Essay - A Revision of the Theory of Modulation and a Renewed Riemannian Analytical Methodology for Extended Tonality

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BITTENCOURT, M. A. A revision of the theory of modulation and a renewed Riemannian analyti al methodolo!y for e"tended ton

A revision of the theory of modulationand a renewed Riemannian analytical methodology

for extended tonality 1

Marcus Alessi Bittencourt 2

(Universidade Estadual de Maringá, Paraná, Brazil)[email protected]

Abstract: This paper presents a proposal for a revision of the theory of modulation in two parallel systems of lassifi ation,a ording to the riteria of modulatory impa t and modulatory fa ture! This proposal is presented together with a series oftheoreti al definitions and e"planations that serve as grounds for the proposal of a renewed #iemannian methodology for theanalysis of nineteenth$ entury e"tended$tonality wor%s! The paper tries to demonstrate the histori al grounds of the analyti alpropositions made, and spe ially highlights their a&ility to graph the harmoni language of nineteenth$ entury e"tended tonality&y means of a few analyti al e"amples ta%en from repertoire!

Keywords: Musi 'nalysis Theory of Modulation Musi tru ture Tonal *armony!

1. Introduction

The present paper intends to anti ipate to the general a ademi pu&li the grounds for a proposal ofan analyti al methodology for Tonal Musi , spe ially for its hromati type from the se ond half of thenineteenth entury! +n luded in these grounds are a series of propositions of ta"onomies, nomen laturesand analyti al sym&ologies that are urrently &eing developed as part of a larger resear h pro e t offormulation of a stru tural model for nineteenth$ entury tonality that aims at finding, e"plaining andgraphing the onstru tive oheren e of the repertoire of the se ond half of the nineteenth entury in a lear,syntheti and pedagogi al way, grounded on a methodology &ased on the riti al revision of histori altheoreti al &i&liography of that time! +n the ourse of this resear h, the need was felt for an e"pressive

return to the original nineteenth$ entury sour es, disse ting musi al te"ts from the repertoire of severaltime periods, re$assem&ling and re$wor%ing the on epts of the nature of hords, the nature of

- This arti le rewor%s in English a paper previously pu&lished in Portuguese at the ournal #evista M.si a *odie /ol! -0, no! -, pp! -01$-12,34-0, 5oi6nia, Brazil, under the title 78 'r a&ou9o de uma Proposta de Metodologia 'nal:ti a para o Tonalismo do ; ulo <+<= uma revis>ota"on?mi a da teoria da modula9>o7!

3 Mar us 'lessi Bitten ourt (&! -@A2) is an 'meri an$Brazilian omposer, pianist and musi theorist &orn in the United tates of 'meri a! *eholds master s and do toral degrees in musi omposition from Colum&ia University in the City of Dew or%, and a &a helor s degree inmusi from the University of >o Paulo, Brazil! *e is urrently a professor of omposition, musi theory and omputer musi at theUniversidade Estadual de Maringá ( tate University of Paraná at Maringá) in Brazil! Email= ma&itten ourtFuem!&r

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ounterpoint, the fun tionality of *armony and the logi of the harmoni progressions! in e the maininterest of this resear h on erns the late #omanti repertoire, the most important sour es have &een, eitherdire tly or mediated &y ontemporary musi ologists, te"ts from the nineteenth entury and &eginning ofthe twentieth entury &y authors su h as 5ottfried Ge&er (-AA@$-H0@), 'nton #ei ha (-AA4$-H0I), Moritz*auptmann (-A@3$-HIH), Jran9ois$Koseph J;tis (-AH2$-HA-), 'rthur von 8ettingen (-H0I$-@34), Carl

Jriedri h Geitzmann (-H4H$-HH4), 'rnold hoen&erg (-HA2$-@1-), *einri h hen%er (-HIH$-@01), andspe ially *ugo #iemann (-H2@$-@-@)! Constru ted from this riti al revision of the nineteenth$ enturytheoreti al imagination, this resear h intends to propose an alternative way to e"plain the ompositionalpro edures of E"tended Tonality, formalized in the form of a tru tural Model whi h will &e presented tothe most possi&le e"tent as a logi al dedu tive system demonstrated 7in geometri al order7, as in pinoza sEthi s, in the most o& e tive and pedagogi al way possi&le! Gith this resear h, the goal is not to solely

reate a pedagogi al and theoreti al &ody of te"ts for the tea hing of Musi al tru turing, &ut also tore uperate for the ontemporary s holars part of the musi al theoreti al imagination of the nineteenth

entury, a &ody of wor% whi h is reasona&ly un%nown to the ontemporary traditional methodologies ofmusi al instru tion, despite its utmost importan e for understanding nineteenth$ entury repertoire!

' ma or part of this resear h involves the finding of re urring paradigmati traits in the histori alrepertoire and the resear h and (re)formulation of models whi h are a&le to e"plain, lassify, andinterrelate su h pra ti es, their origins, properties, developments, and transmutations! The typologies

reated &y these proposed models generate a system of sym&ols and terminologies that are used assemanti points of a ess to the typologi al ontent! This sym&ology and terminology &e ome the ma orand &asi tools for the pro ess of formalization and summarization of the omprehension and analyti almapping of Musi , &e oming the very vehi le of ommuni ation and pedagogi al transmission of the

on epts studied! +t is relevant to remar% that these typologies of models of musi al stru tural elements,identified &y their terminologies and sym&ologies, living and surviving through the times in severallineages of theoreti al wor%s, that form after all the very musi al world of imagination of musi ians!

' warning is due here that there are some serious pitfalls in this pro ess of theoreti al re reation! 'first pro&lem is that the on epts, theories, and &asi terminologies of Musi Theory are not unified and arevery often presented differently and sometimes onfli tingly &y different theorists! Jor e"ample, + suggestthe reading of the interesting omparison made &y Lama ois (in L'M'C8+ , -@H2, p! 301) of ideasregarding the %inship of tonalities formulated &y #iemann, Di%olai #ims%y$ orsa%ov (-H22$-@4H), and/in ent d +ndy (-H1-$-@0-)! More spe ifi ally, the matter of terminologies is a &ig pro&lem, for a spe ifiterm may have &een not only defined differently &y different authors, &ut it may have &een used to signifythings that are a&solutely unrelated! The solution for this pro&lem as%s either the redefinition of ertainterms that &e ause of their widespread usage annot &e a&olished N thus for ing the theorist to pi % a

spe ifi side, so to spea%, in relation to his hoi es N, or even the reation of a&solutely new terminologies,with the hope that a different argon will not in ur in onfusions and misunderstandings!' se ond pro&lem omes from the pertinent realization that for the effe tiveness of a musi $

theoreti al wor% there is the need for a ertain temporal distan e from the o&server (the theorist) and theo&served (the omposers of a ertain time period and style and their repertoire)! +t is only with thisdistan ing that it is possi&le to separate that whi h is simply an isolated mannerism of an author from thatwhi h is a ommon pra ti e of a ommunity of authors! The pro&lem is that, with this distan e, what isgained in vision is lost in pro"imity with the living musi al ommon pra ti e of that time! This happens

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&e ause that pra ti e is not entirely preserved in the histori al theoreti al te"ts= these usually tended toformalize only the most ompli ated aspe ts of the musi al language of the period, while matters of more

ommon omprehension were omitted for the la % of need for the e"pression of things whi h were moreommonly and generally understood (see the ommentary on the theoreti al wor% of Tin toris in

KEPPE ED, -@@3, p! @$-4, for e"ample)! My opinion is that perhaps the &est attitude for the resolution of

these pitfalls is to a ept that may&e we will never &e a&le to find all the missing pie es of this puzzle, andfo us our attention to the re urring aspe ts found in the very musi al te"ts, e"plaining them in a way that isrelevant to the needs of our own times, &ut without inventing theoreti al novelties that are distant from thepast world of imagination of that time period! This an &e a omplished through the riti al study of thatwhi h has &een partially preserved in the histori al treatises, always having in mind that those te"ts annot&e ta%en entirely without riti simply &e ause they la % a ertain temporal distan e from their o& e t ofstudy!

Ghile the definitive formalization of this resear h is &eing prepared in the form of a treatise, someof its most important aspe ts are &eing presented to the a ademi ommunity separately in the form ofindividual papers! This present paper rewor%s a previous te"t of mine (B+TTEDC8U#T, 34-3) in a morepre ise, orre t and onsidera&ly e"panded and revised way, and it presents an important fragment of mywor% on erning a pedagogi al (re)organization of the Theory of Modulation N a 7re&oot7, so to spea%, ofthis most important theme of Musi Theory N, with re astings of theoreti al definitions and a ta"onomyprepared a ording to the on erns and orientations previously des ri&ed!

This wor% is strongly &ased in a former proposal for a nomen lature and analyti al sym&ology (seeB+TTEDC8U#T, 34-0), whi h was developed for this resear h as a &asi tool for stru tural analysis! Thista"onomy of modulation, together with the aforementioned sym&ologi al reform, forms the main &asis forthe analyti al methodology proposal whi h ma%es possi&le this &igger resear h of a stru tural model fornineteenth$ entury tonality!

2. Some (re)definitions and basic concepts

2.1. Harmonic Structures

' *armoni tru ture, from a generalized viewpoint, is defined in this theoreti al proposition as aspe ifi olle tion of different notes to whi h we attri&ute a musi al meaning that does not hange aftersuffering operations of rotation, o tave displa ement or shuffling of its onstituent notes, and that serves asa onte"tualizing element for everything that happens harmoni ally and melodi ally during the spe ifisli e of time in whi h it lives! +n the onte"t of repertoire from &efore the twentieth entury, the idealharmoni stru ture is a olle tion of different notes that an &est produ e a high Toni ity (the measure ofthe pro&a&ility of the e"isten e of a ommon fundamental for the olle tive freOuen y spe trum of thesounding notes N see B+TTEDC8U#T, 34-- p! 13$10, 8ETT+D5ED, -HII, p! 3A$01, and the notion of7tonalness7 in P'#DCUTT, -@H@, p! 31$3I and 1H$1@), and a low #oughness (sensory dissonan e aused&y the &eatings resulting from the intera tion of the partial freOuen ies of the olle tive spe trum of thesounding notes N see P'#DCUTT, -@H@, p! 31 and 1H$1@, and *E M*8 TL, -HA1, p! 3AH)! These ideal

onditions happen when we divide the diapente interval (0=3) at the point of a sesOuiOuarta interval (1=2),in other words, this o urs with the ma or triad and its inversion, the minor triad (see #+EM'DD, -@40,

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pg! I, and B+TTEDC8U#T, 34--, p! 13$10)! 's models of perfe tion and onsonan e (a om&ination ofhigh toni ity and low roughness), the perfe t ma or and minor triads have for enturies populated the worldof imagination of omposers and serve as main onte"tualizing element for the simultaneity of pit hes (see#+EM'DD, -@@3 p! HA$HH)! +t is worth mentioning here that it is possi&le to add other notes to the triadiharmoni stru tures of the dominant and su&dominant that, &y adding tension to its &asi triad, &oost the

fun tional definition of those stru tures= the sevenths and ninths in the dominant stru ture, and the ma orsi"th in the su&dominant stru ture (see #+EM'DD, -@40, p! 11$I-)!

Figure 1. Comparison table between the analytical symbologies of the traditional Scale-Degree Theoryand the Functional Theory, in three different versions !ittencourt"s, #iemann"s, and $%ler"s&.

+n this proposal for an analyti al methodology, the harmoni stru tures are represented a ording tothe proposal for a sym&ologi al reform of the #iemannian Jun tional *armony des ri&ed inB+TTEDC8U#T, 34-0! This sym&ologi al reform ta%es as point of departure the original sym&ols reated&y #iemann (see M+C E ED, -@AA, p! A1$A@, and #+EM'DD, -@40) and its simplifying revision &y*ermann 5ra&ner (-HHI$-@I@) (see M+C E ED, -@AA, p! @3$@2) in its usual translation from 5erman(see 8E #EUTE#, -@H4, and B#+ 8 ', 344I), together with a series of reOuisites and spe ifineeds defined &y the envisioned analyti al methodology (see B+TTEDC8U#T 34-0, p! 03$00)! Jigure -shows a omparison ta&le &etween the sym&ologies of the traditional ale$Qegree Theory and the

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Jun tional Theory, &oth in the reformed sym&ology version used in this resear h and ompared to thehistori al fun tional sym&ols &y #iemann and Gilhelm MRler (-@43$-@AI)! The referen es used for thista&le were= GEBE#, -H1-, and 5'U Q+D, 3442 (s ale$degree notation) B+TTEDC8U#T, 34-0(Bitten ourt s fun tional notation) #+EM'DD, -@40, #E*Q+D5, 3440, and M+C E ED, -@AA(#iemann s fun tional notation) ' M8TTE, -@@H (MRler s notation)!

2.2. The Tonal Formula as a Master Framework

5reat musi theorists su h as *einri h hen%er and *ugo #iemann sear hed after the idea of aJundamental tru ture (a 7 Ursatz 7, in hen%er s argon) whi h would serve as a master framewor% forevery pie e of musi al fa&ri ! +nspired &y the hen%erian model (the 7 Urlinie 7 atta hed to the7Bassbrechung 7 or 7GrundBassbrechung 7, linearly forming a +$/$+ see 'TL, -@01, p! 0-2, andP'D *U# T, 344H, p! 12) and &y the #iemannian model (his 7 große Cadenz 7, the +$+/$+ in on un tionwith +$/$+ see M88DE , 3444, p! H2), the present proposal for a model of a fundamental stru ture is

&ased on the u&iOuitous +$+/$/$+ progression whi h gives rise to the three &asi traditional adentialmodels= the full aden e, the half aden e, and the de eptive aden e (also %nown as false or interruptedaden e)! The hoi e &ehind the usage of the term 7formula7 instead of 7 aden e7 aims at eliminating the

terminal onnotation of the term aden e, dire ting the fo us of our attention &eyond the instants of thephrase endings, and asting the notion of a onstru tive model whi h is to &e respe ted &y all harmoniprogressions prior to the twentieth entury!

imilar to hen%er s opinion, in whi h tonality is seen as an e"pansional horizontal transformationof the perfe t triad (see 'TL, -@01, p! 0-0$0-1), the present formalization of the on ept of aJundamental tru ture also represents a tra e tory of e"pansion and ontra tion of the same initial sound(the toni ), &ut des ri&ed in a manner whi h is not e"a tly melodi ally linear as in hen%er s theories! The

prin iple here is to define the idea of a entral tonal pla e (the aforementioned initial sound, the toni ) &ymeans of an initial movement of departure from that pla e, followed &y a refutation of this departure, witha logi al returning ondu tion to the &eginning! Traditionally, the idea of tonal enter (thesis, affirmation)is asso iated with the harmoni fun tion of Toni , the idea of departure of the enter (antithesis, onfli t)with the fun tion of su&dominant, and the idea of refutation of departure and agent of the returning

ondu tion to the enter (synthesis) is asso iated with the fun tion of Qominant (see M88DE , 3444, p!H2$H1, and C*8EDBE#5, -@@1, p! 0--)!

The present proposal of definition of the idea of Jundamental tru ture, whi h is learly of#iemannian inspiration (see #E*Q+D5, 3440, p! 2A), is therefore fun tionally represented &y theprogression T$ $Q$T, whi h &rings a&out e"a tly this motion of departure from an ar&itrary enter and

onseOuent return to it! +nitially, this model gives rise to the three &asi versions of that whi h is herealled a Tonal Jormula, one version representing a omplete 7landing7 motion towards the enter, another

representing a model in whi h the return to the enter is only suggested &ut not ompleted, and yet anotherin whi h the ondu tion to the enter is surprised &y a 7landing7 in a pla e different than e"pe ted! Thus,the s hemati formalization of these &asi versions of the tonal formula is as follows=

• The Complete Tonal Jormula, whi h is &ased on the traditional full aden e=ST" $S " $Q$T • The +n omplete Tonal Jormula, whi h is &ased on the traditional half aden e= ST" $S " $Q

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• The Qe eptive Tonal Jormula, whi h is &ased on the traditional de eptive aden e=ST" $S " $Q$7surprise 7, in whi h the 7surprise7 an &e implemented with any harmonistru ture whi h is different than the main toni , prefera&ly stru tures whi h ontain areasona&le amount of resolution tones for the tensions of the dominant!

+n these presented s hemes, the &ra %ets 7S 7 mean that the respe tive area of the formula isoptional and it an &e omitted the 7"7 denotes the possi&ility that the respe tive area of the formula may&e implemented &y a su&stitute stru ture of the main harmoni stru ture in Ouestion, su h as a si"thsu&stitute (#iemann s parallelklang , or relative su&stitute in the translation from 5erman) or a leading$tonesu&stitute (#iemann s leittonwechselklang , or anti$relative su&stitute) (see #+EM'DD, -@40, p! A- andp! H4)! Qo note that a ording to these given definitions, the parti ipation of the main toni triad in aformula is not indispensa&le!

Donetheless, the tonal formula is not a simple ommon adential progression! +t is a &asi prin iple,a master framewor%, whi h an &e stret hed in time &y the a tion of stru tural prolongations atta hed tothe main harmoni stru tures of the formula as prefi"es or suffi"es, and &y means of melodi prolongationsof the main stru tural tones of the harmoni stru tures, namely the u&iOuitous foreign or non$stru turaltones and thus the framewor% &looms towards the very foreground surfa e of the musi al fa&ri ! +t is li%ethis that #iemann on eived the prin iple of 7Musi al ogi 7 (see #E*Q+D5, 3440, p! 2A)! +n a ertainway, here are on oined &oth the main idea of prolongation, whi h was of utmost importan e to hen%er,and the prin iple of *armoni Jun tion!

Eventually, it is possi&le to admit the use of su&stitute harmoni stru tures also for the formulaareas of the dominant in the omplete and in omplete formulas, with the parti ipation of the E"traordinaryQominants and the #egnante! The #egnante is the inverted dominant, a dualist term &y 8ettingen (see8ETT+D5ED, -HII, p! IA), whi h is represented &y the triad of iv, always as a minor hord! The regnantehas the same fun tional properties of the dominant, &ut with the natural resolutions and voi e leading in aninverted way! +t is represented in the proposal for an analyti al sym&ology here used &y the reverse of thefun tional sym&ol for the dominant (see B+TTEDC8U#T, 34-0, p! 0I)! The use of mirrored versions ofthe letters Q e to denote the #egnante and the upra$#egnante (the triad of v, always as a minor triadsee 8ETT+D5ED, -HII, p! IA) was in a ertain way inspired in the sym&ology &y igfrid arg$Elert(-HAA$-@00) (see M+C E ED, -@AA, p! @4)!

+n this proposed model, an E"traordinary Qominant is a su&stitute harmony of the main dominant,whi h is onstru ted and effe tively onne ted to the toni harmony a ording to pro edures &ased on thevoi e$leading s hemes found in the most ommon aden es, spe ially the traditional de eptive aden esand the adential si"$four (see B+TTEDC8U#T, 34-0, p! 0@$24)! 8f great importan e for the study of the

repertoire of the se ond half of the nineteenth entury, the on ept of this olle tion of a essorydominants is inspired &y ideas for unorthodo" inter onne tions &etween hords via &asi models of voi eleading suggested &y arl Jriedri h Geitzmann in his Harmoniesystem of -HI4 (see #UQQ, -@@3, p! I1$I@)! 's a matter of fa t, su h onne tions also deserved hen%er s attention (see the ases oftoni alization in C*ED E#, -@12, p! 3I1$3AI)! Jigure 2 in ludes e"amples of the use of e"traordinarydominants and regnants!

+n this sense, the tonal formula represents e"a tly this implementation a t of a tonal enter, ane essary a t of demonstration of a tonal pla e whi h models in a &a %wards way the 7ha&its of the ear7

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postulated &y 5ottfried Ge&er (GEBE#, -H1-, p! 021)! Thin%ing in this manner, without the performan eof a tonal formula there an t &e the idea of a tonality! Thus we posit the proposition that every harmonistru ture of a musi al wor% from &efore the twentieth entury parti ipates inelu ta&ly of a tonal formula,and it is duly inserted in this formula and onte"tualized &y it!

+n this present proposal for an analyti al methodology, a tonal formula is represented &y mar%ing

its e"tension with a horizontal line! 8ver this line we will pla e the fun tional sym&ols of the instantiatedharmoni stru tures and under it the sym&ol of the tonal region defined &y the formula (see figure 3),a ording to the proposal for a sym&ologi al reform of the #iemannian Jun tional *armony presented inB+TTEDC8U#T, 34-0, and also a ording to a series of additional onventions that will &e e"plainedfurther down this paper!

Figure '. ()ample of the analytical notation of tonal formulas.Fr*deric Chopin 1+1 -1+ &, $a/ur0a p. 2 n3 ' in 4 minor composed between 1+5 -51&, measures 2 to 1'.

2. . Tonal !e"ion and Harmonic Field

Comprehensively wor%ed out &y hoen&erg (see C*8EDBE#5, -@I@, p! -@$02), the on ept ofTonal #egion is defined in this proposal for a model of tonality as a diatoni tonal pla e (ma or or minor)whi h is instantiated in the per eption of the listener &y means of the performan e of a tonal formula, inone of its three &asi versions (see last item)! The on ept of region is of the utmost importan e to the

omprehension of the prin iple of Monotonality (see C*8EDBE#5, -@I@, p! -@), a ording to whi h amusi al wor% would &e stru tured in one and only one tonality, &eing this tonality understood not as asimple diatoni pla e &ut as a universe of regions hierar hi ally oriented a ording to relationships of

pro"imity and separation around one entral region whi h at the end gives name to that tonality (see thediagrams in C*8EDBE#5, -@I@, p! 34 and 04)!

The *armoni Jield of a region is defined in this proposal as the olle tion of harmoni stru turesthat are availa&le for the implementation of tonal formulas in that region! +t in ludes initially the harmonistru tures of toni (+), su&dominant (+/), and dominant (/), with the further addition of their respe tivesi"th su&stitutes and leading$tone su&stitutes (see #+EM'DD, -@40, p! A- and p! H4)! +n the minor mode,the presen e of the dominant, whi h is always a ma or triad, auses still the need for the addition of themodal parallels of the su&dominant (+/ instead of iv) and of the dominant (v instead of /) to the harmoni

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field, also together with their si"th and leading$tone su&stitutes! This is done to implement the diatonipassages referent to the melodi as ending and des ending versions of the minor s ale, respe tively! The

olle tion of notes whi h is defined and presented &y the main pillar triads of the tonality forms theQiatoni pa e (or ale) of the region in Ouestion! +t is interesting to remar% here that this idea of formingthe ma or and minor s ales out of the main triads of +, +/, and / is Ouite old and it has &een mentioned &y

theorists su h as *einri h Christoph o h (-A2@N-H-I) (see #UQQ, -@@3, p! 3A$3@)!+n this proposal for an analyti al methodology, onsidering the on ept of monotonality, a region is

sym&olized &y the fun tional sym&ol that the toni triad of this region would have in the harmoni field ofthe entral tonal an hor (for the on ept of 'n hor, see item 0!-!3!), with the further addition of ahorizontal dash a&ove its sym&ol (see B+TTEDC8U#T, 34-0, p! 2-)! This horizontal dash serves todistinguish the sym&ol used for regions from the ones used for harmoni stru tures (see figure 3)!

. # ta$onomical (re)creation of the Theor% of Modulation

Modulation is one of the most important stru turing aspe ts of Gestern Musi , and its importan efor the study of *armony and Musi tru ture is a notorious fa t! +n a wide and general sense (and + herestress the presen e of the word 7general7), this present proposal defines the term 7Modulation7 as &eingevery pro edure that effe ts the mi"ture of harmoni stru tures issued from different harmoni fields! This&rutal generalization may initially &e seen as a &it sho %ing, &ut it nonetheless proves to &e e"tremelyuseful, for it is from this very generalization that the present ta"onomi al proposal springs!

.1. The Modulator% Impact as a criterion for classification

in e every harmoni field serves the purpose of implementing the formation of a spe ifi tonalenter, a modulation (in the aforementioned definition) involves a ertain per eption of displa ement of

tonal enter! The intensity of this tonal enter displa ement, whi h we will here all Modulatory +mpa t,depends on the e"a t manner in whi h this modulation is implemented, and the measurement of this impa tserves as a most important lassifying riterion for modulation, with a signifi ant impa t in the analyti al

omprehension of musi al stru tures! This ta"onomi al proposal re y les parts of on epts already wor%edout &y several nineteenth$ entury theorists (su h as Jran9ois$'uguste 5evaert (-H3H$-@4H) apudL'M'C8+ -@H2, -@HH, -@@4), &ut with a redefined, e"panded and or re$delimited terminology!

Considering the diverse ways in whi h a modulation an impa t our per eption of tonal enter, andalso onsidering the 7 Principium nertiae 7 posited &y 5ottfried Ge&er (see GEBE#, -H1-, p! 000$002),there are the following s enarios in whi h the modulatory pro edure, in in reasing order of modulatoryimpa t= a) does not effe tively defy the authority of the tonal enter in effe t &) reates a temporarydeviation of tonal fo us &ut in a way that always promotes a periodi al return to a &asi re urring enter,whi h a Ouires the status of a Tonal 'n hor ) auses a transport of this tonal an hor to another pla e (andthis is the traditionally most a epted meaning of the term 7modulation7) d) destroys the feeling for atonal an hor &y promoting enter transfers from one point immediately to another without the mediatingre ourse of a re urring tonal an hor and e) o&literates the feeling of a enter &y means of a progressionmade out e" lusively of harmoni stru tures that individually form pra ti ally alone their own tonal

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formula, ea h one pointing towards a different region! Thus, a ording to its modulatory impa t, amodulation will &e lassified in 1 &asi types, namely= the Toni ization, the +ntratonal Modulation, theE"tratonal Modulation (or Modulation Proper), the Crossing, and the Gander!

.1.1. The Tonici&ation

Comprehensively wor%ed out in detail &y several theorists (see 7toni alization7 in C*ED E#-@4I, p! 31I$3I-), the Toni ization is defined in this present model as a modulatory pro edure in whi h aharmoni stru ture &elonging to the harmoni field of the region a tivated &y the tonal formula isprefi"ally prolonged via another harmoni stru ture &orrowed from the harmoni field whose toni is thevery stru ture &eing prolonged! *ere the mi"ture of elements from different harmoni fields is smoothenough so that a real displa ement from the tonal enter esta&lished &y the formula does not happen! More

ommonly, this type of modulation involves the use of applied dominants, traditionally also %nown asse ondary dominants (see B#+ 8 ' 344I, p! I0), a type of prefi"al stru tural prolongation that

introdu es a harmoni stru ture &y means of its own dominant, in other words, &y means of the dominantfrom the harmoni field whose toni is the 7toni ized7 element (namely, the stru ture &eing prefi"allyprolonged)! +t is important to stress the fa t that the toni ized stru ture does not lose the harmoni fun tionthat it performs inside the tonal formula that a tivated the original region! +t is pre isely &e ause of thisthat we an say the toni ization is a pro edure with zero modulatory impa t, for it does not defeat the7 Principium nertiae 7 (GEBE#, -H1-, p! 000$002)!

+n this present proposal for an analyti al methodology, the toni ization is indi ated using thesym&ol for the region of the &orrowed harmoni field, pla ed in parentheses underneath the fun tionalanalyti al notation of the toni izing harmoni stru ture, a ording to the sym&ology proposal des ri&ed inB+TTEDC8U#T 34-0, p! 2-$23 (see figure 0)! in e every sym&ol in parentheses always refers to aregion, it is deemed unne essary to use the horizontal dash a&ove that sym&ol!

Figure 5. ()ample of the analytical notation of tonici/ations.Fr*deric Chopin, $a/ur0a p. 2 n3 ' in 4 minor, measures 15 to 16.

9




.1.2. The Intratonal Modulation

The +ntratonal Modulation is defined in this present model as a modulatory pro edure in whi htonal formulas in different neigh&oring regions alternate in a manner to reate the impression of thee"isten e of a entral, &asi and re urring region, the Tonal 'n hor, whi h imposes itself as the effe tive

enter!*ere, the modulation has enough impa t to dislodge the tonal enter to a neigh&oring region, &ut

only in a temporary fashion, from where it soon returns to the an hor region! This is the most hara teristi&ehavior of a traditional tonal musi al fa&ri , &e oming the &asi model for tonal sta&ility! +t is as if theother neigh&oring tonalities were to serve as 7advan ed outposts7 of the entral tonality, stret hing itsdomains and helping to delimit its frontiers via the proposition of diatoni ontrasts! +t is at this point thatthe notion of a 5reater$Tonality is onstru ted (a 7Tonality with apital T7), a universe of tonalities inwhi h these are hierar hi ally organized around a single main tonality a ording to relationships ofpro"imity and separation! +ndeed, this is the same spirit &ehind the prin iple of Monotonality des ri&ed &y

hoen&erg ( C*8EDBE#5 -@I@, p! -@$34)!

This term 7intratonal modulation7 has already &een used &y 5evaert (see L'M'C8+ -@HH, p!-00$-02), although in a different sense (given the musi al e"amples offered &y Lama ois), and more in the

onte"t of that whi h in this present model is alled 7toni ization7! The idea of 7intratonal7 here e"plained%eeps 5evaert s idea of an a tion that results in a &rief and temporary transferen e of tonal enter, &ut now

onsidering the onte"tualizing element of hoen&erg s 7monotonality7! +n other words, this transferen eof enter is &rief &e ause there e"ists a lear onte"tual intention of %eeping a tonal an hor, and it ispre isely &e ause of this that 7every digression from the toni is onsidered to &e still within the tonality,whether dire tly or indire tly, losely or remotely related7 ( C*8EDBE#5 -@I@, p! -@)!

+n the present proposal for an analyti al methodology, the intratonal modulation is indi ated &y thevery region sym&ol whi h is in luded underneath the horizontal line that delimits the tonal formula! Thefun tional sym&ol used must spe ifi ally reveal the %in relationship that the region of the analyzed formulamaintains with the tonal an hor of the musi se tion under onsideration (see figure 3)! +f the tonal an horof the musi se tion under onsideration is different from the main tonality of the wor%, one should furtheradd, inside parentheses and &elow the &asi sym&ol for the region, another notation ontaining thefun tional sym&ol that spe ifi ally reveals the %in relationship maintained &etween the tonal an hor of themusi fragment in Ouestion and the main tonality of the wor%! +n figure 2, for e"ample, whi h shows the&eginning of the se ond theme from the first movement of the Dinth ymphony in Q minor &y 'ntonBru %ner (-H32$-H@I), the regions of the formulas in ' ma or, E ma or, ' minor, 5 ma or, and J sharpma or are identified in relation to ' ma or, whi h is the tonal an hor of the passage in Ouestion! Thenotations inside parentheses underneath the notation for the region reveal the relationship that the an hor(' ma or) maintains with the main %ey of the movement (Q minor)!

10




Figure . ()ample of analytical notations for intratonal and e)tratonal modulations.

4nton !ruc0ner 1+' -1+ 6&, 7inth Symphony, in D minor 1++2- 6&, first movement, measures 2-1 +.

.1. . The '$tratonal Modulation.

The E"tratonal Modulation, in this model, refers to the pro edure of displa ement of the entrala"is of intratonal modulation (the tonal an hor) to another pla e! +t would have &een possi&le to all thistype 7Modulation Proper7, onsidering that this term denotes that whi h theorists su h as hoen&erg

ommonly on eive as the true meaning of the word 7modulation7=

11




78ne should not spea% of modulation unless a tonality has &een a&andoned definitely and for a onsidera&le

time, and another tonality has &een esta&lished harmoni ally as well as themati ally7!

( C*8EDBE#5, -@I@, p! -@)!

*ere the modulation impa t is apa&le to dislodge the tonal an hor to another lo ality, aroundwhi h new sessions of intratonal modulation develop! This is yet another term that revisits ideas &y5evaert (see L'M'C8+ -@H2, p! -@2), although with a re ast and enlarged meaning!

+n the present proposal for an analyti al methodology, the e"tratonal modulation is graphed &y thatvery sym&ol lo ated inside parentheses underneath the &asi notation for the region, a ording to thee"planation ontained in the last item! The e"ample from figure 2 indi ates that an e"tratonal modulationhas o urred, for the tonal an hor of the theme under dis ussion is different from the main tonality of themovement (Q minor)!

This analyti al sym&ology allows for the differentiation of situations in whi h the same tonalitywould represent, in different se tions of a wor%, different regional relationships a ording to the tonalan hor of ea h onte"t! Jor e"ample, in a musi al fragment an hored in C ma or, ' minor would representthe region of the relative toni of that an hor, &ut if the an hor migrates to 5 ma or, the same ' minorwould then represent the region of the relative su&dominant of the new main region! + am of the opinionthat it is very important for the stru tural omprehension of Musi that these distin tions &e learlyilluminated and graphed &y the analyti al sym&ology!

.1. . The rossin"

The word Crossing here used is inspired &y the notorious 5erman musi al term 7 !urch"#hrung 7(see 5#8/E, -HA@, p! 2A3), to whi h it tries to propose an English translation! +n this model, the Crossingdenotes a pro edure in whi h formulas in different regions su eed ea h other without a on ern for themaintenan e of a tonal an hor, or &etter put, a pro edure in whi h the passage &etween regions happensfrom a point to another point in su h a manner as to &e meaningful, in prin iple, only the relationship&etween the new region and the last heard one! The rossing represents the &asi model for tonalinsta&ility, and it is a&ove all traditionally used in the onstru tion of Qevelopment se tions andTransitions!

+n the present proposal for an analyti al methodology, the rossing is notated &y pla ing underneaththe horizontal line whi h delimits the formula not one &ut two sym&ols for the region, separated &y thesign for diatoni (homograph) or enharmoni eOuivalen e (see B+TTEDC8U#T, 34-0, p! 23)! The firstsym&ol, to &e pla ed to the left, indi ates in two levels (in other words, one simple sym&ol and one insideparentheses immediately &elow) the relationship that the new region maintains with the last one heard,pla ing the sym&ol for that relationship a&ove the sym&ol for the last heard region, this last one inside theparentheses! The se ond sym&ol, to &e pla ed to the right, indi ates in only one level (in other words,without a notation in parentheses) the relationship that the new region maintains with the main tonality ofthe wor% as a whole! The purpose of this method is twofold= -) the first notation reveals that whi h ismeaningful in the modulatory motion &y rossing, namely, the point$to$point relationship and 3) the

12




se ond notation serves to allow the identifi ation of the new region in ust one level in a future formula,thus avoiding the in onvenien e of presenting the ne"t relationship sym&ol with a notation in three or morelevels! This an &e a omplished &e ause with the proposal for a fun tional sym&ology here adopted it isalways possi&le to des ri&e with only one level any %ind of relationship &etween two regions (see figure 1and B+TTEDC8U#T 34-0, p! 2-)!

Figure 8. Table of relationships between the 0ey of C ma9or and all the other regions,measured in the most direct way possible adapted from !:TT(7C ;#T, ' 15, p. 1&.

Jigure I illustrates the analysis pro ess for a fragment of rossing! +n this e"ample, the analyti alnomen lature tries to reveal and graph in sym&ols the seOuen ed repetition of the same point$to$pointmodulatory operation, always moving towards the region of the Tr o of the last one heard, thus ausing amotion through an as ending y le of minor thirds (or &etter put, via hromati mediant relationships)!

Figure 6. ()ample of the analytical notation of a crossing. Fran/ Schubert 12 2-1+'+&,<iano Sonata in 4 minor D. + 8 p. ' 1+'8&, first movement, measures 168-122.

13




.1.*. The +ander

+n this present proposal, the Gander is defined as a radi al pro edure in whi h the feeling for atonal enter is o&literated &y means of a progression ontaining mainly dominant harmoni stru tures, ea hone pointing to a different region or even to several different regions, as in the ase of the use of dominants

with enharmoniza&le symmetri al stru ture (see 7vagrant harmony7 in C*8EDBE#5, -@I@, p! 22)! +n aertain way, the o&literation of enter happens &e ause of the Oui % hanges and the e" ess and

multipli ity of enter indi ations! +n short, this on ept is &asi ally analogous to the idea of 7rovingharmony7 developed &y hoen&erg (see C*8EDBE#5, -@I@, p! 0)! Jun tionally, every dominant in thepassage forms alone its own in omplete tonal formula, and in a ertain way it is &e ause of this that, in this

ase, the seOuen e of hords annot 7unmista%a&ly e"press a region or a tonality7 ( C*8EDBE#5, -@I@,p! 0)! Donetheless, the end result of this is that as an ensem&le the harmoni stru tures end up &y forming ahuge prolongation of the dominant area of a formula, in the manner of a seOuen e of e"traordinarydominants, ust waiting for one of the dominants to resolve into its e"pe ted target, thus halting the pro essof wander! This pro edure is generally of short duration and it usually appears as a radi al intensifi ation

of the modulatory movements of a rossing, spe ially inside Qevelopment se tions! +ts ending isommonly and naturally a half aden e whi h re onOuers, if not a tonal an hor, at least a temporary enter

suita&le for the start of new rossing pro edures!The wander generally involves the onne tion of several dominants via Chromati Transformation,

also %nown as Transformational 'ffinity (see B+TTEDC8U#T, 34-0, p! 23$20)! Two harmoni stru turesare related &y Transformational 'ffinity if a variant of the first stru ture (with added ninth, with

hromatized fifth, with root omission, et !) is identi al to a variant of the se ond, either in a diatoni(homograph) manner, or in an enharmoni one (see B+TTEDC8U#T, 34-0, p! 23)! This pro edure is

apa&le of easily transforming a dominant into another &y means of a om&ination of hromati anddiatoni parsimonious voi e$leading motions, what is ertainly an evolution of the 7law of the shortestway7 prea hed &y Bru %ner (see C*8EDBE#5, -@AH, p! 0@)! This %ind of relationship (indi ated in thesym&ology proposal here adopted &y the sign 7V7) is spe ially found &etween the main dominant and thedominants of the relative and anti$relative su&stitutes of the toni , either in a dire tly diatoni way or in a&orrowed one! This fa t leads us to &elieve that su h hord progressions were 7learned7, so to spea%, fromthe onne tions &etween dominants traditionally pra ti ed in de eptive tonal formulas whi h involved atoni ization of the surprising element, as in the seOuen es S/A N /A vi N vi , S/A N /A iii N iii , fore"ample, as well as their modal variants, either dire t or &orrowed! u h types of onne tion &etweendissonant hords, made possi&le &y the reative maintenan e of &asi natural paradigms of voi e leading,were already rather tentatively mapped &y nineteenth$ entury theorists su h as Geitzmann (see 7de eptiveprogression7 in #UQQ, -@@3, p! I1$II) and #ei ha (see 7e" eptional resolutions of dissonant hords7 in#E+C*', -H@4, p! -12, and #E+C*', -H04, p! 3$1)!

+n the present proposal for an analyti al methodology, the wander is graphed using the sameprin iples utilized in the notation of rossings, &ut with the horizontal line indi ating not e"a tly thefrontiers of the tonal formulas &ut instead indi ating tonal onte"tualization areas for the graphed harmonistru tures! Thus, depending on what is most onvenient for the analysis of the spe ifi musi fragment athand, the harmoni stru tures will &e onte"tualized= a) in the region last formed &efore the wanderingpro ess, or in the region finally onOuered at the end of the wander, with the ontradi tory dominants

14




mar%ed as if they were toni izations without targets or even mar%ed as e"traordinary dominants or &) inseveral little formula stages, inasmu h as it is possi&le to de ode and graph the typi al paradigmati originsof the progressions &etween dominants or even ) in a om&ination of all the aforementioned prin iples!Ghat should &e %ept in mind is that graphing e"a tly whi h are the enters pointed at &y the dominants isnot of mu h importan e and is rarely pertinent! The important thing is to graph the logi and manner in

whi h the dominants onvert into ea h other, mapping this a t of prolongation of a tonal insta&ility tensionuntil the re onOuering of a new enter! The e"ample from figure A shows an e"ample of wander in whi h itis possi&le to o&serve learly those onne tions &y hromati transformational affinity, noting spe ially the

onne tions of dominants a ording to the voi e leading found in the typi al toni ized de eptive aden esS/A N /A vi N vi (measures -0A$-0H and -2-$-23) and S/A N /A iii N iii (measures -20$-22)!

Figure 2. ()ample of the analytical notation of a wander.Schubert, <iano Sonata in 4 minor D. + 8 p. ', first movement, measures 152-1 .

.2. omplementar% classification of modulations accordin" the criterion of Facture

To omplete this pro e t of a ta"onomy of modulations there is still the Ouestion of theidentifi ation of the Modulatory Ja ture! *ere the riterion for lassifi ation is the manner in whi h the&order area &etween two tonal formulas is managed, namely, the way in whi h the mi"ture andintrodu tion of elements from different harmoni fields is demonstrated and onstru ted W hen e the name7fa ture7! This riterion of fa ture is always o&serva&le in all the previously seen types of modulationa ording to the modulatory impa t riterion!

.2.1. Modulation ,ia -i,oted Facture' first &asi strategy to reate a onne tion &etween regions is the use of a mediating

element W traditionally %nown as a 7pivot7 W whi h is shared &y the two harmoni fields involved!trategi ally pla ed at the &order area &etween the two tonal formulas, this mediating element is a

harmoni stru ture whi h operates simultaneously and orre tly W in other words, it parti ipates bona "ideof the unfurling of the tonal formulas in whi h it is in luded W &oth as the ending of the last heard formulaand as the &eginning of the ne"t one (in the ase of toni ization, we an onsider the pairtoni ization toni ized$element as a 7mi ro$formula7)! u h strategy of inter onne tion &etween regions,

15




performed &y means of a harmoni element that undergoes, so to spea%, a fun tional hange of meaning,was already understood and des ri&ed in detail at the &eginning of the nineteenth entury! Jor e"ample,5ottfried Ge&er e"plains that 7one and the same spe ies of fundamental harmony may o ur on more thatone degree of a %ey, and indeed may &elong at one time to one %ey, and at another time to another7(GEBE#, -H1-, p! 3H@), whi h wor%s out the on ept of 7 $ehrdeutigkeit 7 (multiple meaning) &y 5eorg

Koseph /ogler (-A2@$-H-2) (see Q'M C*#8QE#, 344H, p! -11$-1I)! Created &y /ogler and Ge&er (seeBE#D TE+D, 3443, p! AAH$AHH), the analyti al te hniOue of the traditional s ale$degree harmony tries todemonstrate and illuminate e"a tly this Ouestion!

.2.1.1. -i,oted Facture with a irect -i,ot

+n this proposal, a pivot harmoni stru ture is onsidered to &e of the 7dire t7 type whenever it&elongs to &oth the simple harmoni fields involved! By simple harmoni field, in this model, we mean theset of &asi harmoni stru tures of a tonality (the pillars T, , and Q, also in luding, when in the minormode, the modal variants X e QY of melodi use mentioned in item 3!0!), plus their si"th (relative)su&stitutes and leading$tone (anti$relative) su&stitutes!

+n this model, if the pivot e"ists named with identi al spelling in &oth the simple harmoni fieldsinvolved, it is alled a Qire t *omograph Pivot, or more simply, a Qiatoni Pivot! ' modulation wor%edout using this type of pivot is the very same traditionally %nown as 7diatoni modulation7 (seeL'M'C8+ , -@H2, p! 343 or B#+ 8 ', 344I, p! AA)! The tri % here is that one progresses in a diatonimanner through one region until it rea hes the mediating element, from whi h it then pro eeds alsodiatoni ally &ut now in the new region! +n other words, this pro edure prevents a dire t lash &etween the

onfli ting elements of the diatoni spa es of the two regions involved! This dou&le fun tion of the pivotharmoni stru ture must &e identified in the analyses &y means of two fun tional mar%ings for the pivotharmoni stru ture, separated &y the sign for diatoni eOuivalen e (Z) (see B+TTEDC8U#T, 34-0, p! 23)!The figure H, &etween the measures -4@ and --4, gives an e"ample of this idea, analyti ally graphing therole of dire t diatoni (or homograph) pivot that the triad of Q minor performs in the &order area &etweenthe formulas in Q minor and E$flat ma or!

Figure +. ()ample of the analytical notation of a pivoted modulation with a direct diatonic pivot,and of a non-pivoted modulation via a process of chromatic transformation.

Schubert, <iano Sonata in 4 minor D. + 8 p. ', first movement, measures 1 -11 .

16




Jigure I also in ludes e"amples of diatoni pivots on measures -IH, -AI and also -A3, disregardingthe enharmony &etween the regions of 5$flat minor and J$sharp minor! +t is here important to remar% thatin ases of toni ization it is a &it superfluous and irrelevant to mar% down pivots in the analyti al notations&e ause in su h ases there is no real proposal for a hange of region, something that would hallenge or&rea% the omprehension of the tonal formula! u h mar%ings would only loud the analyti al fa t to &e

graphed (namely, the very own toni ization)!+t is also possi&le to a omplish the same aforementioned logi using as a pivot a harmoni

stru ture that e"ists in &oth simple harmoni fields &ut named with different spellings! +n this ase, thepivot is alled a Qire t Enharmoni Pivot! ' modulation performed with this type of pivot is &asi ally thesame one %nown traditionally as 7enharmoni modulation7 (see L'M'C8+ , -@@4, p! 11 or B#+ 8 ',344I, p! H4)! The dou&le fun tion of the dire t enharmoni pivot must &e identified in the analyses &ymeans of the sign for enharmoni eOuivalen e ( ≅ ) (see B+TTEDC8U#T, 34-0, p! 23)! Jigure @ shows ane"ample of a dire t enharmoni pivot (measure 30) whi h ma%es use of the notorious enharmonieOuivalen e &etween diminished seventh tetrads that lie a minor third apart, thus proposing a 7short ut7whi h interrupts and shortens a seOuen e of dominants in a y le of fifths!

Figure . ()ample of the analytical notation of a direct enharmonic pivot.Fr*deric Chopin 1+1 -1+ &, $a/ur0a p. 2 n3 ' in 4 minor, measures '' to '8.

.2.1.2. -i,oted Facture with a /orrowed -i,ot

+n this model, a pivot harmoni stru ture is of the 7&orrowed7 %ind whenever it does e"ist in &othharmoni fields involved &ut only if we onsider the e"tended versions of those fields, whi h in ludeadditions aused &y pro esses of modal &orrowing (a on ept already s%et hed in the se tion 7relationshipto the minor su&dominant7, for e"ample, in C*8EDBE#5, -@AH, p! 333)!

+n this pro ess of e"pansion, to the simple harmoni field of a region will &e added new harmonistru tures whi h are o&tained &y means of om&inations of operations of modal polarity inversion (ma orto minor or vi e$versa), of si"th su&stitution and of leading$tone su&stitution, in several different orderings,performed on the &asi pillar harmoni stru tures of the tonality (T, , and Q, in luding the modal variantsof the minor mode mentioned in item 3!0!)! u h om&inations of operations, whi h where alreadypartially mapped &y #iemann in the nineteenth entury (see #+EM'DD, -@40, p! HH$-4I), have as end

17




result the onte"tualization and in lusion in the same harmoni field of all -3 ma or triads and all -3 minortriads, in a hierar hi al manner ran%ed in de reasing relationships of pro"imity, a ording to the num&er ofoperations involved= the &igger the num&er of operations needed to produ e the new harmoni stru turefrom one of the &asi pillar triads of the tonality, the more hierar hi ally distant is the result!

' ording to this logi , the e"tended harmoni field of a region will &e stru tured in four or&its of

relationships in de reasing order of hierar hi al pro"imity! The first or&it ontains the very own &asipillars of tonality plus the resultant harmoni stru tures of the appli ation on the pillars of ust oneoperation of si"th or leading$tone su&stitution W that is, in this or&it revolves the very own simple harmonifield of the region! The se ond or&it ontains the resultant stru tures of the appli ation on the pillars of oneoperation of modal polarity inversion, optionally followed &y one additional operation of si"th or leading$tone su&stitution W that is, in this or&it revolves the simple harmoni field of the modal parallel region! Thethird or&it ontains the resultant stru tures of the appli ation on the pillars of one operation of si"th orleading$tone su&stitution, followed &y one operation of modal polarity inversion! The fourth or&it ontainsthe resultant stru tures of the appli ation on the pillars of a seOuen e of three operations, in this order= oneoperation of modal polarity inversion, one of si"th or leading$tone su&stitution and another one of modalpolarity inversion W that is, the same prin iple whi h generated the third or&it is applied on the elements ofthe se ond or&it!

#egarding the pro ess of onstru tion of pivoted modulations, this e"pansion pro edure impli atesin a situation in whi h it is always possi&le to find pivot stru tures &etween any two tonalities, howeverodd and distant the proposed tonal relationships may sound! This ends up &y providing the lues for theunderstanding of Geitzmann s statement that 7&etween two onsonant triads onne ted with natural voi eleading there an always &e shown an inner onne tion or relationship7 (*'##+ 8D, -@@2, notes frompg! -)! +t is easy to realize here that the onseOuen e of su h logi is the formation of the very ownnineteenth$ entury on ept of E"tended Tonality!

+n this model, similarly to the ase of dire t pivot types, if the &orrowed pivot stru ture e"istsnamed with identi al spellings in &oth harmoni fields involved only if we onsider their e"tendedversions, it is alled a Borrowed *omograph Pivot! +f the pivot e"ists named with different spellings in&oth harmoni fields involved only if we onsider their e"tended versions, it is alled a BorrowedEnharmoni Pivot! The dou&le fun tion of these pivots must also &e identified in the analyses &y means ofthe signs for diatoni eOuivalen e (Z), in the ase of homography, and for enharmoni eOuivalen e ( ≅ ), inthe ase of enharmony! Jigure -4 gives an e"ample of a &orrowed homograph pivot of the se ond or&it,

onne ting two minor tonalities in hromati mediant relationship (' minor and C minor) &y means of anelement from the harmoni field of C ma or, or rather, &y means of a mem&er of the se ond or&it ofrelationships of the e"tended harmoni field of C minor! Jigure 3 (measure H) also in ludes an e"ample of

a &orrowed homograph pivot!

18




Figure 1 . ()ample of the analytical notation of a borrowed homograph pivot of the second orbit.Fr*deric Chopin 1+1 -1+ &, $a/ur0a p. 2 n3 ' in 4 minor, measures 18 to 1+.

.2.2. Modulation ,ia 0on -i,oted Facture

' se ond strategy to perform the onne tion &etween regions is the use of a pro ess oftransformation that onverts the last harmoni stru ture of the first formula into the first stru ture of thene"t formula &y means of a hromati transformation pro edure (see item 0!-!1!), or rather, &y means of a

om&ination of hromati and diatoni parsimonious voi e$leading motions! Ghile the strategy used in thepivoted modulation is, &y the use of a ommon harmoni stru ture, to strive to avoid a dire t lash &etweenthe onfli ting elements of the different diatoni spa es involved, in the non$pivoted fa ture we valuepre isely the e"posure of that lash &y means of a pro ess of 7melodi distortion7, whi h at the same timeintrodu es a onfli t W in the form of a diatoni ontrast to the first region heard W and an uneOuivo alimpulsion towards the on retization of a motion towards another harmoni field! This idea is &asi allyanalogous to the on ept traditionally %nown as 7 hromati modulation7 (see L'M'C8+ , -@H2, p! 34H orB#+ 8 ', 344I, p! H4)!

The o urren e of this method of un tion &etween regions an &e identified in the analyses via thesign for transformational affinity (V), inserted &etween the two harmoni stru tures that form the &orderarea of the two formulas involved (see B+TTEDC8U#T, 34-0, p! 23)! Jigure H, &etween the measures --0and --2, e"emplifies this idea of modulation via non$pivoted fa ture, analyti ally graphing the onversionof the E$flat ma or triad (toni in the first formula) into the C ma or triad (dominant in the se ond formula),&y a pro ess of hromati transformation! +t is again important to remar%, and for the same aforementionedreasons, that the analyti al notation of this relationship is usually irrelevant in ases of toni ization!Donetheless, sometimes su h mar%ings may &e pertinent, as in the ase of wander found on figure A!

. Final ommentar%

Currently in the pro ess of formalization in a treatise of larger s ope and &readth still underpreparation, these analyti al tools and the theoreti al on epts here presented have demonstrated

onsidera&le pedagogi al usefulness in lasses of *armony and Musi 'nalysis, and they have always&een pra ti ed in a omparative fashion and ounterposed to the lassi tools of the ale$Qegree Theoryand the traditional Jun tional *armony! This pedagogi al wor% has served as a testing ground for the

19




development of this proposal of an analyti al methodology, and it has &een always showing W or evenfor ing W new paths to follow, e"posing faults and nearsightednesses to &e orre ted, this as we verify theeffi ien y (or not) of the stru tural disse ting a tion of this proposed theoreti al model on the wor%s fromthe nineteenth$ entury repertoire! +t is therefore natural that these propositions &e su&seOuently modifiedand ad usted in a near future! +n this paper, + tried to demonstrate the histori al &asis of this present

proposal for a Ta"onomy of Modulation, spe ially emphasizing its a&ility to graph the harmoni languageof the nineteenth$ entury e"tended tonality!

*. /iblio"raphic !eferences

BE#D TE+D, Qavid G!! Dineteenth$ entury harmoni theory= the 'ustro$5erman lega y! +n= %he Cambridge History o" &estern $usic %heory , ed! Thomas Christensen! ), p! AAH$H--! Cam&ridge= Cam&ridge University Press,3443!

B+TTEDC8U#T, Mar us 'lessi! #eimagining a #iemannian sym&ology for the stru tural harmoni analysis of-@th$ entury tonal musi ! 'e(ista )*rte+ , Curiti&a, n!3, p!04$2H, 34-0!

[[[[[[[[[[! %et hes for the foundations of a ontemporary e"perimental treatise on *armony! +n= ,nais do -ncontro nternacional de %eoria e ,n lise $usical , p! 2A$1@! >o Paulo= UDE P$U P$UD+C'MP, 34--!

[[[[[[[[[[! Um 7#e&oot7 para a Teoria da Modula9>o, om o +mpa to Modulat\rio omo rit;rio de lassifi a9>o!+n= ,nais do ) -ncontro de $usicologia de 'ibeir/o Preto , p! 3-1$33I! #i&eir>o Preto$ P= Universidade de >oPaulo, 34-3!

B#+ 8 ', Cyro! Princ0pios de harmonia "uncional ! >o Paulo= 'nna&lume, 344I!

Q'M C*#8QE#, Qavid! %hinking about Harmony1 *istori al Perspe tives on 'nalysis! Dew or%= Cam&ridgeUniversity Press, 344H!

5'U Q+D, #o&ert! Harmonic Practice in %onal $usic ! Dew or%= G!G! Dorton ] o!, 3442!

5#8/E, 5eorge (ed!)! , !ictionary o" $usic and $usicians , /ol! -! ondon= Ma millan and Co!, -HA@!

*E M*8 TL, *! 2n %he 3ensations o" %one as a Physiological Basis "or the %heory o" $usic ! ondon=ongmans, 5reen, and Co! -HA1!

KEPPE ED, ! Counterpoint1 The Polyphoni /o al tyle of the i"teenth Century! Dew or%= Qover Pu&li ations,-@@3!

'TL, 'dele T!! *einri h hen%er s Method of 'nalysis! %he $usical 4uarterly , 8"ford University Press, /ol!3-, Do! 0, pp! 0--$03@, -@01!

8E #EUTE#, *ans Koa him! Harmonia 5uncional N introdu9>o ^ teoria das fun9_es harm?ni as! >o Paulo=#i ordi Brasileira, -@H4!

*'##+ 8D, Qaniel! Harmonic 5unction in Chromatic $usic1 a #enewed Qualist Theory and an ' ount of +tsPre edents! Chi ago= University of Chi ago Press, -@@2!

' M8TTE, Qiether de! ,rmon0a ! Bar elona= +dea Boo%s, -@@H!

M+C E ED, Gilliam C! Hugo 'iemann6s %heory o" Harmony1 ' tudy! in oln= University of De&ras%a Press,-@AA!

M88DE , evin! *ugo #iemann s Qe&ut as a Musi Theorist! 7ournal o" $usic %heory , Qu%e University Press,/ol! 22, Do! -, pp! H-$@@, 3444!

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8ETT+D5ED, 'rthur von! Harmoniesystem in !ualer -ntwicklung ! eipzig= G! 5laser, -HII!

P'D *U# T, Tom! 3chenkerGU !- = a &rief hand&oo% and we&site for hen%erian analysis! Dew or%=#outledge, 344H!

P'#DCUTT, #i hard! Harmony1 ' Psy hoa ousti al 'pproa h! Berlin= pringer$/erlag, -@H@!

#E*Q+D5, 'le"ander! Hugo 'iemann and the birth o" modern musical thought ! Cam&ridge= Cam&ridge UniversityPress, 3440!

#E+C*', 'nton! Corso di Composizione $usicale ! Mil>o= 5! #i ordi ] C!, -H@4!

8888888888. , 9ew %heory o" the 'esolution o" !iscords according to the $odern 3ystem ! ondon= #! Co %s ]Co!, -H04!

#+EM'DD, *ugo! Harmony 3impli"ied : or, The theory of the tonal fun tions of hords! ondon= 'ugener ] Co!,-@40!

[[[[[[[[[[! +deas for a tudy 78n the +magination of Tone7! 7ournal o" $usic %heory , /ol! 0I, Do! - ( pring), pp!H-$--A, -@@3!

#UQQ, #a hel Eloise! arl Jriedri h Geitzmann s *armoni Theory in Perspe tive! PhQ dissertation, Colum&iaUniversity in the City of Dew or%, Dew or%, -@@3!

C*ED E#, *einri h! Harmony ! Chi ago= University of Chi ago Press, -@12!

C*8EDBE#5, 'rnold! %heory o" Harmony ! Ber%eley= Univ! of California Press, -@AH!

[[[[[[[[[[! 3tructural 5unctions o" Harmony ! Dew or%= G!G! Dorton ] o!, -@I@!

[[[[[[[[[[! %he $usical dea and the ;ogic< %echni=ue< and ,rt o" ts Presentation ! Dew or%= Colum&iaUniversity Press, -@@1!

GEBE#, 5ottfried! %he %heory o" $usical Composition< treated with a (iew to a naturally consecuti(e arrangemento" topics , /ol! +! ondon= Messrs! #o&ert Co %s and Co!, -H1-!

L'M'C8+ , K! %ratado de armonia< /olume -! Bar elona= a&or, -@H2!

[[[[[[[[[[! %ratado de armonia< /olume 3! Bar elona= a&or, -@HH!

[[[[[[[[[[! %ratado de armonia< /olume 0! Bar elona= a&or, -@@4!

Documents

Essay - A Revision of the Theory of Modulation and a Renewed Riemannian Analytical Methodology for Extended Tonality