SYSTEMATIC COMPOSITION AND INTUITION IN A CONCERTO FOR ORGAN AND ORCHESTRA. James Timbrel Worlton, B.M., M.M. Dissertation Prepared for the Degree of

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1 SYSTEMATIC COMOSITION AND INTUITION IN A CONCETO O OGAN AND OCHESTA ames Timel Wolton, BM, MM Dissetation eaed o the Degee o DOCTO O MUSICAL ATS UNIVESITY O NOTH TEXAS August 200 AOVED: oseh Klein, Mao oesso David Schaz, Mino oesso Cindy McTee, Committee Meme ames C Scott, Dean o the College o Music C Neal Tate, Dean o the oet B Toulouse School o Gaduate Studies

2 Wolton, ames Timel, Systematic Comosition and Intuition in a Conceto o Ogan and Ochesta Docto o Musical Ats (Comosition), August 200, 10, 25 examles, eeences, 2 titles Histoically, comoses have used methods in addition to insiation in iting music egadless o the souce mateials they used, comoses ultimately ely on thei musical sensitivity to inom the comositional decisionmaking Discuses the otational asects o decimals that ae ceated om cetain ime-nume denominatos, and ocuses on the ime nume 17 Shos ho these decimals can e tansomed y conveting them to dieent nume ases Looks at the Golden ootion and its use in ceating omal stuctues Examines comositional and aesthetic issues aising om using nume seies to geneate the itches, hythms, and sections in the Conceto o Ogan and Ochesta This ocess o comosition eveals musical gestues that may not have een discoveed using moe intuitively ased aoaches to comosition Shos ho musical sensitivity as necessay in shaing the numeically deived mateial in ode to ceate aesthetically satisying music

3 Coyight 200 By ames Timel Wolton ii

4 ACKNOWLEDGEMENT I ould like to thank my athe, David T Wolton, o his assistance ith the details o ogan egistation His knoledge and exeience heled me to choose times that ae oth idiomatic to the ogan and aoiate to the conceto iii

5 TABLE O CONTENTS ACKNOWLEDGEMENTiii Chate 1 INTODUCTION1 Comute-assisted Algoithmic Comosition My Aoach to Algoithmic ocesses Enoyment o Discovey oint o Vie as a Listene Comositional Aoach o the Conceto 2 MATHEMATICAL BASIS O THE CONCETO7 eiodic Decimals Exanding the Gou o Numes The Golden atio STUCTUAL LAN12 ootions and Mas To Simultaneous G Mas Intenal Stuctue o Each Movement Movement I Movement II Movement III Vaiances in the End esult Movement I: Cadenza Movement III: Coda GENEAL MAING OCEDUES19 Music om Numes Numes as itch Numes as hythm Numes as Othe Quantiiale Musical aametes Comositional esonsiilities 5 STAIGHTOWAD MAING2 ecussion and the Binay Sequence The Oiginal Main Theme o Movement I The Ne Oening o Movement I iv

6 Chod ogessions in Movement II Stings at the Beginning o Movement II hythm in the Wind Chods and in the Ogan The Climax o Movement II (at 1: hythm) CONVOLUTED MAING The Climax o Movement II (at 2: itch) The Genesis o Chod A and its Descendants The Cadenza in Movement I Summay 7 INTUITION AND OCEDUE IN MOVEMENT III2 Evolution o the assacaglia Theme eiting Movement III ealizing the Need o Change The ocess otuitous Discoveies CONCLUSION AENDIX A AENDIX B50 AENDIX C57 AENDIX D7 AENDIX E9 WOKS CONSULTED7 SCOE7 v

7 CHATE 1 INTODUCTION Thoughout the histoy o music, comoses have utilized exta-musical souces to enhance the comositional ocess The ancient Geeks elieved that the mathematics contolling the motion o celestial odies as a om o music musica mundana ( music o the shees ) In the Middle Ages, comoses o the as nova and as sutilio ote isohythmic motets and masses Duing the eighteenth centuy, Haydn, Mozat and C E Bach comosed using mathematical inciles om as cominatoia Comoses in the tentieth centuy elied on mathematics to ovide ode in ost-tonal music O couse, not all exta-musical souces have een mathematical Nineteenth-centuy comoses equently de on liteatue and natue o insiation egadless o the exta-musical inluences they used, these comoses had to decide ho comletely thei souces and comositional systems ould detemine the music, and ho much eedom they ould allo themselves to modiy hat thei souces oduced Because comoses cae aout ho thei music sounds, thee must come a oint at hich they shit thei ocus aay om the systems they have een using toad the sounds they ae ceating In my exeience, this shit does not occu only once in the couse o comosing a iece, ut many times This oscillation o ocus eteen system and sound allos a comose to continually eigh that hich the system is oducing against his o he desied sound In ode to ocus on hat ill e head, comoses must emloy thei musical sense to shae the inished ok Musical sense (intuition), as it alies to comosition, is the comose s ealization that hat he o she has itten is accetale and comlete This ealization tanscends any seies o ocedues that the comose may have used in 1

8 develoing the music; and it comes as an insight, ithout conscious thought Intuition develos though study and imitation, as ohn Winso suggests: at o a comose s taining is the study o ast masteieces essentially a veal-analytical ocess that includes memoizing ules, studying scoes, and so oth But the taining also involves iting in the styles o ast mastes This hels young comoses to intenalize the undelying inciles, to execise thei intuition y doing hat ast comoses have done In this ay, comoses develo the aility to think in the same ay that thei edecessos thought 1 Evey comose exeiences moments o musical intuition dieently, and I elieve this individuality inluences a comose s style In this ae I exlain ho I aoached the comosition o the Conceto o Ogan and Ochesta, and the ay intuition inomed my comositional systems My avoite souce o non-musical stimuli is numes, and asic elements o numeical elationshis and attens Most oten I have used ime numes and the Golden atio as insiation, ut I have also used elements o chaos theoy, aticulaly Michel Henon s stange attacto 2 Many o my mathematical ocedues, and the ay they oduce music, could e accomlished on a comute Hoeve, I geneally ee to ok them out y hand I ind that this method acilitates the oscillation eteen system and sound o moe oadly, eteen system and intuition The olloing section discusses the use o comutes in comosition, and my aoach to algoithmic comosition Comute-assisted Algoithmic Comosition Algoithmic comosition is the use o comutes to make decisions that comoses have histoically made themselves Buce L aco illustates the 1 ohn Winso, Beaking the Sound Baie: An Agument o Mainsteam Liteay Music (Ne Yok: Wite s Shocase, 200), 9 2 In my Henon Imovisation See ames Gleick, Chaos: Making a Ne Science (Ne Yok: enguin Books, 199),

9 ationale ehind the use o comutes in comosition: Ceativity comes in to lavous: genius and had ok While the ome may oduce moe insied music, e do not ully undestand it and theeoe have a slim chance o eoducing it The latte esemles an iteative algoithm that attemts to achieve some otimal unction o meit, and is theeoe moe easily ealisale as a comute ogam ichad Oton exands the deinition to include all musical comosition Some musicians have diiculty ith the concet o algoithmic comosition, so at the stat I might tun the notion aound and say that I elieve that all comosition is algoithmic comosition, hethe the comoses ealise this o not He says that in the ocess o lanning a comosition, the comose eoms to dieent tasks: geneating souce mateials, and ceating the set o ocesses that the comose ill use to dive the music though time He agues that these ocedues can e made ust as ell though the use o a comute 5 With egad to algoithmically-geneated music, aco asks hethe o not thee is a dieence eteen a comose ecognizing something that he o she likes and comosing something desiale He descies to oosing vieoints to this question ist, some eole think that at is communication, and so anything that communicates to anothe is at; at is the hysical equivalent o hey look at this Those on the othe side think that only hat comes diectly om the hands o an atist is o the atist 7 In othe ods, at can only come diectly om a human aco oes no anses to this dilemma, ut says that the moe closely an algoithm elects a Buce L aco, Algoithmic comosition as a model o ceativity, Oganised Sound: An Intenational ounal o Music Technology 1, no (Deceme 199): 157 ichad Oton, Design stategies o algoithmic comosition, Contemoay Music evie 15, no - (199): 9 5 Iid, 9-0 aco, Algoithmic comosition, 15 7 Iid

10 comose s methodology, the less question thee is that the ok is authentic and o the comose My comositional method alls in eteen the to extemes My Aoach to Algoithmic ocesses Enoyment o Discovey I do not tyically ely on a comute hen I comose, ut I do emloy ocedues that a comute could e ogammed to eom I ee to ok them out mysel and to decide on a moment-y-moment asis ho to aly the nonmusical data to comosition; in othe ods, ho to ma the data on to musical aametes 9 Unde the oad umella given ealie y Oton, aguing that my method o comosition is not as algoithmic as music hich is geneated entiely y a comute is moot But unde the geneally acceted deinition (the use o comutes to make comositional decisions) I ould not qualiy as an algoithmic comose I ee to comose this ay ecause o the oy o discovey I enoy deciding ho to convet nume attens into music at o the excitement o comosition comes om seeing hat can e ceated om the ules o a system, and hethe o not they have musical meit ( hey look at this ) o me, this is a eading at o the comositional ocess O B Hadison, in talking aout omula oety, said Why should intelligent atists accet omulas that oce them to say things they might not otheise say in ays that they most cetainly ould not use unless oced to y the omulas Beyond the act that they lieate, the aeal o the omulas is ecisely that they oce the ite out o edictale aths; that is, they encouage in act, they comel discovey 10 Iid 9 Hoeve, in my Henon Imovisation(2000) I did use a comute to geneate notes, hythms, and dynamics, hich I then had to shae in a eal-time eomance O B Hadison, Disaeaing Though the Skylight (Ne Yok: enguin Books, 199),

11 This alies to music as ell, and eesents the ay I look at using systematic aoaches to comosition oint o Vie as a Listene As a listene to music, I ee to kno hat lies ehind the sounds oth in tems o ogam as ell as technique This eeence has evolved as I have leaned moe aout music Hoeve, non-musician listenes tend to e inteested in the ogammatic elements o a iece athe than its technical asects Theeoe, as a comose I ty to ite music that is successul o a listene ho does not have a knoledge o my systems I do not cae hethe o not they can ecognize the technical ocedues that ent into the ok I agee ith ohn Winso hen he says that it does not matte hat techniques a comose uses as long as the musical esult is coheent and aeas inevitale and devoid o atiice 11 Comositional Aoach o the Conceto The ith o Chales Wuoinen s ive geneal ut not univesal suggestions o comosition advises: A comositional method exists only to ite ieces It is not saced, and hen the iece has eached, though alication o the method, a suicient degee o comleteness, it ill egin to asset its on ights and needs These may oten seem to contadict the oiginal method o call o changes in the ok s design Do not hesitate hen such a situation aises I the method has seved long enough to allo the ok it has oduced to contadict it, it has moe than ulilled its unction 12 My Conceto o Ogan and Ochesta evolved though the alication o systems that ill e examined in the emainde o this ae I did not constuct an algoithm to 11 Winso, Beaking the Sound Baie, 7 12 Chales Wuoinen, Simle Comosition (Ne Yok: Longman, 1979; eint, Ne Yok: Schime Books, 19), 1 5

12 contol these systems and ceate a inished iece, even in ough om My intent as to alance the ocedues I used ith my musical intuition, hich ould then detemine the inal shae o the music In Chate 2 I exlain the mathematical asis o my ogan conceto the otational asect o some decimals that ae ceated hen the denominato o its action is a ime nume Chate deals ith the ay I calculated the omal divisions o the ok In Chates though 7 I discuss issues elated to tuning numes into music; om geneic ossiilities though some o the moe comlex and convoluted occuences in the conceto

13 CHATE 2 MATHEMATICAL BASIS O THE CONCETO eiodic Decimals A teminating decimal eventually aives at a emainde o 0 o examle: 1/2 = 050; / = 0750; 7/20 = 050 All ininite (non-teminating) decimals can e classiied as eithe eiodic o iational Numes such as i = and the golden nume (eesenting the Golden atio) = 1109 ae iational ecause thei sequence o digits neve eeats A ational nume contains hole numes in the numeato and denominato; and a ational nume hose decimal does not teminate, ceates a eiodic decimal o examle: 1/ = 0 hee the eeats indeinitely ith a eiod o 1; 5/11 = 055 hee the 5 eeats indeinitely ith a eiod o 2; and 11/12 = 091 hee the eeats indeinitely ith a eiod o 1 and a delay o 2 The eiodic asect o these tyes o decimals lays an imotant ole in geneating the music o the Ogan Conceto; aticulaly those that have a ime nume in the denominato ime Numes as Denominatos When cetain ime numes (heeate laeled ) aea as the denominato o a action (a/, hee a is equal to o geate than one, ut less than ) they ceate otating eiodic decimals o many values o the length o the eiod is - 1 7

14 Conay and Guy call imes ith this chaacteistic long imes 1 o examle, using 7 as the denominato ceates an ininitely eeating decimal ith a eiod o : 1/7 = /7 = /7 = and so on, u to /7 = In this examle the six digits (ie, - 1), 1257, otate ith each successive value o a o cetain othe values o this is not tue, that is, the otating eiod is some othe length o examle, o = 1 thee ae to distinct otating eiods o six decimal laces the otating eiod is ( - 1) / 2 Othe eiodic lengths ae also ossile o lage values o 2 o the Ogan Conceto I have chosen the long ime = 17 to use as Each a ceates a distinct odeing o the set o sixteen values, , that I can then aly to musical comosition I chose 17 ecause it oes moe mateial to ok ith than = 7 Also, I did not ant to ok ith a eiodic length othe than - 1, and 17 is the next long ime ate 7 Exanding the gou o numes In addition to the otational asect o the set o digits ceated in a/17, I as inteested in ays o tansoming the set in ode to ceate elated sets o vaious lengths In ode to illustate ho I accomlished this, I must ist exlain the concet o caousel numes My method o augmenting the asic set o numes equied that they ecome 11 1 ohn H Conay and ichad K Guy, The Book o Numes (Ne Yok: Coenicus, 199), 2 Iid, 157-

15 caousel numes Caousel numes ae ceated y extacting the seies o numes om its decimal context (y emoving the leading 0 and decimal oint) This does not change the elationshi eteen the digits in the nume since these numes ehave in the same ay as thei decimal counteats o examle: e have seen ho the digits in a/7 otate o dieent values o a When the six numes stand alone, 1257 (as oosed to 01257), they ill also otate hen multilied y any nume u to the nume o digits in the caousel, ut not eyond o examle: 1257 x 2 = x = x = x 7 = x = 1125 This concet alloed me to exand the ool o numes y conveting them to dieent nume ases ithout the comlications o conveting the decimals o examle: 2/17 = The decimal otion, hen extacted, ecomes the caousel nume and hen conveted to ase 12 it ecomes tee7te75t2t Gay Klatt, Caousel Numes: A Lead-in to Nume Theoy, The Math oum, [Intenet site]; Availale om htt:mathoumog/olando/klatt7olandohtml; Intenet; accessed Octoe 2001 The idea o conveting the decimals o a/17 into its equivalents in othe ases as neve a at o the oiginal lan o this comosition 9

16 (hee t and e stand o 10 and 11, esectively) 5 One advantage o this ocedue is the aility to educe the vaiety o digits in the nume Whee the convesion om ase 10 to ase 12 ceated moe availale digits e lace (telve instead o ten), y conveting to ase, o examle, the only availale digits ae 0, 1, 2: Oviously, ith ee availale digits the nume as a hole lengthens Hoeve, one signiicant yoduct o changing nume ase this ay is the loss o the otational elationshi eteen values o a that aeas in ase 10 Using dieent nume ases this ay aealed to me moe than ocessing the numes ith a modulus A modulus ould only oduce one value o any inut, such as the 9 in the olloing ocess: 1257 is conguent to 9 modulo 12 In othe ods, 1257 / 12 = 1190 ith a emainde o 9 o the comosition I as moe inteested in each individual digit in the caousel numes as it elated to the othes When comosing, one can use these numes o many dieent musical ocesses such as successive intevals in a melodic line, asolute itch-class values (C = 0, C-sha = 1, etc), duational values (hythm), stuctual ootions, vetical intevals ithin a chod, and so on Any to o moe o these ocesses may also occu simultaneously (this ill e discussed in moe detail in Chate ) Hoeve, o the stuctue o the conceto I used the Golden atio The Golden atio The geneal inteest in the Golden atio lies in its alication to a myiad o 5 o digits lage than 9 in ase 11 to ase 20 I used the olloing lettes: t=10, e=11, L=12, h=1, =1, i=15, x=1, v=17, g=1, n=19 See also Aendix C o the comlete set o ase convesions 10

17 dieent situations Maio Livio asks: What do the delightul etal aangement in a ed ose, Salvado Dali s amous ainting Sacament o the Last Sue, the magniicent sial shells o mollusks, and the eeding o aits all have in common The anse is that they all elate to the Golden atio Euclid ovided the ist clea deinition o the Golden atio aound 00 BC 7 The simlest ay to visualize the Golden atio (G) is to conside a line segment AB I the segment AB is divided, C, such that the atio eteen AB and AC is the same as the atio eteen AC and CB, then C divides the segment accoding to the G (See Ex 21) Mathematically it is deined as (1 + 5) / 2 This oduces the iational nume My inteest in the Golden atio o this ok as to ovide stuctual oints o imotance that logically elated to one anothe and to the hole I discuss this asect o the comosition in Chate A C B Ex 21 Line segment divided y the Golden atio Maio Livio, The Golden atio: The Stoy o hi, the Wold s Most Astonishing Nume (Ne Yok: Boaday Books, 2002), 2 7 Iid, ichad A Dunla, The Golden atio and ionacci Numes (Singaoe; ive Edge, N: Wold Scientiic, 1997),

18 CHATE STUCTUAL LAN Beoe discussing the stuctual lan o the conceto I must oint out that all o the calculated timings seved only as a oint o deatue in iting the ok While comosing I tied to match the stuctue that I ill descie hee, ut I ound that musical intuition dictated some eedom ithin this lan Also, in the old o acoustic instumental eomance, these indicated timings could neve e achieved exactly, due to the imeect natue o a hunded musicians oking togethe As a comose I do not think that this inevitaility negated the value o the stuctual lan Béla Batók, o examle, indicated timings in the scoes o many o his oks, hich he cetainly kne ould not e eectly attainale y all eomes o me at least, uilding the stuctue o the conceto this ay ensued that the tenty minutes o sound that I as ceating had a logical ameok ootions and Mas Alication o the Golden atio to the Movement Lengths I decided that the ok should theoetically last 1,201 seconds, o tenty minutes and one second (20 01 ) Using this ime nume as the total duation alloed each o the thee movements to have a ime duation also: 599 seconds (9 59 ) o the ist movement; 229 seconds ( 9 ) o the second movement; 7 seconds ( 1 ) o the thid movement In addition to eing all imes, these duations also elate to each othe y the Golden atio (G) To illustate: I calculated the length o the thid movement y multilying the duation o the ist movement y the golden mean: 599 x 01 = 7012 (the neaest imes ae 7 and 7) olloing the same ocedue ith this ne duation, the length o the second 12

19 movement ecame: 7 x 01 = 2051 (the neaest imes ae 229 and 2) To Simultaneous G Mas To make smalle divisions ithin each movement I ceated to indeendent layes o mas, one o the ogan and one o the ochesta, again using the G The divisions in the ochesta s laye came om successive multilications o the esultant duations y 01 To claiy: eginning ith the total duation, I calculated the ist oint this ay: 1,201 x 01 = 7221 (ounded to neaest ime: 7) the next oint similaly: 7221 x 01 = 59 (ounded: 57) and so on o eleven moe iteations I then mioed this sequence toads the end o the iece, alays ounding to the neaest ime, so the comession o duations ould also occu at the end ounding the numes esulted in an imeectly symmetical sequence o oints thoughout the length o the iece, since thee ae ee oints that ae ime in the aea o 1,201 (See Aendix B o the stuctual mas o each movement) The ocess ust descied oduced the ogan s laye o divisions also, ut ithin each movement individually, and ithout the mioing By only alloing the ocess to un in one diection, I could decide hethe the values ould incease o decease toads the end o each movement Theeoe, the ist and thid movements comess and the second movement exands One inteesting esult o comaing the to layes is that the to main oints in the ochesta s ma coincide ithin ou seconds to oints in the ist and second movements in the ogan s ma These coincidences suggested ossile mao aival oints in the music Aat om these to connections eteen the layes, and ou at the end o 1

20 the ok (in Movement III; see Aendix B), the layes emained indeendent The ist o these unctues, in the ist movement, aeas eteen 7 7 and 7 1, leaving 2 1 eoe the scheduled doule a I decided that this unctue should mak the eginning o the cadenza This lan oked successully since the length o the cadenza in elation to the est o the movement sounded aoiate The second coincidence eteen the to layes aeaed at the G o the second movement (hich is also the G o the hole iece), and seves to initiate the climax Oiginally I thought I ould kee the ogan and the ochesta mateials indeendent, like thei esective mas As I got uthe into the comosition, hoeve, this oved to e imactical I anted moe cohesion eteen the soloist and the ochesta, so the to layes ended u contiuting to the hole musical textue, egadless o thei oiginal ailiation Intenal Stuctue o Each Movement Movement I In the ochestal ma o G oints, the ist six occued so close togethe that they could not seve to mak section changes that ould e musically aoiate o the eginning o the iece These aea in the timani as lo -sha attacks Susequently, evey oint in the ochestal ma maks a change o section Hoeve, since the last to sans eoe the cadenza make u hal o the time o the movement, and I anted some smalle units to ok ith, I needed a logical ay to sudivide them The ist san that I divided ( ) as sixty-ou seconds long I sectioned it y multilying the duation y the smalle at o the G (02): x 02 = 2 (o 25 ounded) This ceated a unit o tenty-ive seconds olloed y a unit o thity-nine seconds 1

21 The next san ell eteen 17 and 2 (1 50 in duation) To divide it I used a unction that ould give me successively smalle sections: N = 17 + (01 x d), hee d is the cuent duation eing divided, and 17 maks the oint at hich these sudivisions ould egin The esults ould indicate oundaies eteen sections on the time ma o examle, the ist sudivision as ceated in this manne: 17 + (01 x 110) = 209, o 21 (the neaest ime) That let sixty-eight seconds, hich I also sudivided: 17 + (01 x ) = 21502, o 211 Ex 1 shos this ocess gahically (01 x ), then 17 + (01 x 110), then 2 ounded to neaest ime ounded to neaest ime Ex 1 Sudivisions eteen 17 and 2, Movement I I used this same aoach in the next san (om 2 to 57, hich is 17 long) ut modiied the unction to ead: N = 57 - (01 x d) The dieence hee is that the 57 eesents the end o the time to e divided, and that the oduct is sutacted om it At the end o this ocedue I had ceated ive units, olloing hich ould egin the cadenza (see Ex 2) At 57 the ochesta eaches its aival eoe the cadenza, and holds that chod o ou seconds, at hich time the ogan s cadenza egins 1 The ma o the 1 These ou seconds ae the dieence eteen the closely coinciding G oints in Movement I (see Aendix B) 15

22 ogan om this oint to the end shos six units ased on the calculations descied ealie o the to seaate mas Since they comess in size toads the end, they seved as good eeences o changing the mateial thoughout the cadenza y oviding a sense o acceleation (See Aendix B) 57 - (01 x d), hee d = 17, 10,, successively, 2 then ounded to neaest ime Ex 2 Sudivisions eteen 2 and 57, Movement I Movement II The second movement has only one G oint in its ochestal ma This oint eesents the G o the entie duation o the ok, and as such, it lays an imotant ole in the movement in that it maks the eginning o the climax The ist thee G oints in the ogan ma aea too close togethe to seve as sectional divisions, ut unction ell as oints o change in the ue stings The emaining oints in the ogan s ma indicate ne sections though metic modulations Movement III In the thid movement I had lanned on using the to mas to ovide oints o emhasis o the ogan and the ochesta I also kne that the movement ould e a assacaglia The gound ass ended u eing exactly 20 long, hich meant that I could it eighteen o them in the allotted time ith 1 let ove This shote it ecame 1

23 the intoduction The ass and sixteen vaiations ollo, leaving the last 20 unit as the coda I kne ealy on that 20 ould e too shot o the coda; as it tuned out, it exanded to aout 1 1 Ealy in the comosition o the movement I also intended to incooate the G oints om the to mas Hoeve, seveal months late, ate ecomosing seveal o the vaiations, I ealized that it ould e imactical to go ack and imose a conlicting stuctue on the music Theeoe, the to mas do not inluence the stuctue o this movement Vaiances in the End esult At this oint in the discussion, it ould e intiguing to examine ho the inal om o the music diveged om the caeully lanned stuctual mas Thee ae to mao instances hee the music ended u longe than lanned, these occuing at the end o oth the ist and thid movements Othe than that the music emained close to its oiginal lan Movement I: Cadenza The cadenza came togethe aily easily accoding to the lan, ut I elt thee needed to e a deatue om the ma in ode to ceate a satisying end to the movement o the ist to units o the cadenza ( and ) the temo is MM=0 It aeaed that this temo ould ok o the est o the cadenza also Hoeve, ate comosing the next thee units (22,, and 10 ), hich as the edals solo, it seemed to me that the music moved too quickly To make it ok I deceased the temo to MM=52 and let all o the notes intact, hich inceased the oveall length o this section In addition, the movement as not comlete at the end o the last unit; it still equied iteen moe eats (o aout 17 ) o the conclusion to sound logical In total, the cadenza ge y 25, om a lanned 2 1 to 2 That as accetale to 17

24 me, hoeve, since lanned stuctues must e ale to e changed to seve the good o the hole Movement III: Coda As in the ist movement, the changes to the oiginal lan o the last movement came at the end Hoeve, in this instance I kne ate ceating the ma that the ending ould equie moe time than I had allotted I had no systematic ay o extending the duation, unlike in the ist movement, so I comosed until it sounded comlete As a esult, the 20 allotted to the coda ge to aout 1 1 As I have alluded to, my divegence om the oiginal calculated lan came aout o aesthetic easons The music did not have enough oom to evolve musically in the igid, caeully lanned stuctue ohn Winso ans against olloing a system too stictly: Systematic aoaches to comosition can, indeed, oduce elegant stuctues, ut they ae not likely to e musical stuctues Even i a systematic aoach is tentatively alied, the comose s choices must e temeed y aual sensitivity What e discove uon election is that the musical ea imaily a unction o the atten-ecognizing and atten-geneating intuitive mode does inom ou choices ith ode ode that is as imessive as any that can e imosed y mathematical omulae and hich is decidedly moe musical 2 Even though he seems to elieve that comoses should not utilize systematic aoaches to comosition, his assetion that thei musical ea must inom comositional choices suots my comositional methodology 2 Winso, Beaking the Sound Baie,

25 CHATE GENEAL MAING OCEDUES In using geneative ocesses duing the ceation o a musical ok, the comose must detemine ho the esults ill ecome music ehas the most asic ay to do this is to ceate a one-to-one coesondence ith the vaious musical elements that ae eing geneated o me this equently oves insuicient, esecially ove lage duations (seveal hases o hole sections) I ee to limit a aticula maing scheme to single hases o eiods, ut as I ill sho late, they occasionally detemine longe sans also ist, I ill intoduce geneal ocedues o conveting data to music, ith an emhasis on hat I used in comosing the conceto Music om Numes Numes as itch At some oint in the comositional ocess the comose needs to decide ho to convet the geneated data (numes in this case) into itches To ossiilities come to mind ist, the numes can eesent seciic itch classes Deending on hethe the esultant music should e diatonic, chomatic, o some othe collection, the comose can tansom the data to it the size o the desied set o itches The second ossiility involves using the numes as distances eteen successive itches (ie, intevals) To geneate itch classes, the comose can eithe aly a modulus, o convet the numes to a ase othe than 10 o a diatonic set, ocessing numeic data mod 7 ovides only the values 0 though no matte hat the size o the oiginal o examle, i an algoithm oduced a 5, ocessing it mod 7 ould oduce a 5 19

26 (5 / 7 = ith a emainde o 5), hich could then seciy the ith element o the diatonic set ( A in the case o C mao, hee C = 0 ) The same 5, i conveted to ase 7, can also oduce a non-diatonic set In ase 7, each lace in the outut can contain only a 0 though, in this case esulting in 15 As itch classes (hee C = 0 ), this outut ecomes the sequence C-sha, E, -sha, These itches could then e used melodically o hamonically As mentioned in Chate 2, I chose to ok ith nume ases athe than moduli ecause I as inteested in the decimals as stings o individual numes My oiginal idea as to see ho I could ceate a hase ith one decimal sting (o moe accuately, one emutation o the elated caousel nume) Since I as comosing in a chomatic context, I equently conveted stings o digits om ase 10 to ase 12 o examle: the asic eeating segment o 1/17 is I emoved the decimal oint and the zeo to its let, hich let the caousel nume (the leading zeo is necessay) and conveted that to ase 12: 055ee59e15 (See Chate 2, note 5 o an exlanation o the lettes in the nume) This ocess esulted in a sting o 15 itch classes, each o hich could e ositioned in any octave and still conom to the system Anothe method o geneating itch om numes is to use the numes as intevals, hee 1 = mino second, 2 = mao second and so on 1 Whee using the numes as itch classes gives the comose choice o octave lacement, y using the numes as intevals the comose has the choice o inteval diection (ie, u o don) Conceivaly, this choice could also e detemined y an algoithm, although I ee to make those decisions mysel; it allos me to shae the line moe intuitively 1 This method seciies the distance eteen to successive itches It does not seciy inteval classes, although a system o inteval classes could e ceated as ell 20

27 Examle 1 comaes the outut o the to methods, oth using 1/17 ase 10 as the souce mateial They oth oduce inteesting lines, hoeve the one ceated om inteval maing aeas in the conceto hile the itch-class maing does not 1/17 ase 10 as intevals 1/17 ase 10 as itch classes n n Ex 1 Comaison o inteval and itch-class maing in a melodic line As I comosed the iece I ound that I geneally eeed melodic lines and chods that came om using intevals ove those that came om using itch classes I elt that the esulting lines o chods elt moe oganic, hile the itch-class method sounded too atiicial Hee, musical intuition temeed hat could have ecome an imossile system otunately, maing intevals oked ell; otheise my musical intuition may have equied a comletely dieent aoach to iting the conceto Numes as hythm As ith maing algoithmic outut o itch, numes can have a one-to-one elationshi ith duational values By using sixteenth notes as the asic unit, a 1 ecomes one sixteenth-note; ecomes an eighth-note (to sixteenths) and so on o examle, i a comose took the 15 om the outut o the hyothetical algoithm mentioned ealie and alied it also to the hythm o the geneated itches (om the convesion to ase 7), the esult ould e a gestue like hat aeas in Ex 2 Any 21

28 othe duations could also e used as the asis, such as eighth-note tilets; o dieent duations could e mixed o this system to geneate longe hases the comose ould need to have a lot o numes o maing Examle shos the ceation o a hase using the decimal o 1/17 ase 10 o itch and duation maing, and laced in a metic context itch: 5 ase 7 (15) as itch classes hythm: 15 as sixteenth notes n Ex 2 Comination o itch and hythm maing itch: 1/17, ase 10: ; numes as intevals, diections ee (0) (0) hythm: 1/17, ase 10; numes as sixteenth notes [1] Ex hase geneated ith a long sting o numes Using altenate nume ases ceates moe hythmic ossiilities, and also allos the comose to limit the nume o duations in a aticula hase In ase 10 thee ae nine ossile duations lus a zeo (hich is useul o ests) 2 In ase, hoeve, thee ae only thee ossile duations lus many moe zeos This eduction o ossile duations lets the comose ceate moe hythmically integated hythms The melody in Ex uses the numes om 2/17 as a asis, ith intevals 2 Nine is a signiicant amount, consideing that Mozat, o examle, used only to duations, quate and eighth, o most o the ist theme in the ist movement o his Symhony no 0 22

29 coming om ase 5 and hythmic duations om ase This esents a moe homogeneous hythmic and intevallic chaacte than it ould using lage ases In the Ogan Conceto, utilizing the dieent nume ases ovided a ealth o mateial o hythmic geneation itch: 2/17, ase 5: ; numes as intevals hythm: 2/17, ase : ; numes as sixteenth notes 1 n 1 7 Ex Melody using dieent nume ases Numes as Othe Quantiiale Musical aametes Nume seies can also ovide maing ossiilities o othe musical elements o examle, numes could detemine the sizes o stuctual comonents such as hases o sections hen maed as eats o seconds, o some othe unit that the comose desies In his ook Simle Comosition Chales Wuoinen exlains ho comoses can aly the telve-tone system o comosition to om The ocedue ith these numes ould e simila (I did not ceate the omal divisions o the conceto this ay, hoeve Instead, I used the golden atio calculations that I descied in Chate ) Anothe maing ocedue could detemine the lacement o accents o dynamics Hee the comose could indicate hee accents o dynamic changes ill occu y using the numes to indicate duations eteen changes In the conceto it See Chales Wuoinen, Simle Comosition (Ne Yok: Longman, 1979; eint, Ne Yok: Schime Books, 19), Chate 11 2

30 elates closely to my ocess o geneating hythm (see Chate 5 o a moe detailed exlanation) A seies o numes could also deine tansosition levels o lage sections, such as o the eetitions o the gound ass in a assacaglia In the conceto I detemined the tansositions o the assacaglia theme in Movement III y using /17 ase ( ) as itch-classes I alloed mysel to sustitute one o the unused itch-classes (, 9, t, e) o one o the values in each ai o eeated numes (11,,, 00) olloing this ocedue the ass tansosition ecame 2917te o the theme and the sixteen vaiations Comositional esonsiilities Excet o the use o nume systems to oganize the vaiations o Movement III, my choice o a seciic decimal and nume ase at any given time in the music aose om actical and aesthetic consideations This meant that I chose them ased on the amount o time that needed to e illed (o hythmic and melodic consideations), and hat sounded good (usually ate seveal oo choices) Only in the assacaglia did the numes detemine hich actions I ould use long in advance o all othe action and ase choices it took a lot o tial and eo to ceate usale mateial But that is eally no dieent om comosing uely om intuition in my exeience I should note that maing the same nume seies could oduce a myiad o distinct ieces To illustate, comae Ex, hich ended u eing athe diatonic, ith Ex 5 I used the same set o numes o intevals and hythm in each examle and only vaied the inteval diection, hich oduced adically dieent esults I ased the choices I made in Ex on a cetain sound that I as looking o, hile those o Ex 5 ee moe andomly chosen This elects the ocess o the hole 2

31 comosition As I have mentioned eoe, I allo mysel the eedom to susend any system that I set u i I eel that the system is inadequate in some ay Musical intuition dives the decision to evise the ocedues, o to adust the outut eely until the esults ae accetale itch: 1/17, ase 10: ; numes as intevals 5 hythm: 1/17, ase 10; numes as sixteenth notes n Ex 5 Altenate vesion using the same ocedue as Ex It has oaly ecome aaent that I give mysel quite a it o eedom in deciding hat to use and ho to aly it While a lot o hat I have discussed so a could e ogammed into a set o instuctions o a comute, the emainde o this ae ill demonstate that the inished om o the conceto could not have esulted om a sel standing seies o algoithms My method o comosing is ee than seial comosition in that I eak the ules (in this case my on ules) moe equently than stict seial comosition allos As ith any comositional method, I stive to ceate music that is satisying in and o itsel o me, the limitations inheent in any comositional system add inteest to the ocess o comosing, i not to the esulting music as ell 25

32 CHATE 5 STAIGHTOWAD MAING In Chate I shoed some ossile maings o nume seies into elements o music As I as comosing the conceto, I alays stove to use as e stes as ossile in geneating the inished music, and in some cases I succeeded in ust one o to stes Hoeve, those eesent the minoity o cases Most o the itch geneation ent though seveal hases to aive at usale mateial In this chate, hoeve, I ill sho some o the moe staightoad maings that ecame at o the conceto ecussion and the Binay Sequence The simlest alication o geneated mateial involved the ecussion, since a itch dimension equently as not needed o it I ound that the dieent ase 2 (inay) seies oked ell o this uose, and I comosed a lot o ecussion mateial this ay, aticulaly the intoduction o the thid movement In comosing this at I anted to ty sueimosing seveal inay steams Stating ith 1/17, I decided to use as many additional actions as I could Ate vetically aligning the nume steams I noticed that 1/17 and 2/17 had exactly the same sequence o zeos and ones (ut oset y the leading zeo that as necessay o all o the ase convesions o 1/17) Uon close insection I noticed that o the actions ith the numeatos 1, 2,,, and 1 the inay sequence o digits as the same, ith an exta zeo added o each highe action (see Ex 51) 1 This conguence oved to e useul ecause I could assign the 2/17 and the /17 sequences to the same ecussionist laying to dieent instuments at the same time, and the /17 and 1 These elationshis occu ecause the numeatos ae elated y B n (B to the nth oe), hee B is the nume ase and n is any hole nume 0 and lage See Aendix C o a moe detailed exlanation o the mathematics involved 2

33 /17 sequences to anothe ecussionist Examle 52 shos ho the music looked eoe editing In this instance all o the ones in the sequence ae sixteenth note attacks and all o the zeos ae sixteenth note ests 1/17 ase 2: /17 ase 2: /17 ase 2: /17 ase 2: /17 ase 2: Ex 51 elationshi eteen numeatos elated y oes o 2 hythm: 7/17 hythm: to, 1/17; ottom, 5/17 hythm: to, 2/17; ottom, /17 hythm: to /17; ottom, /17 Tim Timales Tom-toms Snae D Teno D Ex 52 hythmic geneation om inay sequences I also maed anothe laye onto the oening ecussion textue At ist I anted to take the action that geneated each instument and use the same inay sequence o accents Hee the zeos in the sequence ould indicate skied notes 27

34 (that ae aleady in the textue), and nothing ould e counted ove the ests Hoeve, I did not like this aangement once it as comleted ecause thee ee too many accents A lage ase o each action solved the olem I assigned ase to geneate all o the accents, using the actions 7/17 o the timani, 5/17 o ecussion 1, 2/17 o ecussion 2, and /17 o ecussion Zeos meant no accent on a aticula note; a one indicated an accent Since ase also includes the values o to and thee, I decided that a to ould mean no accent on the second at o the digit, so that a 2 ould e identical to the sequence 1,0 Likeise, a 1,0,0 and 2,0 and ould all mean the same thing I ignoed all o the exta values at the end o the ase stings See Ex 51 o the ay the accents maed on to the hythms The Oiginal Main Theme o Movement I Examle shos a melody I geneated using 1/17 ase 10 o oth inteval content and hythm It took many adustments to the intevallic diections o me to aive at its inal om Once it as comlete, I elt that it needed to aea somehee in the conceto o a hile I consideed it to e the main thematic mateial o Movement I Eventually, ate tying unsuccessully to incooate it into the movement, I egan iting the movement again and let it out Hoeve, I did not ant to aandon it altogethe, ecause I liked it and it seved as an excellent model o this ocess o comosing It ust as not consistent ith the diection the conceto as taking I eventually decided that I ould kee the contou o the melody (the intevals and thei diections), discad the hythm, and then see ho I could extend its content This esulted in the set o tansomations as shon in Aendix D I include the aove examle in this discussion ecause it illustates the validity o oth Wuoinen s and Winso s somehat oosing vies on comosing ith 2

35 systems I had to allo the integity o the conceto as a hole cay a highe ioity than my attachment to a eect secimen The Ne Oening o Movement I I have shon ho I used numes to ceate melodies and thei hythms, ut I also used the system o lage sections, notaly the oening o Movement I Hee, a nine-meme chod uilds u one itch at a time (the constuction o the chod itsel is discussed in the next chate) I detemined the timing o the itch enties ith 1/17, ase 10 ist, I dealt ith the disceancy eteen the nume o elements in the chod and the sixteen laces in the decimal The decimal includes all ten digits (zeo though nine) and most o them aea tice I assigned the digit 9, hich only aeas once, to a ecussion event, and each o the emaining digits to one o the memes o the chod Stating ith the loest note o the chod, I numeed them all uads, and assigned thei ist enty to aea hen thei eesentative nume aeas in the decimal (see Ex 5) The nume I assigned to each itch also eesents the duation in eighth notes eoe the next enty aeas The one excetion is the oening lo -sha ( 0 ), hich lasts o eight eighth-notes eoe the next enty o those digits that occu moe than once I alied an ochestational comonent to add inteest to the textue This ocess esulted in a slo, intoductoy oening to the movement Because it sounds intoductoy, I decided to undemine that ecetion y giving it moe imotance thoughout the movement This had no elationshi to the algoithmic ocesses I used, ut came om musical intuition 29

36 Chod A ecussion event Ex 5 Chod element enty ode at the eginning o Movement I Chod ogessions in Movement II Stings at the Beginning o Movement II Anothe elatively staightoad maing aeas in the constuction o the chods at the eginning o the second movement Ate tying a e sixteen-digit nume seies that had ceated a lot o dulicate itch classes, I ound that 1/17, ase 9 oduced a usale seies o chods I used the numes to detemine the intevals eteen each chod meme om to to ottom, excet o the last chod A common E in the ist thee chods and a common G-sha in the last to link them togethe as a ogession (see Ex 5) This method o ceating chods oved so successul that I geneated nealy all o the chod ogessions in the iece using simila ocedues itch: 1/17, ase 9: ; numes as intevals n 1 Ex 5 Chod constuction at the eginning o Movement II 0

37 hythm in the Wind Chods and in the Ogan To ceate the hythm o the ind chods in Movement II (mm 21-2), I alied 12/17 ase 2 as sixteenth notes (hee 1 = attack and 0 = est) to the six chods I had ceated o this at The chods each eceived a dieent nume o digits om the inay sting (, 7,, 9, 11, 12) Hoeve, this does not indicate the nume o hits each chod eceived, since many o the digits ae 0s The ig hole toads the end o the seies (eight 0s togethe) needed to e illed It seemed to me that the ogan could e-ente at this oint, so I eely ooed om the ogan at in mm 52-5 This led seamlessly to the oint hee the ogan icks u on the hythmic idea that the inds ust inished I then took the chods om the ochesta and ut them in the ogan in evese ode and ith dieent tansositions and common tones I also emedded a line ased on the mateial om mm I needed sixty-ive sixteenth-notes to ill the section u to the temo change, so the hythm o the chod hits (seaate om the emedded line) came om 12/17 ase 11 Hee, the numes eesent the time, in sixteenth-notes, eteen each aeaance o the chod As eoe thee ae six chods, o a total o iteen hits, so each chod gets a dieent nume o digits om the sting They ollo the atten 2,, 2, 1,,, hich I had eely detemined The Climax o Movement II (at 1: hythm) The climax o Movement II involves a laye o staightoad maing and a laye o convoluted maing The olloing discussion exlains ho the method o hythmic maing seves as a oundation o the moe comlex ocess o itch geneation, hich is coveed in Chate hythmically, the climax o Movement II is staightoad Unlike othe sections o the conceto, I did not kno eoehand ho long this section should e I kne 1

38 hee the climax ould stat, ut since it initiates the last section o the movement, and I did not ant it to ill the entie section, I needed a ay to detemine its length By using 7/17 ases 2,, and in succession, ith the eighth note as the asic unit, this section ecame 25 long 2 The maing ocedue o all ases as as ollos: 0 = ecussion event, 1 = ochestal event (non-ecussion), 2 = ogan manuals, = ogan edals When 2 and aeaed I sudivided them so that consisted o 1 and 2, and consisted o 1, 2, and (see Ex 55) The diectness o this system esults in its eing moe audile than most o the ocesses involved in the conceto At the time I came u ith this lan, I tusted that the system ould oduce something usale I only had one chod, hich I kne ould stat the section, so at that oint I needed to geneate itches (see the ist section o Chate ) As the itch content o the section came togethe, I ealized that the ocedue o geneating this section had oved to e successul ec 5 7 Og Och (1 2 ) (1 2) 2 N (1 2) 2 (1 2) (1 2 ) (1 2 ) (1 2) 2 (1 2) 2 1 (1 2) 2 7/17 ase : Ex 55 hythmic maing in the climax o Movement II 2 The entie climax is in duation Ate 25 thee is an inteution, lasting o 1, hich is olloed y a estatement o the 7 eceding the inteution 2

39 CHATE CONVOLUTED MAING The mateial coveed in this chate eesents moe comlex alications o the nume systems descied ealie I ould say that none o the mathematical ocesses that I discuss hee ae diectly audile in the music (even less so than the mateial in the evious chate) Hoeve, thei inaudiility does not detact om thei comositional imotance to the conceto The Climax o Movement II (at 2: itch) The comlex ocedues that I olloed to geneate the chod sequence in this section ae oveshadoed y the staightoad hythmic maing, and its audiility, as shon at the end o Chate 5 Nevetheless, the maing o the itch mateial oved to e equally successul I used the initial chod om the ist movement ( chod A exlained in the next section) to egin the climax o this movement, and tansosed it u a semitone so the ass ould e a G This as an intuitive decision to highlight oth the itch and the eginning o the ne section, since G had not eceived any eight so a in the ass To extend the sequence, I constucted to additional chods ased on 7/17 ase 9, hich it nicely into the ase 2 hythm The thee chods o this ist at no aeaed 5, 11, and times, esectively, to ill the tenty-ou 1s in the ase 2 sequence Seven moe chods ee needed to get though the hythm, so I made thee om /17 ase 9 (as intevals, stating at the to) These I laced in the tentyto ochesta hits that comise the 7/17 ase hythmic section (mm0-), still ith the lo G edal oint They aea 9,, and 5 times, esectively (moe o less eely chosen) o the 7/17 ase hythm (mm 7-9) I emloyed the ou chods

40 om the eginning o the movement (in the stings), ut ith a ne tansosition, and ne elationshis in the ogession (see Ex 1) o thei tenty-one necessay hits I decided on the sequence 7,, 5, and, esectively o 7/17 ase 2 otion o 7/17 ase otion o 7/17 ase otion Comae ith ex 5 n (+) 2 (+2) "chod A" 7/17 ase 9, ist hal (+) (+2) (7+) (5+12) 7/17 ase 9, second hal N /17 ase 9, ist thid /17 ase 9, second thid n (+2) (1+2) (7+12) /17 ase 9, inal thid n n dieent common tones than the ogession at the eginning o the movement Ex 1 Ochestal chod constuction o the climax o Movement II This chod ogession came aout lagely though intuition Cetainly an autonomous algoithm could have oduced the maing o the itches into these chods But I ould not have knon ahead o time, in ogamming the algoithm, hich itches to leave out (as indicated y instances such as + ) I ould have had to edit the outut heavily in ode to get the chods to this state Also, I oganized the chods into an accetale ogession much moe easily y oking ith encil and ae than I ould have i I had had to ceate a ogam to do it In shot, this illustates that, o me, oking ith a elatively minimal amount o extenal data and a lot o musical sense oduces esults moe eiciently than elegating most o the decisions to a system o the ogan s chods duing the ase and ase hythmic sequence, I inveted the diection o the intevals in the ochestal chods (see Ex 2) When the ase otion egins, the ogan edals ente the textue heneve thee is a and the edal notes come om an aoiate itch om chods in the manuals (see Ex