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1 ( )

2 Hz

3 FM i

4 39 40 ii

5 E F A A B B C D D (A) p1-m (A) p1-m2 (2) iii

6 4.21 (B) p1-m (C) p1-m (D) p1-m (1) p1-m (2) p1-m (3) p1-m iv

7 1 1.1 [2 4] FFT [1] ?? 5 6 1

8 [5] required tone air Player flow buzzing Lip Instrument tone reflected wave 2.1: 2

9 (a) (b) 2.2: (a): (b): 2.2 [5] 2.2(a) 2.2(b) [6] 1 ) [2 4] 3

10 a -x0 x 2.3: [5] [7, 8] 2.3 x a a = b(x + x 0 ) γ (2.1) γ b x 0 4

11 f n f n c 4(l + x 0 ) { (2n 1) + β } γ(γ + 1) (2.2) [9] c l β γ < γ > γ = 1 f n nc 2(l + x 0 ) γ γ 0.7 [10] [10,11] (0.7,2,3,4,...) f 0 [5] (2.3) n 2 n/12 [12]

12 n b n (n = 1,2,3) b n = 2 n 12 1 (2.4) b (2.5) b (2.6) b (2.7) 5 4 ˆb 4 b 1 + b < ˆb (2.8) 6 7 b 2 + b < ˆb (2.9) b 1 + b 2 + b < ˆb (2.10) b b (2.11) b (2.12) b 3 = ˆb 4 b (2.13) (2.14) 4 : b 1 + b (2.15) 5 : b 1 + b 3 = ˆb (2.16) 6 : b 2 + b (2.17) 7 : b 1 + b 2 + b (2.18) 6

13 2.1: 1 0 position piston piston piston b / : 2 2 / : 3 / : 3 on/off p1-m r R Keefe Benade [13] B = r/r Z 0 = ρc/πr 2 δz ( ) 1 δz 2I 2 = 1 (2.19) Z 0 πb π ( ) Bcosθ I = cosθ ln dθ (2.20) 0 1 Bcosθ 7

14 2.4: p m p1-m

15 ( 2.5: 2.6: [5] 2.7 V C = V ρc 2 (2.21)!" #$ % & '! 2.7: 0 9

16 ρ c l c S c R Z IN L = ρl c S c (2.22) Z IN = R + Z p + jωl 1 ω 2 LC + jωc(r + Z p ) (2.23) Z p ω 0 = 1 LC (2.24) Z p ω 0 Z p (2.23) 10

17 3 3.1 [5, 14] [15] 3 [1] 11

18 3.1: 3.2: 12

19 wave hanning window FFT log10 20log 10 IFFT power spectrum cepstrum Peak Picking peak profile FFT cepstrum window spectrum envelope 3.3: 13

20 3.2 p m f pm [Hz] f pm = ma 4 2 p (3.1) A 4 A 4 = 440 Hz f pm F s = 48 khz 2 12 f f = F s 11.7Hz (3.2) 212 [1] N f A P 0 x(i) = Ae 2π j(p 0+i f /F s ) = Ae 2π j(p 0+i f /N) (i = 0,1,2,...,N 1) (3.3) ( f = N f F s ) (3.4) f ( ) 2πi w(i) = 1 cos N G(k) G(k) = (3.5) N 1{ w(i)x(i)e 2π jik/n} (3.6) i=0 cos(t) = (e jt + e jt )/2 w(i) = Ae 2π jp 0 N 1{ e 2π j( f k)/n 1 i=0 2 e2π j( f k+1)/n 1 } 2 e2π j( f k 1)/N (3.7) 14

21 a 1 N 1 a i = an 1 i=0 a 1 (3.8) e jt 1 = 2 j sin 1 2 e jt/2 (3.9) G(k) = Ae 2π jp 0 sin π( f k 1) N cos π( f k) N sin π( f k) N sin 2 π N sin π( f k+1) N G(k) k k max k max 1 2 f k max sinπ( f k))e π j( f k) 3.10 G(k max 1) G(k max ) r r = G(k max 1) G(k max ) cos π( f k max +1) = cos π( f k max ) N sin π( f k max 1) N N sin π( f k max +2) N (3.10) (3.11) (3.12) N θ 1 (3.13) cosθ 1 (3.14) sinθ θ (3.15) 3.12 π( f k max 1 N r π( f k max +2 N = f k max 1 f k max + 2 f = k max + 1 2r ( 1 + r f = k max + 1 2r ) Fs 1 + r N (3.16) (3.17) (3.18) 15

22 s = G(k max 1) G(k max ) (3.19) f = k max 1 2s 1 + s ( f = k max 1 2s ) Fs 1 + s N (3.20) (3.21) r s G(k max ±1) f 3.10 A = G(kmax ) N π( f k max) sinπ( f k max ) ( f k max 1)( f k max + 1) (3.22) 16

23 m cm 48 khz DAT 4.1 (A) (1) (A1) (A1) : (a) (b) A YTR-8335UGR 1 17D(inner gold) B V. Bach 180ML 2 1A C YTR-8335HS 3 Schilke 8A4 D YTR-800GS

24 A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 4.2: A1 A2 A3 B1 B1 C1 C1 C1 C1 C1 D1 D1 18

25 Player Trumpet 0.9m Mic DAT 4.1: 1 1 khz 0.6 khz 2 19

26 4.2: 1 4.3: 2 20

27 4.4: 3 4.5: 4 21

28 4.6: 5 4.7: 6 22

29 4.8: 7 4.9: 2 23

30 (A1) E 4 F 4 A 4 D 5 24

31 4.10: p2-m3 p7-m4(e 4 ) 4.11: p1-m3 p6-m4(f 4 ) 25

32 4.12: p3-m4 p7-m5(a 4 ) 4.13: p2-m4 p6-m5(a 4 ) 26

33 4.14: p1-m4 p5-m5(b 4 ) 4.15: p4-m5 p7-m6(b 4 ) 27

34 4.16: p3-m5 p6-m6(c 5 ) 4.17: p2-m5 p5-m6(d 5 ) 28

35 4.18: p1-m5 p4-m6(d 5 ) p1-m (A) p1-m2 (A1) (A1) (1) (C) (C) (D) 29

36 4.19: (A) p1-m2 4.20: (A) p1-m2 30

37 4.21: (B) p1-m2 4.22: (C) p1-m2 31

38 4.23: (D) p1-m p1-m5 (1) (2) (3) (3) (3) 32

39 4.24: (1) p1-m5 4.25: (2) p1-m5 33

40 4.26: (3) p1-m5 34

41 5 FM (A1) p1-m6 5.1 FFT (A1) p1-m ms 1/0.1 = 10 Hz 7Hz [16] 35

42 5.1: (A1) p1-m6 5.2: (A1) p1-m6 36

43 Hz 37

44 6.2 38

45 OB 39

46 [1],. FFT., J70-A, 5 pp , [2],,,.. 16, pp , [3] Kaneko, Y., Mizuhara, S., Mizutani, K., and Nagai, K. Artificial Lips for Automatic Trombone Blower. Proceedings of The First Asian Pacific Conference on Biomechanics Sponsored by The Japan Society of Mechanical Engineering, T , pp , [4],,,.., MA , pp , [5] Fletcher, N.H., and Rossing, T.D. The Physics of Musical Instruments. Springer-Verlag New York, [6] Yoshikawa, S. Acoustical Behavior of Brass Player s Lips. Journal of Acoustic Society of America, 97, pp , [7] Benade, A.H. On Woodwind Instrument Bores. Journal of Acoustic Society of America, 31, pp , [8] Benade, A.H., and Jansson, E.V. On Plane and Spherical Waves in Horns with Nonuniform Flare. Acustica, 31, pp , [9] Benade, A.H. Fundamentals of Musical Acoustics. Oxford University Press, New York, [10] Young, F.J. The Natural Frequencies of Musical Horns. Acustica, 10, pp , [11] Pyle, R.W. Effective Length of Horns. Acustica, 57, pp , [12] Smithers, D., Wogram, K., and Bowsher, J. Playing the Baroque Trumpet. Scientific American, 254(4), pp , [13] Keefe, D.H., and Benade, A.H. Wave Propagetion in strongly curved ducts. Journal of Acoustic Society of America, 74, pp ,

47 [14].., [15],.., [16]. FM AM pp

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