THROUGH NATURE TO ETERNITY: A WORK FOR WIND ENSEMBLE AND A QUANTITATIVE STUDY OF CHROMATICISM: CHANGES OBSERVED IN HISTORICAL ERAS AND INDIVIDUAL COMPOSERS DISSERTATION Presented in Partial Fulillment o the Requirements or the Degree Doctor o Musical Arts in the Graduate School o The Ohio State University By Daniel E. H. Perttu, B.A., M.M., M.A. ***** The Ohio State University 2007 Dissertation Committee: Proessor Thomas Wells, Co-Adviser Proessor Donald Harris, Co-Adviser Proessor David Huron Proessor Russel Mikkelson Aroved by Advisers Graduate Program in Music
ABSTRACT This dissertation consists o two coonents: an original coosition entitled Through Nature to Eternity, and a music-theoretical investigation entitled A Quantitative Study o Chromaticism: Changes Observed in Historical Eras and Individual Coosers. The remarks that ollow seciically concern the latter ortion. Music historians have observed inormally that Western music became increasingly chromatic between roughly 100 and 1900. This view is tested ormally, and the results are shown to be consistent with the standard view. Music historians have similarly assumed that the music o major coosers such as Mozart and Beethoven became increasingly chromatic over their resective lietimes. Measurements o chromaticism in both theme-based and ous-based sales are shown to be inconsistent with these intuitions. At ace value, the results o this study airm that Western art music has become more chromatic over time, but that ive major coosers' use o chromatic tones changed little (quantitatively) over the course o their careers. Reercussions or stylistic interretations o these coosers are discussed. ii
Dedicated to Melinda Craword iii
ACKNOWLEDGMENTS I am grateul to my advisers, Proessors Thomas Wells and Donald Harris, or their suort and guidance o my work in coosition. I areciate their encouragement and enthusiasm. The research in the second ortion o this dissertation was carried out using the acilities at the Cognitive and Systematic Musicology Laboratory at The Ohio State University. My thanks to Proessor David Huron or his advice and assistance. iv
VITA July 2, 1979....................... Born Parma, Ohio 2001............................. B.A. Music and English, Williams College. 200............................. M.M. Orchestral Conducting and Bassoon Perormance, Kent State University. 2004............................. M.A. Music Coosition, Kent State University. 2004 resent...................... Graduate Teaching Associate, The Ohio State University PUBLICATIONS 1. Perttu, Daniel. A Quantitative Study o Chromaticism: Changes Observed in Historical Eras and Individual Coosers. Eirical Musicology Review 2, no. 2 (Aril, 2007): 47-4. 2. Perttu, Daniel and Fetters, Elizabeth. Recommended Bassoon Solos. The Instrumentalist 8, no. 11 (June, 2004): -40.. Perttu, Daniel and Fetters, Elizabeth. An Annotated Bibliograhy o Pedagogical Reertoire and Literature or the Bassoon. Double Reed 27, no. 1 (Sring, 2004). FIELDS OF STUDY Major Field: Music Areas o Ehasis: Coosition and Theory Other Fields: Orchestral Conducting, Bassoon Perormance, and English Literature v
TABLE OF CONTENTS Page Abstract... ii Dedication... iii Acknowledgments...iv Vita... v List o Tables... vii List o Figures... viii SECTION 1: Through Nature to Eternity: A Work or Wind Ensemble... 1 Program Note... 2 Instrumentation... Through Nature to Eternity... 4 SECTION 2: A Quantitative Study o Chromaticism: Changes Observed in Historical Eras and Individual Coosers... 41 Chaters: 1. Introduction... 42 2. Method... 44 2.1 What is chromaticism?... 4 2.2 Musical sale... 4. Results... 48 4. Changes in chromaticism in individual coosers works.... Chromaticism in Mozart s oeuvre... 8.1 Results... 0. Discussion... 1 Notes... Reerences... 4 vi
LIST OF TABLES Table Page 1 Correlations between Date o Coosition and Percent Scale Tones or Themes by Five Coosers... 4 vii
LIST OF FIGURES Figure Page 1 An illustration o the ercent scale tones resent in themes rom the our historical eras that corise the common ractice eriod in Western art music...0 2 An illustration o the ercent scale tones resent in themes in time units artitioned by bidecade... 1 An illustration o the mean ercentage o scale tones or themes by Beethoven, Haydn, and Mozart... viii
SECTION 1 THROUGH NATURE TO ETERNITY: A WORK FOR WIND ENSEMBLE 1
PROGRAM NOTE All that lives must die, Passing through nature to eternity. Shakeseare, Hamlet, Act I, Scene ii This work reresents my ersonal engagement with the reality o lie s transience; it deicts our struggles within the natural world against the dread o ceasing to exist, but closes with a vision o eternity. 2
Piccolo 1 st Flutes 2 nd Flutes 2 Oboes English Horn Clarinet in Eb 1 st Clarinets in Bb 2 nd Clarinets in Bb rd Clarinets in Bb Bass Clarinet in Bb Contra-alto Clarinet in Eb 2 Alto Saxohones Tenor Saxohone Baritone Saxohone Bassoon 1 Bassoon 2 (doubles on Contrabassoon) 4 Horns in F Truets in C 4 Tenor Trombones Bass Trombone 2 Euhoniums 2 Tubas Tiani Percussion 1: Marimba, Brake Drum, Ratchet INSTRUMENTATION ratchet brake drum Percussion 2: Percussion : Percussion 4: Percussion : Vibrahone, Xylohone, Water Gong, Lion s Roar in out lion's roar water gong Crotales, Triangle, Crash Cymbal, Glockensiel Chimes, Tam-tam (medium) Bass Drum, Susended Cymbal triangle crash cymbal susended cymbal bass drum Har Contrabass Score in Concert Pitch: Piccolo sounds one octave higher; bass clarinet, contra-alto clarinet, contrabassoon, and contrabass sound one octave lower.
Score in Concert Pitch Contrabass Clarinet in Eb Maestoso (q = ) m Bassoon 1 m Contrabassoon m iù Horns 1 & 2 in F iù Horns & 4 in F Truets 1 & 2 in Bb m 1. harmon mute stem in m Truets & 4 in Bb 1. harmon mute stem in Truet & in Bb 1. harmon mute stem in metal straight mute m Tenor Trombone 1 & 2 metal straight mute m Tenor Trombone & 4 metal straight mute metal straight mute m m Bass Trombone metal straight mute m Euhonium 1 Tubas 1 & 2 1. a2. m 1. Tiani (E, Ab, C, F) medium sticks s Percussion 1 marimba medium-sot mallets m Percussion crotales Percussion 4 chimes Percussion bass drum Har Contrabass D C Bb Eb Fb Gb Ab 4 q m m
Cl. 1 Cl. 2 Cl. B. Cl. Cb. Cl. A. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 Hn. & 4 Tt. 1 & 2 Tt. & 4 Tt. & Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. 1 Perc. Perc. 4 Perc. H. Cb. A 12 m take bow bowed marcato marcato iù iù oen m harmon mute stem in harmon mute stem in oen oen oen oen oen a2. marcato q m D# C# B E F G A m m iù h iù m m harmon mute stem in sus. cym. D C (chimes) m Db C Bb Eb F G Ab
Cl. 1 Cl. 2 18 m Cl. B. Cl. Cb. Cl. A. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 Hn. & 4 Tt. 1 & 2 Tt. & 4 Tt. & Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. m m m m m m m m a2. m m m m m m m m m m m m m m legato m legato m m m m 1. Ti. Perc. 1 m change F to G Perc. (bowed crot.) to triangle triangle Perc. 4 H. Cb. m Fb Gb h
Picc. Fl. 1 Fl. 2 Ob. 1 Ob. 2 Eng. Hn. Eb Cl. Cl. 1 Cl. 2 Cl. B. Cl. Cb. Cl. A. Sax. Bsn. 1 Bn 2/Cbsn. Tt. 1 & 2 Tt. & 4 Tt. & Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. 1 Perc. Perc. H. Cb. B Allegro barbaro (q = 120) 2 7 a2. brake drum a2. m m m m m m m m m m m take bassoon soli oen oen oen oen hard mallets iù iù iù iù to crash cym. (bass dr.) sz e m (tgl.) 7
Picc. Fl. 1 Fl. 2 Ob. 1 Ob. 2 Eng. Hn. Eb Cl. Cl. 1 Cl. 2 Cl. B. Cl. A. Sax. T. Sax. B. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 Hn. & 4 Tt. 1 & 2 Tbn. 1 & 2 Tbn. & 4 0 m m oco a oco m oco a oco oco a oco m m m oco a oco m m oco a oco m m oco a oco m oco a oco m oco a oco m oco a oco C oco a oco a2. m a2. m m oco a oco m m m m m bassoon m soli soli m iù iù B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. 1 m change Ab to G m m m m m Perc. 2 lion roar Perc. 4 tam-tam wooden stick Perc. (bass dr.) sz 8
Picc. Fl. 1 Fl. 2 Ob. 1 Ob. 2 Eng. Hn. Cl. 1 Cl. 2 Cl. B. Cl. A. Sax. T. Sax. B. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 Hn. & 4 Tt. 1 & 2 Tt. & 4 Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 7 9: iù 9: 9: iù 9: iù iù iù iù iù iù iù m m iù m m iù 9: 9: 9: 9: 9: iù iù 9: 9: 9: 9: 9: 9: 9: 9: 9: m Euh. 2 Tba. Ti. Perc. 1 Perc. 2 Perc. iù m br. dr. lion roar m Perc. 4 H. to chimes Db C Bb Eb Fb G A 9
Picc. Fl. 1 Fl. 2 Ob. 1 Ob. 2 Eng. Hn. Eb Cl. Cl. 1 Cl. 2 Cl. B. Cl. Cb. Cl. A. Sax. T. Sax. B. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 Hn. & 4 Tt. 1 & 2 Tt. & 4 Tt. & Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. 1 Perc. Perc. 4 Perc. H. Cb. D 4 E m m m m m b m m, sub. m m m, sub. m m m, sub. m m b, sub. m m m iù marcato marcato iù iù a2., subito m marcato iù marcato m m m iù m, subito m m m m iù m, subito m m, subito m m m iù m m m iù m, subito m crash cym. chimes (bass dr.) sz leggiero m leggiero leggiero leggiero iù iù m marcato marcato iù m m m marcato iù iù change G to F change C to C# m m brake drum 10
Picc. Fl. 1 Fl. 2 Ob. 1 Ob. 2 Eng. Hn. Eb Cl. Cl. 1 Cl. 2 Cl. A. Sax. Hn. 1 & 2 4 iù m iù m iù m iù m iù m m iù iù m iù m iù m iù m iù m oco a oco iù macato e oco a oco iù macato e Hn. & 4 Tt. 1 & 2 Tt. & 4 Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 iù m iù m iù oco a oco iù macato e iù oco a oco iù macato e oco a oco iù macato e oco a oco iù macato e oco a oco iù macato e m oco a oco iù macato e m oco a oco iù macato e m m oco a oco iù macato e oco a oco iù macato e m Euh. 2 Tba. Ti. Perc. 1 iù m oco a oco iù macato e oco a oco iù macato e Perc. 2 Perc. Perc. triangle xylohone gliss. (chromatic) sz gliss. (chromatic) gliss. (chromatic) (chrom.) 11
Picc. Fl. 1 Fl. 2 Ob. 1 Ob. 2 Eng. Hn. Eb Cl. Cl. 1 Cl. 2 Cl. B. Cl. Cb. Cl. A. Sax. T. Sax. B. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 Hn. & 4 Tt. 1 & 2 Tt. & 4 Tt. & Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. 1 Perc. 2 Perc. Perc. 4 Perc. F q = 120 1 q = 0 bells u!! bells u!! bells u!! bells u!! a2. iù ratchet to erc. 4's tam-tam (tam-tam) (chrom.) (chrom.) (chrom.) triangle (triangle) glissando* chimes chime hammers *2-way "white note" gliss on body o chimes sz sz sz sz sz 12
Eng. Hn. G 70 q = 0 à la "Moldau": Andante con moto Cl. 1 Cl. 2 Cb. Cl. two layers two layers sere two layers two layers sere Bsn. 1 Bn 2/Cbsn. take contrabassoon Hn. 1 & 2 Hn. & 4 Tt. 1 & 2 Tt. & 4 Tt. & Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. 1 Perc. 2 Perc. Perc. 4 Perc. H. Cb. to marimba sz 1 crotales bowed n
Fl. 1 7 < x = x > < x = x > Eng. Hn. m Cl. 1 sere Cl. 2 sere B. Cl. Cb. Cl. Bsn. 1 Bn 2/Cbsn. contrabassoon m Ti. change E to C, F to A, and C# to D Perc. 2 vibes with bow n () Perc. H. m Cb. 14
Fl. 1 Cl. 1 Cl. 2 B. Cl. Cb. Cl. 80 m m iù as i leading Bn 2/Cbsn. Perc. 1 Perc. 2 Perc. H. Cb. m (vibes -- bowed) n (crot. -- bowed) n 1
Picc. oco a oco accel. 84 (q = 72) m Fl. 1 b b m 9: :4 Cl. 1 9: :4 Cl. 2 B. Cl. Cb. Cl. A. Sax. T. Sax. Bsn. 1 m Bn 2/Cbsn. m Perc. 1 Perc. 2 n water gong Perc. n n Perc. 4 H. Db C B E F Gb Ab Cb. 1
Picc. H q = 0 88 Ob. 1 m Ob. 2 Eng. Hn. m m Eb Cl. Cl. 1 Cl. 2 B. Cl. Cb. Cl. clearly articulated A. Sax. clearly articulated T. Sax. clearly articulated Bsn. 1 Bn 2/Cbsn. Perc. 1 Perc. 2 (water gong) Perc. 4 medium marimba mallets H. Cb. 17
Picc. 92 m tutti slur continuously until m. 99; stagger breathe Fl. 1 tutti slur continuously until m. 99; stagger breathe tutti slur continuously until m. 99; stagger breathe Fl. 2 tutti slur continuously until m. 99; stagger breathe Ob. 1 Ob. 2 Cl. 1 Cl. 2 Cb. Cl. m A. Sax. T. Sax. B. Sax. reely Bsn. 1 m m Bn 2/Cbsn. m Tbn. 1 & 2 harmon mute stem in (glisses) Perc. 2 h. Perc. (crot.) hard lastic beater m Cb. m 18
Fl. 1 Fl. 2 Eb Cl. 9 oco a oco oco a oco oco a oco oco a oco Cl. 1 Cl. 2 b m Cl. m b dim. m reely B. Cl. reely b m dim. Cb. Cl. oco a oco A. Sax. oco a oco T. Sax. oco a oco B. Sax. Bsn. 1 oco a oco Bn 2/Cbsn. oco a oco Tbn. 1 & 2 (glisses) Perc. 2 vibes sot mallets H. Cb. "Gushing Chords"* *Slide brusquely between lowest and highest notes in the direction o the arrow D C# B# E# F# G Ab oco a oco 19
Picc. Fl. 1 Fl. 2 Eng. Hn. Eb Cl. Cl. 1 Cl. 2 Cl. B. Cl. Cb. Cl. A. Sax. T. Sax. B. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 Hn. & 4 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. 2 Perc. Perc. 4 Perc. H. Cb. 98 I Maestoso, con brio (q = 92) 9: 9: b tutti a2. marcato a2. a2. tam tam 9: 9: oco a oco oco a oco oco a oco (bass dr.) b (crot.) struck; medium stick m tutti tutti tutti tutti tutti marcato marcato marcato iù iù iù 20
Picc. Fl. 1 Fl. 2 Ob. 1 Ob. 2 Eng. Hn. Eb Cl. Cl. 1 Cl. 2 Cl. B. Cl. A. Sax. T. Sax. B. Sax. Bsn. 1 Hn. 1 & 2 Hn. & 4 Tt. 1 & 2 Tt. & 4 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. Perc. 4 10 a2. iù iù m iù m m m m crash cym. chimes marcato Perc. H. D C Bb E F G A ** **Gliss u to highest A 21
Fl. 1 Fl. 2 Ob. 2 Cl. 1 Cl. 2 Cl. B. Cl. Cb. Cl. A. Sax. T. Sax. B. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 Hn. & 4 Tt. 1 & 2 Tt. & 4 Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. Cb. 109 iù iù m m m m m m m m m 22
Picc. Fl. 1 Fl. 2 Ob. 1 Ob. 2 Eng. Hn. Eb Cl. Cl. 1 Cl. 2 Cl. B. Cl. Cb. Cl. A. Sax. T. Sax. B. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 Hn. & 4 Tt. 1 & 2 Tt. & 4 Tt. & Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. 1 Perc. 4 Perc. Cb. J b 11 b b b b b b b b b b b b b b bells u!!!! dim. bells u!!!! dim. bells u!!!! bells u!!!! bells u!!!! dim. dim. dim. dim. dim. dim. dim. ratchet dim. (gliss as beore) tam-tam to chimes; take chime (b.d.) dim. dim. dim. dim. bells u!!!! bells u!!!! dim. dim. 2
Picc. Fl. 1 Fl. 2 Cl. 1 Cl. 2 B. Cl. Cb. Cl. T. Sax. B. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 120 K Grave; misterioso (q = 4) oco rit. dim. m dim. n Hn. & 4 Tt. 1 & 2 Tt. & 4 Tt. & Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 take lugelhorn n n n n n n n n n n Euh. 2 Tba. Ti. Perc. 1 Perc. H. Cb. marimba medium mallets dim. dim. n crotales sere change C to Eb m dim. oco rit. 24
Ob. 1 L 129a teo m m Eng. Hn. m Cl. 1 B. Cl. s m Cb. Cl. m A. Sax. m sz s B. Sax. Bsn. 1 Bn 2/Cbsn. m Hn. 1 & 2 m Hn. & 4 Tba. m Perc. 1 Perc. 4 tam tam m Perc. sus. cym. scrae with triangle beater m H. Cb. m 2
Ob. 1 M 140 oco rit. Poco iù mosso (q = ) Eb Cl. Cl. 1 m A. Sax. Bsn. 1 Bn 2/Cbsn. r m m Hn. 1 & 2 Hn. & 4 s Tt. 1 & 2 cu mute m molto legato Tbn. 1 & 2 m molto legato Tbn. & 4. Euh. 1 m Euh. 2 Tba. r m m Perc. 2 water gong h m Perc. crotales Perc. 4 chimes m Perc. bass drum H. m Cb. r m oco rit. m 2
Ob. 1 11 m Eng. Hn. Eb Cl. Cl. 1 Cb. Cl. T. Sax. reely m oco a oco dim. Bn 2/Cbsn. m Tt. 1 & 2 to metal straight mute metal straight mute Tbn. 1 & 2 m Tbn. & 4 Euh. 1 Euh. 2 Tba. (ake glissando) m (ake glissando) Perc. 2 vibes hard mallets sz lion roar m m Perc. (crot.) Perc. 4 H. m Db C Bb E# F# G Ab Cb. m 27
Cb. Cl. 19 N Maestoso T. Sax. b n Bsn. 1 Bn 2/Cbsn. m Hn. 1 & 2 m Hn. & 4 m m Tt. 1 & 2 oen metal straight mute oen Tbn. 1 & 2 metal straight mute oen metal straight mute oen Tbn. & 4 metal straight mute oen B. Tbn. metal straight mute oen Euh. 1 Tba. Perc. 1 marimba medium-sot mallets Perc. 2 water gong h h Perc. 4 chimes Perc. bass drum H. Cb. e * * *Gliss to some low, undetermined itch 28
Ob. 1 171 m Cb. Cl. B. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 m iù Hn. & 4 m iù Tt. & 4 m Tbn. 1 & 2 m m Tbn. & 4 m B. Tbn. m Euh. 1 Tba. Ti. Perc. 2 hard sticks Perc. 4 H. Cb. 29
Ob. 1 Eb Cl. Cl. 1 Cl. 2 Cl. B. Cl. O 179 tutti oco a oco oco a oco oco a oco oco a oco oco a oco Cb. Cl. oco a oco m oco a oco A. Sax. T. Sax. B. Sax. Bsn. 1 a2. oco a oco a2. a2. oco a oco oco a oco Bn 2/Cbsn. Tt. 1 & 2 oco a oco m m oco a oco iù a2. m m oco a oco oco a oco Tt. & 4 1. s s m oco a oco oco a oco (m) oco a oco (m) oco a oco Tbn. 1 & 2 m Tbn. & 4 m oco a oco a2. m oco a oco B. Tbn. Euh. 1 m m oco a oco Euh. 2 m Tba. Ti. a2. oco a oco m oco a oco Perc. glockensiel oco a oco Cb. m oco a oco 0
Picc. Fl. 1 Fl. 2 Eb Cl. Cl. 1 Cl. 2 Cl. B. Cl. Cb. Cl. A. Sax. T. Sax. B. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 Hn. & 4 Tt. 1 & 2 Tt. & 4 Tt. & Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. 2 Perc. Perc. 4 Perc. H. Cb. < q = q. > accelerando P Frenetic (q = 12) 18 2. a2. soli marcato lion's roar xylohone to xylohone m m m triangle m tam-tam m m sz D C B E F Gb Ab sus. cym. bass drum wooden sticks 1
Picc. Fl. 1 Fl. 2 Eb Cl. Cl. 1 Cl. 2 Cl. B. Cl. 194 8: 8: 8: 8: 8: 8: 8: 8: 8: 8: 8: 8: 8: Q 8: 8: 8: 8: 8: m 8: m 8: 8: 8: m 8: m m m m Cb. Cl. m A. Sax. T. Sax. B. Sax. 8: 8: 8: 8: 8: m 8: m Tt. 1 & 2 Tbn. 1 & 2 Tbn. & 4 B. Tbn. m Ti. Perc. 2 m 8: 8: Perc. m sz m sz Cb. m 2
204 iù m Cl. 1 iù m iù m Cl. 2 iù m iù m Cl. iù m B. Cl. 8: 8: Cb. Cl. iù m A. Sax. 8: 8: T. Sax. 8: 8: B. Sax. Hn. 1 & 2 1. Tt. 1 & 2 2. Tbn. 1 & 2 m B. Tbn. Ti. Perc. 1 marimba hardest mallets 8: 8: take ratchet Perc. 2 lion roar H. Cb. iù m
Picc. Fl. 1 Fl. 2 Ob. 1 Ob. 2 Eng. Hn. Eb Cl. Cl. 1 Cl. 2 Cl. B. Cl. Cb. Cl. A. Sax. T. Sax. B. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 Tt. 1 & 2 Tt. & 4 Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. 1 Perc. 4 H. Cb. R 21 soli m soli m soli m a2. a2. a2. soli m iù m m leggiero m leggiero iù m m leggiero m leggiero marcato marcato marcato Change Eb to F#, A to C, and F to F# ratchet chimes 4
Picc. Fl. 1 Fl. 2 Ob. 1 Ob. 2 Eng. Hn. Eb Cl. Cl. 1 Cl. 2 Cl. B. Cl. Cb. Cl. A. Sax. B. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 Hn. & 4 Tt. 1 & 2 Tt. & 4 Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. 2 Perc. 4 Perc. H. Cb. S 22 m m m 1. 1. harmon mute stem in harmon mute stem in harmon mute stem in harmon mute stem in m m (b. d.) sz m Db C# Bb Eb Fb G A m m m m m m m m m m xylohone 8: 8: 8: 8: 8: 8: 8: 8: 8:
Ob. 1 2 8: Ob. 2 8: Eb Cl. 8: Cl. 1 8: 8: Cl. 2 8: 8: Cl. 8: 8: B. Cl. 8: 8: A. Sax. 8: T. Sax. B. Sax. Bsn. 1 Hn. 1 & 2 Tt. 1 & 2 1. iù Tbn. 1 & 2 Tbn. & 4 Euh. 1 Ti. Perc. Cb.
Picc. Fl. 1 Fl. 2 Ob. 1 Ob. 2 Eng. Hn. Eb Cl. Cl. 1 Cl. 2 Cl. B. Cl. A. Sax. T. Sax. B. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 Hn. & 4 Tt. 1 & 2 Tt. & 4 Tt. & Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. 1 Perc. 2 Perc. Perc. 4 Perc. H. Cb. 24 7: b 7: 9: b 7: 9: a2. m m a2ṁ T Savage!!!! b 7: 9: b 7: 9: b 7: 9: b 7: 9: b 7: 9: b 7: b 7: 9: b 7: 9: b 7: 9: 7: b 9: b 7: b 7: m 7: 7: 7: 7: 7: 7: 7: 7: 7: 7: 7: 7: 7: 7: 7: 9: 9: 9: 9: 9: 9: 9: 9: 9: 9: 9: 9: 9: 9: 9: a2. m oen oen oen oen m m ratchet to erc. 4's tam-tam triangle m chimes chime hammers (b. d.) (b. d.) sz sz sz sz sz sz * * Db C Bb Eb Fb G A *Gliss u to indeinite high itch b 9: 9: 9: 9: a2. a2. 7
Picc. Fl. 1 Fl. 2 Ob. 1 Ob. 2 Eng. Hn. Eb Cl. Cl. 1 Cl. 2 Cl. B. Cl. Cb. Cl. A. Sax. T. Sax. B. Sax. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 Hn. & 4 Tt. 1 & 2 Tt. & 4 Tt. & Tbn. 1 & 2 Tbn. & 4 B. Tbn. Euh. 1 Euh. 2 Tba. Ti. Perc. 1 Perc. 2 Perc. Perc. 4 Perc. Cb. 24 q = 4 U Largo lugubre (q = 4) stagger breathe stagger breathe stagger breathe stagger breathe stagger breathe stagger breathe stagger breathe marimba sotest mallets use erc 4's tam-tam water gong n (chimes) sot sticks sz sz sz sz molto legato with anguish 8
Cl. 2 27 V Cl. stagger breathe B. Cl. s stagger breathe Cb. Cl. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 sz Tt. 1 & 2 cu mute molto legato m r Tbn. 1 & 2 r m B. Tbn. s stagger breathe Euh. 1 Euh. 2 stagger breathe Tba. Perc. 1 n Perc. 2 water gong n Perc. 4 tam-tam to erc. 's susended cymbal sus. cym. scrae with triangle beater m Perc. Cb. s 9
Cl. 2 W 281 Meno mosso (q = 44) calando al ine Cl. B. Cl. Cb. Cl. Bsn. 1 Bn 2/Cbsn. Hn. 1 & 2 2. Tt. 1 & 2 r Tt. & 4 cu mute B. Tbn. Euh. 1 Euh. 2 Tba. Perc. 1 Perc. crotales Perc. 4 tam-tam strike with triangle beater chimes let all tones vibrate Perc. n H. Cb. 40
SECTION 2: A QUANTITATIVE STUDY OF CHROMATICISM: CHANGES OBSERVED IN HISTORICAL ERAS AND INDIVIDUAL COMPOSERS 41
CHAPTER 1 INTRODUCTION Perhas one o the best generalizations one might oer about the history o Western music is that it has exhibited a broad trend toward increased use o chromaticism since about 100. Few music scholars would disagree with the contention that the music o an early common-ractice-eriod cooser such as Monteverdi is less chromatic than that o a late common-ractice-eriod cooser such as Mahler. An inormal coarison between notated assages rom the 17th century and the late 19th century would reveal a greater use o chromatic tones in the latter eriod. Not only is it the case that art music in general is resumed to have become more chromatic, but it is teting to hyothesize that the music o some individual coosers aears to show an increase in the use o chromaticism over time. For exale, many musicians consider the later works o Beethoven as exhibiting greater chromaticism than his earlier works. Given how widely held this latter view is, there is merit in testing such intuitions eirically. To this end, three studies are described. In the irst study, a large sale o musical themes sanning our centuries o Western instrumental music was studied to determine whether any increase in chromaticism could be detected. In addition, a 42
second theme-based study was conducted to determine whether a coarable increase in chromaticism can be observed in the works o ive individual coosers, namely Johann Sebastian Bach, Franz Joseh Haydn, Wolgang Amadeus Mozart, Ludwig van Beethoven and Johannes Brahms. Finally, in the third study, the entire oeuvre o one cooser, Mozart, was saled. To anticiate the results, one will see that yes, Western instrumental music does broadly exhibit an increase in chromaticism over time. However, with the ossible excetion o Brahms, the results will be inconsistent with the view that the music o these coosers shows any increase in chromaticism over the course o their lives at least according to the quantitative methods used in this study. 4
CHAPTER 2 METHOD From a ractical ersective, it is iossible to test directly the claim that all Western music has become more chromatic over the course o history. The basic terms o this claim must be limited in order to allow or a manageable research roject. Seciically, the meaning o the concet chromaticism must be clariied and the totality o Western music must be restricted to a reasonable subset or sale. What is Chromaticism? Chromaticism may be deined in dierent ways. The deinition ound in The New Grove Dictionary o Music and Musicians rovides a standard characterization: In melodic and harmonic analysis the term chromatic is generally alied to notes marked with accidentals oreign to the scale o the key in which the assage is written (Dyson & Drabkin, 200). Most music theorists would use this as a oint o dearture or an extended discussion o the iortance o chromaticism in relation to musical organization including both vertical (harmonic) and horizontal (melodic) structure. 1 For the uroses o this study, however, these more extended ideas will not be considered and the New Grove deinition will be taken at ace value. In this study, a note will be 44
oerationally considered chromatic i its itch ails to conorm to the revailing key signature. This deinition has a number o otential italls which must be acknowledged at the outset. For exale, it is ossible that a assage modulates to a dierent key without a commensurate change o key signature. Consequently, an accidental may render a note consistent with this revailing (modulated) key yet be construed as chromatic in light o the key signature. Another diiculty with this oerational deinition is that it ails to distinguish degrees o chromaticism. For exale, a chromatic assing tone might be viewed as less indicative o chromaticism than a structural chromatic tone. In this roject, general trends will be distilled rom a large volume o music. Analyzing each non-scale tone in order to classiy it as structural or non-structural would be iractical. Although this oerational deinition o chromatic ails to cature all o the nuances o chromatic theory, it may nevertheless rovide a ractical index that characterizes the broad trends. This study consists o three arts. In the irst art, test the general intuition that music has become more chromatic over the common-ractice eriod in Western art music is tested using a large sale o themes. That is, it is redicted that the roortion o out-o-key tones will increase over the course o Western common-ractice music making. Part o the motivation or this art o the study is to test whether the sile oerational deinition o chromaticism is suicient to show a general musical trend. In the second art, this same methodology will be alied to examine the music o individual coosers. That is, the question o whether there is evidence o an increase in 4
chromaticism over the lietime o seciic coosers will be raised. Because the irst two arts tested only themes, a third study, which examines the entire oeuvre o one o these coosers namely Mozart will be conducted Musical Sale For the initial study, a re-existing database consisting o the colete themes rom the Barlow and Morgenstern Dictionary o Musical Themes (1948) was chosen. This dictionary consists o roughly 10,000 musical themes drawn rom instrumental Western art music works. These themes themselves relect Harold Barlow and Sam Morgenstern s intuitions o the rincial melodic assages that characterize individual works. For shorter works, the Barlow and Morgenstern collection tyically identiies one or two themes. For longer works, many themes are commonly identiied. In their Preace, Barlow and Morgenstern suggest that, rather than err on the side o brevity, their collection includes almost all o the themes the average and even the more erudite listener might want to look u (. xi). Little analytic descrition is rovided as to how they selected the assages they did. As it turns out, the Barlow and Morgenstern collection is structured in a way that reduces the conounding eect o modulation identiied earlier. In Western musical ractice, it is common or rimary themes to be written in the home key o the work as a whole, while secondary themes are oten notated in the dominant (or relative major). In their Dictionary o Musical Themes Barlow and Morgenstern ortuitously notate secondary themes in the local (i.e., modulated) key. Consequently, both rimary and 4
secondary themes are reresented in a key-aroriate context. As a sale source or study, the Barlow and Morgenstern largely sidestes the roblem o modulation. The Dictionary o Musical Themes does not exlicitly include dates o coosition. This inormation was added to the database using established music reerence tools. Unortunately, reerence works tyically identiy the date o ublication rather than the date o coosition. For osthumously ublished works, the date o ublication can be much later than the date o coosition. For exale, in some cases, irst ublication did not occur until hal a century ater the cooser s death. In order to avoid these conounds, all themes whose reorted date was later than the death-date o the cooser were eliminated rom consideration. It must be acknowledged that this rocedure may introduce a bias to the saled music by under-reresenting works in the later years o a cooser s career. 2 A related roblem with dates is that many dates are seciied as a range (such as 184-18). A range may be given because the work actually took several years to coose, or because the work was ublished as art o a collection that was roduced over a eriod o several years, or because scholars are genuinely uncertain o the date o coosition. For these works, the date o coosition was oerationally deined as the mid-oint in any seciied range. Given these conditions, only,809 o the 10,000-odd themes available in the Dictionary Musical Themes were used in this study. The average length o the saled themes was roughly 19 notes. 47
CHAPTER RESULTS In examining the relationshi between two variables, an aroriate statistical method is to coare the roortion o non-scale tones with the year o coosition. Pearson s coeicient o correlation rovides a useul summary statistic. Correlation coeicients range between 1 and +1. A correlation o +1 indicates numerical values that rise and all in recise synchrony with each other. A correlation o 1 indicates numerical values that rise and all in colete oosition: a rise in one set o numbers corresonds to a all in the other set o numbers. A correlation near zero indicates that there is no statistical relationshi between the two numerical sets. The calculation method used in Pearson s correlation coeicient is insensitive to the magnitude o the numerical values, so (or exale) it makes no dierence whether dates are exressed in years, months, days, or seconds. In the case o scale tones, it was redicted that as the year o coosition increases, the revalence o scale tones decreases, resulting in a negative correlation. For the,809 themes available rom the Barlow and Morgenstern collection, the correlation between the reorted year o coosition and the revalence o scale tones was ound to be r= 0.1. While the magnitude o this correlation would seem to be modest, it roves to be statistically 48
signiicant (<0.0000000000000001). This means that the robability o observing the relationshi between these two variables by chance is less than one chance in a trillion. These results are consistent with the hyothesis that Western music has exhibited greater chromaticism over time. Rather than execting an increase in chromaticism with the assing o each successive year, historians would tyically think o musical ractices in terms o revailing stylistic eriods. This suggests that the chromaticism data be reconsidered with resect to historical eras. Identiying the beginning and ending dates or a stylistic eriod is raught with both concetual and ractical roblems. Nevertheless, historians rovide conventional dates, such as those shown in Figure 1. Figure 1 reorts the ercentage o scale tones distributed over our conventionally-deined stylistic eriods. As can be seen, the average ercentage o scale tones declines with each successive eriod. Note, however, that there is a negligible dierence between the Baroque and Classical eriods in the mean ercentage o scale tones used in my sale o themes. On the other hand, the dierence between the Classical and Romantic eriods is statistically signiicant (t=.2; <0.0001). 49
Scale Tones (Mean %) 9 94 92 90 88 8 84 82 80 Baroque (100-170); 8 Themes Classical (171-1820); 1110 Themes Romantic (1821-1910); 47 Themes Modern (1911- );794 Themes Figure 1. An illustration o the ercent scale tones resent in themes rom the our historical eras that corise the common ractice eriod in Western art music. As is exected, the ercentage o scale tones decreases over time; i.e., the quantity o chromaticism increases. 0
Scale Tones (Mean %) 98 9 94 92 90 88 8 84 82 80 78 1700-1719 (7 Themes) 1720-179 (288 Themes) 1740-179 (18 Themes) 170-1779 (118 Themes) 1780-1799 (9 Themes) 1800-1819 (89 Themes) 1820-189 (92 Themes) 1840-189 (41 Themes) 180-1879 (82 Themes) 1880-1899 (1207 Themes) 1900-1919 (74 Themes) 1920-199 (474 Themes) 1940-199 (110 Themes) Figure 2. An illustration o the ercent scale tones resent in themes in time units artitioned by bidecade. Once again, the results are consistent with the traditional intuition that Western music became increasingly chromatic over time. Although the oerational deinition o chromaticism seems limited, it nevertheless is able to cature broad trends about coosers use o chromatic tones. One diiculty with looking at articular time eriods is that coosers who coosed in very dierent styles oten lived at the same time. For 1
exale, the lives o Haydn and Beethoven overlaed or 9 years. It might be useul, thereore, to ocus on the ractices o individual coosers. 2
CHAPTER 4 CHANGES IN CHROMATICISM IN INDIVIDUAL COMPOSERS WORKS An iortant question is whether the works o articular coosers exhibit increasing amounts o chromaticism over the course o their careers. In looking at individual coosers, statistically generalizable results are ossible only with a reasonably large sale o music. Given the small number o notes in each theme, hundreds o themes would be necessary in order to yield reliable results. In the Barlow and Morgenstern database, ive coosers are well reresented: J.S. Bach (27 themes), F.J. Haydn (212 themes), W.A. Mozart (182 themes), L. van Beethoven (1 themes), and J. Brahms (81 themes). Table 1 reorts the correlation coeicients between the date o coosition and the ercent scale tones or the themes rom each o these ive coosers.
Cooser Correlation Statistical Scale Tones Number o Coeicients (r) Signiicance () (Mean %) Themes (n) Bach +0.017 0.91 9.74 27 Beethoven -0.02 0.29 92.9 1 Brahms -0.00 0.1 89.8 81 Haydn +0.001 0.494 94.98 212 Mozart -0.009 0.42 94. 182 Table 1: Correlations between Date o Coosition and Percent Scale Tones or Themes by Five Coosers None o the correlations reorted in Table 1 reaches statistical signiicance as conventionally understood. (It might be noted, however, that there is a 7 in 8 chance that the music o Brahms does exhibit an increase in chromaticism albeit weak.) With the ossible excetion o Brahms, these results suggest that these coosers did not tend to use more chromatic tones in themes as their careers rogressed. This result is not consistent with many musicians intuitions, esecially regarding the music o Beethoven. Part o the surrise here is that Beethoven is widely regarded as a transitional igure, straddling the classical and romantic eriods. His later works, esecially the late string quartets, are oten viewed as exhibiting a more adventurous or colex character. Since a statistically signiicant dierence between the chromaticism exhibited in the Classical and Romantic eriods was reorted, one might have exected to see a similar change in Beethoven s music. Aart rom broad stylistic eriods (such as the Baroque), music scholars oten suggest that coosers themselves exhibit relatively stable coositional 4
eriods. For exale, in the case o Igor Stravinsky, marked changes in his coositions ily three distinct ersonal stylistic eriods. Agreement about such eriods is by no means universal. Nevertheless, reerence works (such as the New Grove Dictionary o Music and Musicians) will oten describe coosers by reerring to articular eriods. It is ossible that the aarent lack o chromatic change in Beethoven s music is an artiact o insensitivity to the dierent stylistic eriods. To this end, the data can be re-analyzed or individual coosers using the coositional eriod inormation claimed in the New Grove. Figure rovides a breakdown or three o the ive coosers studied, or whom the New Grove describes as having resumed stylistic eriods.
Scale Tones (Mean %) 98 97 9 9 94 9 92 91 90 Early Beethoven (1770-1802); 24 Themes Middle Beethoven (180-1812); 200 Themes Late Beethoven (181-1827); 87 Themes Haydn at Esterhaza (171-1790); 70 Themes Haydn in London (1791-179); 117 Themes Late Haydn (179-1809); 0 Themes Early Mozart (17-1771); 1 Theme Mozart in Salzburg (1772-1781); 4 Themes Mozart in Vienna (1781-1788); 12 Themes Late Mozart (1788-1791); Themes Figure : An illustration o the mean ercentage o scale tones or themes by Beethoven, Haydn, and Mozart, broken down by stylistic eriods deined by the New Grove Dictionary o Music and Musicians. Although there are dierences in the averages between the three coosers, there is no aarent decrease over time or any o them. In the case o Beethoven esecially, these results do not aear to conorm with commonly held views. It is iortant to recall that the results o the irst and second studies are based on sales o musical themes. By using themes, the data reresent only a single voice and do not include bass and inner voices that may relect colex harmonic atterns that
could be deemed chromatic. It may be the case, or exale, that a cooser such as Beethoven tends to create redominantly diatonic themes that are nevertheless resented in chromatic harmonic settings. On the other hand, it is ossible that themes might exhibit a higher degree o chromaticism (coared to suorting voices) due to the increased resence o non-harmonic tones. 7
CHAPTER CHROMATICISM IN MOZART S OEUVRE In order to address these otential conounds, a third study was carried out that makes use o a dierent saling method. Instead o saling rom selected musical themes, a better aroach might involve saling rom the entire corus o a cooser s oeuvre. Obviously, saling rom a cooser s entire corus can be somewhat labor intensive. So, in this third study, Mozart s oeuvre was chosen. A useul oerational deinition o an oeuvre might be all the ieces contained in ublished collected works excluding instrumental arts, ragments, sketches, works artially attributed to other coosers, or arrangements o works by other coosers. In this third study, the saling goal was to make all o the notes within the cooser s oeuvre equally likely to be saled. This would include both thematic and accoanimental assages, not just themes. In order to secure a reasonable amount o statistical ower, it was resolved a riori to sale roughly 1,00 notes. Mozart s oeuvre as it has been deined consists o 18,417 ages o the Neue Mozart Ausgabe (19-1991), so a single note was selected rom every 12th age (aroximately 18,417/1,00). 8
The method o saling roceeded as ollows. Without looking at the notated age, a stylus was randomly ositioned on the age, and the notehead closest in radial distance to the stylus osition was identiied. Notes in ercussion arts were excluded rom the sale since these notes either do not reresent recise itches or only reresent tonic and dominant (in the case o tiani). Since the saling was based on noteheads, tied notes were roortionally more likely to be saled; or exale, a single note onset might be sustained across several measures, and so be notated as a series o N noteheads. I resent, the chromatic inlection o a tied note was maintained. In lacing a stylus on a age, it is unlikely that this can be done randomly. In articular, it is likely that there is a mental bias to osition the stylus near the middle o the age. In the case o orchestral works, only one system sometimes aears on a single age. This means that lacing the stylus near the center o the age might result in a tendency to sale inner-voice arts rather than uer- or bass arts. In order to alleviate this tendency, a mental strategy was used in which three successive sales were biased toward the to, middle, and bottom o the age. Another issue relates to the horizontal ositioning o the stylus. The let side o a age is more likely to bias the sale toward sonorities associated with the beginning o a work, and the right side o a age is more likely to bias the sale toward sonorities associated with the end o a work. Also, since ages have blank let and right margins, the let side o the age will tend to avor selecting notes rom the beginning o a measure, while the right side o the age will tend to avor selecting notes rom the end o a measure. In order to avoid these ossible biases, all saled notes were drawn rom near 9
the horizontal middle o the age. In other words, successive sales roceeded rom the to center, to the middle center, to the bottom center. Using this rocedure, 111 notes were saled rom the 18,417 ages o the Neue Mozart Ausgabe that constitute Mozart s oeuvre as deined above. Each note was characterized as either belonging to, or alien to, the revailing key. For each saled note, the nominal date o coosition as reorted in the Neue Mozart Ausgabe was recorded. When two successive dates were listed, the lesser o the two was chosen. When a larger range o dates was given, an average o the beginning and ending dates was used, with the mean date being rounded u to the next year i needed. Results As in the earlier studies, the dates o coosition or each saled work were correlated with the roortion o scale tones. Once again, a negative correlation would be consistent with an increase in chromaticism over time. Calculating Pearson's correlation coeicient or the 1,11 data oints resulted in a ositive correlation o r=+0.01277 (=0.0 N.S.). In the irst instance, the data are skewed in the oosite direction rom what was redicted. Moreover, this result is not statistically signiicant. 0
CHAPTER DISCUSSION Although the oerational deinition o chromaticism was able to demonstrate that Western music does indeed increase in chromaticism over time, it showed that the chromaticism resent in themes written by ive major coosers does not aear to increase over the course o their lives. Moreover, the colete oeuvre o Mozart similarly aears to show no increase in chromaticism at least in terms o quantity. This result ilies that the quantity o chromaticism that these individual coosers use is a relatively stable stylistic trait. It is ossible, however, that the quality o chromaticism in the works o any o these coosers does change over the course o their careers. Further eirical research would be needed to address this hyothesis. In addition, a marked dierence in the results o the theme-based and oeuvrebased sales must be acknowledged. The overall ercent o scale tones in this sale o Mozart s music was ound to be 90.9%, which is notably lower than the coarable ercentage o 94.% or the Barlow and Morgenstern collection o Mozart themes. This dierence ilies that musical themes are notably more diatonic than ull textures in the case o Mozart. It is unknown whether a similar attern would be evident or other coosers. In light o the dierences in saling or Mozart, it is ossible that Bach, 1
Haydn, Beethoven, and Brahms tended to coose themes that remained airly diatonic throughout their lives, but that the harmonic settings or their works became increasingly chromatic. This hyothesis awaits urther eirical research. 2
NOTES 1 No list o reerences can do justice to the breadth and deth o writings on the subject o chromaticism. A artial list might include Rameau, 1722; Hindemith, 1942; Mitchell, 192; Fux, 19; Forte, 1980; Baker, 199; and Boatwright, 1994. 2 In general, coosers exerience greater renown in their later years and so have little diiculty getting their works ublished. Oten, osthumous ublications reresent overlooked works rom earlier eriods o a cooser s lie that come to light ater the cooser s death. Consequently, one should not necessarily assume that the elimination o ost-death-date ublications results in an under-reresentation o later works.
REFERENCES Barlow, H., & Morgenstern, S. (1948). A Dictionary o Musical Themes. New York: Crown Publishers. Baker, J. (199). Chromaticism in Classical Music. In: C. Hatch & D. W. Bernstein (Eds.), Music Theory and the Exloration o the Past. Chicago: The University o Chicago Press,. 2-08. Boatwright, H. (1994). Chromaticism: Theory and Practice. Fayetteville, NY: Walnut Grove Press. Dyson, G., & Drabkin, W. (200). Chromatic. In: L. Macy (Ed.), Grove Music Online, (accessed Feb. 1, 200), htt://www.grovemusic.com Eisen, C., & Sadie, S. (200). Mozart () (Johann Chrysostom) Wolgang Amadeus Mozart. In: L. Macy (Ed.), Grove Music Online, (accessed Aug. 2, 200), htt://www.grovemusic.com Forte, A. (1980). Generative Chromaticism in Mozart s Music. Musical Quarterly, Vol.,. 49-8. Fux, J. (19). Gradus ad Parnassum. A Facsim. o the 172 Vienna Ed. New York: Broude. Hindemith, P. (1942). The Crat o Musical Coosition. New York: Associated Music Publishers, Inc. Huron, D. (199). The Humdrum Toolkit: Reerence Manual. Menlo Park, CA: Center or Couter Assisted Research in the Humanities. Kerman, J., Tyson, A., and Burnham, S. (200). Beethoven, Ludwig van. In: L. Macy (Ed.), Grove Music Online, (accessed Aug. 2, 200), htt://www.grovemusic.com Mitchell, W. J. (192). The Study o Chromaticism. Journal o Music Theory, Vol.,. 2-1. 4
Mozart, W. A. (19-1991). Neue Ausgabe sämtlicher Werke: in Verbindung mit den Mozartstädten, Augsburg, Salzburg, und Wien / herausgegeben von der Internationalen Stitung Mozarteum, Salzburg. Kassel, Germany: Bärenreiter. Rameau, J. (1722). Traité de l harmonie. Trans. by P. Gossett as Treatise on Harmony. New York: Dover Publications, 1971. Webster, J. (200). Haydn, (Franz) Joseh. In: L. Macy (Ed.), Grove Music Online, (accessed Aug. 2, 200), htt://www.grovemusic.com