SERAPH FOR PIANO AND STRING ORCHESTRA BRAD STARK. B. Mus., The University of Toronto, M. Mus., The University of Western Ontario, 2005

Transcription

1 SERAH OR IANO AND STRING ORCHESTRA y BRAD STARK B Mus, The University o Toronto, 200 M Mus, The University o estern Ontario, 200 A THESIS SUBMITTED IN ARTIAL ULILLMENT O THE REQUIREMENTS OR THE DEGREE O DOCTOR O MUSICAL ARTS in THE ACULTY O GRADUATE STUDIES (Music) THE UNIVERSITY O BRITISH COLUMBIA (Vancouver) April 201 Brad Stark, 201

2 Astract Seraph is a iteen minute composition or piano and string orchestra In addition to traditional musical techniques such as variation and passacaglia, the ork employs a distinctive approach to phrase structure, orm, harmony, and compositional technique An ascending natural minor scale serves as the main theme hile other melodies, textures, harmonies, and motives interact in counterpoint ith the theme In addition, the theme undergoes its on developmental transormations and modiications, and it inluences a distinctive harmonic language eaturing extended, non-traditional chords and progressions that rarely repeat Musical phrases and secondary melodies are unusual in length and mostly avoid expectations o cadence, hile metric instaility occurs through requent meter changes In addition to variation technique, the musical structure eatures episodes that contrast in compositional design through the development o less prominent motives and dierences in approach to harmony The piano part as partly composed through the use o transcried improvisation, hich serves as a asis or the harmonic and motivic structures heard throughout the composition The piano part also eatures complex rhythmic divisions and technical demands or the perormer hile interacting ith the orchestra in a variety o textures As a hole, the ork possesses several eatures hich contriute to an original style and aesthetic ii

3 Tale o Contents Astract ii Tale o Contents iii Acknoledgements iv Dedication v 1 Style and Aesthetics 1 Aesthetic Basis and Motivations or the Composition 1 Contemporary Inluences and Style 2 The Inluence o Concerto Design Transcried Improvisation and its Inluence on Compositional Design 2 Musical Structure and orm 21 hrase Grouping and Design 22 Variations, Episodes, and the Inluence o assacaglia 2 Variations: Examples and Compositional Technique 12 Development o the Ascending Scale-Motive 12 Development o Susidiary Motives 1 2 Episodes: Examples and Compositional Technique 1 2 Notated iano Improvisation 18 Harmony 19 1 Harmonic Language: Chords 19 2 Harmonic Language: Chord rogressions 21 Orchestration 2 Melodic Decoration 2 Exchanging Material 2 Exchanging Suordinate and Superior Roles 2 Conclusions 2 Biliography 29 Seraph or iano and String Orchestra: ull Score 0 iii

4 Acknoledgements I oer my enduring gratitude to the aculty o music, sta, and my amily or their support I oe particular thanks to Dr Stephen Chatman, hose guidance and continuous support has helped me to complete my thesis I ould also like to thank ohn Roeder and Dorothy Chang or their revie and valuale eedack on my thesis document I d also like to thank student advisor Rayne Graham or her invaluale support and guidance And inally, special thanks are oed to my amily and my ie ung Hee, or their support iv

5 Dedication Dedicated to my ie ung Hee, my daughter Kaelyn, and my son Lucian v

6 1 Style and Aesthetics Aesthetic Basis and Motivations or the Composition Although the title or Seraph came late in the compositional process, the idea o a heaven-like orld as a asis or inspiration and musical decision making rom the eginning Seraph is descried as a memer o the highest order o angels, 1 and there are several aesthetic eatures in the music that are inluenced y the concept o Seraph and a heaven ased orld The ascending scale-motive, shon in example 21-1, possesses an approaching heaven musical symolism It egins as a ass line and continues to gradually ascend oth in the theme itsel and later on in the composition Continuing to ascend in the variations, it reaches a high register in the violins in variation eight, hich can e seen in example 1-1 It is also symolic that the original orm o the scale-motive in Seraph does not contain distinctive rhythms typical o normal themes and motives This design as chosen ecause the use o a traditional theme or motivic design ould etray a don to earth symolism In comparison, many traditional themes and motives rom musical orks are oten associated ith characters, nature, and emotions, like the amous theme o Beethoven s th symphony, hich represents ate 2 The lack o rhythmic motives in the scale theme 1 The quoted deinition or Seraph is taken rom 2 Beethoven s secretary Anton Schindler rote: [Beethoven] himsel provided the key to these depths hen one day, in this author s presence, he pointed to the eginning o the irst movement and expressed in these ords the undamental idea o his ork: Thus ate knocks at the door! olly, Constance Beethoven as I Kne Him; London: aer and aer, 19; as translated rom Schindler s Biographie von Ludig van Beethoven, 180 1

7 o Seraph and reve durations characteristic o renaissance church music ere chosen to symolize a heaven ased orld The goal as to mostly avoid the typical treatment o motives and themes since this ould suggest a human-like symolism The concept or a variation design ased on the scale-motive as also chosen since multiple contrasting themes could counteract the symolism o a singular, heaven ased orld The use o a modal scale, as seen in renaissance church music, is also symolic, ut it also aects aesthetic decisions or harmony Chords are chosen to support the atmosphere and mood o the composition and many contain perect iths The use o this interval in harmony and motives in the ork is also symolic as the perect ith as an important musical interval in medieval church music Similarly, the approach to counterpoint in Seraph symolically reerences renaissance church music There ere also other musical design decisions hich support the symolism o a heaven orld, including melodies that are extended to unusual lengths, unpredictale meter changes, and loing textures that continuously evolve rom the scale-motive and avoid arupt textural changes Contemporary Inluences and Style Seraph demonstrates a distinctive aesthetic through its approach to harmony, intervallic relationships, phrase length, orm, and compositional technique hile the overall aesthetic design o Seraph is not intentionally derived rom or inspired y other contemporary compositions, there are identiiale inluences and comparale orks or example, the approach o uilding musical compositions through very simple 2

8 musical materials can e seen in the music o Arvo ärt Like the music o ärt, Seraph is ased on a simple scale-motive and the harmonic language is related to this material Hoever, Seraph diers rom ärt s early music through its complex, nonrepetitive harmonic style This harmonic variety in the composition is made coherent through the emphasis o certain harmonic intervals such as the perect ith and perect ourth as ell as the consistent use o pitches derived rom the scale-motive Another ork that shares similarities in compositional design ith Seraph is Conessional or Cello and Orchestra (199) y Christos Hatzis Both compositions eature the metamorphosis o intervallic content and harmony inluenced y motivic material As Hatzis descries, his ork develops the intervallic and melodic content o [a chant melody] its constant metamorphosis account or most o the melodic and harmonic material Hoever, Seraph diers in approach rom Conessional through its use o transcried improvisation and a harmonic language uilt on non-triadic harmonies Another contemporary concerto that has similarities to Seraph in approach to composition is L arre des songes, Concerto or Violin and Orchestra (198) y Henri Arvo ärt (19-) or example see compositions such as ratres or Violin and iano (1980), Spiegel im Spiegel (198), Cantus in Memory o Benamin Britten or strings and ells ( ), and other compositions Christos Hatzis (19-), Conessional or Solo Violoncello Orchestra (199), romethean Editions Christos Hatzis Conessional Solo Violoncello Orchestra, romethean Editions, accessed anuary 201

9 Dutilleux A rench composer inluenced y such composers as Ravel and Deussy, his music oten eatures impressionistic, non-tonal harmony Although the harmonic language o Seraph is not directly inluenced y the compositions o Dutilleux or other composers, the stylistic approach and conception o harmony has a similar aesthetic This can e seen in comparing the harmonic style o Seraph ith L arre des Songes; hile chord selection is governed y some principles and intervallic content, it is not a strict system as seen in some kinds o contemporary music As in the music o Dutilleux, Seraph s approach to harmony is partly intuitive rather than purely systemic In addition, as Allan Kozinn descries, Dutilleux s music is cloaked in a harmonic language that emraces dissonance ithout making it the point The harmonic style o Seraph can e descried in similar terms Another less ovious eature in common to Seraph is descried y Dutilleux, ho said o his violin concerto that he elt incapale o riting a ravura piece 8 Both L arre des songes and Seraph eature challenging solo parts, ut neither explicitly develops the soloist s part or virtuosic display at the expense o the orchestral accompaniment Henri Dutilleux (191-), L'Arre des Songes, Concerto or Violin and Orchestra or Orchestra, Schott Music (HL9009) Alan Kozinn, In a Retrospective Evening, a Celeration, Too, The Ne York Times, une 28, Quoted in, ohn Henken, L arre des songes, Concerto or Violin and Orchestra, accessed anuary, 201

10 The harmonic style and use o solo piano in rometheus (19) y Alexander Scriain can e identiied as another inluence on Seraph 9 Interestingly, Scriain called his ork a symphony rather than a concerto, even though it strongly eatures the piano Like Seraph, the piano solo o rometheus is not intended or virtuosic display as the relationship eteen piano and orchestra is largely one o partnership rather than o conlict The late harmonic style o Scriain is also an inluence through his non-triadic approach to harmony, although Seraph derives harmonies rom diatonic scales rather than synthetic scales The Inluence o Concerto Design Seraph can e descried as a concerto, ut it also possesses eatures hich are not typical o the traditional concerto orm Hoever, these eatures are also true o many contemporary orks, including the violin and cello concertos y Hatzis and Dutilleux already discussed or example, there is no cadenza section or passages o overtly virtuoso display in Seraph In act, the piano has no solo sections in the composition at all although it is very sparsely accompanied in the opening measures Nevertheless, there are also eatures o a more traditional concerto style; piano and orchestra alternate in dominant and susidiary roles in the music, as ell as equal partnership Another aesthetic dierence rom traditional concerto design in Seraph is that the music is rarely a competition eteen soloist and orchestra; the piano solo and orchestra ork together rather than vying or attention 9 Alexander Scriain ( ), oem o Ecstasy and rometheus: oem o ire, Dover ulications (A-2811)

11 Transcried Improvisation and its Inluence on Compositional Design Another signiicant and unusual aesthetic eature o Seraph is the use o transcried piano improvisation as a asis or the design o the composition The piano part heard in the opening ars o the composition ere recorded and transcried This transcried improvisation serves to inluence harmony, motives, and musical textures later in the composition Aesthetically, it means that the composition is partly a spontaneous creation that is also developed through traditional, non-improvised means

12 2 Musical Structure and orm 21 hrase Grouping and Design Seraph oten eatures melodies o extended length that avoid cadential ormulas, although some occasional reerences to traditional models o phrase design are present Example 21-1 The scale-motive, seen in example 21-1, oten lasts longer than 0 seconds The motive is oten developed ith continuous textures in the variations and counter melodies are oten o similar length in design Another notale technique in Seraph is the overlapping o sections and textures An example o this technique can e oserved in examples 21-2 and 21- Example 21-2

13 Example 21- The melody shon in example 21-2 is presented ith clear rhythmic and pitch motives resemling a classical period structure ith antecedent and consequent Hoever, the sense o cadential arrival on D in measure 2 overlaps ith a ne texture in the piano part, as shon in example 21- hile rhythmic cadential eects are oten avoided in Seraph, examples o more typical phrase structures are occasionally ound Consider the opening chords o the piano part, hich rhythmically suggests cadence It develops as a period-like structure containing antecedent (measures 1-) and consequent (measures -8), shon in example 21- Example 21-8

14 The antecedent and consequent phrases are closed y a cadential ormula involving durational accent and pitch Hoever, the harmonic language o this phrase is consistent ith the style o Seraph or example, the tonic and dominant relationship o the classical antecedent and consequent are not strictly olloed as the antecedent ends on the sudominant scale degree instead o the dominant Hoever, a suggestion o the classical style period is present in the inal chord ith the tonic scale degree ass line hich gives the phrase a sense o harmonic arrival and closure It should e noted that the inal chord o the consequent phrase does not contain a third (ie or natural), hich is a departure rom traditional consequent phrase harmony Aside rom these approaches to phrase structure, Seraph employs extended melodic grouping structures hich last or entire variations An example o this is seen in the violin I line rom variation nine, shon in example 21- Example 21- Here, the Violin I part decorates the scale-motive in descending motion The rhythmic design suggests continuous orard motion, although there is some repetition and development in this extended melody or example, the irst three ars are repeated in 9

15 measure 20 In measure 208, the melody egins ith a similar contour, ut in measure 211, develops ith repetitions o the rhythmic igure present in measure 20 In addition to the continuous orard motion o the melody present in this variation, the ass line and accompaniment also suggest continuous orard motion through the similar use o continuous rhythm 22 Variations, Episodes, and the Inluence o assacaglia The large scale orm o Seraph consists o a series o sections hich are either variations or episodes Variations strongly eature, modiy, or develop the main scalemotive hile episodes develop less prominent material such as countermelodies and unction to provide contrast ith the variations hereas traditional variation orms oten call or sections o proportionate length and the metamorphosis o individual parameters such as melody, harmony, phrase structure, rhythm, etc, the variations o Seraph continuously introduce ne material in counterpoint to the original scale-motive In other ords, the scale-motive is oten treated as ackground material rather than as a traditional melody or theme This style o variation is typically associated ith passacaglia, although there are also some dierences in Seraph The scale-motive is initially developed like a passacaglia: it appears as a ass line that serves as a asis or developing ne musical materials Unlike the passacaglia, the idea itsel is varied in dierent ays including length, rhythm, pitch, and register, as the theme is not alays present as a ass line olloing the initial variations The passacaglia design is most evident in the opening and irst variations As ould e expected in this style o composition, the scale-motive hen it

16 appears in the ass line undergoes only minor alternations Hoever, most o the composition does not eature the scale-motive as a ass line, indeed it does not return to this arrangement until the inal variations Another signiicant ormal eature o Seraph is that the variations and episodes are not alays equal in duration as the scale-motive itsel undergoes extensions and modiications Example Duration o Variations in Seconds hile there are small tempo changes in the composition, Seraph consists o a series o variations and episodes that range rom 2 to 0 seconds 11

17 2 Variations: Examples and Compositional Technique Development o the Ascending Scale-Motive The scale-motive is developed using dierent techniques including melodic decoration, rhythmic decoration, extension, ragmentation, inversion, as ell as contrapuntal devices The scale-motive undergoes rhythmic and melodic decoration throughout the composition In the theme section and irst three variations, the ass line carries the theme and undergoes only small rhythmic alterations ith each statement In variation our, a ragmented version o the theme is signiicantly decorated and the register is transerred up to the Violin I section as seen in example 2-1 Example 2-1 In this variation lasting 1 measures, the scale-motive has een reduced to only the irst ive notes (D-E--G-A), marked ith oxes This underlying pitch skeleton is decorated ith ne melodic and rhythmic motives as ell as requent meter changes The irst three notes o the theme (D-E-) are audily present due to their metrical position and pitch emphasis throughout the irst three ars, although the use o mixed meter generates rhythmic unpredictaility Starting in measure 8, a descending version o motive is also present (A-G--E-D) and is marked ith dotted slurs The inal D is not shon in the example 12

18 Example 2-2 In variation ive, shon in example 2-2, the theme is extensively decorated in the violin II part ith elaorate melodic and rhythmic decorations The scale-motive is indicated in the example ith dotted slurs In addition, the theme is modiied y extension and inversion eginning in ar hen C is reached Instead o continuing to the high D as expected, the line reverses direction and descends as shon ith dotted slurs until arriving at D at measure 0 olloing this, the decorated theme continues to descend rom the high D Due to the meter changes, rhythmic design, thematic extension, and decoration, the theme is strongly disguised ithin a contrapuntal texture in this variation (ars 8-0) Example 2- In variation nine, the violins decorate an inverted version o the scale-motive ith added rhythms and intervals, shon ith oxes in example 2- Later in the variation, the theme is urther extended and altered 1

19 Development o Susidiary Motives In addition to the scale-motive, characteristic intervals and motives are presented and developed throughout the composition In the theme and opening variations or example, the intervals o the perect ith and perect ourth are requently used to generate motives and pianistic textures The chords o the right hand rom the piano part consistently eature the interval o iths and ourths in the opening o the composition as seen in example 21- Another example o material ased on perect iths and perect ourths is shon in example 2-, here the right hand o the piano part similarly develops these intervals Example 2- In the olloing variations, these intervals are similarly used to invent textures or the piano part Example 2-1

20 In variation six, the perect iths and perect ourths are used to create a rhythmic texture in the piano part, shon in example 2-, that is developed throughout this variation In variation nine, the piano part also utilizes the perect ith and ourth intervals to develop a ast moving, roken chord texture A clear example o this can e seen miday through the variation in oth hands o the piano part in ar 21 shon in example 2- Example 2- In addition to the perect iths and ourths in this texture, a melodically decorated version o the scale-motive in contrary motion is oserved in the let hand ith alternating hal-note and quarter-notes 1

21 2 Episodes: Examples and Compositional Technique Example 2-1 Seraph also contains sections called episodes hich contrast rom variations through the development o secondary motives and approach to harmony Example 2-1 shos episode one, in hich some o these compositional approaches are evident Material rom the opening ars o the piano part (see example 21-) appears in the string orchestra, the irst restatement o material other than the scale-motive in the composition In addition, this episode contrasts in harmony ith the variations through its destailization o tonal center as it shits rom a B Lydian scale to an natural minor scale hile the scale-motive is suggested, the compositional approach oserved in this episode contrasts ith the variation sections 1

22 Example 2-2 Example 2- Episode three takes to ars o transcried improvisation rom the piano part o variation three (example 2-2), and develops it into an entire section o the composition (example 2-) In addition, episode three modulates and unctions as a ridge to variation six in the scale o E natural minor The asence o the ascending scalemotive in this episode results in a contrasting section 1

23 2 Notated iano Improvisation As has een riely discussed, the piano part rom ars 1 through ere improvised and later transcried to orm the theme and opening three variations In improvising, the intent as to use the ascending scale-motive as a rameork or developing harmony, motives, and textures In addition, the improvisation as planned to increase in rhythmic and textural complexity as it unolded The strings parts, composed ater the piano part as transcried, complement the harmonic style and textures o the improvisation The strings unction to accompany the ritten-out piano improvisation in these opening variations through increasing counterpoint and harmonic density as the music progresses Only minor changes ere made to the transcried piano part in an attempt to preserve a sense o ree improvisation Hoever, other sections o the ork ere initially sketched out irst, ut underent signiicant revisions throughout the composing process in response to the improvisation 18

24 Harmony 1 Harmonic Language: Chords The theme o Seraph is an ascending D natural minor scale, hich is also used to develop the harmonic language o the composition Apollon Musagète y Stravinsky is a composition hich similarly uses a diatonic scale as a asis or unconventional harmony Importantly, there are some principles ehind chord selection in Seraph; many o the harmonies are constructed ith ourths and iths instead o thirds and extended ith ths, 9ths, etc Some examples o these harmonies, all uilt over a D ass note, are shon in example 1-1 Example 1-1 The eleven harmonies shon are characterized y their variety, as only the Dsus and Dm chords are reused The chords also sho consistency in intervallic content as ell Igor Stravinsky, Apollon Musagète or String Orchestra, Boosey Hakes, 19 19

25 as pitch collection or instance, six o these eleven D chords omit, the chordal third that ould appear i this music ere traditionally tonal Another intervallic consistency is that many o these chords use the perect ith and ourth intervals aove the ass Nine chords contain the perect ith interval (A) aove the ass, hile six chords contain the perect ourth In addition, many other intervals are commonly used such as 9ths and ths An analysis o the structures o chords over ass E in the composition similarly demonstrate the harmonic variety and intervallic construction o chords ound in Seraph as shon in in example 1-2 Example 1-2 Unlike the chords over D, some o the chords over E contain notes that are oreign to the D natural minor mode or example, measure 0 contains an E maor seventh chord, although it is used in a modulation passage rom episode 1 Hoever, some o the E pitches have een altered to E in a e o the D minor variations, resulting in modal mixture As ith the chords over D, chords over the ass note E are similarly diverse; eight o these nine chords are dierent In terms o intervals, the perect ith is normally unavailale since the mode hich possesses a B, except here E is the 20

26 ass note urthermore, the ourth against the ass is contained ithin our o these nine chords The use o the third is ar more common in these chords ith seven out o nine occurrences Like the chords harmonized over D, these chords demonstrate the requent use o added intervals such as ths and 9ths A similar analysis o other harmonisations in Seraph also demonstrates a diversity o harmonisations and intervallic similarity o chord constructions 2 Harmonic Language: Chord rogressions hile traditional tonal systems rely on hierarchy here certain scale degrees and harmonies have more importance and unction (eg tonic and dominant), Seraph treats scale degrees largely as equal in eight and importance This is partly ecause the scale-motive itsel largely treats each scale degree to e o equal importance Although many sections in the music are homophonic, large sections o the composition are polyphonic in texture or example, the opening ars o the piano improvisation demonstrate a homophonic texture in the composition Example

27 The harmony rom the opening measures is shon in example 2-1 in a simpliied arrangement that demonstrates the voice leading and common tones implied in the progressions The irst three chords share all tones in common except the ass; D chords ith the third replaced y a second In the ourth chord over G, some ne pitches are introduced resulting in a G minor ninth chord, ut some notes are also common rom the previous chords As this is a simpliication o the progressions, these common tones are only implied, ut it is noteorthy that D and A are common to the maority o these chords Another oservation is that D is only asent over ass C, hile A is present in every chord in this passage Example 2-2 In variation three, the progressions are more triadic in construction in comparison to the opening chords, as shon in example 2-2, ut the use o common tones over a changing ass line is also oserved in the irst three measures o this passage In variation telve, the harmonies eature complex constructions urther demonstrating the distinctive style o Seraph 22

28 Example 2- Many o these chords, shon in example 2-, contain ive or six individual pitch classes, creating a thick harmonic texture in this passage that can e considered to e the climactic section o the composition 2

29 Orchestration The piano and orchestra relate to each other in dierent textures and arrangements throughout the composition including melodic decoration, exchanging material, and exchanging suordinate and superior roles Melodic Decoration An example o a melodic decoration is seen in variation six hen the piano part increasingly elaorates the violin I melody as shon in example 1-1 Example 1-1 The right hand o the piano part increasingly adds rhythmic and melodic decorations to the violin I line, hich ecome increasingly complex as the variation unolds 2

30 Exchanging Material An example o exchanging material eteen the piano and strings can e seen in examples already discussed or example, the opening piano music as seen in example 21- is later reinterpreted y the strings in episode one, shon in example 2-1 Exchanging Suordinate and Superior Roles The piano and strings also exchange suordinate and superior roles throughout the composition The piano part is most dominant in its role in the opening and ending variations o the composition as seen in examples 21-, 21-, 2-, and elsehere in the composition Example 1-2 2

31 In contrast, the strings have the superior role in variation eight here the piano is silent and in variation ten, here the piano accompanies the Violin I melody, shon in example 1-2 2

32 Conclusions Seraph s overall tonal plan, orm, and treatment o the scale-motive is outlined in example 1-1 The theme, thirteen variations and ive episodes explore contrasting textures and compositional techniques as discussed As also shon in the example, the scale-motive undergoes signiicant metamorphosis throughout the composition The introduction o E /D as modal mixture as seen in episode one and variation ive results in shits to scales here this pitch is later emphasized in episode to, three, and variation six In addition, episode our strongly empahsizes D to the exclusion o D or most o the section, utilizing the pitch collection A-B-C-D -E--G Episode ive and the inal three variations serve to revisit earlier variations ith ne arrangements and ring the composition to a close 2

33 Example

34 Biliography Dutilleux, Henri L'Arre des Songes or Violin and Orchestra Mainz: Shott Music, 198 Hatzis, Christos Conessional or Solo Violoncello Orchestra romethean Editions, 199 ärt, Arvo Cantus in Memory o Benamin Britten or Strings and Bells ien: Universal Edition, 1981 ratres ür Violin und Clavier ien: Universal Edition, 1980 Spiegel im Spiegel or Violin and iano ien: Universal Edition, 198 Scriain, Alexander oem o Ecstasy and rometheus: oem o ire Ne York: Dover, 199 Stravinsky, Igor Apollon Musagète or String Orchestra Boosey Hakes, 19 29

35 B Violin I Violin II Viola Cello Doule Bass iano q = B I 9 Seraph or iano and String Orchestra Brad Stark ( ) Brad Stark Score 0

36 I 1 B I 20 r r ruato B I 2 B r LH RH 1

37 28 LH RH cresc cresc I B cresc cresc cresc cresc I 1 B R Div Div 9 R R I B p p p p p p p Non Div p Non Div p Div p p p 2

38 9 Unison 9 9 n n Div I B Div I B I M n n n n n n n n n n n n B n n n n n n n n n

39 I 9 9 Unis Unis Unis B I 8 B I B Divisi Unison Non Div

40 I 8 B I B Div Div Unis Unis I B Div Unison Non Div

41 8 9 p I 9 p 9 p B 9 p 9 p 9 pizz p I 9 B I B

42 I Non Div 12 8 B pizz I B I 9 9 B

43 I 9 B 9 9 I 8 B pizz arco I 8 B r r Ÿ~~~~~~~~~~~ Ÿ~~~~~~~~~~~ Ÿ~~~~~~~~~~~ 8

44 I 9 B Ÿ~~~~~~~~~~~~~~~ Ÿ~~~~~~~~~~~~~~~ Ÿ~~~~~~~~~~~~~~~ ruato Ÿ~~~~~~~~~~~~~~~ Ÿ~~~~~~~~~~~~~~~ 1 Ÿ~~~~~~~~~~~~~~~ 1 1 I 9 Ÿ~~~~~~~~~~~~ Ÿ~~~~~~~~~~~~ Ÿ~~~~~~~~~~~~ B q = I 8 B

45 I 8 B I B I 8 - B

46 I I g B B I () 121 B

47 I 12 B I g 9 9 B I g 9 9 n n n n n n n n n n n n n n n n n n n n n n n n B n n n n n n n n n n n n n n n n n n 2

48 I q = 12 1 > > > > > > B > > > > > > > > > > > > > > > > > > > > > > > > > > > > > 8 > > > I 8 > > > > > > 8 > > > B 8 8 > > > > > > 8 8 > > > > > I > > 8 > > > B > > 8 8 > > 8

49 I B > > > > > > > > > I 8 > > > 9 > > > > 8 B 8 8 > > 9 9 > > I 8 > > > 9 > > > B 8 8 > > 9 9 > > 8 9

50 19 > > > 8 > > > 9 > > I B > > > > > > > > > > > > > > > > 9 > I 8 B 8 > 9 9 > > > I () B n n n π n n n n n n π n n n π n n n π n n n π n n n π

51 18 I B 19 () I B pizz π 191 () I B arco pizz π arco

52 I 201 B 20 q = p p p p p p I B 212 () I B

53 21 () I B 221 I B 22 n n n I n B n n n 8

54 q = I B 2 I B I 2 B n n

55 22 8 I I 8 > > > > > > 8 Div 8 B Div > > > 8 B I B > > > > > > 9 9 0

56 21 > > > I B I 2 B > > > n n I B Non Div n Non Div n n n n 1

57 I Div B Non Div Div I Non Div Div B 2 M Unis I Unis B Unis 2

58 I I () 2 B R 8 > U U > 8 B Div Unis U U U U Div Div I q = 28 ƒ ƒ Div ƒ Div B ƒ Div ƒ ƒ n n

59 B I 29 Unison B I 299 > > > > > > > pesante rit U 1 U U U U U U