Analysis of Groove Feelings of Drums Plays 47 56340 19 1 31
Support Vector Machine (SVM) 4 SVM SVM 2 80% 100% SVM SVM SVM 4 SVM 2 2 SVM 4
1 1 1.1........................................ 1 1.1.1............................. 1 1.1.2................. 3 1.1.3......................... 4 1.2................................... 4 1.3................................... 5 2 6 2.1............................. 6 2.1.1............................... 6 2.1.2............................... 7 2.1.3............................. 9 2.2......................... 11 2.2.1..................... 11 2.2.2................. 12 2.3 Support Vector Machine............................. 16 2.3.1 SVM.................... 16 2.3.2...................... 19 2.3.3............... 21 2.3.4 SVM................................ 23 3 24 3.1............................... 24 i
3.2.................................. 25 3.3.............................. 28 3.4 SVM................................. 32 3.5................................... 32 3.5.1................. 32 3.5.2.......................... 35 3.5.3...................... 37 3.6................................. 37 3.7 SVM.......... 39 3.8................. 41 4 SVM 43 4.1...................................... 43 4.2 SVM 4........ 44 4.2.1.................................. 44 4.2.2........................ 45 4.2.3.................................. 45 4.2.4........................ 55 4.3 SVM 4.......... 57 4.3.1.................................. 57 4.3.2........................ 58 4.3.3........................... 58 4.3.4.................................. 59 4.3.5............... 61 4.4 SVM 2.................. 63 4.4.1.................................. 63 4.4.2........................ 64 4.4.3........................... 64 4.4.4.................................. 65 ii
4.4.5................... 75 5 77 5.1....................................... 77 5.2.................................... 80 82 85 86 iii
1.1 [2]......................... 2 1.2................. 5 2.1.............................. 8 2.2 8................................ 10 2.3 16............................... 10 2.4 8 /100BPM/ [10]................................... 14 2.5 8 /100BPM/ [10]................................... 14 2.6 8 /100BPM/ [10]........ 15 2.7 8 /100BPM/ [10]........ 15 2.8 2 17 2.9............ 20 2.10 2 [15]..................... 22 3.1........................ 27 3.2.............................. 29 3.3 1................................. 30 3.4 2................................. 30 3.5 3................................. 30 3.6 4................................. 30 3.7 [18]............................ 34 3.8................... 35 iv
3.9...................................... 36 3.10.. 36 3.11..... 39 3.12................. 41 4.1 1.................................... 49 4.2 1.5.................................... 49 4.3 2.................................... 50 4.4 2.5.................................... 50 4.5 3.................................... 51 4.6 3.5.................................... 51 4.7 4.................................... 52 4.8 4.5.................................... 52 4.9 1 53 4.10 3 53 4.11 2........................................ 54 4.12 4........................................ 54 4.13 1........................................ 69 v
4.14 1.5...................................... 69 4.15 2........................................ 70 4.16 2.5...................................... 70 4.17 3........................................ 71 4.18 3.5...................................... 71 4.19 4........................................ 72 4.20 4.5...................................... 72 4.21 1.. 73 4.22 3.. 73 4.23 2. 74 4.24 4. 74 vi
3.1...................... 31 3.2............................ 40 4.1 SVM............. 46 4.2 SVM................... 47 4.3............ 48 4.4..... 55 4.5 SVM 4. 60 4.6 SVM.................. 61 4.7 SVM SVM................. 62 4.8 SVM 2. 66 4.9 SVM................... 67 4.10............................. 68 vii
1 1.1 1.1.1 1.1.2 1.1.3 1.2 1.3 1.1 1.1.1 1.1 1
1 1.1 1.1: [2] MIDI Musical Instrument Digital Interface MIDI [1] MIDI 2
1 1.1 DTM DeskTop Music 1 1.1.2 1 DTP DeskTop Publishing 3
1 1.2 1.1.3 Led Zeppelin [3] 1.2 1.2 1.2 2.2.2 4
1 1.3 1.2: 1.3 2 Support Vector Machine (SVM) 3 4 Support Vector Machine 3 5 5
2 2.1 2.2 2.3 Support Vector Machine (SVM) 2.1 2.1.1 2.1.2 2.1.3 2.1.1 2.1 6
2 2.1 Bass Drum Snare Drum 1 Floor Tom Tom-Tom 2 Hihat Cymbal Ride Cymbal Crash Cymbal 3 3 3.3 3 2.1.2 2.2 2.2 8 8 1 3 2 4 1 Snare 2 High Tom Low Tom 7
2 2.1 2.1: 8
2 2.1 1 3 2 4 8 16 16 16 2.3 2.1.3 8 16 3 3 A B B 9
2 2.1 2.2: 8 2.3: 16 10
2 2.2 2.2 2.2.1 Friberg [4] 2 8 1 2 1 2:1 3:1 8 Friberg 3.5:1 1:1 Waadeland [5] [6] 11
2 2.2 2.2.2 Draguna [7, 8, 9] [10, 11, 12, 13] 8 4 2 2 4 2.1.2 2.2 8 100BPM 5 4 5 Beat Per Minute 1 1 2 120BPM 12
2 2.2 2.4 2.5 2.4 2.5 16 2.4 2.5 2.4 2.5 2.6 2.7 2.6 2.7 100% 2.6 2.7 2.6 2.7 13
2 2.2 2.4: 8 /100BPM/ [10] 2.5: 8 /100BPM/ [10] 14
2 2.2 2.6: 8 /100BPM/ [10] 2.7: 8 /100BPM/ [10] 15
2 2.3 Support Vector Machine 2.3 Support Vector Machine Support Vector Machine (SVM) 1992 Vaptik [14] 2 SVM SVM SVM 2.3.1 SVM 2 f : X R n R x = (x 1,, x n ) T f(x) 0 f(x) x X f(x) = w T x + b (2.1) f(x) = 0 X 2 w b 2 2.8 2.8 SVM 2 SVM 6 2.8 1 x 1,, x n y 1,, y n x i A y i = 1 B y i = 1 6 16
2 2.3 Support Vector Machine 2.8: 2 w, b min w T x i + b = 1 (2.2) i=1,,n w T x i + b min i=1,,n w (2.3) 1 w w b min w w 2 (2.4) ) y i (w T x + b 1 (2.5) 17
2 2.3 Support Vector Machine 1 w α i ( 0) L(w, b, α) = 1 2 w 2 n i=1 { )} α i y i (w T x i + b (2.6) L b = L w = 0 n α i y i = 0 (2.7) i=1 n w = α i y i x i i=1 (2.4)(2.5) α (2.8) max α n α i 1 n n α i α j y i y j x T i x j (2.9) 2 i=1 i=1 j=1 α i 0, Karush-Kuhn-Tucker n α i y i = 0 (2.10) i=1 ) α i (y i w T x i 1 = 0 (2.11) α 0 SV (2.9) α w w b w = αi y i x i (2.12) x i SV b = y k w T x k (x k SV ) (2.13) 18
2 2.3 Support Vector Machine f(x) f(x) = w T x + b = αi y i x T i x + b (2.14) x i SV αi y i x T i x + b = 0 (2.15) x i SV 2.3.2 2.3.1 (2.5) w, b C-SVC [14] ξ i 0, i = 1, 2,, n (2.16) (2.4) (2.5) 1 n min w 2 w 2 + C ξ i i=1 (2.17) ) y i (w T x + b 1 ξ i (2.18) (2.17) C C C C 19
2 2.3 Support Vector Machine 2.9: α i (2.9) (2.10) max α n α i 1 n n α i α j y i y j x T i x j (2.19) 2 i=1 i=1 j=1 0 α i C, n α i y i = 0 (2.20) i=1 α i > 0 7 ξ i 2.9 2.9 SV (2.19) (2.9) (2.15) 7 α i > 0 20
2 2.3 Support Vector Machine αi y i x T i x + b = 0 (2.21) x i SV C C 3.7 C 2.3.3 2.3.2 φ(x) x = (x 1, x 2,, x n ) φ(x) = (φ 1 (x), φ 2 (x),, φ N (x)) (2.22) φ : R n R N N N n 2.10 2 1 φ : R 2 R 3 (x 1, x 2 ) (z 1, z 2, z 3 ) = (x 1 2, 2x 1 x 2, x 2 2 ) (2.23) 21
2 2.3 Support Vector Machine 2.10: 2 [15] 3 2.10 2 φ K(x, z) x, z K(x, z) = φ(x) T φ(z) (2.24) K K (2.15) x i SV α i y i K(x i, x) + b = 0 (2.25) φ φ(x) T φ(z) K(x, z) = ( γ x T z + r) d, γ > 0 (2.26) 22
2 2.3 Support Vector Machine Radial Basis Function (RBF) ( K(x, z) = exp γ x z 2), γ > 0 (2.27) ( d K(x, z) = tanh γ x T z + r) (2.28) γ, r, d 2.3.2 C SVM 3.7 2.3.4 SVM SVM 2 SVM SVM SVM 23
3 3.1 3.2 3.3 3.4 SVM 3.5 3.6 3.7 SVM 3.8 SVM 3.1 2 2.2.2 2 24
3 3.2 Support Vector Machine (SVM) SVM 3.4 3.6 3.2 3.1 3 4 4 25
3 3.2 SVM 4 SVM SVM SVM SVM 2 2 SVM 4 2 26
3 3.2 3.1: 27
3 3.3 3.3 2.2.2 [10, 11, 12, 13] 2 Digidesign Pro Tools 1 [17] Pro Tools 3.2 44.1 KHz 16bit WAVE 2 2 100 BPM / 132 BPM 4 8 4 3.3 3.6 3.3 1 3.4 2 2.5 1 DAW 28
3 3.3 3.2: 3.5 3 3.5 3.6 4 2 4 8 8 1 1 17 3.1 3.1 13 14 2 4 29
3 3.3 3.3: 1 3.4: 2 3.5: 3 3.6: 4 30
3 3.3 3.1: (BPM) 1 1 100 2 1 100 3 1 132 4 1 132 5 2 100 6 2 100 7 2 132 8 2 132 9 3 100 10 3 100 11 3 132 12 3 132 13 4 100 14 4 100 15 4 100 16 4 132 17 4 132 31
3 3.4 SVM 3.4 SVM SVM 2.3.4 SVM 2.2 SVM 2 3.5 3 1. 2. 3. 3 3.5.1 WAVE 32
3 3.5 WAVE [18] 1. FFT 2. FFT 0 0.5 2 0 DC 0.5 1 3. 2 IFFT x a (n) r(n) h(n) h(n) 3.7 A(n) A(n) = r(n) 2 + h(n) 2 (3.1) π/2 [19] 3.8 33
3 3.5 3.7: [18] 34
3 3.5 3.8: 3.5.2 1. 3.5.1 2. n 2n + 1 3. 2 k n = 250 k = 3 3.9 3.9 3.10 35
3 3.5 3.9: 3.10: 36
3 3.6 3.5.3 1. w 2. h 3. 4. 5. 4 6. w = 1000 h = 100 3.6 3.1 37
3 3.6 N t 1, t 2,, t N t d 3.11 2 m(t, d) = N [t k {t + (k 1)d}] 2 (3.2) k=1 m t d m t d 2 m t = 0 (3.3) m d = 0 (3.4) t = N N 4(N + 1) t k 6 kt k k=1 k=1 N(N + 1) (3.5) d = N N 12 kt k 6(N + 1) t k k=1 k=1 N(N + 1)(N 1) (3.6) 2 N N (3.5) (3.6) t k kt k 2 k=1 k=1 n k = k=1 n(n + 1), 2 n k 2 = k=1 n(n + 1)(2n + 1) 6 38
3 3.7 SVM 3.11: (3.5) (3.6) t d 3.7 SVM SVM 2.3.2 C 2.3.3 SVM C overfitting C (cross validation) n (n-fold cross validation) 1. SVM 39
3 3.7 SVM 3.2: 1 A, B, C D 2 A, B, D C 3 A, C, D B 4 B, C, D A = 75% C 2. n 3. n 1 SVM 4. 1 SVM 1 5. 3 4 6. SVM A,B,C,D 4 4 3.2 C 40
3 3.8 3.12: 3.3 17 17 3.8 SVM SVM SVM SVM 3.12 2 p 1 p 2 p 1 2 SVM 41
3 3.8 3.12 p 1 +5p 2 5 = 0 p 1 1 p 2 5 p 2 42
4 SVM 3 4.1 4.2 SVM 4.3 4.2 SVM 4.2 SVM 4.4 SVM 4.1 CPU : Intel Pentium 4 3.40GHz 2 RAM : 2.00GB OS : Microsoft Windows XP Service Pack 2 C++ Python SVM SVM LIBSVM [20] 43
4 SVM 4.2 SVM 4 4.2 SVM 4 17 SVM 4.2.1 1. SVM 2. SVM 3. SVM 4. SVM 1 1.5 2 2.5 3 3.5 4 4.5 44
4 SVM 4.2 SVM 4 1 3 2 4 SVM y i y i = 1 y i = 1 4.2.2 1 17 4 4 4.2.3 4.2.3.1 SVM 2.3.2 45
4 SVM 4.2 SVM 4 4.1: SVM C % 1 0.5 76.4706 1.5 512.0 82.3529 2 0.125 70.5882 2.5 0.5 88.2353 3 16.0 100.0000 3.5 2.0 94.1176 4 8.0 70.5882 4.5 4.0 88.2353 1 2.0 88.2353 3 0.25 88.2353 2 4.0 82.3529 4 512.0 76.4706 C 2 5, 2 4,, 2 15 C 17 C C 4.1 C 4.2.3.2 4.1 C SVM SVM [ 1, 1] 4.2 4.2 46
4 SVM 4.2 SVM 4 4.2: SVM % 1 82.3529 1.5 100.0000 2 76.4706 2.5 94.1176 3 100.0000 3.5 94.1176 4 88.2353 4.5 94.1176 1 88.2353 3 88.2353 2 82.3529 4 100.0000 2 76.4706% 1.5 3 4 100.0000% 80% 90% 4.2.3.3 4.3 f(x) f(x) = a 1 x 1 + a 2 x 2 + a 3 x 3 + a 4 x 4 + b = 0 (4.1) x 1, x 2, x 3, x 4 47
4 SVM 4.2 SVM 4 4.3: a 1 a 2 a 3 a 4 b 1-0.4379-1.2721 0.2892 0.4440-0.2892 1.5-7.3730-22.3474 25.6994-0.0244-1.6728 2-0.1434-0.3420 0.6253 0.5826-0.0708 2.5 0.2476-0.8413 1.2883 0.6664-0.2601 3 0.7195-3.7472 4.1272 1.0184-2.8756 3.5-1.3156-1.7463 1.0883 0.7699 0.1078 4-0.1106-0.6649 3.7733-1.1612-1.1827 4.5 0.7351-0.6975 2.9755 0.7320-0.8135 1-2.0616 0.8125 0.7273 0.7666-0.0832 3-0.3267-0.0192 1.0971 0.3093-0.0639 2-2.1500 0.3907 1.4855-1.0190 0.3718 4 0.2693-3.2189 40.4558-1.9122-24.0406 x 1 x 2 x 3 x 4 3.8 4 4.1 4.12 4.1 4.12 48
4 SVM 4.2 SVM 4 Coefficient (Absolute Value) of the Separating Hyperplane 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.1: 1 Coefficient (Absolute Value) of the Separating Hyperplane 30 25 20 15 10 5 0 Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.2: 1.5 49
4 SVM 4.2 SVM 4 Coefficient (Absolute Value) of the Separating Hyperplane 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.3: 2 Coefficient (Absolute Value) of the Separating Hyperplane 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.4: 2.5 50
4 SVM 4.2 SVM 4 Coefficient (Absolute Value) of the Separating Hyperplane 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.5: 3 Coefficient (Absolute Value) of the Separating Hyperplane 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.6: 3.5 51
4 SVM 4.2 SVM 4 Coefficient (Absolute Value) of the Separating Hyperplane 4 3.5 3 2.5 2 1.5 1 0.5 0 Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.7: 4 Coefficient (Absolute Value) of the Separating Hyperplane 3 2.5 2 1.5 1 0.5 0 Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.8: 4.5 52
4 SVM 4.2 SVM 4 Coefficient (Absolute Value) of the Separating Hyperplane 2.5 2 1.5 1 0.5 0 Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.9: 1 Coefficient (Absolute Value) of the Separating Hyperplane 1.2 1 0.8 0.6 0.4 0.2 0 Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.10: 3 53
4 SVM 4.2 SVM 4 Coefficient (Absolute Value) of the Separating Hyperplane 2.5 2 1.5 1 0.5 0 Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.11: 2 Coefficient (Absolute Value) of the Separating Hyperplane 45 40 35 30 25 20 15 10 5 0 Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.12: 4 54
4 SVM 4.2 SVM 4 4.4: 1 2 2 8 0 2 2 5 1 4 3 2 3 1 6 4 6 2 2 2 4.2.4 4.1 4.12 4.4 4.4 1 8 4 4 4.7 4.5 4.8 3 4.10 4 4.12 2.2.2 1 2 2 55
4 SVM 4.2 SVM 4 1 4.9 2 4.11 1 1.5 4.2 56
4 SVM 4.3 SVM 4 4.3 SVM 4 17 SVM 4.2 SVM 4.3.1 1. SVM 2. SVM 3. SVM 4. SVM 5. SVM SVM 4.2 1 1.5 2 2.5 3 3.5 4 4.5 57
4 SVM 4.3 SVM 4 1 3 2 4 SVM y i 4.2 y i = 1 y i = 1 4.3.2 1 17 4.2 4 4 4.3.3 SVM 2.3.3 2.3.3 58
4 SVM 4.3 SVM 4 Radial Basis Function (RBF) ( K(x, z) = exp γ x z 2), γ > 0 (4.2) RBF [21] RBF γ (2.28) RBF [22] RBF SVM C RBF (4.2) γ 4.3.4 4.3.4.1 4.2.3.1 SVM C RBF (4.2) γ 2 C : 2 5, 2 4,, 2 15 γ : 2 15, 2 14,, 2 3 C γ 17 C γ 59
4 SVM 4.3 SVM 4 4.5: SVM 4 C γ % 1 128.0 0.001953125 76.4706 1.5 64.0 0.0625 88.2353 2 1.0 0.125 76.4706 2.5 16.0 0.015625 88.2353 3 16384.0 0.00048828125 100.0000 3.5 2048.0 0.00048828125 94.1176 4 8192.0 0.00048828125 70.5882 4.5 1.0 0.5 94.1176 1 2048.0 0.00048828125 88.2353 3 8.0 0.015625 88.2353 2 1024.0 0.001953125 82.3529 4 4096.0 0.03125 82.3529 C γ 4.5 RBF C γ 4.3.4.2 4.5 C γ SVM 4.6 4.6 1 2 82.3529% 3 4.5 4 100.0000% 80% 90% 60
4 SVM 4.3 SVM 4 4.6: SVM % 1 82.3529 1.5 94.1176 2 82.3529 2.5 94.1176 3 100.0000 3.5 94.1176 4 88.2353 4.5 100.0000 1 88.2353 3 88.2353 2 82.3529 4 100.0000 4.3.5 4.2 4.2 SVM 4.6 SVM 4.7 4.7 SVM SVM 2 4.5 1.5 4 SVM 4.2 SVM 80% 90% SVM 61
4 SVM 4.3 SVM 4 4.7: SVM SVM SVM SVM % % 1 82.3529 82.3529 1.5 100.0000 94.1176 2 76.4706 82.3529 2.5 94.1176 94.1176 3 100.0000 100.0000 3.5 94.1176 94.1176 4 88.2353 88.2353 4.5 94.1176 100.0000 1 88.2353 88.2353 3 88.2353 88.2353 2 82.3529 82.3529 4 100.0000 100.0000 62
4 SVM 4.4 SVM 2 4.4 SVM 2 4.2 2 17 SVM 2 4.4.1 1. SVM 2. SVM 3. SVM 4. SVM 5. SVM 4.2 4.3 1 1.5 2 2.5 3 3.5 63
4 SVM 4.4 SVM 2 4 4.5 1 3 2 4 SVM y i 4.2 4.3 y i = 1 y i = 1 4.4.2 1 17 4.2 2 2 4.4.3 SVM 2.3.3 4.3.3 64
4 SVM 4.4 SVM 2 Radial Basis Function (RBF) ( K(x, z) = exp γ x z 2), γ > 0 (4.3) RBF 4.3.3 SVM C RBF (4.3) γ 4.4.4 4.4.4.1 4.2.3.1 4.3.4.1 SVM C RBF (4.3) γ 2 C : 2 5, 2 4,, 2 15 γ : 2 15, 2 14,, 2 3 C γ 17 C γ C γ 4.8 RBF C γ 4.4.4.2 4.8 C γ RBF SVM SVM 65
4 SVM 4.4 SVM 2 4.8: SVM 2 C γ % 1 256.0 0.00048828125 82.3529 1.5 4096.0 0.5 82.3529 2 32.0 0.015625 76.4706 2.5 1.0 4.0 70.5882 3 32.0 0.015625 70.5882 3.5 128.0 0.015625 76.4706 4 8192.0 0.125 64.7059 4.5 8.0 1.0 100.0000 1 2048.0 0.001953125 58.8235 3 8.0 2.0 76.4706 2 1024.0 0.5 52.9412 4 256.0 0.0625 64.7059 4.9 4.13 4.24 4.9 3 70.5882% 1.5 4.5 100.0000% 1.5 4.2 SVM 100.0000% 70% 80% 4.2 SVM SVM SVM 2 66
4 SVM 4.4 SVM 2 4.9: SVM % 1 82.3529 1.5 100.0000 2 76.4706 2.5 88.2353 3 70.5882 3.5 76.4706 4 88.2353 4.5 100.0000 1 70.5882 3 94.1176 2 88.2353 4 82.3529 4.4.4.3 4.13 4.24 4.13 4.24 4.13 4.24 4.10 4.13 4.24 1. 2. 1000 1000 3. RBF f(x) 67
4 SVM 4.4 SVM 2 4.10: ( f(x) 0.1 ) (f(x) < 0.1 ) (f(x) > 0.1 ) f(x) = x i SV ( αi y i exp γ x i x 2) + b (4.4) f(x) 0.1 f(x) 4.4.1 y i f(x) < 0 f(x) > 0 68