/LMAR=0/XLINE=10/FONT#0=METL/FONT#1=METLI/FONT#2=METS/FONT#3=MUS[HHA,LCS]
␈↓ α≠␈↓␈↓ ε81␈↓ 
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␈↓"β␈↓ α≠␈↓␈↓ ¬jCHAPTER I


␈↓"β␈↓ α≠␈↓␈↓ ¬HINTRODUCTION


␈↓"β␈↓ α≠␈↓␈↓ α{Except␈αpossibly␈αfor␈αthe␈αworks␈αof␈αa␈αfew␈αcomposers␈αovercome
␈↓ α≠␈↓by␈α⊃a␈α⊃theoretical␈α⊃bent,␈α⊃music␈α⊃has␈α⊃always␈α⊃been␈α⊃written␈α⊃from␈α⊃the
␈↓ α≠␈↓point␈αof␈αview␈αthat␈αthe␈αproof␈αis␈αin␈αthe␈αhearing␈α--␈αthe␈αear␈αbeing␈αthe
␈↓ α≠␈↓final␈α∂arbiter.␈α∂ Of␈α⊂course,␈α∂we␈α∂all␈α∂know␈α⊂that␈α∂the␈α∂ear␈α∂is␈α⊂simply␈α∂the
␈↓ α≠␈↓receptor␈α
of␈α
sounds␈α
and␈α
that␈αwhat␈α
we␈α
mean␈α
when␈α
we␈α
say␈αa␈α
musical
␈↓ α≠␈↓passage␈α∩"sounds␈α⊃wrong"␈α∩is␈α⊃that␈α∩it␈α⊃includes␈α∩elements␈α∩which␈α⊃our
␈↓ α≠␈↓minds␈α⊂are␈α⊂unable␈α⊂to␈α⊃accept␈α⊂in␈α⊂terms␈α⊂of␈α⊂the␈α⊃musical␈α⊂conventions
␈↓ α≠␈↓which␈α∂we␈α∂know␈α∞or␈α∂can␈α∂conceive␈α∞of.␈α∂ It␈α∂is␈α∞obvious␈α∂that␈α∂any␈α∞one,
␈↓ α≠␈↓isolated␈α⊂note␈α⊃sounds␈α⊂neither␈α⊃right␈α⊂nor␈α⊃wrong.␈α⊂ For␈α⊃that␈α⊂matter,
␈↓ α≠␈↓those␈α
well␈α
acquainted␈αwith␈α
20th-century␈α
music␈α
would␈αhesitate␈α
to
␈↓ α≠␈↓apply␈α∞"right"␈α
or␈α∞"wrong"␈α
to␈α∞even␈α
a␈α∞fairly␈α
large␈α∞number␈α∞of␈α
notes,
␈↓ α≠␈↓produced␈α≠either␈α~melodically␈α≠or␈α~harmonically.␈α≠ "Rightness"␈α~or
␈↓ α≠␈↓comprehensibility␈α≤is␈α≤really␈α≤found␈α≤only␈α≤in␈α≥complete␈α≤musical
␈↓ α≠␈↓passages␈α
or␈α
phrases.␈α
 The␈α
phrase,␈αor␈α
even␈α
the␈α
whole␈α
piece,␈α
will␈αbe
␈↓ α≠␈↓insufficient,␈α∂however,␈α∂if␈α∞the␈α∂listener␈α∂has␈α∞not␈α∂established␈α∂at␈α∞least
␈↓ α≠␈↓some␈α∞backlog␈α
of␈α∞experience␈α∞in␈α
hearing␈α∞music␈α
of␈α∞the␈α∞same␈α
general
␈↓ α≠␈↓style.

␈↓"β␈↓ α≠␈↓␈↓ α{Because␈αof␈αthe␈αtremendous␈αbacklog␈αof␈αexperience␈αin␈αhearing
␈↓ α≠␈↓music␈α∂based␈α∞on␈α∂triadic␈α∞tonal␈α∂harmony,␈α∞almost␈α∂everyone␈α∂who␈α∞has
␈↓ α≠␈↓grown␈α∂up␈α∂during␈α∂the␈α∂last␈α⊂fifty␈α∂to␈α∂one␈α∂hundred␈α∂years␈α⊂in␈α∂contact
␈↓ α≠␈↓with␈α∩western␈α⊃culture␈α∩is␈α⊃well␈α∩acquainted␈α⊃with␈α∩the␈α⊃conventions
␈↓ α≠␈↓established␈α∪by␈α∪music␈α∪composed␈α∪from␈α∪the␈α∪time␈α∪of␈α∪the␈α∪late␈α∩17th
␈↓ α≠␈↓century␈α_to␈α_the␈α_early␈α_20th␈α_century.␈α_ Comparatively␈α_abruptly,
␈↓ α≠␈↓17th-century␈α
composers␈αestablished,␈α
from␈αthe␈α
implications␈α
of␈αthe
␈↓ α≠␈↓earlier␈α∀modal␈α∪procedures,␈α∀the␈α∪bases␈α∀for␈α∪the␈α∀tonal␈α∀system.␈α∪ This
␈↓ α≠␈↓tonal␈α
system␈αproved␈α
to␈αhave␈α
possibilities␈αvast␈α
enough␈α
to␈αintrigue
␈↓ α≠␈↓the␈αbest␈αmusicians␈αfor␈αalmost␈αthree␈αhundred␈αyears.␈α Then,␈αin␈αturn,
␈↓ α≠␈↓the␈α∞latest␈α∂implications␈α∞of␈α∂tonality␈α∞gave␈α∂rise␈α∞to␈α∂the␈α∞20th-century
␈↓ α≠␈↓tendency away from the necessities of tonal harmony.
␈↓ α≠␈↓␈↓ ε82␈↓ 
k


␈↓"β␈↓ α≠␈↓␈↓ α{Although,␈αstrictly␈αspeaking,␈αthe␈αterm␈α"atonal"␈αcan␈αbe␈α
applied
␈↓ α≠␈↓to␈α∀some␈α∃20th-century␈α∀music,␈α∃its␈α∀use␈α∃should␈α∀be␈α∃frowned␈α∀upon
␈↓ α≠␈↓because␈α!it␈α!seems␈α!to␈α imply␈α!a␈α!lack␈α!of␈α!organization.␈α  Since
␈↓ α≠␈↓contemporary␈α⊗music␈α⊗has␈α↔transcended␈α⊗tonality␈α⊗and␈α↔depends␈α⊗on
␈↓ α≠␈↓clear␈α∩organization␈α∪as␈α∩much␈α∪or␈α∩more␈α∩than␈α∪earlier␈α∩music,␈α∪a␈α∩more
␈↓ α≠␈↓positive␈αword,␈α"contextual",␈αis␈αpreferred.␈α The␈α␈↓↓particular␈αcontext␈↓,
␈↓ α≠␈↓as␈α∞established␈α
by␈α∞the␈α
consistent␈α∞use␈α
of␈α∞the␈α
basic␈α∞elements␈α∞in␈α
each
␈↓ α≠␈↓piece,␈α⊂seems␈α∂to␈α⊂have␈α⊂replaced␈α∂the␈α⊂role␈α⊂of␈α∂the␈α⊂tonal␈α⊂center.␈α∂ Thus
␈↓ α≠␈↓Stravinsky␈α∞can␈α∞use␈α
triads␈α∞and␈α∞diatonic␈α
scales␈α∞"atonally"␈α∞(i.e.,␈α∞in␈α
a
␈↓ α≠␈↓manner␈α∩outside␈α⊃the␈α∩realm␈α⊃of␈α∩functional␈α⊃tonal␈α∩harmony)␈α∩and␈α⊃be
␈↓ α≠␈↓found␈α∃to␈α∃follow␈α⊗basic␈α∃procedures␈α∃remarkably␈α∃similar␈α⊗to␈α∃those
␈↓ α≠␈↓followed␈αby␈αSchoenberg␈αin␈αhis␈αuse␈αof␈αsingle␈αseries␈αof␈αnon-diatonic
␈↓ α≠␈↓intervals.

␈↓"β␈↓ α≠␈↓␈↓ α{Only␈α↔now,␈α⊗when␈α↔the␈α⊗procedures␈α↔of␈α↔functional␈α⊗harmony
␈↓ α≠␈↓have␈α∩clearly␈α∩outlived␈α∩their␈α∩usefulness␈α∩as␈α∩the␈α∩primary␈α∩basis␈α⊃for
␈↓ α≠␈↓musical␈αorganization␈αfor␈αserious␈αcomposers,␈αdo␈αwe␈αseem␈αto␈αbe␈αable
␈↓ α≠␈↓to␈α#form␈α#consistent␈α"views␈α#concerning␈α#the␈α#purely␈α"musical
␈↓ α≠␈↓significance␈αof␈αtonal␈αharmonic␈αprogressions.*␈αAs␈αwe␈αmight␈αexpect,
␈↓ α≠␈↓beginnings␈α⊂were␈α⊂made␈α∂in␈α⊂this␈α⊂kind␈α⊂of␈α∂thinking␈α⊂just␈α⊂at␈α⊂the␈α∂time
␈↓ α≠␈↓when␈αthe␈αdemise␈αof␈αfunctional␈αharmony␈αbecame␈αassured.␈α
 In␈α1906
␈↓ α≠␈↓Schoenberg␈αcomposed␈αhis␈α␈↓↓Kammersymphonie␈↓,␈αOp.9,␈αwhich␈αcarried
␈↓ α≠␈↓tonality␈αto␈αwhat␈αwas␈αnearly␈αits␈αfarthest␈αextreme,␈αand␈αin␈αthe␈αsame
␈↓ α≠␈↓year␈α⊗Heinrich␈α⊗Schenker␈α↔published␈α⊗␈↓↓Harmony␈α⊗(or␈α↔New␈α⊗Musical
␈↓ α≠␈↓↓Theories␈α∂and␈α∂Phantasies␈α∂by␈α∂an␈α⊂Artist)␈↓,␈α∂the␈α∂first␈α∂of␈α∂his␈α⊂group␈α∂of
␈↓ α≠␈↓highly␈α
influential␈α
works␈α
that␈α
brought␈α
to␈α
the␈α
fore␈α∞the␈α
realization
␈↓ α≠␈↓that␈α∞music␈α∂was␈α∞much␈α∂more␈α∞than␈α∞a␈α∂series␈α∞of␈α∂isolated␈α∞progressions
␈↓ α≠␈↓and modulations.

␈↓"β␈↓ α≠␈↓␈↓ α{In␈α∀retrospect,␈α∪we␈α∀can␈α∀now␈α∪see␈α∀(or␈α∀hear)␈α∪that␈α∀the␈α∀era␈α∪of
␈↓ α≠␈↓tonality␈α
was,␈α
in␈α
a␈α
sense,␈α∞an␈α
era␈α
of␈α
monotonality.␈α
 We␈α
see␈α∞that␈α
the
␈↓ α≠␈↓concept␈α∪of␈α∩modulation␈α∪is␈α∩best␈α∪considered␈α∩in␈α∪relative␈α∪terms␈α∩and
␈↓ α≠␈↓that␈α∞virtually␈α∞all␈α∞music␈α∞was␈α∞intuitively␈α∞written␈α∞with␈α∞a␈α∞view␈α
to

␈↓"β␈↓ α≠␈↓__________
␈↓"β␈↓ α≠␈↓*␈α↔This␈α↔is␈α↔not␈α↔to␈α↔imply␈α↔that␈α↔music␈α↔based␈α↔on␈α↔functional␈α⊗tonal
␈↓"␈↓ α≠␈↓harmony,␈α∞written␈α∞in␈α∞the␈α∞tonal␈α∂era,␈α∞has␈α∞lost,␈α∞or␈α∞ever␈α∞will␈α∂lose␈α∞its
␈↓"␈↓ α≠␈↓vitality.
␈↓ α≠␈↓␈↓ ε83␈↓ 
k


␈↓"β␈↓ α≠␈↓large-scale␈α∀tonal␈α∀unity,␈α∪the␈α∀exceptions␈α∀being␈α∪nearly␈α∀all␈α∀in␈α∪the
␈↓ α≠␈↓realm␈αof␈αoperatic␈αor␈αdramatic␈αmusic.␈α Very␈αuseful␈αin␈αthis␈αregard␈αis
␈↓ α≠␈↓Schenker's␈αterm␈α"tonicization";␈αi.e.,␈αto␈αcreate␈αon␈αa␈αtone,␈αother␈αthan
␈↓ α≠␈↓the␈αoriginal␈αtonic,␈αa␈αtemporary␈αtonic␈αfunction␈αwhich␈αplays␈αonly␈αa
␈↓ α≠␈↓secondary␈α⊂role␈α⊂with␈α∂regard␈α⊂to␈α⊂the␈α∂basic␈α⊂tonic.␈α⊂ Thus␈α∂modulation
␈↓ α≠␈↓might␈αbe␈αsaid␈αto␈αto␈α
be␈αtonicization␈αon␈αthe␈αlargest␈α
scale.␈α However,
␈↓ α≠␈↓Roger␈α~Sessions␈α~has␈α~pointed␈α~out*␈α~that␈α~a␈α~reasonable␈α≠basis␈α~for
␈↓ α≠␈↓differentiating␈α∞between␈α∂these␈α∞two␈α∞terms␈α∂lies␈α∞in␈α∂the␈α∞examination
␈↓ α≠␈↓of␈α⊃the␈α⊃structure␈α⊃of␈α⊃a␈α∩piece␈α⊃of␈α⊃music␈α⊃and␈α⊃the␈α⊃comparison␈α∩of␈α⊃the
␈↓ α≠␈↓larger␈α*harmonic␈α)movements␈α*with␈α)the␈α*harmonic␈α)details.
␈↓ α≠␈↓Modulation␈α∩is␈α∪movement␈α∩␈↓↓to␈↓␈α∪a␈α∩new␈α∪musical␈α∩area;␈α∪tonicization␈α∩is
␈↓ α≠␈↓movement ␈↓↓within␈↓ a single musical area.
␈↓"β␈↓ α≠␈↓␈↓ ∧K_______________


␈↓"β␈↓ α≠␈↓␈↓ α{While␈α∂form␈α∂and␈α∂the␈α∂larger␈α∞aspects␈α∂of␈α∂harmony␈α∂will␈α∂also␈α∞be
␈↓ α≠␈↓dealt␈α∂with,␈α⊂the␈α∂main␈α⊂body␈α∂of␈α∂this␈α⊂handbook␈α∂will␈α⊂treat␈α∂problems
␈↓ α≠␈↓that␈αarise␈α
within␈αunified␈α
areas␈αof␈α
pieces.␈α The␈αrelationships␈α
within
␈↓ α≠␈↓a␈α∩single␈α∩tonal␈α⊃area,␈α∩even␈α∩though␈α⊃quite␈α∩complex,␈α∩can␈α∩usually␈α⊃be
␈↓ α≠␈↓grasped␈α--␈αup␈αto␈αthe␈αpoint␈αwhere␈αtonality␈αis␈αdestroyed.␈α However,
␈↓ α≠␈↓when␈α∩the␈α∩large-scale␈α∩harmonic␈α∪movements␈α∩of␈α∩a␈α∩long␈α∪work␈α∩are
␈↓ α≠␈↓highly␈α
complicated,␈αit␈α
becomes␈αnearly␈α
impossible␈αfor␈α
even␈αthe␈α
best
␈↓ α≠␈↓musicians␈α⊗to␈α⊗follow␈α⊗all␈α∃the␈α⊗functional␈α⊗relationships.␈α⊗ It␈α∃seems
␈↓ α≠␈↓doubtful␈α∀that␈α∀composers␈α∀have␈α∪ever␈α∀expected␈α∀their␈α∀audience␈α∪to
␈↓ α≠␈↓grasp␈αsome␈αof␈αthe␈αlargest␈αrelationships␈αin␈αanything␈αmore␈αthan␈αthe
␈↓ α≠␈↓most␈α∀general␈α∪terms.␈α∀ These␈α∀relationships␈α∪␈↓↓are␈↓␈α∀present,␈α∀but␈α∪their
␈↓ α≠␈↓significance␈α⊗seems␈α⊗subtly␈α↔different␈α⊗in␈α⊗kind␈α⊗from␈α↔those␈α⊗found
␈↓ α≠␈↓␈↓↓within␈↓␈α
the␈α
various␈α
unified␈α
sections␈αof␈α
a␈α
long␈α
work.␈α
 (An␈αextreme
␈↓ α≠␈↓example:␈α∞␈↓↓Parsifal␈↓␈α∞begins␈α∞in␈α∞A␈↓βF␈↓␈α
and␈α∞ends,␈α∞hours␈α∞later,␈α∞in␈α∞the␈α
same
␈↓ α≠␈↓key.)

␈↓"β␈↓ α≠␈↓␈↓ α{We␈α
have␈αbeen␈α
conditioned␈α
to␈αexpect␈α
rather␈α
specific␈αthings␈α
in
␈↓ α≠␈↓a␈α∂piece␈α∞of␈α∂music␈α∞once␈α∂we␈α∞are␈α∂presented␈α∞with␈α∂any␈α∞small␈α∂group␈α∞of
␈↓ α≠␈↓recognized␈αrelationships.␈α When␈αan␈α
expected␈αpattern␈αis␈αbroken,␈α
we

␈↓"β␈↓ α≠␈↓__________
␈↓"β␈↓ α≠␈↓*␈α⊃Roger␈α⊃Sessions,␈α⊃Harmonic␈α⊃Practice,␈α⊃Harcourt␈α⊃Brace,␈α⊃New␈α⊂York,
␈↓"␈↓ α≠␈↓1951.
␈↓ α≠␈↓␈↓ ε84␈↓ 
k


␈↓"β␈↓ α≠␈↓have␈α
learned␈α
to␈α
be␈α
especially␈α
wary␈α
for␈α
the␈α
ultimate,␈α
even␈α∞if␈α
long
␈↓ α≠␈↓delayed,␈αreturn␈αto␈αthe␈αpattern.␈α Or,␈αif␈αa␈αpiece␈αbegins␈αwith␈α
elements
␈↓ α≠␈↓that␈α≡are␈α∨juxtaposed␈α≡in␈α≡a␈α∨manner␈α≡new␈α≡to␈α∨us,␈α≡we␈α∨seek␈α≡a
␈↓ α≠␈↓␈↓↓retrospective␈↓␈αjustification␈αfor␈αthe␈αopening.␈α All␈αcomprehension␈αof
␈↓ α≠␈↓music␈α⊃is␈α⊃based␈α⊃on␈α⊃the␈α⊃listener's␈α⊃ability␈α⊃to␈α⊃relate␈α⊃what␈α⊃has␈α⊃gone
␈↓ α≠␈↓before␈α∃with␈α∀what␈α∃is␈α∀momentarily␈α∃at␈α∀hand␈α∃and␈α∀with␈α∃what␈α∀he
␈↓ α≠␈↓expects␈α
(or␈α
knows)␈α
is␈α
coming.␈α
 One␈α
reason␈α
we␈α
can␈α
understand␈α
on
␈↓ α≠␈↓first␈α∞hearing␈α
a␈α∞tonal␈α∞work␈α
that␈α∞is␈α
new␈α∞to␈α∞us␈α
is␈α∞because␈α∞at␈α
almost
␈↓ α≠␈↓any␈α∀given␈α∀point␈α∪we␈α∀need␈α∀concern␈α∪ourselves␈α∀with␈α∀a␈α∪relatively
␈↓ α≠␈↓small␈α
number␈α
of␈α
alternatives␈α
as␈α
to␈α
what␈α
will␈α
happen␈α∞next.␈α
 Since
␈↓ α≠␈↓tonal␈αmusic␈α
is␈αbased␈α
on␈αconventions␈α
most␈αof␈α
us␈αhave␈αassimilated␈α
in
␈↓ α≠␈↓childhood,␈α∩it␈α⊃might␈α∩be␈α⊃said␈α∩that␈α∩at␈α⊃this␈α∩time␈α⊃in␈α∩our␈α∩culture␈α⊃no
␈↓ α≠␈↓adult really ever hears such a piece for the first time.

␈↓"β␈↓ α≠␈↓␈↓ α{However,␈α
it␈α
is␈α
only␈α
when␈αa␈α
musical␈α
phrase␈α
is␈α
complete␈αthat
␈↓ α≠␈↓we␈αcan␈αhope␈αto␈αgrasp␈αthe␈αtrue␈αimplications␈αof␈αthe␈αvarious␈αparts␈αof
␈↓ α≠␈↓the␈α∪phrase.␈α∩ Likewise,␈α∪it␈α∩is␈α∪only␈α∩when␈α∪a␈α∩piece␈α∪or␈α∪movement␈α∩is
␈↓ α≠␈↓ended␈α
that␈α
we␈αare␈α
presented␈α
with␈αall␈α
the␈α
facts␈αand␈α
are␈α
then␈αable␈α
to
␈↓ α≠␈↓receive␈α
the␈α
full␈α
impact␈α
of␈αthe␈α
work.␈α
 For␈α
these␈α
reasons,␈αharmonic
␈↓ α≠␈↓␈↓↓functions␈↓␈α∩can␈α⊃never␈α∩be␈α∩studied␈α⊃in␈α∩a␈α⊃vertical␈α∩sense.␈α∩ The␈α⊃chords
␈↓ α≠␈↓themselves␈α⊂are␈α⊃vertical␈α⊂occurrences␈α⊃but␈α⊂the␈α⊃harmonic␈α⊂functions
␈↓ α≠␈↓exist␈α⊗only␈α⊗in␈α⊗the␈α⊗horizontal␈α⊗presentation␈α⊗of␈α⊗series␈α⊗of␈α⊗chords.
␈↓ α≠␈↓When␈α
we␈α
are␈αattempting␈α
to␈α
ascertain␈αthe␈α
function␈α
of␈α
a␈αparticular
␈↓ α≠␈↓chord,␈α∃we␈α∃must␈α∃look␈α∀(or␈α∃listen)␈α∃both␈α∃forward␈α∃and␈α∀backward.
␈↓ α≠␈↓Composers␈α⊃always␈α⊃have␈α⊃particular␈α⊂goals␈α⊃in␈α⊃mind␈α⊃and␈α⊃only␈α⊂after
␈↓ α≠␈↓these␈α∞goals␈α∂are␈α∞achieved␈α∂can␈α∞we␈α∂detect␈α∞the␈α∂specific␈α∞justifications
␈↓ α≠␈↓for␈α
the␈α
harmonic␈α
means␈α
used.␈α That␈α
it␈α
is␈α
valuable,␈α
either␈αas␈α
listener
␈↓ α≠␈↓or␈α∪performer,␈α∪to␈α∪be␈α∩completely␈α∪aware␈α∪of␈α∪these␈α∪processes␈α∩seems
␈↓ α≠␈↓obvious.␈α≠ For␈α≠a␈α≠composer,␈α≠even␈α≠though␈α≠he␈α≠uses␈α≠nothing␈α≠of
␈↓ α≠␈↓functional␈α
tonality␈αin␈α
his␈αwork,␈α
high␈αawareness␈α
of␈αtonal␈α
processes
␈↓ α≠␈↓is␈α⊃even␈α⊂more␈α⊃valuable,␈α⊂in␈α⊃that␈α⊂it␈α⊃may␈α⊂help␈α⊃him␈α⊃develop␈α⊂insight
␈↓ α≠␈↓into␈α∞the␈α
most␈α∞basic␈α∞factors␈α
of␈α∞the␈α∞art␈α
--␈α∞factors␈α∞which␈α
transcend
␈↓ α≠␈↓the␈α⊃special␈α⊂conventions␈α⊃of␈α⊃tonality.␈α⊂ Perhaps␈α⊃analysis␈α⊃can␈α⊂never
␈↓ α≠␈↓teach␈αanyone␈αanything␈α
about␈αmusic␈αthat␈α
he␈αdoes␈αnot␈αalready␈α
grasp
␈↓ α≠␈↓in␈α⊂some␈α⊂intuitive␈α⊂manner,␈α⊂but␈α⊂it␈α⊂can␈α⊂help␈α⊂develop␈α⊂a␈α∂vocabulary
␈↓ α≠␈↓for the expression and consideration of these intuitions.
␈↓ α≠␈↓␈↓ ε85␈↓ 
k


␈↓"β␈↓ α≠␈↓␈↓ α{The␈α⊂approach␈α⊂to␈α⊂analysis␈α∂to␈α⊂be␈α⊂presented␈α⊂on␈α⊂the␈α∂following
␈↓ α≠␈↓pages␈α→will␈α→consider␈α→harmonic␈α→factors␈α→almost␈α~exclusively.␈α→ Of
␈↓ α≠␈↓course,␈α
it␈αis␈α
never␈αpossible␈α
to␈α␈↓↓fully␈↓␈α
appreciate␈αthe␈α
role␈αof␈α
harmony
␈↓ α≠␈↓when␈α⊃it␈α⊃is␈α⊃isolated␈α⊃and␈α⊃studied␈α⊃by␈α⊃itself.␈α⊃ In␈α⊃fact,␈α⊃harmony␈α⊃can
␈↓ α≠␈↓hardly␈αbe␈αsaid␈α
to␈αexist,␈αin␈αa␈α
musical␈αsense,␈αapart␈αfrom␈α
the␈αmelodic
␈↓ α≠␈↓and␈α~rhythmic␈α→factors␈α~which␈α→project␈α~it.␈α→ While␈α~these␈α→factors
␈↓ α≠␈↓certainly␈αcannot␈α
be␈αignored,␈α
no␈αattempt␈α
will␈αbe␈α
made␈αhere␈αto␈α
offer
␈↓ α≠␈↓any␈α
consistent␈α
method␈α
of␈α
analyzing␈α
them.␈α
 Because␈α
music␈α
is␈α
made
␈↓ α≠␈↓up␈α∞of␈α
a␈α∞multiplicity␈α
of␈α∞events␈α∞that␈α
move␈α∞in␈α
time,␈α∞it␈α∞is␈α
impossible
␈↓ α≠␈↓to␈αspeak␈αof␈αall␈αof␈αthe␈αvarious␈αelements␈αsimultaneously.␈α Due␈αto␈αthe
␈↓ α≠␈↓limitations␈α∪of␈α∪verbal␈α∪expression,␈α∪each␈α∪aspect␈α∪of␈α∪music␈α∀must␈α∪be
␈↓ α≠␈↓dealt␈α⊂with␈α⊂separately,␈α⊂the␈α∂final␈α⊂synthesis␈α⊂being␈α⊂extraverbal␈α∂and
␈↓ α≠␈↓unique␈αwith␈αeach␈αindividual.␈α Nor␈αwill␈αacoustical␈αjustifications␈αof
␈↓ α≠␈↓harmonic␈α⊂functions␈α⊂be␈α⊂treated␈α⊂in␈α⊂any␈α⊂detail.␈α⊂ This␈α⊃latter␈α⊂subject
␈↓ α≠␈↓has␈α∪been␈α∩discussed␈α∪in␈α∪several␈α∩volumes␈α∪--␈α∩with␈α∪notable␈α∪lack␈α∩of
␈↓ α≠␈↓success.

␈↓"β␈↓ α≠␈↓␈↓ α{Harmonic␈α_functions␈α→will␈α_be␈α→considered␈α_as␈α→occurring␈α_on
␈↓ α≠␈↓various␈α↔"levels",␈α↔and␈α↔maintaining␈α↔a␈α↔clear␈α_distinction␈α↔between
␈↓ α≠␈↓these␈α
levels␈α
will␈α
be␈α
seen␈α
as␈α
a␈α
highly␈α
problematic␈α
aspect␈α
of␈α
analysis.
␈↓ α≠␈↓The␈αuse␈αof␈αthe␈αtraditional␈αnames␈αfor␈αthe␈αharmonies␈αon␈αthe␈αvarious
␈↓ α≠␈↓scale␈α⊂degrees,␈α∂other␈α⊂than␈α⊂tonic,␈α∂subdominant␈α⊂and␈α⊂dominant,␈α∂will
␈↓ α≠␈↓generally␈α∀be␈α∃avoided.␈α∀ The␈α∀Roman␈α∃numerals␈α∀are␈α∀preferred␈α∃as␈α∀a
␈↓ α≠␈↓direct␈α~and␈α~immediate␈α~indication␈α~of␈α~the␈α~relative␈α~positions␈α→of
␈↓ α≠␈↓harmonies.␈α≡ The␈α≥words␈α≡␈↓↓note␈↓,␈α≥␈↓↓tone␈↓␈α≡and␈α≥␈↓↓pitch␈↓␈α≡will␈α≡be␈α≥used
␈↓ α≠␈↓interchangeably.␈α~ ␈↓↓Key␈↓␈α≠and␈α~␈↓↓tonality␈↓␈α≠will␈α~serve␈α≠as␈α~synonyms;
␈↓ α≠␈↓however,␈α
the␈αword␈α
␈↓↓tonic␈↓␈αwill␈α
have␈αmany␈α
meanings.␈α We␈α
speak␈αof
␈↓ α≠␈↓a␈α
basic␈αtonic␈α
(or␈αmain␈α
key),␈αa␈α
temporary␈αtonic␈α
(or␈αsupplementary
␈↓ α≠␈↓key),␈αa␈αtonic␈αnote␈α(the␈αfirst␈αnote␈αof␈αa␈αkey's␈αscale),␈α
etc.␈α Individual
␈↓ α≠␈↓notes␈αwill␈αbe␈αreferred␈αto␈αby␈αcapital␈αletters␈α(e.g.,␈αF,␈αA,␈αC);␈αtonalities
␈↓ α≠␈↓will␈α
appear␈α
as␈α
underlined␈α
letters,␈α
capitals␈α
for␈α
major␈α
and␈α
lower␈α
case
␈↓ α≠␈↓letters␈α⊃for␈α⊂minor␈α⊃(e.g.,␈α⊂␈↓&B␈↓βF␈↓␈↓)αβ,␈α⊃␈↓&D␈↓)αβ,␈α⊂␈↓&f␈↓βS␈↓␈↓)αβ,␈α⊃␈↓&g␈↓)αβ).␈α⊂ The␈α⊃word␈α⊂␈↓↓function␈↓␈α⊃is␈α⊂used
␈↓ α≠␈↓mainly␈αin␈α
the␈αspecial␈αsense␈α
herein␈αdeveloped;␈αi.e.,␈α
the␈αfunction␈αof␈α
a
␈↓ α≠␈↓chord␈α∞is␈α∞usually␈α∞dependent␈α∞upon␈α∞the␈α∞interval␈α∞relation␈α∞of␈α∞its␈α
root
␈↓ α≠␈↓to␈α∂the␈α∂tonic␈α∂note.␈α∂ Also,␈α∂a␈α∂functional␈α∂chord␈α∂is␈α∂one␈α∂which␈α∂plays␈α∞a
␈↓ α≠␈↓truly␈α≠harmonic␈α≠role␈α≠and␈α≤has␈α≠a␈α≠noticeable␈α≠influence␈α≤on␈α≠the
␈↓ α≠␈↓␈↓ ε86␈↓ 
k


␈↓"β␈↓ α≠␈↓harmonic␈α∞movement.␈α∞ Other␈α∂uses␈α∞of␈α∞this␈α∂word␈α∞will␈α∞be␈α∂defined␈α∞as
␈↓ α≠␈↓they appear.

␈↓"β␈↓ α≠␈↓␈↓ α{Generally␈α∞speaking,␈α∞every␈α∞note␈α∂of␈α∞a␈α∞piece␈α∞should␈α∂be␈α∞related
␈↓ α≠␈↓in␈α∞some␈α∞way␈α∞to␈α∞the␈α∞harmonic␈α∞functions.␈α∞ It␈α∞is␈α∞essential␈α∂that␈α∞each
␈↓ α≠␈↓note␈α→be␈α→examined␈α→with␈α→regard␈α→to␈α→its␈α→potentially␈α→chordal␈α_or
␈↓ α≠␈↓auxiliary␈α∩role.␈α∪ Related␈α∩to␈α∩this␈α∪is␈α∩the␈α∩problem␈α∪of␈α∩"contrapuntal
␈↓ α≠␈↓chords",␈α∀passing␈α∪chords␈α∀without␈α∪harmonic␈α∀function.␈α∀ There␈α∪are
␈↓ α≠␈↓situations␈α⊗where␈α↔chords␈α⊗(usually␈α⊗in␈α↔first␈α⊗inversion)␈α↔have␈α⊗no
␈↓ α≠␈↓functional␈α
role␈α∞but␈α
rather␈α∞appear␈α
as␈α∞a␈α
continuous␈α∞parallel␈α
motion
␈↓ α≠␈↓from␈αone␈αfunctional␈αchord␈αto␈αfinally␈αanother.␈α (See␈αthe␈αopening␈αof
␈↓ α≠␈↓the␈αfinale␈α
of␈αBeethoven's␈α
␈↓↓Sonata␈αin␈αC,␈↓␈α
Op.2,␈α#3.)␈α
In␈αa␈αcertain␈α
sense,
␈↓ α≠␈↓some␈αchords␈α(particularly␈α␈↓#
␈↓α␈↓#
6␈↓#␈αp␈↓#v4␈↓#␈↓␈αchords)␈αthat␈αoccur␈αin␈αpassages␈αwhere
␈↓ α≠␈↓the␈α↔outer␈α↔voices␈α↔move␈α_in␈α↔a␈α↔simple␈α↔stepwise␈α↔manner␈α_may␈α↔be
␈↓ α≠␈↓considered␈α∂as␈α∂passing␈α∞chords.␈α∂ However,␈α∂in␈α∞most␈α∂cases␈α∂the␈α∞choice
␈↓ α≠␈↓of␈αnotes␈αfor␈αthe␈αdetails␈αof␈αthe␈αinner␈αparts␈αis␈αdictated␈αprimarily␈αby
␈↓ α≠␈↓considerations␈αof␈αharmony␈αrather␈αthan␈αcounterpoint.␈α The␈αconcept
␈↓ α≠␈↓of␈α∀contrapuntal␈α∀chords␈α∀must␈α∃often␈α∀be␈α∀taken␈α∀into␈α∃account,␈α∀but
␈↓ α≠␈↓almost␈αalways␈αas␈αan␈αextension␈αof␈αthe␈αprinciples␈αguiding␈αthe␈αuse␈αof
␈↓ α≠␈↓auxiliary notes.

␈↓"β␈↓ α≠␈↓␈↓ ∧	___________________


␈↓"β␈↓ α≠␈↓␈↓ α{In␈α∞developing␈α
any␈α∞"formula"␈α∞for␈α
harmonic␈α∞analysis,␈α∞it␈α
must
␈↓ α≠␈↓always␈αbe␈αremembered␈αthat␈αwhile␈αmusic␈αexists␈αas␈αa␈αspecific␈αseries
␈↓ α≠␈↓of␈αevents,␈αthe␈αlistener␈αrarely␈αconcentrates␈αto␈αthe␈αpoint␈αof␈αhearing
␈↓ α≠␈↓every␈α
note.␈α
 More␈α
important,␈α
the␈α
relationships␈α
between␈α
the␈αnotes
␈↓ α≠␈↓are␈αrarely␈αheard␈αin␈αexactly␈α
the␈αsame␈αway␈αtwice.␈α This␈αis␈α
especially
␈↓ α≠␈↓true␈α∞in␈α∞so-called␈α∞"wandering"␈α∞or␈α∞"roving"␈α∞passages.␈α∞ The␈α
creation
␈↓ α≠␈↓of␈α⊂consistent␈α⊂means␈α⊂of␈α⊂accounting␈α⊂for␈α⊂the␈α⊂possible␈α⊂variations␈α⊂in
␈↓ α≠␈↓hearing␈α∩would␈α∩call␈α∪for␈α∩an␈α∩exhaustive␈α∩study␈α∪of␈α∩the␈α∩role␈α∪of␈α∩the
␈↓ α≠␈↓levels␈α∂of␈α∂harmonic␈α∂function␈α∂in␈α∂relation␈α∂to␈α∂the␈α∂complex␈α∂levels␈α∂of
␈↓ α≠␈↓rhythmic␈α⊂and␈α⊂melodic␈α⊂occurrences.␈α⊂ Clearly␈α⊂an␈α⊂understanding␈α⊂of
␈↓ α≠␈↓the␈α↔implications␈α↔of␈α↔harmonic␈α⊗functions␈α↔is␈α↔prerequisite␈α↔to␈α⊗the
␈↓ α≠␈↓formulation␈α⊃of␈α⊂an␈α⊃approach␈α⊂to␈α⊃the␈α⊂complete␈α⊃piece␈α⊂of␈α⊃music␈α⊂that
␈↓ α≠␈↓␈↓ ε87␈↓ 
k


␈↓"β␈↓ α≠␈↓gives␈α∂an␈α∂undistorted␈α∂view␈α⊂of␈α∂the␈α∂piece␈α∂and␈α∂yet␈α⊂remains␈α∂flexible
␈↓ α≠␈↓enough␈α⊃to␈α∩admit␈α⊃the␈α⊃possible␈α∩variants␈α⊃in␈α∩individual␈α⊃perception.
␈↓ α≠␈↓The␈α∪graphing␈α∀of␈α∪these␈α∪functional␈α∀implications␈α∪will␈α∀not␈α∪always
␈↓ α≠␈↓present␈α
a␈α
simple␈αpicture.␈α
 It␈α
will␈α
generally␈αbe␈α
simple␈α
or␈αcomplex␈α
in
␈↓ α≠␈↓relation␈α∀to␈α∀the␈α∃music␈α∀it␈α∀is␈α∃representing.␈α∀ To␈α∀strive␈α∃overly␈α∀for
␈↓ α≠␈↓simplicity␈αin␈αanalysis␈αis␈αto␈αforget␈αthat␈αeven␈αthe␈α
simplest␈αfour-bar
␈↓ α≠␈↓phrase␈α_in␈α_Mozart␈α_is␈α_made␈α_up␈α_of␈α_acoustical␈α_and␈α↔psychological
␈↓ α≠␈↓relationships that, if all were recounted, would stagger the mind.