Hybrid Musical Instruments: Artistically Inspired, Mathematically Explored

Transcription

1 Hybid Msical Insmens: Aisically Inspied, Mahemaically Eploed APPROVED: Thesis Dieco Honos Commiee Membes Dean o he College Vice Pesiden o Academic Aais

2 Hybid Msical Insmens: Aisically Inspied, Mahemaically Eploed Thesis Pesened o he Honos Commiee o McMy Univesiy In Flilmen o he Reqiemens o Undegadae Depamenal Honos in Mahemaics By Tammy Wene Apil 10, 008

3 Acknowledgemens I wold like o hank he mahemaics depamen o all hei had wok and encoagemen hogho he wiing pocess. A special hank yo goes o Cindy Main o he eos o psh me hogh lae nighs and npodcive days. I also hank he a depamen o giding me hogh and allowing me o ceae many les o gahe daa. I wold also like o hank my iends Lindsey Ra, Kenley Meye, Laen Yeae, and Elizabeh Richey o ping p wih my high sess amosphee his pas yea, bibing me wih ice ceam, and saying p wih me lae ino he nigh on mliple occasions. I don know wha wold have happened wiho hose gils by my side. 3

4 Absac Msical insmens ae iniging boh mahemaically and aisically; hei oms and he msic hey podce ae beail. Thosands o yeas ago hmans ecognized he beay o msical sond b i is only in ecen imes ha scholas have begn o eploe he mahemaical beay o msic. This hesis is ocsed on he mahemaics o ceamic insmens. This emphasis is inspied by my medim o choice in a, which is clay. Clay is iniging becase o he opponiy o change dieen vaiables in each insmen. Thee ae seveal qesions ha his hesis will aemp o answe: Wha eqaions goven a amily o insmens? How do eqaions govening ypical insmens change wih hybids, insmens ha combine aspecs om mliple amilies in one? Wha do hese eqaions model? This hesis sas by eplaining whee mah is ond in msic. Then i saes some basic physics eqaions, hei se, how o se hem, and wha hey model. The onedimensional wave eqaion, which govens he eqencies o wind and sing insmens, is hen eploed. The solions ae deived and an eplanaion o hei se is given. Impoan esls ae eploed. The same is done o he wo-dimensional wave eqaion, which govens many pecssion insmens. This hesis will also seek o answe he above posed qesions o his ype o insmens. 4

5 Conens 1 Mahemaics and Msic 6 1.1Hamonics, Oveones, and Ocaves 6 1. Scales, Beas, and Mee..9 Classes o Msical Insmens and he Dieenial Eqaions Govening Thei Feqencies 14.1 Winds.14. Sings 0.3 Dms.5 3 Ceamics 33 4 Hybid basics and Ideas o eqaions modeling hybids 38 5

6 Chape 1 Mahemaics and Msic Mah can be ond in mos aspecs o msic. Some aspecs o msic ha have been inclded in mahemaical analysis ae: hamonics, scales, beas, hyhm, mee, and sond. Once one has a gasp o he mahemaics o he above-menioned componens o he msic o each class o insmens, eploing he highe ode mahemaical eqaions ha model hybid msical insmens is a naal ne sep in his analysis. 1.1: Hamonics, Oveones, and Ocaves Hamonics ees o he dobling, ipling, and ohe whole nmbe mliples o basic eqencies (Josephs 57). The hamonics ond in msic om any insmen ae eqenly analyzed mahemaically. Seveal es se he wod oveone in discssing hamonics. An oveone is deined as any highe eqency vibaional mode o a sysem eclding he ndamenal, which is he lowes eqency mode (Whie 115). Thee ae seveal oveones, which ae podced om an insmen. The is oveone is he second hamonic, he second oveone he hid hamonic, and so on. When a msician moves p an ocave, he eqency o he noe being played is dobled. When he msician dops an ocave, he eqency is halved. Fo eample, he ndamenal eqency, o is hamonic, o he noe A is 440 Hez (Hz). Moving p an ocave makes he eqency 880 Hz. Mahemaically, hee is a elaionship beween ceain hamonics and ocaves. Moving p one ocave dobles he eqency o he is hamonic. Moving p wo ocaves om he is hamonic implies ha one wold doble 6

7 he eqency o he is ocave. I have ceaed he ollowing heoem, which will allow o ease in calclaing he eqency o a given ocave. Theoem 1.1.1: I λ is he eqency o a given noe, he eqency o a noe n ocaves highe wold be n. Poo by mahemaical indcion: Le P( n) {I is he eqency o a given noe, he eqency o a noe n ocaves highe n is.} Then show o P (1). Moving p 1 ocave dobles he eqency, so he eqency o a noe one ocave highe is 1. Assme (n) P is e. Then show o P ( n 1). The h ( n 1) ocave is one ocave highe han he h n ocave. Since (n) P is e, he eqency o he h n ocave is n. Moving p one ocave om P (n) dobles he eqency, which means he eqency o he h ( n 1) ocave is n n 1 ( ) P( n 1). Hence, by mahemaical indcion, he saemen is e o he se o naal nmbes N. Each hamonic has a coesponding wavelengh. The second hamonic has 1 he wavelengh o he is. The hid hamonic has 3 1 he lengh o he oiginal wavelengh. This paen epeas. This elaionship, which will be discssed lae, leads o he obsevaion ha eqency is invesely popoional o he wavelengh. In geneal, waves ha have an amplide and peiod can be elaed o sine and cosine waves. I one looks a a geneal wave whee L is he lengh o he sing o ai colmn and he amplide is 7

8 aken o be one, he geneal eqaion o he wave is n sin whee n is he n h L hamonic. This will become moe signiican as he wave eqaions ae discssed. The ollowing able liss he eqencies o he is eigh hamonics o conce A (440 Hz), he coesponding wavelengh and sine ncion. Hamonic Ocave Feqency (Hz) Wavelengh Eqaion sin L sin L sin L sin L sin L sin L sin L sin L Table Fis Eigh Hamonics o Conce A 8

9 1. Scales, Beas, and Mee An ocave is he only ineval ond in all scales (Josephs 78). A scale occs when a paen o whole and hal seps is omed (Band 5). This paen ypically akes on aios beween he noes in he scale. Seveal scales ae common enogh o have been named: he chomaic, he penaonic, he Pyhagoean, he js and he empeed. The chomaic scale moves by hal seps om one one o is ocave, welve seps away (Casellini 136). Since he noes in his scale ae eqidisan, he msician has he choice o which noe hey will sop playing. Anohe eqidisan scale is he whole-one scale (137). I conains si whole seps, which will ake i o he ne ocave. The penaonic scale is sally ond in msic o he wold om aniqiy. This scale ses ive black keys o ive whie keys o a piano played in any ode. Mch o Wesen olk msic is played in his scale (Casellini 9). Pyhagoas developed a msical scale ding his ime (sih ceny BCE). The Pyhagoean scale was a simple ecipe o ceaing new noes om he old ones. He ook he eising aio and mliplied o divided by 3, which is he same as aising o loweing he noe by a ih. Whaeve nmbe was goen om his pocess was eihe dobled o halved. I he nmbe was geae han, i was halved. I i was smalle han 1, i was dobled. (Johnson 7). Polemy epanded pon Pyhagoas heoy. Polemy s sysem was epanded o ive insead o sopping a o. His wok becomes he majo hid. He ond ha i yo pick a noe and aise i by boh a hid (5:4) and a ih (3:), yo obain a majo iad (Johnson 1). Pleasing msic canno be ceaed based om mah. Howeve, msic and msical paens can be mahemaically analyzed. 9

10 Pyhagoas, a mahemaician, based his msic heoy solely on mah. Today, his msic wold sond displeasing o mos. The js scale, o inonaion, ses whole nmbe aios beween eqencies on he scale (Benon 173). This scale is a combinaion o seven whole and hal seps (96). I was noiced ha ceain combinaions o noes wee aally pleasing when sonded ogehe. The majo scale is omed based pon hee noes having a aio o 4:5:6 (Josephs 81). The mino scale is also bil p on a iad. The aio o hese noes is 10:1:15. The majo scale shall now be deived. The ollowing able liss he aios beween noes in a given ocave saing a C, which has a eqency o 64 Hz. Noe C D E F G A B C Raio 1 9/8 5/4 4/3 3/ 5/3 15/8 /1 Decimal Table 1..1 Noe Raios These aios ae sed o calclae he eqencies o he noes in he scale bil om one o he iads. I he noe saing he scale is dieen om C, he same aios will be sed o each o sbseqen noes. C Majo D Majo E Majo C 4 64 D E F G A B C D E Table 1..: Js Majo Scale 10

11 Noice ha some o he eqencies ae spesciped wih a. This denoes a eqency, which has been onded p o down o he appopiae noe. This onding is done when a noe in he majo o be played canno be eached peecly ned o a dieen majo. Conside a msician who needs o play a noe having a eqency o 445 Hz in he D Majo. He inconvenienly has an insmen ned o he C Majo scale. He aives a an appopiae solion by playing he noe in C Majo closes o he eqency o be played, A in his case. The same ollows o he ohe noed eqencies. This adjsmen gives ise o he eqal empeed scale. The eqal empeed scale is omed by eqally dividing he ocave. While he ohe scales accon o a speciic se o noes, he eqal empeed scale allows o a geae vaiey in a msical piece. The change om eigh noes in an ocave o he js scale o welve in he empeed scale oces he aio vale o be ecalclaed. This change can be mahemaically inepeed. One needs o ind he aio beween each o he noes. Since he noes ae eqally spaced, he aio is whaeve nmbe imes isel welve imes eqals wo, o he noe in he ne ocave, o n n n n n n n n n n n n n 1. Obseve. n 1 n 1 1 n So he aio beween each noe is The able below liss he eqencies o he welve noes in he ocave saing a C. 11

12 Noe Feqency (Hz) C 64 C#, D ь 80 D 96 D#, E ь 314 E 33 F 35 F#, G ь 37 G 394 G#, A ь 418 A 44 A#, B ь 468 B 496 C 55 Table 1..3: Eqal Tempeed Scale When wo insmens play dieen scales ogehe, one noices a bea, o paen, o sonds oming. Beas occ when wo waves wih slighly dieing eqencies avel in he same medim (Josephs 3). The pich ha is peceived, ahe han being one noe o he ohe, is a combinaion o he wo. The bea eqency is ond by he dieence o he eqencies, BF 1. The pich peceived is ond sing 1 (Whie 58). Mee is a paen o accened and naccened beas. Thee ae hee common mees: dple, iple, and qadple. The dple has an accened down bea ollowed an naccened bea which is epeaed. The iple mee has an accened down bea ollowed by wo naccened, and he qadple mee has an accened down bea ollowed by hee naccened beas (Pen 5). Now having some knowledge o he mahemaical aspecs o msic, he basics o sond can be eploed and a basic vocablay gained. Sond waves emied om insmens ae longidinal in nae, moving in he same diecion as he medim, ai, 1

13 in which hey ae aveling. Sine ncions ae sed o descibe he moion o sond waves. This is de o he ac ha a sond wave has an amplide, a peiod, and a basic sine (o cosine) wave, Asin( B D), which is bil om a gaph sing ha knowledge. Amplide o a ansvese wave is he disance om he eeence line o he highes o lowes poin on he wave. On a longidinal wave, he amplide is he maimm displacemen o a poin om is eqilibim posiion (Hasing 10). The peiod is he amon o ime in seconds eqied o make one complee vibaion. The peiod can also be ond i he eqency, nmbe o vibaions pe second, is known sing he omla T 1. While he omlas above goven he happenings wihin an insmen o he simples degee, ohes go ino mch moe deph and ae moe accae. The wave eqaion, c 0, will be sed o he eploaion o each amily o insmens pesened in his pape. 13

14 Chape Classes o Msical Insmens and he Dieenial Eqaions Govening Thei Feqencies.1: Winds The aeophone, o wind, amily o insmens obained is name om he way i is sonded: vibaing a colmn o ai by blowing wind ove o hogh he vibaing elemen (Pen). Thee ae wo sides o he wind amily: he woodwinds and he basses. The woodwinds, which ae sonded when wind is blown acoss a mohpiece wih eihe one o wo eeds, have o sbypes o insmens: les, lageole, single-eed, and doble-eeds. Bass insmens podce sond ceaed by he vibaion o a msician s bzzing lips and emboche, he placemen o lips, eeh, and onge. Fles have a hole, which he msician blows acoss, seing he ai molecles ino vibaion as well as he walls o he insmen. Flageoles ae designed o he msician o blow ino a whisle ype mohpiece ha diecs he ai a a beveled lip. Boh single and doble eed insmens ae sonded by a msician blowing ove hin ecangles o cane vibaing agains each ohe. Thee ae seveal geneal obsevaions based on he physical aibes and vaiables in wind insmens. The pich o he woodwinds is invesely popoional o he lengh o he vibaing colmn o ai. The longe he pipe, he lowe he noe. Likewise, he shoe he lengh o pipe is, he highe he noe. Also, becase he be is hinne han i is long, he noe is aily independen o he diamee. The noe also dops 14

15 an ocave when he end o he pipe is closed (Johnson 4). When he pipe is closed, he waves behave as one hal o he ai colmn, which is wice as long as he be. Along he same hogh, he shape o he pipe, whehe cylindical o conical, hadly aecs he pich. The ai colmn wihin he pipe behaves like a seched sing, wiho bonday condiions, moving along he be. In a le, he waves in he colmn o ai move along he pipe. In eed insmens, he eed vibaes, opening and closing de o he oce o ai, which podces sond. The opening and closing is called he Benolli eec. This also occs in side blown les, closed whisles, and ocainas whee ai, no a eed, is he case o he eec. The ype o pipe podcing he sond, eihe closed o open, aecs he ai colmn wihin he insmen and hence he noe he ea heas (Josephs). Fo any pipe, hee ae seveal known popeies: a node 1 always occs a he closed end while a loop oms a he open end, he ai a he open end always vibaes wih maimm displacemen eqal o he amplide o he wave, and he colmn o ai o a closed pipe esonaes wih odd hamonics. The eason he closed pipe esonaes wih odd hamonics is de o he ac ha he closed end oces he displacemen o zeo (Benson 114). One can see he illsaion in he wo images below. Image.1.1 Displacemen o Fndamenal Image.1. Displacemen o Second Hamonic 1 A node is a poin along a sanding wave whee he wave has minimal amplide. A loop is he pa o a sanding wave beween he nodes. 15

16 Noice ha he hoizonal displacemen o he ai colmn ollows he sine waves illsaed in Table In mahemaics hee is a well-known dieenial eqaion called he wave eqaion. I ns o ha he ncion descibing he displacemen o he ai a posiion and ime and he ncion govening acosic pesse boh saisy he wave eqaion. This secion will show how he ncions saisy he wave eqaion. To do his, vaiables o displacemen, ξ, and acosic pesse, p, ae inodced (Benson 99). Le be he posiion along he be and (, ) be he displacemen o ai a posiion a ime. The es vale o pesse is he ambien ai pesse ρ. Acosic pesse p(, ) is meased by sbacing ρ om he absole pesse 3, P (, ) so ha p (, ) P(, ). Hooke s law saes p B (.1.1) whee B is he blk modls 4 o ai. Newon s second law implies p (.1.). Now se eqaions.1.1 and.1. o deive a dieenial eqaion in ems o he displacemen vaiable. Using Eqaion.1.1 p B B B Sbsiing his esl ino Eqaion.1., one aives a 3 The absole pesse o a sysem is he oal pesse o he sysem; he sm o he acosic and ambien ai pesses. 4 The blk modls o ai is deined as he abiliy o esis niom compession. 16

17 17 B Reaanging his eqaion B Leing B c one obains 0 c This is he sandad wave eqaion o displacemen. I is now ime o deive a dieenial eqaion in ems o he acosic pesse, p. In ode o do his, one will ake he paial deivaive o Eqaion.1.1 wih espec o. This gives (.1.3) B B p Since is wice coninosly dieeniable, is mied paials ae eqal. So (.1.4) B B p I hen ook he paial deivaive wih espec o o boh sides o he pevios eqaion. Since he paial deivaive wih espec o o he acosic pesse is wice coninosly dieeniable he mied paial deivaives ae eqal, which gives way o he ollowing:

18 18 B p B B B B p Sbsiing Eqaion.1. gives 1 p B p p B B p Which becomes 0 p B p Leing B c gives 0 p c p This is he sandad wave eqaion o acosic pesse. Obseve ha he wave eqaion is a linea combinaion o he second deivaives: 0 c. As peviosly saed, sond waves ae modeled by sine and cosine eqaions. This means ha one shold be able o ind he solions sing sin( ) c (.1.5)

19 and cos( c ) (.1.6). Conside is sin( c ). ccos( c ), cos( c ) c sin( c ), sin( c ) c c sin( c ) c ( sin( c )) 0 Obseve o sin( c ). ccos( c ), cos( c ) c sin( c ), sin( c ) c c sin( c ) c ( sin( c )) 0 Conside now cos( c ) csin( c ), sin( c ) c cos( c ), cos( c ) c c cos( c ) c ( cos( c )) 0 Likewise o cos( c ) csin( c ), sin( c ) c cos( c ), cos( c ) c c cos( c ) c ( cos( c )) 0 Hence, i has been shown ha sin( c ) and cos( c ) ae solions o he wave eqaion as i govens ai colmns. 19

20 .: Sings The chodophones, moe commonly known as he sing amily, consiss o hee dieen ypes o insmens: insmens wih open sings, eed insmens, and bowed insmens (Johnson). The amily has seveal nivesal eaes ha denoe an insmen as being a membe: aachmen devices o aach boh ends o he sings o he body o he insmen, a shell ha allows vibaion o esonae and ampliy he sond, and a ingeboad nde he sing o conol pich (Pen). Thee ae hee mehods o seing a sing in moion: siking, plcking, o bowing. The wave eqaion o sings, c 0 mahemaically models he posiion o he sing se ino vibaion (Sabiova). Fo each mehod o seing a sing in vibaion, he omla changes slighly. To ndesand he wave eqaion o sings, one ms is ndesand he laws modeling he eqency o singed insmens. Pee Mesenne and Galileo Galilei discoveed hese laws individally aond 1635 (Joseph). They ond ha he eqency o a seched vibaing sing o be modeled by C Ld F whee C is he consan o popoionaliy, L is he lengh o he sing, d is he diamee o he sing, ρ is he sing s densiy, and F is he oce acing on he sing. 0

21 Sa s Calcls book pesens a poblem ha shold help one bee gasp how he eqency o a noe podced om a sing changes. I consides wha happens as he lengh, ension, and linea densiy ae changed. The eqaion ha is o be woked wih is 1 L T (..1) Lengh is he is vaiable o be eploed. Taking he deivaive wih espec o lengh gives L 1 L T. Since he deivaive is negaive, he eqency is deceasing as he lengh inceases. This means ha he longe he sing becomes, he smalle he eqency, and hence he deepe he noe. Similaly, as he sing is shoened, he eqency inceases and he noe becomes shille. I ne consideed eqaion..1 wih espec o ension, T. The deivaive is 1 1. T 4L T Since he deivaive is posiive, he eqency is inceasing as he ension is inceased, and hence, he noe becomes highe. Likewise, as he ension is deceased, so is he eqency and hence he noe, becomes lowe. I las consideed eqaion..1 wih espec o he linea densiy. The deivaive wih espec o linea densiy is 1 4L T. 3 1

22 Since he deivaive is negaive, as he densiy o he sing is inceased, he eqency slows, and hence he noe becomes lowe. Similaly, as he sing s densiy is deceased, he eqency inceases and he noe becomes highe. Waves in sings avel along he sing o is end and back o he ohe. The waves se ino vibaion he molecles in he ai as well as in anohe pa o he insmen ha vibaes moe molecles. A colmn o ai can be se ino vibaion qie easily. I is no necessay o esic, o bond, he colmn o ai nless one is aemping o achieve a paicla sond. A sing howeve, ms be bond in ode o vibae. The lengh beween he bonds can be vaied pessing he sing a dieen poins which changes he lengh o he sing allowed o vibae. Changing he lengh changes he noe podced om he vibaion. Conside a sing insmen ha is se ino moion by being sck. An eample o an insmen o his nae is a piano. Since he sing is ied a boh ends, i gives he ollowing bonday condiions: ( 0, ) ( L, ) 0 whee (, ) is a solion o he wave eqaion, is he hoizonal posiion along he sing a ime. Beoe he sing is sck, i is a es so hee ae he ollowing iniial condiions: (,0) 0 (,0) 0 Combing he iniial and bonday condiions wih he wave eqaion gives he ollowing paial dieenial eqaion sysem:

23 (, ) c (0, ) ( L, ) 0 (,0) 0 (,0) 0 (, ) 0 (..) To solve his sysem I sed he mehod o sepaaion o vaiables. I le (, ) ( ) g( ) whee () is he ncion govening he posiion o he sing and g () is he ncion govening he amon o ime ha had passed. Obseve g and g. Taking he second deivaives gives g and g. This gives g c g 0 which can be eaanged as 1 g. c g Seing his eqaion eqal o a consan, any eal nmbe k gives 1 c g g k This gives he eqaions k and g c kg. This gives he sysem k 0 (0) ( L) 0 A solion has he om ( ) e. This means ( ) e and e. Sbsiing his ino sysem.. gives e ke 0 which leads o he chaaceisic eqaion k 0. The chaaceisic eqaion has solions k, whee k 0 is. 3

24 Using he bonday condiion ( 0) 0, ( 0) A B 0 in ( ) Ae k Be k which gives B A. So k k ( ) A( e e ) Asinh( kl). Since sinh() 0 only i =0, o ( L) Asinh( kl) 0, A is oced o eqal zeo. This holds o all k>0 as hee ae only ivial solions. Conside now he case when k<0. Le k - so wih he bonday condiions (0)=(L)=0. This gives which becomes. The ncion becomes Acos Bsin. Fom above, A=0 which gives Bsin. Using he bonday condiion Bsin L 0. So o nonivial solions o he wave eqaion, nl n o n n. The solion L n coesponds o he amily o waveoms, which ae dependen pon he physical seing on he model. Ren o g c kg and solve o g. Since n gave waveoms, g - kc g 0 is epesenaive o he amplide o each waveom. While solving o, i was shown ha o k>0 hee ae no solions, so only k<0 is consideed. This gives g c g, wih iniial condiions g(0)=1 and g (0)=0. The chaaceisic ncion is ( c). The ncion becomes g Acosc Bsinc. Using he condiion g(0)=1, i is ond ha A=1. To obain B, ake he deivaive so ha g - casin c cbcos c. Obseve ha g (0)= λcb=0, so B=0. Tha gives g() cosc. Fom λ above, nc g( ) cos( ) (..3). L Combining he eqaions o n (,), whee ees o and ees o g gives 4

25 nc n (, g) Bn cos( )sin( ), L L whee B n is a consan and n=1,,3. This eqaion models he elaionship beween waveoms and vibaions. Noice ha he simple he waveom is, he longe he peiod..3: Dms The idiophones, o pecssion, is he amily o insmens, whee each insmen has one disinc pich (Pen). Pecssion insmens podce sond by hei vibaing elemens, like he membane o a dm o he modiied plaes in bells, being sck, bbed, shook, o aled. Membaneophones, o dms, and iangles ae ypical insmens o his amily. The vibaing elemen o a pecssion insmen is mos o, i no he enie, insmen isel. Membaneophones ae dieen om mos o he pecssion amily in ha he vibaing elemen is a membane, a seched leible shee o animal hide, o a synheic vaiaion. Sond is podced when a hand o a sick sikes he seched membane. When eploing he mahemaics o dms, he membane is assmed o be ideal. Tha is, he membane is assmed leible, niom, ininiesimally hick, and seched eqally in all diecions by a oce naeced by he moion o he membane (Josephs). The eqency o vibaion o he membane can be ond sing he eqaion.405 whee is he adis o he membane, τ is he ension o he membane and σ is he aea densiy o he membane. 5

26 While he abiliy o calclae he eqency o vibaion in a membane is sel, one shold be able o descibe wha eqaion(s) he waves ollow a a given posiion as a dm is sck. A vaiaion o he wave eqaion is sed: whee c, (.3.1) (,0) 0 is he Laplacian o and he symbol epesens he spaial coodinaes ha ae appopiae (eihe he Caesian o Pola). I am looking o a solion o he eqaion above ha is in he om, (, ) g( ) ( ). (.3.) The pocess is simila o he mehod o sepaaion o vaiables ha I sed in inding solions o he wave eqaion in elaion o he sing amily. Sbsiing Eqaion.3. ino.3.1 gives Reaanging he eqaion gives g c g g gc Seing each side o he eqaion sepaaely o. gives g (.3.3) gc and. (.3.4) The eqaion 6

27 g gc becomes g gc which can be ewien gc 0 g. The vale o g n epesens he amplide ncion o he n h modes o he wave acoss he membane. Since he dm is iniially a es g n (0)=0 and ollowing he seps in solving o in he sck sing poblem, g n c n sin cn. Taking c n eqal o one, (, ) sin( c n) ( ). Look now a (.3.4). The eqaion becomes 0. A woking eqaion o se o he Laplacian is needed. Conside a cicla membane o ni adis. Take ( ). Tha is he same as, y which becomes yy. Since a cicla membane is being consideed, i makes sense ha ha vaiables ae changed om Caesian coodinaes o hei pola coodinae eqivalen. To do his ake cos and y sin. Take he deivaives wih espec o is. The is deivaive will be o he om y. y 7

28 Fo consisency he noaion y will be sed. The deivaive becomes y cos sin. y The second deivaive is ond o be can be ewien as cos which, when simpliied, becomes So cos y cos sin. y sin y y cos sin sin cos sin. y y y yy yy cos sin cos sin (.3.5). y yy Now ake he deivaives wih espec o θ. So -sin cos y. Tha means [-sin cos ] which is wien y cos sin ( y y ) sin y cos ( y yy y ). Conine o simpliy so ha (cos sin ) sin cos (sin cos ) (.3.6). y Ren o he eigenvale poblem om above: y 0. Ledde ewies he eqaion as 1 1 ( ) 0 (.3.7) (535). I wold like o show ha yy 8

29 1 1 ( ) yy (.3.8). Obseve ha 1 1 ( ) 1 (.3.9) Sbsiing in he appopiae above eqaions (.3.5) and (.3.6) o yields 1 (.3.10) (sin cos ) (sin cos ) yy sin cos. Which, sing igonomeic ideniies 5, edces o y sin cos y 1 (cos sin y ) 1 yy. This means ha 1 1 yy. (.3.11) The le side o he eqaion can be ewien as One conines o edce o Which sbsiing (.3.8) edces o 1 1 ( ) (.3.1) 5 The Pyhagoean ideniy cos +sin =1 was sed. 9

30 Combining (.3.7) and (.3.9) shows (.3.1) is valid. Ren now o he wave eqaion, which is wien as 1 1 ( ) 0. To ind solions o he wave eqaion, one again ses sepaaion o vaiables. Begin wih (, ) R() ( ) whee R. Replacing appopiae vales gives which becomes Disibe so ha Reaanging he ems so ha 1 1 R R R R R R R 0. 1 R R 1 R R and dividing boh sides by R gives: 1 R R 0. R R 1 1 R R R R 1. Mliplying hogh by and seing boh sides eqal o a consan, R R R k. R One now solves o each ncion R and Θ. Conside R is. The eqaion is 30

31 R R R R k. In seing he eqaion eqal o zeo, R R R( k) 0, a paameic Bessel eqaion is obained. Thee is an implici bonday condiion R(0). I saes ha a he cene o he dm, whee he adis is zeo, he amplide o he wave is less han ininiy. R(1)=0 is he ohe condiion. Since he dm being consideed has ni adis, he adis eqaling one ells s ha a he im o he dm hee is no movemen. Fom pevios solions when λ=0, hee ae no nonzeo solions and hence all solions ae ivial. Conside λ>0. The solion becomes R c1j m ( ) cym ( ) whee c 1, c ae consans, J m is he Bessel eqaion o he iniial kind, and Y m is he Bessel eqaion o he second kind. Since he dm head is bonded, c Y m (λ) can be ignoed as i ees o an nbonded eqaion. Tn now o he eqaion o Θ: k. Seing he eqaion eqal o zeo gives Θ θθ +kθ=0. Again conside k>0. To ense ha k>0, le m k. The eqaion becomes Θ θθ +m Θ=0. Thee ae wo addiional condiions o hose above: Θ(π)=Θ(0) and Θ θ (π)=θ θ (0). These condiions ae impoan becase hey eqie ha he membane on he dm be coninos when he adis eqals one. The condiions make sense becase θ=0 and θ=π epesen he same poin on he membane. The solion o he dieenial eqaion becomes acos m bsin m. 31

32 The oal solion o he wave eqaion as i govens seched membanes is R c J m( )( acos m bsin m ) (.3.13). 1 To help visalize wha is happening acoss a membane, as i is se ino vibaion, hink o a ampoline wih a single jmpe. Beoe he individal climbs on, he ampoline is a es. Once he jmpe is in place a he cene, hey sa o se he ampoline ino moion. As he vibaions o he ampoline become geae, he jmpe lies highe ino he ai. Noice ha he ampoline only gives so mch. Wih each jmp on he ampoline, waves ae sen o om he jmpe s locaion and back, mch like a msician beaing a he cene o a dm s membane wih a malle. Visalizing wha is happening acoss a membane played wih moe han one malle o boh a msicians hands is slighly moe diicl. We give he same eample o a ampoline, b now wih wo jmpes. Since neihe can jmp in he same spo, he waves sen acoss he ampoline ae coninally disped. The same happens wih a dm. Each bea ineps he wave paens om he pevios bea. 3

33 Chape 3 Ceamics The c o his pape has been eached. How ae he eqencies o a msical insmen changed when consced om ceamic maeial? Ideally, he maeial om which an insmen is made will no change anyhing. In pacice and wih a ained ea, he eqencies change beween evey insmen and wih each dieen conscion maeial. Beoe delving ahe ino he discssion o how clay aecs an insmen s eqencies, one needs o know moe abo he basic popeies o clay. Thee ae many ypes o clay bodies, each wih hei own popeies ha ae a aco when conscing any piece. A shinkage able o seveal dieen clay bodies is povided. Wha happens when one ype o clay is sed o consc les will also be eploed. The chemical composiion o a clay body will aec wo hings: he amon o shinkage beween each iing and he iing empeae. Relaed o he iing empeae o a clay body is he viiicaion o he clay. A clay body is viiied when i is wae igh wiho he se o a glaze. Viiicaion is impoan as he clay body isel, when sck, gives a highe piched ing. I is hogh ha he viiicaion o he clay body in he insmens will have he same eec. The pecenage a clay piece shinks is also an isse beween dieen clay bodies. In geneal, o calclae he shinkage o a clay body one olls o a slab o clay o even hickness en inches long (his makes calclaions easie). The slab is hen bisqe ied and emeased. Havey Body saes his pocess o calclaing shinkage: 33

34 1. Make seveal bas a leas 1 cenimees long and 1 cenimee hick om he we clay o he consisency yo plan o se.. Scach wo maks on each ba eacly 100 millimees (10 cenimees) apa. 3. Dy he bas compleely. 4. Mease he disance beween he wo maks. Each millimee is 1 pecen o he oal. I he disance now meases 97 mm ae dying, hen dying shinkage was 3 pecen. 5. Fie he bas o whaeve empeae yo ae esing hem a; when cool, mease he disance again. I i now meases 95 mm, he iing shinkage was pecen and he oal shinkage 5 pecen (10). Body s mehod is js one way o calclae shinkage. I se a slighly dieing mehod o calclaion: C b Ca s 100, Cb whee C b is he lengh o clay pio o iing and C a is he lengh o clay ae iing. Eihe mehod allows one o accon o shinkage beoe a piece is ied. I a slab o a speciic size ae a iing is desied, he oiginal clay piece is inceased by p 100 s, whee p is he pecenage o a piece emaining ae a shinkage o s pecen: T L ( 1 p) whee T is he size o he piece beoe iing, L is he size o he piece needed ae a iing, and p is ha o he paagaph above. Any clay piece can easily go hogh wo o moe iings wih each iing shinking he piece. Shinkage calclaions shold be done 34

35 o each clay body beween each iing. Evenally an oveall pecenage o shinkage will be obained. To help ndesand he isse o shinkage, I have olled o a sample slab o each clay body and consced seveal side blown les o vaying lenghs om a single clay body. The les wee eded as a hollow be and allowed o im p ovenigh. One end was hen closed and he moh and inge holes wee made. They wee allowed o dy slowly o abo a week so hey cold be bisqe ied. The les wee hen decoaed so ha hey wee ncional and glaze ied. A able showing he calclaed shinkages o he slabs and each lengh o le is inclded below. Lengh (inches) Shinkage % Clay Type We Bisqe Balo Wallow Rak Pocelain Longhon Whie High Fie Whie Sdio Table 3.1 Shinkage Pecenages o Clay Bodies Longhon Whie Fles Fle Lengh We (in) Bisqe (in) Shinkage (%) Glaze (in) Shinkage (%) Toal Shinkage (%) I II III IV V VI VII VIII IX X XI XII XIII XIV XV blank Aveage Toal Shinkage 5.0 % 35 Table 3. Shinkage Table o Lowie Fles

36 We n now o ceamic models o insmens and how hey help s ndesand he mah peviosly eploed. The les om Table 3. will help s hea wha he onedimensional wave eqaion models. Dms o vaying shapes, sizes, and dmheads will help s o hea wha he wo dimensional wave eqaion models. The basic physics eqaions govening waves will be consideed and applied o he les. Each le, becase o hei vaying lenghs and vaying levels o viiicaion, has a dieen eqency. Applying physics o he dms will be diicl as he eqaions govening he waves o a wo dimensional sace ae mch moe complicaed. The dieence in he pich o a dm is head mch easie han calclaed. A deep esonaing boom is head when he ovesized dm Boome is sck. A highe piched hd sonds om he smalles dm. Vaios componens o ceamic insmens have been discssed in ems o mah. Wha physically happens ding he ceaion o he insmens and wha wee some o he poblems enconeed? No knowing anyhing abo insmens o wha eally wen ino insmen conscion made he ask diicl. As knowledge was gained abo vaios insmens and hei conscion, he challenge became moe abo how o incopoae hose speciic hings ino a clay conepa. Ae he wokings wee iged o, he ne concen was ceaing an insmen o niom hickness. Coil bil insmens ended o have his poblem he mos. The coils may have been he same size b blending and shaping he insmens cold and did change he hickness 36

37 wiho he bilde being awae o i. Eded les also sggled wih niom hickness. The die c sed was no always ceneed and neve sayed in he same place. Conscion o insmens was no he only poblem. Fiing he insmens poved o be poblemaic. A lage insmen, like Boome, had o be moved and loaded wih wo people. I baely i in he kiln. The les had he mos poblems ding iing. Some wee oo long o i in he kiln and wee heeoe ied a angles. Those ha cold i wee ied on shelves ha wee bowed. Boh ways o iing he les cased bowing in he les. The bowing conined in he glaze iing. In he e, he dimensions o he kiln will be kep in mind as pieces ae bil. Glaze iings ae no kind o les. A le ms be glazed eveywhee o be ncional. To be ied, he le ms es on sils. This can be a poblem as les can, and have olled o he sils and sck o he kiln shel. Anohe glaze poblem is ha he glaze can clog he aiway o inge holes. Aside om glaze isses no allowing he le o ncion, hee can be ohe glaze poblems ha wold ende a le dysncional. Some poblems wih glazes ae cawling, shiveing, nde iing, o clogging o inge holes. 6 Chape 4 Fhe Reseach: Hybid Insmens Hybid insmens ae insmens ha combine key eaes om wo o moe amilies. The concep o a hybid insmen came om Bay Hall s Md o Msic. His book eploes seveal aiss who ceae ceamic insmens and pesens mliple hybid 6 Cawling is a glaze deec in which he glaze sepaaes on a piece. This is sally de o wa o oil om he ais s hands. Shiveing is a deec whee a glaze lakes o a piece. 37

38 insmens. Some o he hybid insmens pesened ae Sone Fiddle, Elephonim, Globbla Dm Hon, and Ocaina Dm (Hall). The sone iddle combines a le, a dm, and iddle. Elephonim is pa dm, pa hap. The dm hon is js ha; a combinaion o a dm and a hon. Ocaina dm also names is paens; he ocaina and a dm. Sone Fiddle Elephonim Globbla Dm Hon Ocaina Dm Table 4.1 Skeches o Hybid Insmen I is believed ha a combinaion o pas o eqaions govening ohe amilies o insmens will esl in an eqaion modeling he hybid. Each hybid will have a dieen eqaion based pon he amilies he hybid is composed o as well as how he piece is analyzed. 38

39 Thee ae seveal hoghs abo how he eqaion will be ceaed: as a composiion o he dieen wave eqaions, he podc o he eqaions, o a simple smmaion o he vales o each secion o he amily sing he especive wave eqaions. I, as in a sone iddle, we look is a he le and is eecs on he dm and sing, he iniial condiions o hem will be dieen as opposed o i hey had been a es, hen se ino moion. I we plcked he sing hen blew ino he le, he membane s oiginal posiion is dieen om is esing posiion. I he dm is sck is, i will aec he iniial condiions o he sing. In he mos basic om, i is hogh ha he eqaion govening hybids will be a smmaion o eqaions.1.5,..3, and.3.13, which looks like nc sin( c ) cos( ) c1j m ( )( acos m bsin m ) (4.1). L This pape has looked a and eploed aeas in msic whee mah can be applied, some basic physics eqaions govening eqency, and he wave eqaion as i govens dieen amilies o insmens a he mos basic level. I also povided a vey basic look a some complicaions and concens in sing clay as a conscion maeial o msical insmens. In wiing he pape, he above pose qesions wee answeed and many moe qesions wee aised. As he opic o eqaions o hybids conines, many epeimens will be condced sing measing devices o accacy. In he e, hee may vey well be an eqaion ond ha can cape each hybid s mahemaical essence. 39

40 Bibliogaphy [1] Benson, David. Msic: A Mahemaical Oeing. Cambidge Univesiy Pess. Scoland, UK. 006 [] Band, William E. The Way o Msic. Allyn and Bacon, Inc. Boson [3] Casellini, John. Rdimens o Msic. W.W. Noon & Company, Inc. NY [4] Elmoe, William C. Mak A. Heald. Physics o Waves. McGaw-Hill Book Company. New Yok [5] Feahe, Noman. An Inodcion o Physics o Vibaions and Waves. Edinbgh Univesiy Pess. Edinbgh [6] Hall, Bay. Fom Md o Msic. The Ameican Ceamic Sociey. Weseville, Ohio [7] Heile, W. Elemenay Wave Mechanics. nd ed. Ood Univesiy Pess. London [8] Jenkins, Jean L. Msical Insmens. nd ed. Inne London Edcaion Ahoiy. London [9] Johnson, Ian. Meased Tones: The Ineplay o Physics and Msic. nd ed. Insie o Physics Pblishing. London, England. 00. [10] Josephs, Jess J. The Physics o Msical Sond. D. Van Nosand Company, Inc. New Yok [11] Ledde, Glen. Dieenial Eqaions: A Modeling Appoach. McGaw-Hill Companies, Inc. New Yok [1] Moavcsik, Michael J. Msical Sond. Paagon Hose Pblishes. New Yok [13] Newman, William S. Undesanding Msic. nd ed. Hape and Bohes, Pblishes. NewYok [14] Pen, Ronald. Inodcion o Msic. McGaw-Hill, Inc. New Yok [15] Schödinge, E. Colleced Papes on Wave Mechanics. Chelsea Pblishing Co. New Yok

41 [16] Taylo, C.A. The Physics o Msical Sonds. Ameica Elsevie Pblishing Company, Inc.New Yok [17] Whie, Havey E. Donald H. Whie. Physics and Msic. Sandes College. Philidelphia

42 VITA Tammy Wene Pemanen Addess 434 Biawes San Anonio, Teas 7847 (10) Degee Bachelo o Science in Mahemaics, Bachelo o Fine As in Ceamics; May 008 Majo: Mahemaics, Ceamics Edcaional Insiions Aended James Madison High School: San Anonio, Teas ( ) McMy Univesiy: Abilene, Teas ( ) Oganizaions and Aciviies Camps Aciviies Boad ( ) Mah Clb ( ) Gamma Sigma ( ) Tease ( ) Honos Kappa M Epsilon Honos Mah Faeniy ( ) Kappa Pi Honos A Faeniy ( ) McMy Univesiy Dean s Lis (004, ) 4