Calculation of Inharmonicities in Musical Instruments

Transcription

1 UIUC Physics 406POM/Physics 93POM Physics o Music/Musical Istrumets Calculatio o Iharmoicities i Musical Istrumets As we have discussed i the POM lectures, real -dimesioal musical istrumets do ot have a perect/ideal harmoic sequece o overtoes i.e.,,,3,..., i cotrast to the predictios o the perect/ideal simple irst-order theory o how the istrumet works such irst-order theories eglect/igore higher-order real-world eects such as iite strig stiess, viscous dampig/dissipative eects that do ideed chage/shit/perturb the resoat requecies rom their ideal simple irst-order theory predictios. For ay give -dimesioal musical istrumet, i we measure the requecies o the idividual harmoics o the istrumet, we ca compare them with the perect/ideal simple irst-order theory predictio,,,3,... o the harmoic sequece o overtoes to see how close to perect/ideal the musical istrumet actually is. I geeral, the experimetal results will be close to, but ot precisely idetical with the simple perect/ideal theory predictios. The deviatio o each harmoic rom the perect/ideal predictio,,,3,... is a measure o the iharmoicity o the musical istrumet. I.) Striged Istrumet Iharmoicities: For striged istrumets, the udametal requecy o a ope vibratig strig is ot sigiicatly perturbed rom its perect/ideal theory value o v T L, thus we ca use it as a reerece or the higher harmoics,,,3,.... We ca calculate the requecy ratios R these results to the simple perect/ideal theory predictios The measured requecy ratios Proessor Steve Errede, Departmet o Physics, Uiversity o Illiois at Urbaa-Champaig, IL,,3... ad compare R,,,3,... R should be close to, but will ot be precisely equal to,,,3,... or striged istrumets, e.g. due to iite strig stiess ad, to a lesser extet, due to the eect(s) o viscous dampig/dissipatio eects o the surroudig air i proximity o the vibratig strig. The deviatio o a give measured requecy ratio R theory value R,,,3,... rom its perect/ideal is give by the experimet vs. theory dierece betwee these two quatities: R R R I we the ormalize the above expressio to the theory ratio R the we obtai the ractioal deviatio o a give measured requecy ratio R rom its perect/ideal theory value R,,,3,... :,,,3,...

2 UIUC Physics 406POM/Physics 93POM Physics o Music/Musical Istrumets R R R R R Next, i we multiply this expressio by the theory value o the requecy o this harmoic, the this is equal to the shit/departure o this harmoic s requecy rom its perect/ideal theory value (i Hz): R Hz R Stated aother way: R R R R R The the % deviatios % o the actual/measured harmoics o a striged istrumet rom their perect/ideal theory values are obtaied by multiplyig the above expressio by 00%: R R R % R R Now oe semi-toe i the tempered scale is equal to 00 cets, ad sice there are otes i the tempered scale, the oe octave = 00 cets. The requecy ratio betwee two adjacet otes (i.e. a semitoe) i the tempered scale is a Thus, oe semitoe low rom the requecy is give by the ormula: a semitoe low ad oe semitoe high rom the requecy is give by the ormula: a semitoe high Proessor Steve Errede, Departmet o Physics, Uiversity o Illiois at Urbaa-Champaig, IL

3 The requecy dierece UIUC Physics 406POM/Physics 93POM Physics o Music/Musical Istrumets semitoe low or oe semitoe low rom is give by: a a semitoe semitoe low low Hz The requecy dierece semitoe low or oe semitoe high rom is give by: a a semitoe semitoe high high Hz Notice that semitoe semitoe low high. They re close, but they are ot precisely equal to each other. Suppose the requecy o a ote o the musical scale, e.g. A5 is Hz. The requecy o Ab5 (oe semitoe low) is Hz, dierig rom A5 by Hz. The requecy o B5 (oe semitoe high) is Hz, dierig rom A5 by 5.33 Hz. I order to express our measured experimetal vs. theory requecy diereces semitoe i cets, we eed to a.) irst ormalize to low (i 0 ) or ormalize to (i 0 ), the b.) multiply each o these ratios by 00 cets: semitoe high I 0: # cets low 00 semitoe low I 0 : # cets high 00 semitoe high cets cets The, isertig the ormulas or, semitoe low ad semitoe high i these expressios, we have: 00 I 0: # cets low 00 I 0 : # cets high 00 where: ad: 00,,,3,... cets cets 3 Proessor Steve Errede, Departmet o Physics, Uiversity o Illiois at Urbaa-Champaig, IL

4 UIUC Physics 406POM/Physics 93POM Physics o Music/Musical Istrumets Numerically, these ormulas, or striged istrumet iharmoicities are: I 0 : # cets low cets I 0 : # cets high where: ad:,,,3,... cets II.) Wid/Brass Lip-Reed Istrumet Iharmoicities: For wid/brass lip-reed istrumets, the udametal requecy o a ope vibratig strig is sigiicatly perturbed rom its perect/ideal theory value o v, because o a variety o higher-order eects ot take ito accout i the simple perect/ideal theory e.g. requecy-depedet wavelegth eects at the bell ed o the istrumet; the (logitudial) speed o soud propagatio i the coied/iteral space o the istrumet is also ormally requecy depedet, i.e. v v ad especially so a lower requecies, due to requecydepedet viscous dampig/dissipatio eects. I wid/brass lip-reed istrumets, the udametal requecy v(aka the pedal ote teds to be systematically pulled low, ad sigiicatly so. Thus, we caot use the udametal as a reerece or the higher harmoics,,,3,.... The pedal ote/udametal is also extremely diicult to play o wid/brass lip-reed istrumets, ad also ot with ay great accuracy. The secod harmoic o wid/brass lip-reed istrumets is much less adversely aected tha the udametal (= the irst harmoic), ad also easy to play, accurately so. Hece we ca/will use the secod harmoic as a reerece or the higher harmoics i wid/brass lip-reed istrumets. We ca the calculate the requecy ratios R,,3... compare these results to the simple perect/ideal theory predictios R,,,3,... The measured requecy ratios ad R should be close to, but will ot be precisely equal to,,,3,... or wid/brass lip-reed istrumets, due to the eect(s) o viscous dampig/dissipatio eects o the air iside the bore o ad i cotact with the iside surace(s) o the istrumet. 4 Proessor Steve Errede, Departmet o Physics, Uiversity o Illiois at Urbaa-Champaig, IL

5 UIUC Physics 406POM/Physics 93POM Physics o Music/Musical Istrumets The deviatio o a give measured requecy ratio R simple theory value R,,,3,... theory dierece betwee these two quatities: 5 rom its perect/ideal is give by the experimet vs. R R R I we the ormalize the above expressio to the theory R,,,3,... the we obtai the ractioal deviatio o a give ratio measured requecy ratio R R,,,3,... : rom its perect/ideal theory value R R R R R R R R Next, i we multiply this expressio by the theory value o the requecy o this harmoic, the this is equal to the shit/departure o this harmoic s requecy rom its perect/ideal theory value (i Hz): R Hz R Stated aother way: R R The the % deviatios % o the actual/measured harmoics o a striged istrumet rom their perect/ideal theory values are obtaied by multiplyig the above expressio by 00%: R R R % R R Proessor Steve Errede, Departmet o Physics, Uiversity o Illiois at Urbaa-Champaig, IL

6 UIUC Physics 406POM/Physics 93POM Physics o Music/Musical Istrumets Now oe semi-toe i the tempered scale is equal to 00 cets, ad sice there are otes i the tempered scale, the oe octave = 00 cets. The requecy ratio betwee two adjacet otes (i.e. a semitoe) i the tempered scale is a Thus, oe semitoe low rom the requecy is give by the ormula: a semitoe low ad oe semitoe high rom the requecy is give by the ormula: The requecy dierece a semitoe high semitoe low or oe semitoe low rom is give by: a a semitoe semitoe low low Hz The requecy dierece semitoe low or oe semitoe high rom is give by: a a semitoe semitoe high high Hz Notice that semitoe semitoe low high. They re close, but they are ot precisely equal to each other. Suppose the requecy o a ote o the musical scale, e.g. A5 is Hz. The requecy o Ab5 (oe semitoe low) is Hz, dierig rom A5 by Hz. The requecy o B5 (oe semitoe high) is Hz, dierig rom A5 by 5.33 Hz. I order to express our measured experimetal vs. theory requecy diereces semitoe i cets, we eed to a.) irst ormalize to low (i 0 ) or ormalize to (i 0 ), the b.) multiply each o these ratios by 00 cets: semitoe high I 0: # cets low 00 semitoe low cets I 0 : # cets high 00 cets semitoe high 6 Proessor Steve Errede, Departmet o Physics, Uiversity o Illiois at Urbaa-Champaig, IL

7 UIUC Physics 406POM/Physics 93POM Physics o Music/Musical Istrumets The, isertig the ormulas or, semitoe low ad semitoe high i these expressios, we have: 00 I 0: # cets low 00 I 0 : # cets high 00 where: ad: 00,,,3,... cets cets Numerically, these ormulas, or wid/brass lip-reed istrumet iharmoicities are: I 0: # cets low cets I 0 : # cets high where: ad:,,,3,... cets 7 Proessor Steve Errede, Departmet o Physics, Uiversity o Illiois at Urbaa-Champaig, IL

8 UIUC Physics 406POM/Physics 93POM Physics o Music/Musical Istrumets Legal Disclaimer ad Copyright Notice: Legal Disclaimer: The author speciically disclaims legal resposibility or ay loss o proit, or ay cosequetial, icidetal, ad/or other damages resultig rom the mis-use o iormatio cotaied i this documet. The author has made every eort possible to esure that the iormatio cotaied i this documet is actually ad techically accurate ad correct. Copyright Notice: The cotets o this documet are protected uder both Uited States o America ad Iteratioal Copyright Laws. No portio o this documet may be reproduced i ay maer or commercial use without prior writte permissio rom the author o this documet. The author grats permissio or the use o iormatio cotaied i this documet or private, ocommercial purposes oly. 8 Proessor Steve Errede, Departmet o Physics, Uiversity o Illiois at Urbaa-Champaig, IL