Wavetable Matching of Inharmonic String Tones Clifford So and Andrew Horner Department of Computer Science Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong email:
[email protected] and
[email protected]
Abstract Most previous work on resynthesizing musical instrument tones with wavetable synthesis has assumed the original tone was harmonic or nearly harmonic. This assumption does not hold for string tones. This paper introduces a wavetable matching technique for inharmonic string tones. It separates partials into different classes based on their degree of inharmonicity, and uses wavetable matching for each class. We give results for bowed vibrato violin and cello, plucked Chinese pipa and non-vibrato trumpet tones. Listening tests have found that the new method significantly improves the perceived match of the inharmonic string tones, but a degradation on the harmonic trumpet tone compared to simple wavetable matching. This suggests that the new method is best strictly for inharmonic tones modeling. 0. Introduction The term “wavetable synthesis” is currently often used synonymously with “sampling synthesis” in the music industry. We will use the classical computer music meaning of “wavetable synthesis” throughout this paper, where an oscillator table is loaded with several partials and indexed depending on the fundamental frequency. Several weighted wavetables can be summed to produce spectral changes in multiple wavetable synthesis. When the partials are disjoint (i.e. each partial only appears in one of the wavetables), this is called group additive synthesis [1]-[3]. Recent work has also shown how to match wavetable parameters for resynthesizing instrument tones [4]-[7]. One of the most successful of these methods, wavetable 1
matching, uses additive syntheses of spectral snapshots from the original tone to load the wavetables. This approach is intuitive, and guarantees an exact match at the timepoints of the selected spectral snapshots and excellent matches at neighboring points as well. Figure 1 shows the multiple wavetable block diagram, where the output of each wavetable is scaled by an amplitude envelope and summed into the output signal. Wavetable matching occurs in the frequency domain and determines the spectra of the wavetables, called the basis spectra. Previous work on wavetable matching has used genetic algorithm (GA) optimization to determine the best time points to select the basis spectra. GAs are combinatorial optimization techniques modeled on the idea of evolving solutions to problems through processes analogous to natural selection and breeding [8][9]. These evolutionary processes guide parameter searching using populations of potential solutions. To judge the fitness of a particular combination of basis spectra, the fitness function uses a least-squares solution to find the amplitude envelopes of each basis spectra, and an average relative amplitude error to measure the difference between the original and matched tones: N hars
ξ rel =
1 N frames
N frames
∑ j =1
∑ (b k =1
k, j
N hars
− bk′ , j ) 2 (1)
∑b k =1
2 k, j
where the N frames is the number of analysis frames, N hars is the number of partials, bk′ , j is the kth partial amplitude of the synthesized tone at the jth analysis frame, and bk , j is the original tone’s amplitude on the same partial and analysis frame. A relative amplitude error of zero indicates an exact match, while an error of 0.1 indicates an average relative difference of 10%. The GA uses the relative amplitude error as the fitness value of the candidate basis spectra, mixing and matching basis spectra combinations from the best individuals in the population, returning the best set of basis spectra as the solution. The cost of computing the relative amplitude error for each set of basis spectra can be held down by limiting the match points to a small number of representative test spectra (usually about 20).
2
However, wavetable synthesis makes a basic assumption: the partials are harmonics, i.e., they have frequencies which are at integer multiples of the fundamental frequency ( f k = kf1 ). This causes problems when modeling inharmonic instruments, such as the string instruments. Wavetable matching generates a dynamic spectrum as close to the original as possible. However, it does not take into account the tone’s inharmonicity (the amount of frequency deviation between the partials and integer multiples of the fundamental frequency). The degree of inharmonicity is dependent on the instrument [10]. Figure 2 shows the normalized frequency deviation (NFD) for four acoustic instruments. We define the NFD for a specific partial k as the following: NFDk =
1 N frames
N frames
f k (t ) − 1 kF
∑ t =1
(2)
where F is the fixed fundamental frequency, f k (t ) is the actual time-varying frequency of partial k, and t is the frame number. The expression calculates the plus or minus frequency deviation percentage in each time frame and averages the percentages over the total number of frames. Figure 2 shows that string instruments such as the plucked pipa (G2) and bowed vibrato cello (E2) have larger and more variable frequency deviations than the non-vibrato trumpet (F4) and bowed vibrato violin (C#4), which has nearly constant deviations. It is well known that plucked pipa produce “free vibrations”, which contains inharmonicity; bowed strings produce “forced vibrations”, which are strictly harmonic when averaged over time [15]. In our experiment, however, string tones with vibrato is used, which is different from the situation of “forced vibrations”. Inharmonic behavior is detected in our string tones with vibrato. The cello tone has larger frequency deviations then the violin tone just because of the stronger vibrato width shown in our sample. The trumpet is close to the harmonic assumption of wavetable matching, while the strings are more problematic.
3
Lee and Horner [11] recently introduced a group additive synthesis method for matching piano tones with stretched partials. Piano tones have simpler inharmonicity behavior than the string tones discussed in this paper, because for the piano the amount of average frequency deviation increases in a fairly continuous way from partial to partial. This is unlike the pipa tone (see Figure 2), where a partial may be very flat and its neighbor very sharp. Such deviations require a special model to represent them accurately. This paper introduces an inharmonic wavetable matching technique that classifies partials with similar normalized frequency deviations. The method then uses simple wavetable matching for each class. The next section describes the method. Section 2 gives results for the violin, cello, pipa and trumpet tones. The results show that the adapted method can simulate inharmonic partials, and even possibly reduce the average relative amplitude error compared to simple wavetable matching.
1. Inharmonic Wavetable Matching This section shows how to adapt wavetable matching to allow inharmonic partials. The basic idea is to separate the partials into different classes based on their degree of inharmonicity, and then use simple wavetable matching to find the basis spectra and amplitude envelopes for each class. Figure 3 shows the procedure. First, a short-time Fourier analysis algorithm such as phase vocoder is used to transform the tone from the time domain to the time-varying frequency domain. Bandpass filters are applied on the center of the harmonics of a fundamental frequency [12] in the analysis. This, however, will fail if frequency deviations are so great that the higher harmonic frequencies swing on the edges or outside the ranges of the filters. We use an extension of the McAulay-Quatieri analysis [13] to successfully track the frequency deviations where they are very large (that is, substantial vibrato). Beauchamp [14] gives more details on the applications of these techniques. Next, partials are grouped into different classes according to their normalized frequency deviations NFDk . Ordinary GA-wavetable matching then determines the basis spectra and
4
amplitude envelopes of each class. We also need to find a common normalized frequency deviation for each class ( CNFDi ). To resynthesize the sound, the output signal xi (t ) of each class i is generated using the basis spectra, amplitude envelopes and time-varying fundamental frequency deviation gi (t ) (which is derived from the CNFDi ). Finally, the class signals xi (t ) are summed to form the resynthesized tone x(t ) .
Section 1.1 discusses how to separate the partials into different classes. Section 1.2 shows how to find the basis spectra for each class, and section 1.3 describes how to resynthesize the signal.
1.1
Separation of the Partials into Different Classes
Figure 4 shows a simple method for separating partials with similar frequency deviations into distinct groups. The user specifies a NFD class size (0.002 for this example) which defines a series of equally-spaced bins along the NFD axis. If the NFD of a partial k falls into the bin of class i, partial k belongs to class i. The common NFD for class i, CNFDi is calculated by a weighted average of its partials’
NFD, CNFDi =
∑
k∈Classi
Bk NFDk
∑B
k∈Classi
(3) k
where Bk is the average amplitude of partial k :
Bk =
1 N frames
N frames
∑ bk ,t
(4)
t =1
The CNFDi are weighted to allow louder partials to contribute more. We can calculate the time-varying fundamental frequency of class i as:
5
g i (t ) = f1 (t )(1 + CNFDi )
(5)
f1(t) can be obtained from our McAulay-Quatieri analysis previously done. Dedicated time-varying gi(t) can be calculated by not relating it strictly to the f1(t) like Equation 5, but relates to an arbitrary f1(t) having the time-varying trend similar to the class i. This may give better overall resynthesize resolution, or in the case of weak or missing fundamental frequency. However the Equation 5 employed in our experiment is plausible because our fundamentals are strong and the trend of the frequency deviations are similar in each class i. This gives a reasonable data reduction in resynthesize. On the other hand, if the instrument tone has widely-spread NFDk values and not having enough classes, noticeable artifacts will occur. The artifacts will decrease as the number of the classes is increased. The user should choose a suitable value depending on the spread of the NFDk values.
1.2
Selection of the Basis Spectra
After the partials have been separated into classes, we simply pass the spectrum of each class i to the normal GA-based wavetable matching procedure to determine the basis spectra and amplitude envelopes. So how many wavetables does each class i need? Well, that depends on how many partials are in the class. We set the number of wavetables ji in class i according to the following relation:
ji =
# Hi d , if 1 ≤# H i ≤ dj max j max , if # H i > dj max
(6)
where # H i is the number of partials in class i, d is a division factor ( ≥ 1 ), and j max is
the maximum number of basis spectra allowed in any class ( j max acts as an upper bound
6
for ji , and is chosen depending on the amount of data reduction desired). We will give results for various values of d in the results section. In Figure 4, j max = 5 and d = 2. Note that in Equation 6, we have a special case when # H i = 1 . If the class has only one partial (see Figure 5), we directly copy the original amplitude envelope to the class’s amplitude envelope wi ,1 (t ) and skip the GA-based matching procedure. Since the basis spectrum of the class only contains that partial, it gives a perfect match for that partial and saves the matching processing time. We will also see in Section 2 that one-partial classes help contribute to a lower overall relative amplitude error. 1.3
Resynthesis
To resynthesize the signal, we first synthesize each class i using multiple wavetable synthesis. We use the deviated fundamental frequency gi (t ) for each class rather than the original fundamental frequency f1 (t ) . y i ,l (t ) =
∑a
k∈Classi , l
k ,Classi , l
sin( 2πk ∫ g i (t )dt )
ji
xi (t ) = ∑ wi ,l (t ) y i ,l (t )
(7)
(8)
l =1
where yi,l(t) is the time-varying l-th wavetable for class i, a k ,Classi ,l is the spectrum amplitude of the partial k in to the l-th wavetable for class i, and wi,l(t) is the time-varying weight of the l-th wavetable for class i. Afterwards, we sum up the signals of all the classes xi (t ) to produce the resynthesized signal x (t ) .
x(t ) =
NumClass
∑ x (t ) i =1
(9)
i
7
2. Results Four instruments were matched using the original wavetable matching method and the new method. Instrument tones were recorded from bowed vibrato violin (C#4) and cello (E2), plucked pipa (G2) and trumpet (F4). The quality of the synthesized tones were evaluated in two dimensions: relative amplitude error (defined in Equation 1) and average NFD error (ANFDE). The ANFDE is defined as:
ξ ANFD =
1 N hars
N hars
∑ NFD k =1
k
− CNFDClass ( k )
(8)
where Class(k ) returns the class index of partial k . The quality of the synthesized tone is perceptually close to the original only if both errors are small. First, we consider relative amplitude error alone. Figures 6-9 show the plot of relative amplitude error versus the total number of wavetables for the different instruments. Each graph contains five curves representing the new method with five different division factors d (1.5, 2, 3, 4 and 6), and one curve representing the original method for reference. The points for the new method are obtained by varying the NFD class size. The point A represents a single large class that contains all the partials. The NFD class size decreases along the curve from left to right. The relative amplitude error decreases as the class size is narrowed and the number of wavetables increases. Surprisingly, the new method outperforms the original method when the total number of wavetables is large (we expected the relative amplitude error to increase a bit as a tradeoff for improving the average NFD error). Why? It appears the divide-and-conquer approach of the new method (using wavetable matching on each class) is more effective than optimizing the large number of basis spectra directly (the genetic algorithm is getting stuck on local optima). With more wavetables, the reduced class size of the new method results in more one-partial classes. Matching these crucial partials exactly
8
appears to result in a lower overall relative amplitude error. Although the new method cannot improve the relative amplitude error for smaller numbers of wavetables, the perceived quality improves because the ANFDE has been lowered, while the relative amplitude error is only slightly worse (about 1%). Figures 10-13 show ANFDE plotted versus relative amplitude error for the four tones. Each graph contains five curves representing the new method with different division factors, and one curve representing the original method. The curve representing the original method is a horizontal line across the top since the original method has only a single frequency deviation shared by all the partials. Next, we consider the curves representing the new method. The points from left to right along the curves are obtained by increasing the NFD class size. The point A represents a single large class that contains all the partials. The curves show a good performance on both errors when the class size is small, since the smaller class results in fewer partials in each class, and a class spectrum that is simpler to match. A smaller class size also results in a large number of classes, which increases the frequency deviation resolution. So, what is the best division factor? In general, we recommend a division factor of 3 or 4. A large division factor (such as 6) is undesirable because the relative amplitude error is much higher for a small number of wavetables. On the other hand, a small division factor (1 or 2), results in a higher frequency deviation error. Point B represents what we feel is a good balance point balance point between these factors. Formal listening tests were carried out using the original and two synthesized tones for each instrument, with one synthesized tone from the original wavetable matching method and one from the new method. Point B parameters were chosen for the new method. For the original method, the same number of wavetables were used as in the new method. Five subjects were employed, all under 30 with good hearing and reasonably strong backgrounds in both music and synthesis. The listeners used high quality headphones for the tests (Sony MDR-7506). Each listener heard each synthetic tone a total of five times.
9
A computer program was used to play test sets of an equal number of randomly ordered acoustic and synthetic tones. The subjects are asked to distinguish whether the played tone was acoustic or synthetic. Figure 14 gives the percentage of trials in which the average listener could correctly identify the synthetic tones. 50% represents a threshold of "near indistinguishability" where the user is basically guessing. The figure shows that the new method gives significant improvement on the cello and pipa tones, two instruments with large frequency deviations. For the nearly-harmonic tones of the violin, the new method gives about the same performance as the original method, but a degradation on the trumpet.
3. Conclusion We have introduced a variant of GA wavetable matching that takes time-averaged frequency deviations into consideration. The strategy is different from Lee and Horner’s approach [11] in which the inharmonics are grouped in an adjacent order. Our method groups the inharmonics based on their degree of deviations. Our results show that the new method can improve the frequency deviation resolution while maintaining the relative amplitude error. For division factors of 3 or 4, it gives an even lower relative amplitude error with large numbers of wavetables. Our listening test results show that the new method gives significant improvement for inharmonic string tones but except the near harmonic trumpet tone. This may be explained by the fact that the little inharmonicity in the unstable attack phase of the trumpet is “spread” across the whole tone, in which the sustain phase should be near-harmonic. This counter-example give us a clue that our method is best for modeling highly inharmonic tones like plucked strings and bowed strings with vibrato. For highly harmonic tones like brass instruments, we suggest reverting to the original wavetable matching method.
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Bibliography [1]
P. Kleczkowski, “Group Additive Synthesis,” Computer Music Journal, vol. 13, no. 1, pp. 12-20 (1989).
[2]
N.-M. Cheung and A. B. Horner, “Group Synthesis with Genetic Algorithms,” Journal of Audio Engineering Society, vol. 44, no. 3, pp. 130-147 (March 1996).
[3]
S. Oates and B. EagleStone, “Analytical Methods for Group Synthesis Computer Music Journal, vol. 21, no. 2, pp. 21-68 (1997).
[4]
M.-H. Serra, D. Rubine and R. Dannenberg, “Analysis and Synthesis of Tones by Spectral Interpolation,” Journal of Audio Engineering Society, vol. 38, no. 3, pp. 111-128 (1990).
[5]
A. B. Horner, J. Beauchamp, and L. Haken, “Methods for Multiple Wavetable Synthesis of Musical Instrument Tones,” Journal of Audio Engineering Society, vol. 41, no. 5, pp. 336-356 (May 1993).
[6]
A. B. Horner, J. Beauchamp, “Synthesis of Trumpet Tones Using a Wavetable and Dynamic Filter,” Journal of Audio Engineering Society, vol. 43, no. 10, pp. 799-812 (1995).
[7]
A. B. Horner, “Computation and Memory Tradeoffs with Multiple Wavetable Interpolation,” Journal of Audio Engineering Society, vol. 44, no. 6, pp. 481-496 (1996).
[8]
J. H. Holland, Adaptation in Natural and Artificial Systems. Ann Arbor: The University of Michigan Press.
[9]
D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Reading, MA: Addison-Wesley.
[10]
S. Dubnov and N. Tishby, “Influence of Frequency Modulating Jitter on Higher Order Moments of Sound Residual with Applications to Synthesis and Classification,” Proceedings of the 1996 ICMC, pp. 378-385 (1996).
[11]
K. Lee and A. B. Horner, “Modeling Piano Tones with Group Synthesis,” Journal of the Audio Engineering Society, vol. 47, no. 3, pp. 101-111 (1999).
[12]
R. Maher, “An approach for the separation of voices in composite music signals,” Ph.D. Dissertation, University of Illinois at Urbana-Champaign, pp. 19-28, 49-55 (1986).
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[13]
R. McAulay and T. Quatieri, “Speech analysis/synthesis based on a sinusoidal representation,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-34, pp. 744-754 (1986).
[14]
J. Beauchamp, “Unix workstation software for analysis, graphics, modification, and synthesis of musical sounds,” Presented at the 94th Convention of the Audio Engineering Society, (Abstract) Journal of Audio Engineering Society, vol. 41, no. 5, pp. 387 (1993).
[15]
J. C. Brown, “Frequency rations of spectral components of musical sounds,” Journal of Acoustical Society of America, vol. 99, pp. 1210-1218 (1996).
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List of Figures Figure 1. Multiple wavetable synthesis block diagram (original method). Figure 2. Normalized frequency deviations of partials of the violin, pipa, cello and trumpet. Figure 3. Overview of inharmonic wavetable matching. Figure 4. Classification of partials of the cello E2 tone (NFD class size = 0.002, division factor d = 2, jmax = 5, total number of wavetables needed = 15). Figure 5. Additive synthesis when only one partial is in the class. Figure 6. Plot of Relative Amplitude Error versus Total Number of Wavetables for Violin C#4 tone (79 harmonics). Figure 7. Plot of Relative Amplitude Error versus Total Number of Wavetables for Cello E2 Tone (133 harmonics). Figure 8. Plot of Relative Amplitude Error versus Total Number of Wavetables for Pipa G2 tone (31 harmonics). Figure 9. Plot of Relative Amplitude Error versus Total Number of Wavetables for Trumpet F4 tone (32 harmonics). Figure 10. Plot of Average Normalized Frequency Deviation Error versus Relative Amplitude Error for different Division Factors for Violin C# tone (79 harmonics). Figure 11. Plot of Average Normalized Frequency Deviation Error versus Relative Amplitude Error for different Division Factors for Cello E2 tone (133 harmonics). Figure 12. Plot of Average Normalized Frequency Deviation Error versus Relative Amplitude Error for different Division Factors for Pipa G2 tone (30 harmonics). Figure 13. Plot of Average Normalized Frequency Deviation Error versus Relative Amplitude Error for different Division Factors for Trumpet F4 tone (32 harmonics). Figure 14. Percentage of trials the average listener was able to correctly identify synthetic tones from a test set of both real and synthetic tones.
13
f1 (t )
....
w1 (t )
×
w2 (t )
....
×
wN (t )
×
+ wi (t ) amplitude envelope of ith table at time t
x(t )
f1 (t ) fundamental frequency at time t x(t ) sample output at time t N number of wavetables
Figure 1. Multiple wavetable synthesis block diagram (original method).
14
0.020 0.010
-0.010 -0.020 -0.030 Partials Violin C#4
Pipa G2
Cello E2
Trumpet F4
Figure 2. Normalized frequency deviations of partials of the violin, pipa, cello and trumpet.
15
29
27
25
23
21
19
17
15
13
11
9
7
5
3
0.000 1
Normalized Frequency Deviation
0.030
Original Tone
Short-time spectrum analysis
Time-varying harmonic spectra
Classification of partials based on their NFDk
For each class i
Class i partials
GA-based wavetable matching for class i
Find CNFDi
basis spectra and
gi (t ) = f1 (t ) (1 + CNFDi )
amplitude envelopes
Wavetable synthesis for class i
xi (t )
NumClass
∑ x (t ) i
i =1
x(t ) Figure 3. Overview of inharmonic wavetable matching.
16
Analysis Synthesis
Class +3 Class +2
29
27
Class 0
25
23
21
19
17
15
13
11
9
7
5
3
Class +1 1
Normalized Frequency Deviation
0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.000 -0.001 -0.002 -0.003 -0.004 -0.005 -0.006 -0.007
Class -1 Class -2 Class -3 Partials
Class index NFD range Partials in the class range
-3 [-0.007, -0.005)
-2 [-0.005, -0.003)
-1 [-0.003, -0.001)
2,4
6
1,3,5,9, 13,19,22, 23,24,28
Number of wavetables, 1 1 ji to be determined CNFDi -0.00557 -0.00316
0 [-0.001, +0.001] 8,12,14, 15,16,17, 18,20,21, 26,29
5
j max (=5 here)
-0.00183
+0.00023
+1 (0.001, +0.003]
+2 (+0.003, +0.005]
+3 (+0.005, +0.007]
10,11,25
7,27
2
1
+0.00189 +0.00422
Figure 4. Classification of partials of the cello E2 tone (NFD class size = 0.002, division factor d = 2, j max = 5, total number of wavetables needed = 15).
17
Classi
Basis Spectrum
Amplitude Envelope,
Figure 5. Additive synthesis when only one partial is in the class.
18
wi ,1 (t )
0.030
A
Relative Amplitude Error
0.025 0.020 0.015 0.010
B
0.005 0.000 0
5
10 #wt=#har/1.5 #wt=#har/4
15 20 25 30 Total Number of Wavetables #wt=#har/2 #wt=#har/6
35
40
#wt=#har/3 Original method
Figure 6. Plot of Relative Amplitude Error versus Total Number of Wavetables for Violin C#4 tone (79 harmonics).
19
45
0.025
A
Relative Amplitude Error
0.020
0.015
0.010
0.005
B 0.000 0
5
10
#wt=#har/1.5 #wt=#har/4
15 20 25 30 Total Number of Wavetables
#wt=#har/2 #wt=#har/6
35
40
#wt=#har/3 Original method
Figure 7. Plot of Relative Amplitude Error versus Total Number of Wavetables for Cello E2 Tone (133 harmonics).
20
45
0.140
A
Relative Amplitude Error
0.120 0.100 0.080 0.060 0.040 0.020
B
0.000 0
5
10
15
20
25
Total Number of Wavetables #wt=#har/1.5
#wt=#har/2
#wt=#har/3
#wt=#har/4
#wt=#har/6
Original method
Figure 8. Plot of Relative Amplitude Error versus Total Number of Wavetables for Pipa G2 tone (31 harmonics).
21
30
0.070
Relative Amplitude Error
0.060 0.050 0.040 0.030
A
0.020
B 0.010 0.000 0
5
10
15
20
25
Total Number of Wavetables #wt=#har/1.5
#wt=#har/2
#wt=#har/3
#wt=#har/4
#wt=#har/6
Original method
Figure 9. Plot of Relative Amplitude Error versus Total Number of Wavetables for Trumpet F4 tone (32 harmonics).
22
30
Average Normalized Frequency Deviation Error
0.0008 A
0.0007 0.0006 0.0005 0.0004 0.0003
B
0.0002 0.0001 0 0.000
0.005
0.010
0.015
0.020
0.025
Relative Amplitude Error #wt=#har/1.5 #wt=#har/4
#wt=#har/2 #wt=#har/6
#wt=#har/3 Original method
Figure 10. Plot of Average Normalized Frequency Deviation Error versus Relative Amplitude Error for different Division Factors for Violin C# tone (79 harmonics).
23
0.030
Average Normalized Frequency Deviation Error
0.0035
A
0.0030 0.0025 0.0020 0.0015 0.0010
B
0.0005 0.0000 0.000
0.005
0.010
0.015
0.020
Relative Amplitude Error #wt=#har/1.5
#wt=#har/2
#wt=#har/3
#wt=#har/4
#wt=#har/6
Original method
Figure 11. Plot of Average Normalized Frequency Deviation Error versus Relative Amplitude Error for different Division Factors for Cello E2 tone (133 harmonics).
24
0.025
0.0050
A
Average Normalized Frequency Deviation Error
0.0045 0.0040 0.0035 0.0030 0.0025 0.0020 0.0015 0.0010
B
0.0005 0.0000 0.000
0.020
0.040
0.060
0.080
0.100
0.120
Relative Amplitude Error
#wt=#har/1.5
#wt=#har/2
#wt=#har/3
#wt=#har/4
#wt=#har/6
Original method
Figure 12. Plot of Average Normalized Frequency Deviation Error versus Relative Amplitude Error for different Division Factors for Pipa G2 tone (30 harmonics).
25
0.140
A
0.0010 Average Normalized Frequency Deviation Error
0.0009 0.0008 0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001
B
0.0000 0.000
0.010
0.020
0.030
0.040
0.050
0.060
Relative Amplitude Error #wt=#har/1.5
#wt=#har/2
#wt=#har/3
#wt=#har/4
#wt=#har/6
Original method
Figure 13. Plot of Average Normalized Frequency Deviation Error versus Relative Amplitude Error for different Division Factors for Trumpet F4 tone (32 harmonics).
26
0.070
Tones Original method violin New method violin Original method cello New method cello Original method pipa New method pipa Original method trumpet New method trumpet
Number of Wavetables 18 18 21 21 12 12 12 12
Percentage 60% 60% 76% 32% 88% 24% 48% 68%
Figure 14. Percentage of trials the average listener was able to correctly identify synthetic tones from a test set of both real and synthetic tones.
27
