A System For Pandiatonic Music

 

A SYSTEM FOR CREATING PANDIATONIC PANDIATONIC MUSIC by James Charles Woodward

A Research Paper Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Music

ARIZONA STATE UNIVERSITY May 2009

 

UMI Number: 3357288

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A SYSTEM FOR CREATING PANDIATONIC PANDIATONIC MUSIC by James Charles Woodward

has been approved April 2009

Graduate Supervisory Committee: Jody Rockmaker, Chair Rodney Rogers Glenn Hackbarth

CCEPTED CCEP TED B Y   TH

GR

DU

TE COLLEGE COLLEGE

 

ABSTRACT Many comp ositions in the twentieth twentieth century use the pitches of the diatonic diatonic scal scalee in a unique unique w ay, but pandiatonicism has become so vague a concept that it has has lost significance signifi cance a s a viable term in mo dern mus ic. Theo rists have attemp ted to define define pandiatonicism by the absence of traditi traditional onal elements in passages, but these observations have not perpetuated pandiatonicism as anything more than a superficial description and have produced a definiti definition on of exclusion. The definition of pandiatonicism will become more clear and intelligible if it describes descri bes patterns and elem ents that are are present present in many pand iatonic com positions. It will also become more understandable and communicable if it categorizes different compositional styles and techniques techniques com posers are currentl currentlyy exploring. One such way is is the use of melodic tonality, hierarchy, hierarchy, and tendency tones in pandiatonic melody and the movem ent from from consonance to dissonance then then back to consonance in pandiatonic pandiatonic harmony. If pandiatonicism remains a general and open description of any passage that uses diatonic pitches, whether tertian or non-tertian, with or without a tonal center, then each stylee and technique can be explored and analyzed and a system atic way of creating styl creating pandiatonic music can be developed. Pandiatonicism Pandiatonicism will then regain an importance importance in the theory theory of mo dern m usic by promoting the use of the pitches of a of  a diatonic scale in a way that creates intelligent and artistic musical thought and substance.

in

 

ACKNOWLEDGEMENTS This dissertation dissertation would not have been been possible w ithout the support of  Dr. Jody Rockm aker. The dedication he has shown through returning materials in in a timely fashi fashion on with his suggestions and the quali quality ty of advice in his comments helped the content and and language of this dissertation reach a higher level of professional writing. Additional ackn owledgem ents and thanks are gi given ven to Dr. Rodney Rogers and Dr. Glenn Hackbarth w hose guidance and direct direction ion have m ade this paper more accessible and clear.

IV

 

TABLE OF CONTENTS Page LIST LIS T OF MUSICAL EXAMPLES

vi

CHAPTERS 1

INTRODUCTION

1

2

MELODIC TENSION TENSION AND HARMONIC DISSONANCE

3

A SYSTEM FOR CREATING PANDIATONIC PANDIATONIC MELODY AND

10

HARMONY

19

4

PANDIATONICISM PANDIATONICISM AND THE MELODIC MINOR SCALE

29

5

MODULATION

38

6

CONCLUSIONS

49

REFERENCES

50

v

 

LIST OF MUSICAL EXAMPLES Musical Exam ple

Page

1. Morten Lauridsen,  As B irds Come Nearer,  mm . 2 to 5

5

2.   Aaron Copland,  Appalachian Spring,  mm . 611 to 619

7

Clarifying ng Dissonance and Tensi Tension, on, mm .  1  to 6 3.   Clarifyi

10

4.   Samuel Barber,  Capricorn Concert 178 Concerto, o, Third Mo vement, mm . 169 to 178

12

5.   Arvo Part, Credo from  Berliner Messe, m m .  1  to 11

14

6. Arvo Part,  Credo from  Berliner Messe, reducti  reduction on of m.  1  to 11 with the num ber of diatonic notes present in each sonority 7. Benjamin Britten,  Nocturne from  Serenade,  second caden za 8. Aaron Copland,  In the Beginning, m m .  1  to 13

16 .....18 .....18 ....19 ....19

9. Developing Contrapuntal Lines in Pandiatonicism Pandiatonicism

21

10.   Ninth Chords

24

11. 11 .  Morten Lauridsen,  O Magnum Mysteri Mysterium, um,  m m .  1  to 5

25

12.   Eleventh Chords

27

13 13..  James Woodward, Ave Maria,  mm . 30 to 41

28

14.   Tendency Tones of the Melodic Minor Scal Scalee

29

Concerto, o,  Second Mo vement, mm . 30 to 35 15. 15 .  Samuel Barber,  Capricorn Concert

30

16.   Aaron Copland,  Piano Variations,  mm . 11 to 20

32

17.   James Woodward,  Wa it by the Station, mm.  10 to 19

33

18. 18 .   An Arrangem ent of Ba ch's T wo Part Invention Invention in C minor, mm . 10 to 13 ...34 ...34

19 19..   Contrapuntal Voices in Pandiatonicism Pandiatonicism using Modal Scale Scaless

vi

35

 

Musical Exam ple (continued)

Page

20 .  Bela Bartok, Pentatonic Melody, m m .  1  to 10

36

2 1 .  James Woodw ard, Prayer,  mm . 5 to 16

39

22 .  Samuel Barber,  Capricorn Concerto, Third Mov ement, mm . 151 151 to 159 159

42

2 3 .  Elliott Carter, Piano Sonata,  Second Mo vement, mm . 11 to 22

43

24 .   Ninth Chords and Elevent Eleventhh Chords with Tritones

45

from G major to C major 2 5 .   A Short Pandiatonic Modulation from

46

26 .  A Short Pandiatonic Pandiatonic Mo dulati dulation on from from A Melodic Minor to C Melodic

Minor

47

27 .   Analysis of a Short Pandiatoni Pandiatonicc M odulati odulation on fro from m A Melodic Minor to C

Melodic Minor

.48

vii

 

INTRODUCTION Pandiatonicism has becom e so vague a concept that it has very littl littlee m eaning or use. Music theorists have been unable to agree on the the defini definiti tion on of tthis his term term beyond the use of the pitches of a of  a diat  diatonic onic scale. This single single limitati limitation on m eans any portion of music that appears on a grand staff that does not have accidentals or any piano piece that utilizes only the white keys of the piano, regardless of style or musical content, is considered pand iatonic. For examp le, if every non-d iatonic accidental is taken away fr from om Beethoven's Eroica Symphony and Schoenberg's Chamber Symphony, they will both be categorized categor ized as pan diatonic. Nicolas Slonimsky (1894 - 1995) first coined the term pandiatonicism in 1939 in his book   Music Since 1900.  He asserts "all seven degrees of the diatonic scale scale are used freely freely in dem ocratic eq uality." He also states the "functional "functional impo rtance of the pri primary mary tri triads ads remains undiminished in pandiatonic harmo ny." Slonim sky's explanation ooff pandiatonicism is not substantially different from that of traditional harmony and does not mention linear structures beyond the act of "small children promenading their little fi finger ngerss over the piano ke yboard." W rit ritten ten in 1939, Sl Slonimsky onimsky introduced his nom enclature during a growing trend trend of music that used twelve-tone techniques or did not assert a specific key, mode, or tonal center. cente r. Man y com posers still still used used the notes of the diatoni diatonicc scale, but the methods employed in com posing serial serial and fr free ee chromatic works hav e been the focal point for most discussions.

Slonimsky, Nicolas.   Nicolas Slonimsky: The First Hundred Years (New York: Schirmer Boo ks, 1994), 358.

 

2 Toward the end of his life, even Slonimsky did not think highly of the term and param eters he created for a portion of music utilizing utilizing diatonic notes. In an interview in 1990 given by Richard Kostelanetz, Slonimsky gave this definition for his term: Pandiatonicism Pandiatoni cism - "pan" as in pancake, "dia" as in diarrhea, and "tonicism " as in hair tonic. One professor in Cleveland described it as "C-major that sounds like 2

hell ." I like that definition.

Slonim sky's ow n adm ission of the the ineffectiveness ineffectiveness of this label gives gives good cause for authors of texts on twentieth century music to give little attention to pandiatonicism. T he  New H arvard Dictionary of Music gives substantial descriptions of many twentieth-century compositional topics but gives only this brief description of pandiatonicism. Nicolas Slonimsk y's term term for the predominance in l

some 20  -century music of the pitches of the diatonic scale, as distinct from the chromaticism of  late  19 th th

century music and of 20 -century atonal and twelve tone music. Such music , including some associated with neoclassicism, often employs dissonant diatonic

Richard Kostelantez and Nicolas Slonimsky, Richard Slonimsky,  Conversation with Nicolas Sloni Slonimsky msky about His Composing,   The Musical Q uarterly, Vol. 74, No . 3 (Oxford: Oxford University Press, 1990), 458 - 472.

 

3 harmo nies such as added sixth, seventh, seventh, and ninth chords. 3 This definition states that pandiatonic harmonies incorporate added sixths, sevenths, and ninths, but the the primary contribu tion of this citati citation on is what pandiaton icism is not. It is not chrom atic. It is not atonal or twelve tone. Stefan Kostka gives little mention of pandiatonicism in his widely used  Ma terial terialss

and T echniques of Twentieth Century Mu sic.  Likewise, Bryan Simm's M usic of tthe he Twentieth Century  barely m entions it at at all. Both Kotska and S imm s have altered altered Slonimsky's definition. Simms w rites rites that pandiatonicism pandiatonicism does not have "a clear and stable tonic."4   H e illustrates this technique with the first six measures of   he Sunken Cathedral  from Debussy's first book of Preludes.5   Simms describes describes the opening of this prelude aass usi using ng "all the notes of the of  the C-major scale, but projects projects no tonic note by wh ich a key or mode could be inferred." Although Kotska gives a bit more detail iinn his description, he maintains that there there are multiple multiple approaches and disagreements on pandiatonicim. He define definess it as "a passage that uses only the tones of some diatonic scale but does not rely on traditional traditional harmonic

Randel, Don M ichael, ed. ed.,,   The New Harvard Dicti Dictionary onary of M usic  (Cambridge: The Belknap Press of Harvard Universi University ty Press, 1986), 605 . Simms, Bryan.   Music of the Twen tiet tiethh Century: Style and Structure  (Belmont: Schirmer, 1996), 56. Written in in 1910, De bus sy's first book of  Preludes  was written over twenty years before the creation of the the term pandiaton icism. Even if the pitch collection of these six measures d oes n't articulate articulate C as the tonic, Simm Simm s unfortunately unfortunately ignores ignores C majo majorr as an important tonal center later in the piece.

 

4 progressions and dissonance treatment." treatment." 6   Like the New Harvard D ict ictionary ionary and the Simms definitions, this explanation does not focus on what pandiatonicism is, it focuses on what it is not. Theorists have attempted to define define pandiatonicism pandiatonicism through the absence of traditional tradi tional elements in pass ages, but these observations have not perpetuated perpetuated pandiatonicism pandiatonici sm as anything more than a vague description description and h ave produced an explanation explanati on of exclusion. The defin definit ition ion of pandiatonici pandiatonicism sm will become m ore cl clear ear and intelligible if it describes patterns and elements that are present in many pandiatonic comp ositi ositions. ons. It will also also become more understandable understandable and comm unicable if if it  it categorizes different compositional styles and techniques composers are currently exploring explori ng with sets of diatonic diatonic notes. If pandiatonicism pandiatonicism remains a genera generall and open description of a of  a passage that uses diatonic diatonic pitches, whether tertian or non-tert non-tertian, ian, with or without a tonal center, then each style and technique can be explored and analyzed and a systematic systemat ic way of creati creating ng pandiatonic music can be developed. Pandiatonicism Pandiatonicism wil willl then regain regain an im portance in the the theory of modern m usic by promoting the use of the the pitches of a of  a diatonic scale in ways that create intelligent and artistic musical thought and substance. Through composer's examples and the author's own compositions and arrangements, this paper will identify a few styles and techniques present in pandiatonic compositions and suggest a systematic way of creating similar pandiatonic melody and

Kotska, Stefan.   Materials and T echniques of Twentieth Century Mu sic  (Upper Saddle Rive: Pearson Prentice Hall, Hall, 2006), 108.

 

harmony. These system atic techniques will include include tertian tertian sonorities and a pitch center, but will also highlight non-tertian elements as a way of creating pandiatonic music. Pandiatonic melody can be created through melodic tonality 7 , hierarchy, and tendency tones. M elodic tension in in pandiatonic melody can be achieved through establishing a tonic and a hierarchy of pitches. Unlike traditional tonal melody, a pandiatonic melody can take any shape or contour including large leaps and extreme registers. Pandiato nic me lody will oft often en have a greater emphasis on the the tendency tones of scale degrees 2, 4, 6, and the leading tone. It is is less li likely kely to cycle through triadic arpeggios or skip among notes of  a  chord built by thirds. Instead, a pandiatonic m elody can use leaps that are unusual compared to traditional melody, such as multiple leaps of  a fourth or leaps of a seventh. Example 1: Morton Lauridsen, As Birds C ome Nearer,  m m m. 2 to 5.

m

=E

As birds come

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  Melodic tonality as defined by Rudolph Reti manifests itself "through melody only." A melody in melodic tonality, as compared to classical tonality, can come to a conclusion on the tonic pitch "from "from any point of its course, without the help of of any dominant or any other overtone element, simply through its own innate melodic impulse." Reti, Rudolph.  Tona lity, Atona lity, Panton ality: A Study of Some Trends in Twentieth Century Music   (New York: Macmillan, 1958), 17, 23.

 

6 In the vocal line of  As Birds C ome Nearer  [Ex.  1], 1] , Morton Lauridsen increases increases melodic tension by displacing the tonic note of Db down an octave. The leading tone, emp hasized o n a stronger beat comp ared to the the tonic, is is lef leftt w aiting for a resolution. The jump of a of  a major seventh will become a motive developed later in the piece. Pandiatonic harmon y can prepare or release release dissonance like traditional traditional harmony, but unlike pieces that use functi functional onal harmony , pandiatonic pandiatonic harm ony can use any dissonance m ade availabl availablee through the Combination Combination of diatonic notes. In  Appalachian Spring, Aaron Copland creates a set of variations on a Shaker theme,  T he Gift to Be Simple.  At the climax of this portion of the ballet, the chamb er orchestra of thirteen music ians divides into three contrapuntal lines. These three li lines nes can be seen in the first violin, violoncello, and double bass (these lines have been isolated in Ex. 2). The first violin violin will reach the tonic two beats before the end of this of  this climactic variation. Therefore, the me lodic tension will resolve before the fi final nal note of the theme. The rhythmic nature alone would not be effec effecti tive ve enough to com plete thi thiss broad and grand variation. This variation variation is driven by these three contrapuntal voices and the tertian tertian and nontertian tert ian sonorities they create. Beginning with the octaves on tthe he dow nbeat of m. 612, tthe he three voices reach their mo st significa significant nt harm onic dissonanc e in m. 617. This non-tertian dissonance calls for the eventual return of the consonant pitch of  C  in octaves. To prevent a feeling feeling of closure, Copland m aintains aintains tertian harmony under the first two two tonic notes that end the theme in m. 618. All voices resolve resolve to the pitch pitch center in m. 619.

 

Example 2: Aaron Copland,  Appalachian Spring, m m. 611 to 619.

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The harmony created by three contrapuntal voices fluctuate from consonance to dissonance back to to consonance. This passage passage does not need the dominant triad triad to to achieve achie ve a feeling feeling of closure. Likewise, pandiatonicism does not need the primary triads triads to create harm onic function. Instead, the mo tion from from conso nance to dissonance back to consonance is all that is needed in p andiatonicism. andiatonicism. W hile the Lauridsen and Copland excerpts illustrate illustrate melodic tonality tonality and harmonic dissonance respectively, the proposed systematic systematic technique of creating creating pandiatonicc m usic will be achieved through the com bination pandiatoni bination of these two two elements.

 

8 There is a limit to the the amount of dissonance a composer can create with with only seven diatonic notes. Slonim sky, critical critical of pand iatonicism , states it "alway s sound[s] mild because, with the white keys only, you ca n't create create sharp dissonances as you can in in polytonality." 8   Com pared to the chromatic scale, the major and perfect iintervals ntervals present in the major scale greatly diminish the availabili availability ty of disson ant son orities. An interval vector of the major scale, < 254361 >, reveals only two minor seco nds and one tri tritone. tone. The majority majority of intervals in the major scale are consonant perfect fourths and fi fift fths hs followed followed closely by major seconds and minor sevenths. The melodic minor scale, however, utilizes two different pitches for both scale degree six and seven and has a much larger array of dissonances availabl available. e. The m inor scales have been clearl clearlyy established established in tonality through a hie hierarchy rarchy of pitches and and functional functional harmon y. The natural, harmo nic, and and melodic m inor scal scalee can bbee explored through pan diaton icism . Like the major major scale, these sets of diatonic pitches can be manipulated manipulated int intoo pand iatonic sonorities with with stronger dissonances and greater melodic tensions. If a composition abides only to the single limitation of the use of diatonic notes, the music created can sound static and bland. An attempt to avoid the sameness created by the uniformity uniformity of each pitch can be made through applying other elemen ts, such such as reoccurring rhythmic motives or textural effects, but monotony is the risk of maintaining a single single set of diatonic pitches for for an extended period of time . Pand iatonic co mp ositions

8

Richard Kostelantez and Nicolas Slonimsky,  Conversation with Nicolas Slonimsky about His Composing,  The Mu sical Quarterly, Vo l. 74, N o. 3 (Oxford: Oxford University Press, 1990), 462.

 

9 can create a new tonal center by altering pitches or can modulate to a differ different ent diatonic scale. W hile these modulation s introduce notes outside the established established key, composers can avoid the sound of chrom aticism if only only a single pitch in the diatonic scale is altered altered at a time based on its closely related keys. For exam ple, if the pitch is chosen based based on a clockwise or counterclockwise move around the circle of fifths, a new chromatic note will not sound foreign.

 

MELODIC TENSION AND HARMONIC DISSONANCE It is is necessary to first clarify clarify harmo nic dissona nce, melod ic tension, and a pitch center in pandiatonic music [Ex. 3]. Example 3 : Clarify Clarifying ing D issonance issonance and Tension, Tension, mm .   1  to 6.

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The pitch B in octaves in the first measure is a consonant sono rity. If no other pitch is introduced, it can only be identified identified as a pitch center that does not need to be resolved. The cluster of notes in the second measure is very dissonant. In the common practice period, the resolution of this dissonance could be any consonant sonority. The consonant sonority could be related to the pitches of this dissonant sonority through traditional harmonic function, pandiatonicism, or chromaticism, but the relationship is irrelevant compared to the need for the dissonance to be resolved to a relatively consonant sonority. Tonal m usic of the Western European tradition is goal oriented. The third third and fourthh m easures are an exam ple of melodic tension. Since C major is established, the fourt listener list ener feels melodic tension when the line reaches the pitch B. The listener wants to to hear the arrival of the goal, the tonic pitch C. A convincing progression in pandiatonicism is the combination of harmonic dissonance and melodic tension. In the fifth fifth measure, withholding the final melodic resolution of  C in the right hand creates tension while the placement of  C in the lower

 

11 voice creates a dissonant harmonic interval of a of  a major sev enth. In the last mea sure, the top voice reaches C w hile the lower voice moves dow n to the leading tone, crea creati ting ng a harmonic minor ninth, a dissonant interval that includes more space between the two voices. This additional space gives more clarity clarity to to each voice melodically. The final movement of Samuel Barber's  Capricorn Concerto  [Ex. 4] illustrates illustrates the combination of me lodic tension and harmonic disson ance. In m. 169, Barber returns to the opening motive with the trumpet accompanied by thick and full C major chords in the strings. These ch ords disappear in m. 172 and a flurry of activity ensues. From the interaction of contrapuntal voices, the melodic tension and harmonic dissonance is greatly increased from m m. 172 to 175. The listener listener desires a release and a return to to the strong C major tria ds. Barber releases this tension and dissonance in m. 176 to to finis finishh the movement and the concerto. To create this flurry of activity, Barber has taken multiple contrapuntal lines and displaced their resolu tions. On every downbeat between mm . 171 and and 175, there is is a leading tone tone clash ing with the tonic. In mm . 172 to 175, the leading tone played by the cellos is further accentuated by the preceding leap of  a major seventh from the tonic pitch. This leading tone clashe s with the the violins who have already resolved to C and E. To increase the amount of dissonance and activity in mm. 173 to 175, Barber likewise includes displaced resolutions of scale degree 4 to 3, F to to E. After two contrapuntal voices enter in m.   171, a third enters in m. 172 172 beginning on F. This voice does not clash w ith the other tw o at first, first, but in the second beat of m. 173, it collides collides

 

12 harmonically w ith the the violins who have extended their melodic figure. Barber adds furthe fur therr dissonance and tension in this this regard in m. 17 1744 with the flute and oboe. Example 4: Samuel Barber,  C apricorn Concerto, Third Movement, m m. 169 169 to to 178.

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13 This excerpt moves harmonically from consonance to dissonance back to consonance . It establishes a tonic key and hierarchy of pitches in the fi first rst few few m easures. There are elements of tertian harmony in mm. 172 to 174, but the counterpoint and texture focusing on the dissonant intervals of minor seconds, major sevenths, and the tritone between scale degree s 1, 3, 4, and 7 is the most significant significant element in this excerpt. These final measures are B arbe r's way of creating creating a fresh fresh and new sound in the tonic to to dominant to tonic progression by substituting the dominant with harmonic dissonance and melodic tension. These dissonant contrapuntal lines with displaced resolutions communicate the need for a final resolution. Individual voices do not need to be as independent as those heard in the Barber to createe pandiatonicism. The  Credo in creat  in Arvo Pa rt's Berlin Mass [Ex. 5] uses repeating, rhythmically displaced melodic lines to create harmonic dissonance and melodic tension. In the beginning of the movement, the tenor and bass begin two contrapuntal melodic lines.   In m. 5, the tenor and bass lines continue and the alt altoo and soprano enter in

imitation. The m ovem ent is entirely entirely in E major and the tenor and soprano sing only only the notes of the E major triad. Part changes the meter acco rding to the the numb er of syllables syllables in each word and further highlights the text of this movement with the strings who enter on the accent of every word. Even though the texture is often saturated with diatonic notes, the harmonic dissonance still fluctuates. Part uses a maximum of four four diatonic notes in ever everyy beat of the fir first st four measure s [Ex. 6]. He increases the the number in m. 5 to al alll seven pitches. pitches. As the strings drop out in m. 6, the dissonance from from all seven pitches sub sides but still still

 

14 lingers without a resolution. W hen the stri strings ngs re-enter on the fifth beat of m. 6, the ensemble aga in sounds all seven diatonic pitches. pitches. Part then resolves this dissonance dissonance w ith a more consonant sonority of only five five diatonic pitche s. The dissonance and tension created by many diatonic pitches decreases from mm. 7 to 10 when only two, three, and four notes are used at a time. Example 5: Arvo Part, Credo from B erliner Messe,  m m .  1 to 11.

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The gradual re lease of tension can be further further seen in the numbe r of diatonic notes based on the entrance of   t h e   strings (highlighted in Exam ple 6). In m. 5, the entrance entrance of the strings strings uses all seven diatonic pitches. In m. 6, this sonority resolves to five pitches. The string entrance in m. 7 uses four pitches of the diatonic scale that resolve to three pitche s. Four pitches of the diatonic scale in in m. 8 resolve in m. 10 with two followed followed by three pitches.

 

16 Example 6: Arvo Part,  Credo from B erliner Messe, reduction of m.  1  to 11 with the number of diatonic notes present in each sonority.

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3

While the number of pitches in each beat contributes to the fluctuation of dissonance diss onance and consonan ce, Britten's  Nocturne from the Serenade for Tenor, Horn, and

Strings  utilizes melodic tension to create an implication that must be tonally realized. In the the second cadenza  of Nocturne [Ex. 7], Britten begins in C major preceded by a traditionally functi functional onal domina nt seventh chord. As the cadenza continue s, the lowest pitch, C in the cello, reaffirms reaffirms the listen er's assu mption that the pitch F will be used. Britten however introduces an F# in the horn, a tritone away from the pitch center. The melodic tension of the pitch F# greatly adds to the climax created by the crescendo in the ensem ble. The tendency tones and hierarchy have suddenly shi shifte ftedd for for the listener. listener. If the missing scale degree were F natural natural in the key of  C major, this

 

17 tendency tone would resolve down and not create create a signif significant icant am ount of harmonic dissonance. Instead, tthis his F# suddenly supplies supplies the music with a trit tritone, one, harmonic dissonance between F# and G in the the strings, and and introduces an un expected and needed resolution of a of  a tendency tone upward. Barber's elements of counterpoint, Arvo Part's use of harmony and the number of diatonicc notes in a given beat, and B ritten's display of melodic tension all use diatoni pand iatonicism effectively. Throu gh displacing the resolution of mu ltiple contrapuntal voices and utilizing utilizing extended tertian tertian harmon y, a technique for pandiatonic music m ay be created.

 

Example 7: Benjamin Britten,  Nocturne  from Serenade,  second cadenza. cresc.

PP

-3

=*=^=

W

Blow,

ng

let us

hear

the pur -ple

PPP

re -

poco a poco cresc.

trem.

^

S

^

ply-ing:

3u-gle,

J]3rfrJ]jiir

/

blow; blow;

an -swer,

e-choes

> i - .   JJrfrJl

^

dim.

m l* -  

glens

IP

P 3

»-

^ J. JTj j . hi

^m

dy-ing,

^ffrirfji

3=fc

dy-ing,

^ dy-ing

* ittfifrj ittfifrjjiJjiJt//m.

J

poco  a poco dim.

i

 

A SYSTEM SYSTEM FOR CREATING PANDIATONIC PANDIATONIC MELODY AND HARMONY Pandiatonicism is difficult to perceive in an unaccompanied instrument or voice. Melodic tension a lone is insuffi insufficient cient and harmonic dissonance is best discerned with at least lea st two two voices. As C opland has shown in his variati variations ons on the Shaker Them e, a melodic line can be traditionally tonal or a basic folk tune and still be used in a pandiatonic m anner through additional contrapuntal lines or harm ony. In his choral composition, In the Beginning [Ex. 8], Copland stretches the limits of pandiatonicism with a pitch center in a single voice. Example 8: Aaron Copland,  In the Beginni Beginning, ng,  mm. 1 to 13. mp

3=3

Mezzo-Soprano

* In the be - gin - ning

d heav-en and the earth

and

dark - ness

d

And the

was

God

up - on

ea rth was

the

with -ou t

face

ere

at - ed

the

i^E

^f

and

the

deep

form

of

-

void

The opening solo of  In the Beginning establishes a hierarchy quickly through an emphasis on the leading tone and notes of the of  the tonic triad. triad. Mo ving melodically down to to express the void and formless earth, Copland again visits the leading tone in m. 7. This leading tone however is signifi significantly cantly fa farr removed from from the tonic tonic pitch. A hierarchy of pitches would not have been perceived without the emphasis of A, a stable scale degree

 

20 in D major in m. 9. Additional leading tones in mm. 10 and 11 resolve in mm. 12 and 13 with the tonic pitch. 9 The time necessary to establish and m aintain aintain a hierarchy of pitches to create create melodic tension in an unaccom panied pand iatonic iatonic comp osition osition will likely likely lea leadd to triadic triadic elements of traditional tonality or an over emphasis of the tonic or stable pitches of the key. Copland creates melodic tension in this this work, but tests tests the memory of the the audience on which pitch is the the tonic. Copland has sh own that writing writing a pandiatonic pandiatonic com position for an unaccompanied instrument or voice is possible, but composers have yet to explore this possibility fully. An unaccom panied instrument or voice will will likely likely need a wide range and the abilit abilityy to easily easi ly make leaps in order to avoid traditi traditional onal scale scale passages or skips am ong notes of a of  a chord built by thirds. Harm onic disson ance is also also easier to create and a hierarchy of pitches can be identifie identifiedd much more quickly with multiple voices. As the B arber exam ple illustrated, illustrated, one of the the easiest ways of writing pandiatonicism is through through multiple melodic li lines nes reaching their appropriate resolutions at dif differe ferent nt tim es. Lau ridsen also illustrates illustrates this this in in  As Birds C ome Nearer  [Ex. 1, pg. 5]. When the v oice sings the leading leading tone in m. 2, it clashes with the tonic pitch in the piano. Shifting Shift ing and adjus tments to identical lines lines can achieve this result. This is best realized realized when the tendency tone s of scale degrees 7 and 4 clash with their resoluti resolutions ons of of    1 and 3 sounding in a different voice.

9

 Copland briefly briefly uses the pitch G# in m. 11. The nature of this pitch in pandiatonicism pandiatonicism will be discussed in Chapter 4.

 

21 Example 9: Developing Contrapuntal Lines in Pandiatonicism. n.

Pi

B

sS

**m

^ip

^IFS

c O

\

J

  * d  U U

•rf^F—

E

svr? j

 

m G

Hg

9

m



«»

o

o-



SI

n  ruji

I JW

-«=•*

ii

«»

o

^

F

| 

r l .flpl

r r  r r H •

*

™

»

o

«-

iS

S i£

Example 9A illustrates a melodic line played in octaves with a final resolution of a compound third in the third third measure. A simple rhythmic displacement [Ex. 9B] will create crea te mom entary d issonance, but this dissonance is ineffe ineffecti ctive ve because of the the degree of the displacement, displacement, in this case case by the the second eighth of the second meas ure. Rhythmically Rhythmically

 

22 displacing the two two lines furthe furtherr [Ex. 9C] creates parallel thirds which m akes the dissonances vanish. Diatonically transposing one line a scale degree down creates dissonance and displaces the resolution [Ex. 9D] . The lower voice does not reach the the tonic note until until the the third mea sure. W hile the dissonance does fluctuate fluctuate between major and minor ninths, the parallel motion and constant presence of ninths decreases the dissonan ce's effectiveness. effectiveness. Augmentation [Ex. 9E] creates rhythmic independence, a useful tool in creating contrapuntal pandiatonic ism. Dissonance fluctuates from from a consonant perfect fourth fourth and minor sixth to a dissonant minor ninth that resolves to a consonant major third effectivel effect ively. y. How ever, the parallel motion by step step still still prohibits these two lines from from being independent. Including some skips in the lower voice [Ex. 9F] repeats repeats many pandiatonic aspects from from the other examp les. The lower voice is rhythm ically independent. Dissonance rises and falls more fluidly fluidly than in the other examp les. An initial initial consonant minor sixth leads to harmonies of three sixths followed by sevenths before arriving on the dissonant minor ninth. The resolution is displaced and the me lodic tension is increased increased further fur ther from from the tonic note in the the four fourth th beat of the the first measure. The low er voice adds to the melodic tension by reaching the tonic pitch before sounding the leading tone on the strongest beat of the second measure. The two contrapuntal lines gain further independence when using contrary motion [Ex. 9G ]. The resolution of the third third measure sounds more com plete with a falli falling ng bass, but the downward motion in the first measure brings stability to the rising soprano line.

 

23 The harmonic minor ninth in the second measure is also not as dissonant from an extra octave of space. Exam ple 9H maintains this harmo nic minor ninth and the motion upward in both lines. It obtains further further melodic closure w ith a falling falling bass. The triplet figure also creates rhythmic independence and hardens the consonant sixth on the fourth beat. After choosing a pitch source of diatonic notes, composers of pandiatonic music can derive their desired harmonic palette through upper tertian sonorities. The sound of traditional functional harmony is particularly difficult to avoid with diatonic seventh chords. The major-minor sev enth, for exam ple, was used widely in traditional Western E uropean M usic. Over use of this diatonic seventh seventh may result in music that is too traditional to be pandiaton ic. The major-minor seventh and half diminished seventh also use a large number of tendency tones but do not include any harmonic dissonances beyond the tritone and a major second. Diatonic seventh chords are often too stable to be the highest point of dissonance in a pand iatonic passa ge and not stable enough to be a resolution. The interval vectors of diatonic seventh chords ( for the major-major seventh, for the major-minor seventh, for the minor-minor seventh, and for the half diminished seve nth) show an abundance of major and minor thirds. These seventh chords however can be used to move gradually from a consonant sonority to a dissonance , or from from the highest point of dissonance to a resolution. In  Credo  [Ex. 5, pg. 14], Part uses a diatonic seventh chord on the second beat of m. 7 to gradually reduce the

number of diatonic pitches from mm. 6 to 10.

 

Example 10: Ninth Chords.

l l f 

f f fci

>):tfF f f f f f

# ^ f *«r r f r f P: y ri  •yrf

f r 

f f

*

8

f p^P

f f f

P f

Complete ninth, eleventh, and thirteenth chords add additional thirds and other 10

dissonances to these interval vectors,  but complete upper tertian chords can be counterpro ductive to melodic tension. Exam ple 10 illustrates ninth chords in G M ajor, ajor, first com plete, then with a missin g seventh, then with a missin g fift fifth. h. W hile the comp lete five voice voice ninth chords sound thick and dissonant, the melodic tonality tonality of each pitch could get lost within this thick texture. To aid the clarity clarity of contrapuntal v oices, pand iatonic harmon y can utilize utilize incomplete upper tertian tertian structures. This will also 10

 The interval vector for a complete thirteenth chord on C in C major, , shows an equal number of seconds and thirds.

 

25 emphasize harmonic dissonance. A missing seventh will will maintain the the ratio ratio of seconds to thirds in the interval vector ( for major-major ninth, for minor-major ninth, for for minor-minor ninth, and and fo forr diminished-minor ninth). A missing fifth will create diatonic clusters and increase dissonance with more seconds than thirds thir ds (the com bination o f two sets of thirds thirds a seventh apart will be fo forr two minor thirds, for for two m aj ajor or thirds, and fo forr a major third and a minor third). Even thoug h pand iatonic harmonies have been created, created, there is still still the presence presence of tertian harmony or, at minimum, the interval of  a thi  third rd as the basis of each harmony. To hide tertian elements, these sonorities can be written in an inversion and incorporate seconds or intervals greater than a third between the majority of notes. Example 11: Morton Lauridsen,   O Magnum Mysterium, Mysterium,  m m .  1  to 5.

Soprano

II  i  »

^ mag - num mys - te

ri - um

O

// Alto

iEFf

F=F

ri - um

mag - num mys - te

ri ri  um   - um

mag - num mys - te

Tenor

Bass

g p j^g

^m

O

//

O

a

mag - num mys - te

 

26 The inverted ninth chord with a missing seventh is a favorite sonority of Morton Lauridsen. The first inversion ninth chord, heard at the opening of his choral work  O

Magnum Mysteri Mysterium um   [Ex. 11 ], creates unique qualities qualities in this position. It imm imm ediately establishes a hierarchy of pitches. A major key is iinflecte nflected, d, but the voicing of this this inversion, in particular the minor sixth between bass and tenor, creates an unsettling quality. Likew ise, dissonance is intensif intensified ied by the add ition of the the ninth, not at a distance from from the other chord me mb ers, but directly next to to the tonic pitch of the first first sonority, dividing the tenors. Lauridsen uses the ninth chord again in m. 4 in inversion, without the seventh, to end the opening phras e. The ninth in the the alto resolves to add closure to the phrase, but aann unsettling second inversion maintains just enough tension for the listener to expect the piece to mov e forward. An additional mom entary dissonance in the the third third quarter of m. 1 occurs when the altos jump a fourth to the pitch G, the eleventh of this sonority in D major. Adding the eleventh to a chord will offer an additional variation of harmonic dissonance [Ex. 12]. When the eleventh alone is combined with the major, minor, and diminished triads, it will generate similar results in interval vectors and dissonance as adding the the ninth to these triads. The sonorities of the of the  second system of incomplete eleventh chords will offer a greater emphasis in seconds and fourths and less emphasis on thirds (interval vectors , , and ). Additional variations of four voice harmonies including the eleventh and quartal harmony will all result in a repetition of presented sonorities and interval vectors.

 

27 Example 12: Eleventh Chords. 0  H /f **

«I 4V

ft

«»

£   tt A . " , shows significantly more available available dissonances compared to the the major major scale. The descending form form of the the melodic m inor scale alone provides all harmonic sonorities of the major scale. Dissonance and tension is greatest when two contrapuntal lines of equal importance clash harmonically. An effect effective ive use of pandiatonicism pandiatonicism in melodic minor draws out dissonance and tension through cou nterpoint. In the the first first variation of Copland's  Piano Variations [Ex.  16], two rhythmically unrelated contrapuntal lines are written writt en in the low register of the the piano. Copland confirmed confirmed he is using using "a technique drawn from Schoenberg's Twelve Tone Technique" 11  and the mo tive of the the theme C-C#D#-E forms a [0,1,3,4] pitch class set. This pitch class class set can also be understood in C# minor (C would becom e the leading leading tone, perceived enharmo nically nically as B# ). For example, the end of the theme in m. 11 implies implies the closing closing of  a phrase and cadence in C# minor. This first variation is quite dissonant and incorporates displaced resolutions of this [0, [0,1,3, 1,3,4] 4] pitch class set in C# minor. There are also also brief momen ts of consonance and resolution resoluti on in mm. 13 and 15 when the lower voice sounds an E, creating creating a mom entary entary root and third of the C# min or triad. triad. These resolutions h owe ver are short lived lived as the lower voice mo ves to the the leading tone. There are elements elements of harmonic function function w hen these two voices soun d the root and third (spelled (spelled en harmonically as C) of the of  the dominant

Simms, Bryan.   Music of the the Twentieth Century: Style and Structur Structuree  (Belmont: Schirmer, 1996), 27 3.

 

32 triad in m. 17 before closing in C# minor w ith a dissonant dissonant harm ony of the tonic tonic and leading lead ing tone in m . 20.

12

Example 16: Aaron Copland,  Piano Variations, mm . 11 to 20.

s Piano

s pR

i>

^

sp IP

^

Vg

Hf

¥^

ffl

Pno.

iitU

'tlsLJ   ^

To take full advantage of  the dissonant harmony available in the melodic minor scale, lines can use both forms of scale degree 6 and 7. The au thor 's string quartet, Wait

by the Station   [Ex. 17], is written entirely in in A melod ic minor. The melody utilizes the descending form of the melodic minor scale before before skipping to the lleading eading tone. The harmonic dissonance is heightened through staggered entrances and displaced resolutions, combining both forms of  scale degree 7 in mm. 13 to 15 and both forms of scale degree 6 in m. 15.

Two added tone s, D natural and G natural, are also also present in this short excerpt. excerpt. The nature of these tones will be discussed in Chapter 4.

 

33 Example 17: James Woodward,   Wa it by the St Station, ation, m   m m. 10 to 19.

m

very little to no vibrato Vln.  Vln.   1

^

P

rr   rl»

very {little to no vibrato

Vln. II

^

p

m

Via.

very little to no vibrato

ES

«—^

mp

s

Vc.

pa

*t*

littlee to no vibral very littl

i)

i»

Jli pizz. (sul E)

ln. I

0

ia.

pll 6\

/

Vc.

BE

J Whl Whl  a

i=t

jfee

«¥ '

IQt

mm^m /

£ ^ E OT£>

##p

/ Vln. II

3EE£

0 -

^RP

PPP

SE PPP

An experim ent w ith the creati creation on of pandiatonic dissonance and tension through counterpoint consists of arranging existing existing B aroque com positions. For example, one could create an approp riate third third contrapuntal voice for a two pa rt invention that would amplif ampl ifyy dissonance and consonance. Rhythmically or melodically melodically altering a li line ne alre already ady present will also add to the m elodic tensi tension on of a of  a three or four voice fugue.

 

34 Example 18: An Arrangement of Bach 's Two Part Invention Invention in C minor, mm . 1100 to 13.

In the the auth or's arrangement of Bach 's Two P art Invention Invention in C minor [Ex. 18], the arrival of the consonant sound of the dominant key is preceded by an increase in dissonance in m. 12 through an added contrapuntal voice. The ascending form form of sscale cale degree 6 and 7 is is present in the soprano. To create this dissonance, the added alto voice takes the descending form of the seventh and sixth scale degrees providing greater dissonance and tension before the resolution of  G m inor. Modal scales, including the natural minor scale, are less ef effecti fective ve pitch sources of pandiatonicism . W hile these collections collections are applicable to pandia tonicism , their tendency tones and pitch pitch center w ill require a greater amount of consideration. If the excerpts tthat hat develop contrapuntal voices in pandiatonicism are used [Ex. 9, pg. 21] with a modal collection of pitches [Ex. 19], the pitch center becomes unclear and the resolution is often

 

35 incomplete. The absence of the lea leading ding tone in in Example 19A iinn C Mixolydian result resultss in the loss of the line 's ability to project the tonic pitch. Even thoug h the leading ttone one is stilll present in C Lyd ian in Exam ple 19B, the final resolution is unclear since the line stil could progress further to a sonority projecting G major. Exam ple 19: Contrapuntal Voices in Pandiatonicism Pandiatonicism using M odal Scal Scales. es. A

P0&

B

^m m

line •)^rr   r r

a D

#

^ B

k r *r r

The overall usefulness of a of  a  modal scale in pandiatonicism is dependent upon the placeme nt of the the half steps within the scale. M odes that include a pitch a half step aaway way fr from om the tonic pitch, including the Phrygian m ode, will be the most effecti effective. ve. Me lodic tension can be lost in mixolydian, dorian, and aeolian due to the absence of the leading tone or a half step relati relationship onship involving involving the tonic pitch. These m odes w ill oft often en require reinforcement of the pitch center as a culminating resolution. A com poser m ust assert a tonic pitch while manipulating the linear elements elements of a modal scale in pandiatonic w riting. riting. This can be done through through rhythm ic plac placement ement of the

 

36 tonic pitch or isolating the half steps in tthe he diatonic scale as a pitch from from a secondary dominant. Bela Bartok a ccom plishes both of these effects effects in the repeating pattern pattern of

Pentatonic Melody   [Ex. 20] . If the pentatonic melody in the left left hand is isolated, isolated, it will project a tonal tonal cen ter of E , but to maintain a pitch center and project the mo de of  C Lydian, Bartok reinforce s the tonic pitch in the le left ft hand on the dow nbeat o f every measure. Example 20: Bela Bartok,  Pentatonic Melody,  m m .  1  to 10.

Piano

¥

Moderate

^>—

f J1

*jiij

£=f

w^ w^ p s

^

mw

-,   p i-T

Pno.

^

^

i^-^i

Melodic tension and dissonant sonorities in a mode will often lead the listener back to the major key to the point that even the reinforcement of the pitch center may not suffice. Britten's  Nocturne  [Ex. 7, pg. 18] uses function functional al harm ony and a pedal to establish the key of  C Major. When the horn and tenor tenor reach the pitch F#, the hierarchy of pitches pitches changes. The resolution to G becomes the most necessary melodic realization realization instead of  a resolution to C . The pedal, however, co ntinues to assert C as a tonic pitch. From these these devices, either G Major Major or C Lydian can provide provide a resolution. resolution. Part's  Credo

 

37 [Ex. 5, pg. 16] uses collections of four to seven diatonic notes of the E major scale in mm . 6 to 8 to create create dissonance. The immediate two and three three note harmonic resolution of this this dissonance at tthe he end of the phrase in mm . 9 and 10 is best perceived as having a root of G# or C#. How ever, the root of these sonorities sonorities is not strong strong enough compared to the persistent persistent use of the diatonic notes of the E major scal scalee and the dissonances of m m. 6 to 8. The bass and tenor com plete their their contrapuntal contrapuntal melodic lines in m. 11 with a resolution resoluti on on the tonic and do minant pitch of E. The limited available dissonant harmonies in pandiat pandiatonicism onicism can be constrict constricting, ing, especially especial ly over a long p eriod of time, but the composer can increase the number of available avail able pitches and move the pitch center center through a modulation similar ttoo a traditional traditional pivot chord mod ulation. A pitch outside the given diatonic collection will be introduced, but the goal of a of  a modulation in a pandiat pandiatonic onic com positi position on is to make this pit pitch ch sound diatonicc and not chrom atic. diatoni

 

MODULATION Part's   Credo [Ex. 5, pg. 14] is beautiful, even heavenly, but after two minutes of only the diatonic diatonic notes of E of  E majo  major, r, many ears wil willl become taxed and grow tired of hearing only only seven pitches. The musicians singing singing soprano and tenor may likewise likewise grow tir tired ed of singing only the pitches of the the E major major triad. Wo uld a modulation incorpor incorporati ating ng a chromatic note interrupt the peaceful continuity of this movement? It can be very difficult difficult for a com poser to break aw ay from a single diatonic scale. The introduction of notes outside of the diatonic scale will sound chromatic, breaking the diatonic sound. These chrom atic pitches can oft often en soun d li like ke a  jarring arring event within a  a j soothing soothi ng texture, but there are ways a com poser can mod ulate in pandiatoni pandiatonicism cism by masking the entrance of a chromatic pitch therefore preventing an interruption in the diatonic texture. A pand iatonic comp osition osition in major uses seven different different pitches. To decrease tthe he perception of a of  a foreign, chromatic pitch, the key of this diatonic scale should modulate clockwise or counterclock wise around the circle of ffift ifths. hs. These mo ves af affec fectt o nly one of the seven diatonic notes. For example, the pi pitch tch source of C of  C maj  major or can become F major or G major by altering B to Bb or F to F# respectively. Pandiatonic writing can emphasize the half step step relationship between the leading tone to the tonic and scale degree 4 to its resolution of scale degree 3, but two melodically chroma tic steps in the same direction direction would be an obvious use of notes outside a diatonic scale. To further further decrease the chrom atic sound of of a  a modu lati lation, on, the pitch that will be altered should not be used for a brief  time. he  time.   After the mo dulation, tthe

 

39 altered pitch should be introduced in a different different v oice and introduced thro ugh octave displacement or a cross relation. The author 's  Prayer [Ex. 21] provides a good example. Example   2 1 :  James Woodward, Prayer, mm. 5 to 16. rit.

a tempo •   •>  V

'jJ   ¥ 

J

33 #RP

S pp

p

\P

f3=f

Vln. II

^

*

z*

V Via.

WW-

"fp"

i>??2>

V -J -  :

to>

_HP

pz= unis. H

J

r

i?p

n PP

PPP

g rY <

^

m

as

.B.

Vln.  Vln.  I

pm

8=^=3

r

^^F

#^ ?

^

^

P

dim.

P

dim.

pp\

V Vln. II

mp

Via

m s

S3:

mp

Vc.

mp

D.B.

s

m

-i?p

JP

dim.

^ i>

^

«

i?p

^=~pp

V

_2

mp

pp

 

40 The piece b egins in E major and a rrives in m. 6 on a dominant triad clashing w ith the resolution already present in the bass and cello. The piece continues by going counterclockwise through the circle of  firths, first to A major then D major. D# is heard heard in the viola in m. 7, but in m. 8 is lef leftt out for for three quarter note values and the entrance of the cello draws the attention of the listener. listener. A D natural in the second violin and bass completes the first first mov e to A major. The piece continues to move cou nterclockwise to D major in m. 8 after after the first violin moves beyond G# . After an absence of  this scale degree for a complete measure, G natural is played by the first violin in m. 10. Com posers can also use a brief tonicization to the left left or right of the initial key on the circle circle of fifth fifths. s. This does not need to be achieved through a secondary dom inant or traditional functional functional harm ony. Incorporating notes of the diatonic scale to the lef leftt or right of the the initial key on the circle of fift fifths hs will provide this brief tonicization. There are multiple examples of Copland utilizing these brief tonicizations.   In the

Beginning  [E x. 8, pg. 19] starts starts with the the diatonic pitches of D major. As stated above, the the tonic pitch becomes less clear in this unaccompanied solo before it arrives on the dominant pitch of A in m. 9. Copland then briefly briefly tonicizes A m ajor in mm. 10 and 11 with the the pitch G#. Through a jum p of an octave, Copland returns to D major in m. 12, satisfying the implications of the opening melody. The Copland   Piano Variations [Ex. 16, pg. 32] also includes brief ton icizations from the home key of  C minor to F# minor and and G# minor. The introduction of  G natural, or its enharmonic equivalent of  F double sharp in the left left hand of  m. 18, is a brief tonicization in the key of the dominant pitch, G# minor. In m. 15, Copland introduces the

 

41 pitch D, a note not in C# minor. The pitch F# iinn the right right hand of m. 16 suggests the possibility of briefly briefly using the pitches of F# m inor. While these brief departures from the initial key introduce chromatic notes, Copland uses octave displacement and cross relations to further distance these pitches from from sounding chro ma tic or foreign. In m. 18, the G natural in the lef leftt han d is displaced by an octave from its resolution as a leading leading tone to to G#. The D natural in m. 15 is a cross relation of the D# in the left left hand at the beginning of the measure . The D natural is almost three octaves away from the D# in the low register of the piano. If modulation in pandiatonic writing can incorporate chromatic pitches but linearly mask their entrance, a composer can separate two pandiatonic voices from one initial initi al key into into multiple ke ys. Barber achieves this effect effect in the third third movem ent of the the

Capricorn Concerto   [Ex. 22] . Barber es tablishes a repeating pattern in C major in the cello and bass in m. 151. The three solo instruments, flute, oboe, and trumpet, continue above this pattern with legato pandiaton ic so norities. After arriving on a G maj major or chord on the fourth beat of m. 154, the three solo instruments abandon the pi pitch tch B. While the flute flute and trumpe t do not

play outside the key of  C major, the oboe introduces the pitch Bb in m. 156 and continues in F major. The mu sic reverses this process in mm . 157 to 159, dropping the pitch Bb in the oboe and introducing B natural again. Barber is using pandiatonicism in two diff different erent k eys. All half steps fro from m A ttoo C are briefly present within the first beat of  m.  157. If the harmon y o f m. 156 156 is isolated, isolated, it

 

42 will sound chromatic, but from the careful and gradual introduction of   a  chromatic note, the linear melodic lines do not lose the sound of diatonicism. Exam ple 22: Samuel Barber, Capricorn Concerto, Third Movement, mm. 151 to 159.

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Modulation in pandiatonicism does not need to be a slowly developing process. Moving around the circle of fifths rapidly may risk creating a seemingly more chromatic texture, but the rising melodic tension and dissonant harmony that loosens the diatonic of chromatic nature of pitches is unique to to pandiatonicism . Elliot Car ter's  P iano Sonata [Ex. 23] provides a good example. Example  2 3 : Elliot Carter, Piano Sonata, Second Movem ent, mm. 11 to 22.

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The excerpt begins with the pitches of  F major. In m. 14, Carter abandons Bb for a moment and re-enters with B natural and the notes of  C  major in m. 15. He quickly moves around the circle of fifths, adding F# and C# at the same time in m. 16 while moving from C to G and D Major. To increase the speed speed at which he mo dulates, he again nearly skips a key by avoiding G# and immediately introducing D# in m. 17. A# and E#

 

44 are added in m. 18. B# is added in m. 20 which suggests the arrival of C of C major in m. 21 . Carter does not fully resolve this final arrival and instead reverses the process back through the same keys by quickly adding naturals followed by flats. No voice mov es through direct direct chromaticism chromaticism in tthis his examp le. In al alll of the modulations from F to F# major, time is taken to alter, avoid, and introduce each new scale degre e. Carter mo ves from from B natural ttoo B# only in the fi final nal mo dulation in m. 20. Carterr however introduces the B# by a cross relation. Carte relation. Having established established a systemati systematicc technique for writing pandiatonic harmony, a systematic systemat ic technique can now be explored for m odulating. The interval vector of the notes of the major scale () shows the tritone between s cale degree 4 and 7 as a unique interval for each major scale. From the symmetrical division of the octave, two notes that form a tritone can be found in only two major maj or keys. These two pitches are are the the only not notes es in comm comm on between these two keys. A third note will will therefore determine a key. For exam ple, the the trit tritone one betw een the pitches B and F can be found in the diatonic major scales of  C  major and F# major (enharmonically B and E#). These two pitches are the the only common tones between these ttwo wo diatonic diatonic itional onal note will determine determine the key. If C# is added to the pitches B and F, keys.   Any add iti keys. this sonority could be derived from the pitch source of  F major, not C major. The ninth and eleventh chords in four four v oices [Ex. 10, pg. 24, and E x. 12, pg. 27] that include a tritone app ear in only a single major key [Ex. 24 ]. All other chords are represented represe nted in multiple keys.

 

45 Example 24 : Ninth Chords and Eleventh Eleventh Chords wi with th Trit Tritones. ones. 0  ft.

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