerptbeginswith a
tonic triad in D major, but the next chord is open to some interpretation (remember to include the flute part in your analysis).The pitch classes,in alphabeticalorder,are A, C, D'
Et, F#, G. Bassnotesusually want to be roots,if they can,and the high D in the flute on the
fourth beat reinforces the notion that this might be some sort of G chord. On the other
hand, the G could be the I lth of a completeD 1lth chord: D-Ffi-A-C-Et-G, and the 45
to D6 in the flute seemsto support that. Or perhapsit should be analyzedas a D 9th chord
presentedsimultaneouslywith G, its note of resolution.The uncertaintyis not resolvedby
the Bb major 7th chord that follows, sinceneitherG-Bb nor D-Bb seemsparticularlycompelling as a progression.The remainder of the excerpt is more straightforward: A minor
7th to D minor 7th in m. 3, both slightly ornamented,to a more lavishly ornamentedG
maior triad in m. 4.
The VerticalDimension:Chords and Simultaneities 49
EXAMPLE3-5
Prokofiev:Sonatafor Fluteand Piano,Op. 94 ( | 943),l, mm. l-4
(Music by SergeiProkofiev.
Editedby leon-PierreRompol.Copyright@ I 986 lnternotionolMusicCo.
Copyrightrenewed.lnternotionolcopyrightsecured.All rights reserved.Used by permission.)
Moderato() - so)
-i
::l
-1
\o
D
----
G
BbMT
TERTIAN CHORDS \^/ITH ADDED NOTES
Thoughthe possibilityof a triad'shavinga noteaddeda 6th abovethe root wasrecognized
by theoristsas early as the eighteenthcentury,chordswith addednotes(sometimescalled
chord,sof addition) did not becomean acceptedpart of the harmonic vocabularyuntil the
twentiethcentury.The basicchordsare usually triads,and the addednotes(alwaysfigured
abovethe root) are usually 2nds or 6ths, less frequently4ths.Any triad with an added6th
could alsobe analyzedas a 7th chord,but the contextwill usuallysettlethe issue,as Exas an
ample3-6 illustrates.
Sirrrilarly.
a triad with an added2nd or:lth could be intelpr-eted
EXAMPLE3-6
Ve
Added 6th and lnverted7th Chords
ladd6
'f
G:
samesound-J
incomplete9th or I lth chord, especiallyif voiced with the addednote abovethe triad.
Sincethe root is the samein eithercase.the distinctionis not a crucialone.For all practical purposes,a chord with an added2nd or 4th can be consideredthe sameas one with an
added9th or I lth. SeeIrxample3-7.
50 The VerticalDimension:Chords and Simultaneities
EXAMPLE3-7
1 add2
Chords with Added 2nds and 4ths
1add9
1add4
l addl I
The situation is unambiguous in Example 3-8, where the cadential chords are the
dominant 7th and the tonic with added 6th in Gb Major. And in Example 3-9, the final
chord is clearly a C triad with an added9th. But Example3-10 is more involved.In the first
phrase(mm. l-2) a double pedal point on G4 and 84 (in the middle of the texture)adds
EXAMPLE3-8
Debussy:Preludes,
Book | ( l9l0),"The Girl with the FlaxenHair"
("La fifle aux cheveuxde lin"), mm.23-24
Cedez------- 1J
Pz'1---{
IV V
vi V-------..
-V7
1add6
EXAMPLE
3-9 Debussy:
Preludes,
Bookll ( l9 | 3),"Canope,"
mm.29-33
// Plus lenl
3
trdsdouxet trDsexpressif
The VerticalDimension:Chords and Simultaneities 51
bite to a conventionalprogression.Though severalof the sonoritiescould be analyzedas
added-notechords,the pedal-pointanalysisis just as good, explainingall the dissonances
exceptthe added4th in the passingtonic !. The secondphrase(mm. 3-4) keepsthe B as a
pedalpoint, shifting it down an octavein m. 4. Again, the pedalpoint accountsfor most of
the dissonances,
but the tonic triads are probably bestanalyzedas added-notechords.
EXAMPLE
3-10
Stravinsky:suite
ltalienne(1932),"lntroduction,',
mm. l-4
(acopyrisht
t934by
Eoosey& Howkes(London)Ltd.Copyrightrenewed.Reprintedbypermissiono[Boosey
&Howkes,lnc.)
Allegro mod to
/->
rV6
4
ii6
(add 4)
y7
IV 6
(add 2)
ry6 rf ii6 vg/v v
(add4)
Added notesare a featureof what is sometimescalled "wrong-note style," in which
the listener'sconventionalexpectationsare almost met, but not quite. This was the casein
Example 3-10, where Stravinskyis almost quoting Pergolesi,an early eighteenth-century
composer'The result is often humorous,as in Example 3-l l, where the melody heads
toward a G4 but landsa half-steptoo high (m. 4l), then a half-sreproo low (m. 42), before
finally succeeding(m. 43).
EXAMPLE
3-f I
Gian carlo Menorti:TheTelephone
(1946),mm.4l-43 (pianovocalscore)
(@ CopyrightI 947 (Renewed)by G. Schirmer,lnc.(ASCAP)lnternotionolcopyrightsecured.All rights
Reserved.Reprintedby permission)
--_
g
et?
?
uo
o
52
The Vertical Dimension: Chords and Simultaneities
TERTIAN CHORDS \^/ITH SPLIT CHORD MEMBERS
A specialkind of added-notechord f'eaturesone or mol'echord rnembersthat are "split" by
addinga note a minor 2nd away.Commonexamplesare triadsand 7th chordswith split
3rds,but split roots.5ths.and Tthsalso clccur.Someof the possibilitiesare shownin Example3-12. Tlrereis no standardanalyticalsynrbollor split chordmembers.In this fext an
exclamationpoint will be used.as in the example.The dorninant7th chordwith split 3rd is
a traditional"bhles"chord.whereit is analyzedas a dominant7th with an augmenfecl
9th.
Debussyusesa Db7 (3 !) in Example3-13 to achievea Spanishflavor.The split 3rd (Fb)is
spelledhereas an augmentedgth (Eh).Eitlreranalysisis acceptable.
EXAMPLE
3-I2
Members
4T 4L F:
F (l l )
E XA M PL E3 -1 3
F(l l )
F7(7l l
F7r.t11
D ebussy:P rel udes,
B ook l l (1913)," 1aP uerradel V i no,"mm.9-15
U
The VerticalDimension:Chords and Simultaneities 53
In Example 3-14 Ravelusesa seriesof major triads with split roots.The effect is not
as dissonantas you might expectbecauseof the speedat which the chordsare played.
EXAMPLE3-l4
rllirrors( | 905),"SorrowfulBirds"'m . 15
Ravel:
uro
Aaron Coplandusessplit-3rdchordson C and E in Example3-15. The effect is intensified
by the strings,which play E's that are a quarter-toneflat and G's that are a quarter-tone
a systemusedalso
sharp.Notice the notationthat Coplandusesherefor the qual'ter-tones,
by Alois Hdba,among others.
54
The VerticalDimension:Chords and Simultaneities
EXAMPLE3-15
Copland:Vitebsk(1929),mm. l-2
Fund
(@Copyright
!934bytheAoron
Coptond
forMusic,
lnc. Copyrightrenewed.Reprintedby permissionofEoosey& Howkes,lnc.Solelicensee)
Lento ntolto marcato d - sz )
JfvT"
ffz
ffz
tr"
gffz
tr"
A more complex example was containedin Example 2-1 (p. 26). The accompaniment in m. 984 clearly containsan A major triad on beat 3, accented.Below and aboveit
are Cls (a split 3rd), the higher C leapingup to an F (a split 5th), while an inner voice sustains an Eb (another split 5th). Meanwhile, the singersproduce Bh (split root) and Gfi
(addinga major 7th to the chord).The listenercannotfollow all of this, of course;the aural
effect is one of extremedissonancecompetingwith the soundof a pure triad.l
OP EN .s T H C H OR D S
While a large number of added-notechords are possible,there is only one important
"chord of omission"-that is, a traditionalsonority that is transformedinto somethingunusual by leaving out a note-and this is the triad without a 3rd. Omitting the root or the
5th, or omitting anythingfrom a 7th chord, only resultsin yet anothertraditionalsonority.
But the soundof an open 5th had beenout of style for centuries,exceptfor its occasional
usein two-partcounterpoint.
The sound of open 5ths rapidly becomestiresome,so extendedpassagesbasedon
this chord are rare.Typically they are usedto createan impressionof the Orient or of the
distantpast.In Example 3-16 open 5ths on G and A are usedto introducea chantmelody
in the chorus.The notationof the time signaturesin Example 3-16 is a very practicalone
that a number of twentieth-century composershave adopted.
The Vertical Dimension: Chords and Simultaneities
EXAMPLE3-16
Carl Orff: CorminoBurono(1936),"Verisleta facies,"mm.4-6
55
gopyright
t937
Mainz. Copyrightrenewed.All rights reserved.Usedby permissionof EuropeonAmeriB. Schon! Soehne,
con MusicDistributorsCorporotion,soleU.S.ond Conodionogentfor 8. Schottt Soehne,Moinz.)
)=*
'r
''r ''r
sentpre un poco pesante
m
-J
Ve -
r is le - to fa
Ve
ris le - tn
fa
C I.
CS
tnun
do pro
pt- n0
do pro
pt- na
A
gP
klansvoll
Cor.i Trbne.,Pno.
QUARTAL AND QUINTAL CHORDS
Composersof the twentieth century havenot restrictedthemselvesto tertian sonorities-that
is, to chordal formations basedon stacked3rds. There are essentiallyonly four possibilities:
Chordsbuilt from 2nds (7ths)
Chordsbuilt from 3rds (6ths)
Chordsbuilt from 4ths (5ths)
Chords built from mixed intervals
In this sectionwe will explorequartaland quintal chords-those built from 4ths and 5ths.
Later sectionswill deal with chords constructedfrom 2nds and with mixed-interval chords.
A quartal chord canhave as few as three pitch classes(as in Example 3-l7a) or it can
have several (Example 3-l7b). It is sometimespossible to omit a member of a quartal or
quintal chord (the E4 in Example 3-17b, for instance)without losing its character.Various
56
The VerticalDimension:Chords and Simultaneities
voicings and octaveduplicationsare also used (as in Example 3-l7c), but some arrangementscould destroythe quartalcharacterof the sonority.Quintal chordswork the sameway
(as in Example3-17c1),butthey havea more open and stablesoundand,of course,occupy
more vertical spaceper chord member.Surely a near-recordfor rangemust be held by the
ten-notequintal chord that occursnear the end clf GycirgyLigeti's Melodiesfor Orchestrd
(1971),spanninga rangeof morethanfive octaves,fionr Abl to 86.
EX AMP LE3-17
6
Quartal andQui ntalC hords
la
a
la
la
t.
aa
( a)
A convenientway to describequartaland quintal chords is to use, fbr example,
quartalchord with B as the bottom pitch class.as
"3 x 4 on B" to meiln a three-pitch-class
in Example3-17a. Exarnple3-l7b would be a "7 . 4 on C{." All of the chordsin Exan.rple 3-17c would be "1 x 1 on E," and "-5 x -5on G" would describeExample3-11d.
A static6 x 4 chold on A is the basislbr Examnle3-18.
EXAMPLE3- l8
Howard Hanson:SymphonyNo. 2, Op. 30 ( 1930),l, rehearsal
J (stringsonly)
bypermissionofCorl
(Copyright@l932bytheEostmonSchoolof Music.Copyrightrenewed.Reprinted
Fischer,
lnc.,sole ogent for the EostmonSchoolof Music Publicotions.)
Molto pif mosso
cott sttnlittri
vt.1
l
VI.II
,]
con sordinri
Vla.
I
l
l
The VerticalDimension:Chords and Simultaneities 57
The chords in lJxample 3-19 are almost exclusively quartal, but the analysis is
and voicings.One possibilityis to analyzethe harmonyas
complicatedby arpeggiations
four quartalchords,as rnirrkedon the score.
a: 5X
4onFf
b: 3x
4 on D ("inverted")
c : - 5X
,1on Bb
d: 4x
4onDb
EXAMPLE 3- | 9
(@Copyright
Coptond
t957by theAoron
Copland: Piano Fantasy ( | 957), mm.20-24
lnc.,sole/icensee.)
of Boosey
& Howkes,
Fundfor Music,lnc.;copyright
renewed.
Usedbypermission
IJ"
3
A, 9
Q
WA
d-
e
r
laJ
l,
t,
l-
9t ,
l
+
fre
,,4
3
I
b
-,.
fea. s ft).
c---
l
-d-
A secondapproach(andstill othersarepossible)would be to combinea andb into a
singleT X 4 chordon F{ and to combinec andd into a singleT X 4 chordon Bb.Both of
theseanalysesare supportedby the pedal markings.Notice that the a/b chord containsall
of the pitch classesof the G major scale,whereasthe pitch classesfrom Cb major makeup
the c/d chord. Notice also the oversizetime signaturesCoplandemployshere.Presumably
theseare seenmore readily than are the traditionaltime signatures,certainly when usedin
a conductor'sscore,as in Example3-20.
The last 2l measuresof the movementfrom which Example 3-20 is taken are static
of a quintalchord on Dl,.At
harmonically,consistingfor the mostpart of embellishments
the very end, shown here.the chord turns out to be a 5 x 5 chord on Db.
58
The VerticalDimension:Chords and Simultaneities
EXAMPLE
3-20 PercyGrainger:
Uncolnshire
Posy( 1937),
lll,"RuffordParkPoachers,"
mm.99-103
(ASCAP)
tnternotonotcopvrightsecured
Attrights
::::J,:,:::'^7r::::';";:#::,,::,iSchirmer'tnc
long
3
4
----/
_--l
v
LonS
pp
Quartal and quintal chords are most often made up of perfect intervals, but augmented and diminished 4ths and 5ths may be included. In Example 3-21 eachof the arpeggiated triplet chords,as well as the eighth-notechordsbeneaththem, is a3 x 4 chord,with the
lower 4th augmented.
The VerticalDimension:Chords and Simultaneities 59
EXAMPLE
3-2|
Debussy,
Preludes,
Bookll ( l9l3),"Ondine,"
mm.4-7
?3
nf ----=:-=:_=_
r r-j
]L -
The use of diminished4ths and augmented5ths in quartal and quintal chords can
lead to perplexingquestions,sincetheseintervalssoundlike 3rds and 6ths.An exampleis
AlexanderScriabin's"mystic chord," a sonority that flavors much of his music, although
he seldom usesit in literal fashion.The "mystic chord" is found in at leastthe two forms
shown in Example 3-22. Example 3-22a containsone o4, while Example 3-22b contains
two of them. As long as the voicing is predominantlyquartal,as it is here, it is probably
correct to analyzeboth chords as altered 6 X 4 chords, but other voicings might lead to
other analyses.The diminished scale can serve as the source for the chord in Example
3-22b.If you turn back to Example2-17 , you will seethat this chord is strongly suggested
in varioustranspositionsand voicingsin that excerpt.
EXAMPLE3-22
Scriabin'sMystic Chord
d
(a)
t
(b)
5+ #-€ --i:0_
z:
SECUNDAL CHORDS
The third possibility for chord constructionis the secundalchord, a sonority built from
major or minor 2ndsor from a combinationof the two. Suchchordsmay be voiced as Tths
ratherthan as 2nds,but this is the exception.More often the notesof a secundalchord are
placed adjacentto each other, an arrangementsometimesreferred to by the terms "cluster"
and "tone cluster."
The secundalchord in the secondmeasureof Example 3-21 is voiced as a cluster,
but the arpeggiationsobscurethis somewhat(the chord is Ff-G-A-B-Cfi t. Example 3-23
providesa clearerillustrationof clusters.
60
The VerticatDimension:Chords and Simultaneities
EXAMPLE3-23
t95l (Renewed)
(1915)'ll (copvright@
Charleslves:PianoSonataNo'2 (concord)
by AssociotedMusic
Reprinted
publishers,lnc.(BMt) lnternotionolcopyrightsecured.All rightsreserved'
bYPermission
)
(fasterand faster) --'---
In somekeyboardw orks,speci al notati oni susedtoi ndi catewhet her
or not t he
others require that the cluster be performed
black keys are to be included in the cluster.
3-24, thepianist plays black keys with the
with the forearm or with a board. In Example
l e fth a n dandw hi tel .v.* i .t' theri ght' resul ti ngi nnearl ychromati cclust er sof eight or
nine pitcheseach.
EXAMPLE3-24
(1977)
SofiaGubaidulina:Lamentofor Tubaand Piano
(@ t gg I by MusikverlogHons Sikorski'Homburg')
The VerticalDimension:Chords and Simultaneities 61
In Example 3-25a, the composer suggeststwo ways of playing a "lateral tremolo"
between white and black keys, shown in Example 3-25b.
William Bolcom,No. 5 from l2 New Etudesfor Piono( 1986)'m. | |
Compony.)
(@1988byEdward
B.MorksMusic
EXAMPLE3-25
A littleslower J = c. 70
ffi
ffiu
pqsstonate
-l
The chords that accompany the first violin in Example 3-26 might be explained as
secundalchords,the first one containingthe pitch classesGb, Ab, Bb, C, and D, and the
other chords being transpositionsof the first one. The voicing of the chords,however,is
not as clusters,but as alternatingmajor 3rds and tritones, resulting in the sound of an
incompleteBb 13thchord:
G,4
tritone
c4)
major 3rd
A,3
)
tritone
D3)
major 3rd
Br2 )
62
The VerticalDimension:Chords and Simultaneities
PaulHindemith:StringQuartet No. 3, Op. 22 (192l),V mm.64-66
EXAMPLE3-26
(@B.schott's
1923.@Renewed.AII
rightsreservedUsedby permission
Soehne,Moinz,
ofEuropeonAmericonMusic
DistributorsCorporation,soleU.S.ond Conodionogent for B. Schott! Soehne,Moinz.)
dim. nnlto
..'---./
1=I
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1si.
[.+ft
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ifrF
MIX ED .IN T E R V A L C H OR D S
A mixed-intervalchord is one thatdid not originateas a seriesof 2nds,3rds,or 4ths,:but
insteadcombinestwo or more of those interval types (with their inversionsand compounds,of course)to form a more complexsonority.The possibilitiesarenumerous.
Most mixed-interval
chordsiire subjectsto otherinterpretations-thatis. theycould,
on closer inspection,be arrangedto lcxrk like secundal,tertian, or quartal chords.The
mixed-intervalchord in Example3-27 will be usedto illustratethis. It could be arranged,
irlthoughincorrectly,as:
a secundalchord
a tertianchord
a quartalchord
D-E-F-Gf-A#-B
E-G*-B-D-F-A$
F-B-E-A#-D-G{
S onataN o.4 (1985),11,m.58(@t esl, M M BM usic)
EX AMP L E3-27 GeorgeW al ker:P i ano
>>>
6
i"
JT
b,
The VerticalDimension: Chords and Simultaneities 63
In rnostcasesthe contextwill suggestthe bestanalyticalapproach.For instance,the
sonatafrom which Example 3-27 comesis an atonalwork with few, if any secundal,tertian, or quartal chords,so it is appropriateto call this a rnixed-intervalchorcl,even if that
term is too broad to be very descriptive.
This brings us to the questionof just how one goesabout analyzingand labeling
thesesonorities,a complicatedproblem that has been tackled by various composersand
theorists-notably Paul Hindernith, Howard Hanson,and Allen Forte. Becauseso many
combinationsof intervals are possible,a completely new system of chord classification
had to be devised,and this systemis the subjectof much of Chapter9.
WHOL E. T O NE CHO R D S
Any chord whose memberscould be obtainedfiom a single whole-tonescaleis a wholetone chord.3A numberof suchchordsare possible,of course.A few of them are illustrated in Example 3-28. Such sonoritiesfbr the most part appearedrarely in classicaltonal
harmony,but some whole-tonechords.includirrgthose in Exarrrple3-28. are irt least
reminiscentof traditionalchords.Example3-28b, for instance,is an incompletedominant
7th chord,and Exanrple3-28d is a Fr-ench
chord.but Examples3-28c and
augmentecl-6th
e would haveto be explainedas alteredversionsof sirnplerchords.
EXAMPLE3-28
Whole-ToneChords
*, H! Uo! H*?'i#&I
(a)
(b)
(c)
(d)
(e)
Whole-tonechordswill naturallyoccur in any music that is basedon the whole-tone
scale.For an illustration.turn back to Example 2-6 (p. 25), a whole-toneexcerptby
Dukas.The first measureusesa Frenchar.rgmented-6th
sonority (8fi-E-Ffi-Afi1. When the
Bf movesto E in the next measure,we areleft with only the notesof an incompletedominant 7th chord (E-F{-Af ;. Neither of thesechordsis usedin a traditionalmanner.The last
two measures
whole-tonescale.andthe whole-tonechords.though
arebasedon a ditTerent
present,are more difficult to characterize.
that are not
More interesting,perhaps.is the use of whole-tonechordsin passages
basedprimarily upon the whole-tonescale,becauseherethey provide an unexpectedharmonic color.Again an earlierexample,Example2-10 (p. 28), can providean illustration.
In this casethe pitch environmentis Phrygian,exceptfor the last chord of the secondmeasur-e(Ab-Ff-C-D). The altered tone, Ff. prodncesa whole-tonechord that could be explained as a French augmented-6thchord moving directly to the tonic G, or as a
second-inversion
dominant7th in G with a flatted-5th(Abl. Mol'e whole-tonechordsare
seenin Example 3-29, the first phraseof a work that is atonal until a suggestionof a Gb
tclnalcenterin the final cadence.In this phrasethe measuresaltelnatebetweenWT-l and
WT-0, althougheachmeasureincludesa passingtone that is out of the scale.
64
The Vertical Dimension: Chords and Simultaneities
EXAMPLE
3-29 Scriabin:
Etude,
Op.56,No.4 (1907),
mm.l-4
Finally, look once more at the Hindemith excerpt,Example 3-26. We haveanalyzed
the chordsin this excerptas secundalchords,as mixed-intervalchords,and as l3th chords;
we can see now that they are also whole-tone chords, each one being derived from one of
the two whole-tonescales.
POLYCHORDS
A polychord combines two or more chords into a more complex sonority, but it is crucial
that the listener be able to perceive that separateharmonic entities are being juxtaposed if
the result is to be a true polychord.Any I lth or 13thchord could be explainedas a combination of two simpler sonorities,but this would be an incorrectanalysisif we do not hear
them that way.
EXAMPLE
3-30 ApparentPolychords
In order to be heardas a polychord,the individual sonoritiesthat make up the polychord must be separatedby some meanssuch as registeror timbre. In Example 3-3 I the
first and last chordscould easily be heardas I I th chords,but in the rest of the phrasethe
different registersand the pervading contrary motion between the two chordal units result
in an unambiguouspolychordal texture.Persichetticoncludesanotherof his works, the
Symphony fbr Band, Op.69 (1956), with a spectacularpolychord that combines four
registrally distinct sonorities:Bb major, A major 7th, B major 7th, and F major with an
added9th. The resultingpolychordcontainsall twelve pitch classes.
The VerticalDimension:Chords and Simultaneities 65
EXAMPLE3-3 |
Vincent Persichetti:LittlePionoBoo( Op. 60 ( 1953),"Prologue,"mm. l-4
(@ 1954 Hkan-Vogel,lnc.
Usedby permissionof the publisher.)
Adagio pesante
Stravinsky'sfamous"Petrushkachord" combinestwo triads a tritone apart:C major
and Ffl major.This polychord is seenin Example 3-32, along with anotherpolychord: F$
major/G major. In the orchestral version, the ascendingfigures are played on a piano, the
descendingoneson clarinets.
EXAMPLE3-32
( | 9 | | ), SecondTableau (@Copyright
t9 t2 byHawkes
& Son
Stravinsky:Petrushko
(London)Ltd.Copyrightrenewed.Reprintedby permissionof Boosey& Howkes,lnc.)
espr.
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3
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3
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The constituentsofpolychords are usually tertian triads or 7th chords,but all of the
other kinds of sonoritiesdiscussedin this chaptercould also conceivablybe susceptibleto
polychordaltreatment.For instance,near the end of Example 2-1 (p.26) thereis a 3 x 4
chord on E in the upperstaff of the accompanimentand a DL major triad in the lower staff.
The only requirement for a polychord is that the listener be able to perceive the chords as
separateentities.There will inevitably be ambiguouscases,however,where one listener
hearsa polychord and anotherhearsa single complex sonority.
The use of a polychord does not necessarilyindicate that a passageis polytonal.
Polytonalitywill be discussedin more detail in Chapter5.
The VerticalDimension:Chords and Simultaneities
SUMMARY
The harmoniesof tonal music were limited for the nrostpart to tertiantriads and 7th
chords.In contrast,composersof the twentiethcenturyhavef-eltfree to make use of
any conceivablecombination of pitches.In the most sirnple terms, there are four
possibilitiesfor chord construction:
Secundalchords(alsotone clusters)
Tertianchords(including gths,etc.)
Quartalchords(also quintal chords)
Mixed-intervalchords
Tertian chords,the most traditionirl of the four types,have been subjectedto some
new variations:
Added notes
Split chord members
Open 5ths
One specialcase,especiallyimportantin the early part of the century:
Whole-tonechords
And finally. the possibility of juxtaposing two or more aurally distinguishable
sonorities:
Polychords
It is liequently the case that a particular sonority is open to more than one
interpretation.This is particularly true with mixed-intervalclrords,many of which
can be arrangedto resemblesecundal,tertian,or quarlalchords.The studentmust be
sensitiveto the context and the voicing in attemptingto choosethe best analytical
approach.The three chords in Example 3-33, though containing the same pitch
classes,obviously must be analyzeddifferently.
EXAMPLE3-33
aa
Three DifferentChords Containingthe SamePitchClasses
6 g n g E#r
The VerticalDimension:Chords and Simultaneities 67
NOIES
1. Another approach to chords with added notes and split chord members is taken by
Bryan Simms in Music of the TwentiethCentury,pp. 55-58. He identifiesnine pairsof
"triadic tetrachords"-that is, four-note chords that contain a major or minor triad.
The first chord of eachpair is a major triad plus one of the other nine notes of the chromatic scale,while the secondchord of each pair is the mirror inversion of the first.
2. Some writers use the term compound chord.
3. Some theoristsuse the term whole-tonedominant for whole-tonechords that have a
traditional dominant function.
EXERC'SES
PortA: Fundamentals
l. Review the nine chord types in the Summary section.Then find one example of each
type in the example below.
2. Make up one example of each of the nine chord types and notate them on staff paper.
Try not to duplicate any of those found in the text. Label each chord.
3. Find the doubly chromaticmediantrelationshipin Example3-9.
4. Name the scalesusedin the following excerpts:
(a) Example 3-3, treble-clefmelody
(b) Example 3-9, last four measures(without the Eb; C is tonic)
(c) Example3-15
(d) Example3-21
(e) Example 3-26, cello only (missingan A)
(fl Example3-27
Part B: Anolysis
l. Two different types of tetrachordsare used in Example 2-8 (p. 27), one in 8a and one
in 8b. Identify thesechord types.
2. Debussy:Preludes,Book I, "The Engulfed Cathedral,"m. 1-5
There are three planes to the texture ofthis excerpt. One is the static three-notechord
68
The VerticalDimension: Chords and Simultaneities
in the highest register,anotheris the quarter-notechords,and the third is the threenote chordsin the lowestregister.What kind of chord do all threeplanesmake useof'J
To what scaledo the quarter-notechordsbelong'/
Profonddrnent
calme (Dansunebrumedoucement
sonore)
/
1- \-J 4 - 1 I
t . . J -2 1
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€>.
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(195l ), Il l
3 . S tr avi nskyThe R ake' sP rogress
What kind of sonoritypredominates
in this passage'?
N IC K
Wcll.
I. r'
then.
i\p'b
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fu
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i
My heart-
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rvild
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\ilh
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The VerticalDimension:Chords and Simultaneities 69
4. Debussy:TheJoyou,s
Isle (1904),ntrn. 152-55
And in this one?
:-
tr .t.Zt..3..
l r.r-.a..1..
-'
t
T-
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, ----
l
--L
sempre
cresc.
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P]
5. Ravel:"Minuet on the Name of Haydn" ( l909), mm. 50-54
The chordshereare tertian.Label the six bracketedchordswith Rornannumerals.
Lent
rall.
?
-)
\
I+
-o
ab
c
d
C--
I
70
The VerticalDimension:Chords and Simultaneities
6. Ravel:Sonatina(1905),I, mm. 79-84
This excerpt contains several sonorities that could be analyzed as added-notechords.
Find them. as well as an unconventionalGermanausmented-6thchord.
a Tempo
The Vertical Dimension: Chords and Simuttaneities 71
7. CharlesIves:Violin SonataNo.4 (1915),II
Explain or discusseachof the five labeledchords.
2P
L.H.
(Copyright@ I 942 (Renewed)by Associotedllusrc Publlshers,/nc. (BMt) lnternationol copyright secured.All
righr reserved.Reprintedby permission.)
8. Alban Berg: "Warm Is the Air" ("Warm die Liifte"), Op. 2. No. 4 (1910), mm.20-25
This is a complicated and intriguing excerpt from early in Berg's compositional
output.Our analysiswill be concernedonly with the piano part, while recognizingthat
the relationshipbetweenthe voice and the accompanimentwould have to be considered in a completeanalysis.
(a) There are at least two ways to approach chords a-f analytically. one is to
understandhow they are "generated"-that is, how they come about.what sort of
patterndoesthe bassline underchordsa-f fbllow?
(b) Meanwhile,the right hand in chordsa-f movesa three-notechord down chromatically. How would you classifythat chord?
(c) The secondway of dealing with chords a-f is to analyzeeach individually. The
last of them could be analyzedas a dominant7th with split 3rd: B7(3!). The others appear to be mixed-interval chords, but closer inspection reveals that they
could also be analyzed as tertian chords. which chords out of a--e could be
analyzedas incomplete dominant 7th chords with split 3rds'?
72
The VerticalDimension:Chords and Simultaneities
(d) And which could be analyzedas incomplete13thchords?
(e) Of the remainingchords(g-l), list any that are identicalto or are transpositionsof
one ofthe earlierchords(a-f).
(0 What singlecategorywould bestdescribechordsg, i, and k?
(g) What patternseemsto emergein chordsg and i?
(h) Considering your answer to the preceding question, what note seems to be
missinefrom chord k?
kurzer
Halt
sehr ruhig
Der
Ei - ne
stirbt,
da
p,A
ne - ben
der
An
-)
,.,prr,rT(,'--l-,
or:
i=
Li
I'
j
Lk
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ff
f").
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The Vertical Dimension: Chords and Simultaneities
73
FURTHER
REqD'NG
LeoN. Tbchniquesof TwentiethCenturyComposition.SeeChapter6, Chord Structure.
Der-r-rN.
Kosrrn. Srsr.eN.e,NpDoRorrHvPRvr.rs.Tonal Harmony with an Introduction to TwentiethCentury Music. See pp. 482491.
Ppnsrcuprrr. VlNcsNr. Twentieth-Century Harmony. See Chapter 3, Chords by Thirds;
Chapter 4, Chords by Fourths; Chapter 5, Added-Note Chords; Chapter 6, Chords by
Seconds;Chapter7, Polychords;and Chapter 8, Compound and Mirror Harmony.
Ptsron, Wnlren. Harmony. Seepp. 499-507 in Chapter31, Scalarand ChordalTypes.
Rersor'nc,Hon,qcs. "The Vertical Dimension in Twentieth-Century Music," in Gary
Wittlich, ed.,Aspects of Twentieth-CenturyMusic. Seepp. 322-372.
Srurrrs,BnvnN R. Music of the TwentiethCentury. Seethe section titled "Triads and Triadic
Extensions"in Chapter3.
Srernra, Gnec A. Bridge to 20th Century Music. SeeChapters 11 and 12,Expanded Chord
Vocabulary I and II.