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Page 00000391 The timbre space of the classical guitar and its relationship with the plucking techniques Nicola Orio DEI - University of Padova Padova, 35131, Italy firstname.lastname@example.org ABSTRACT In this paper it is presented an investigation of the timbre space of the classical guitar, with nylon strings and plucked by fingers. To highlight the relationship among the plucking techniques and the obtained timbre, a set of forty guitar sounds were sampled and analyzed. Samples were obtained by applying four different plucking techniques with a number of different degrees. All the samples had the same pitch and the same loudness. Three parallel analyses were performed: timefrequency analysis, psychoacoustic parameterization, and Mel Frequency Cepstrum Coefficients extraction. Using Principal Component Analysis it was reduced the redundancy of the information. Analyses highlighted: (i) which are the main acoustic parameters related to the different plucking techniques, in good agreement with the theory on the guitar sound production; (ii) that it is possible to recognize the plucking technique through sound analysis. 1. INTRODUCTION The act of playing a musical instrument consists in the production and control of sound. Since acoustic musical' instruments are able to produce a great variety of different sounds, which differ in their characteristics of pitch, loudness, and timbre, great part of musicians' training is spent in learning the relationship among the playing techniques and the obtained sound. While pitch and loudness have a clear definition and it is, in general, easy to point out their relationship with the playing technique, the same is not true for timbre. First of all, timbre is a sound characteristic hard to study, because of its intrinsic complexity: it has a multidimensional feature, which varies in time. Moreover, even the common definition of timbre is somehow elusive: "timbre is the characteristic that permits to recognize differences in two sounds when pitch and loudness are equal". From this definition appears that the word timbre can be used both to distinguish sounds produced by different musical instrument, and to differentiate sounds produced by the same instrument that have a different colors: we can refer to these colors as timbre nuances. Some works were carried out on the timbre of different musical instruments. Gray  developed a study about the relationship between the timbre characteristics of a set of sound samples and the subjective impressions of a group of listeners. Other works focused on the creation of a timbre space through sound parameterization and mapping techniques, like Neural Nets  and multivariate statistical analyses . In this work the analysis is focused on the timbre nuances produced by a single musical instrument, when the playing technique is varied. That is, when timbre depends on how a given note is played. It was chosen to use, as a test instrument, an acoustic nylon string guitar, plucked by fingers with nails, like in the common technique taught in Music Conservatories. Two reasons suggested that the acoustic guitar could be a good testbed for this kind of analysis. From the one hand, the guitar can produce a great variety of different timbre nuances, but at the same time it has a typical sound, always recognizable by listeners; hence timbre variations are inside some perceptual boundaries. From the other hand, guitar timbre nuances mainly depend on the interaction between the string and the right hand fingers, which is easily observable and measurable. Moreover the guitarist has a number of different plucking techniques that he can apply independently one from the other, as well as in different combinations. 2. SOUND SAMPLES Sounds were obtained by a classical Spanish guitar, a First Class Ramirez of the 1988, with nylon strings. Sounds were digitally recorded at a sampling rate of 48 kHz, with a DAT connected to an AKG cardiod microphone positioned at a distance of 20 cm from the guitar hole. All the notes were played on the same string, the fifth, on the fifth fret: thus pitch was D at 150 Hz, with a loudness of - 10 dB that corresponded to a mezzoforte. The different timbre nuances were obtained by slowly changing the plucking technique. The variations of the plucking technique were made considering a normal position and inclination of the finger, seven centimeters away from its vibrating center and orthogonal to the guitar body. Samples were obtained by changing respectively: * the finger position along the string. Twentyeight samples were recorded, each one moving the finger of 1 cm from the 12th fret to the bridge; the finger had the normal inclination. * the inclination between the finger and the string, ICMC Proceedings 1999 - 391 -
Page 00000392 moving the finger in a plane orthogonal to the guitar body and parallel to the string. Seven samples were recorded, each one changing the inclination of 15~, from -45~ to +45~; the finger was in the normal position along the string. * the inclination between the hand and the string, moving the finger in a plane orthogonal to the guitar body and to the string. Seven samples were recorded, each one changing the inclination of 300, from -90~ to +90~; the finger was in the normal position along the string. * the degree of relaxation of the plucking finger. Seven samples were recorded, each one progressively decreasing the finger's tension, from strappato (maximum tension) to appoggiato (minimum tension); the finger was in the normal position and inclination on the string. A number of different sounds, obtained by mixing the different techniques, were added to these samples. Moreover they were sampled also sounds obtained by changing the rotation of the finger around its axis; since the differences among samples were not relevant, both perceptually and acoustically, they were neglected. 3. TIMBRE PARAMETRIZATION All the samples were analyzed, using time-frequency techniques, psychoacoustical parameterization, and Mel Frequency Cepstral Components extraction and mapping through Principal Components Algorithm. Time-Frequency Analysis Each guitar sound were analyzed in ten different portions of the signal, spaced of 100 ms and beginning at the onset time, in order to test if there are relevant differences in timbre depending on the evolution of the signal. The Short Time Fourier Transform was used: the signal was windowed with a Hamming window 1024 points long. The degree of inharmonicity of the sounds was the first measured parameter. As it is known a certain inharmonicity, with slightly sharper harmonics, is typical for the non-ideal string. The variations of this parameter were found significantly relevant only in relation to the finger position along the string. Posit. 2nd 9th 16 23rd 30th 0 2.008 9.133 16.258 23.312 30.368 9 2.037 9.145 16.305 23.321 30.459. 18 2.038 9.157 16.289 23.311 30.401 27 2.056 9.422 16.716 24.023 31.185 Table 1: Harmonic-to-fundamental ratio for some of the samples, obtained by moving the finger along the string; distance are measured in cm from the center of the string. In Table 1 it is quoted the harmonic-to-fundamental frequency ratio for some of the harmonics. It can be seen that this ratio increases when the finger plucks the string closer to the bridge. Other techniques were not found related to this parameter: the variations of inharmonicity are about 1/10 smaller and they probably are affected by noise, since a random trend was observed. The measure of the harmonics amplitude shows that this parameter is related to the changes in all the different techniques. In particular moving the finger towards the bridge enhances the higher harmonics, coherently with the theory on plucked strings. Moreover the normal position has a high-pass spectrum if compared to the samples obtained by changing both finger and hand inclinations. Only the degree of finger's relaxation seems to have a slight correlation with the harmonics amplitude. Attack time of the fundamental were found a good discriminant for the normal position in respect to hands movements. It has an attack time of 6.7 ms, while all the others are inside the range from 3.6 to 4.3 ms. This can be explained considering that an hand inclination greater that zero (in both direction) excites the resonant mode orthogonal to the guitar body, which transmits its energy to the bridge faster than the resonant mode parallel to the body (that is the only one directly excited when the string is plucked in the normal position). Changes in other techniques were not significantly related to the attack time, even if it was observed that samples obtained with finger at 4 cm, or less, to the bridge have a very fast attack time, from 2.4 to 2.7 ms. Psychoacoustical Analysis Another kind of analysis were carried out, for the extraction of the so-called psychoacoustical parameters . Two parameters were calculated, the Center of Gravity of the Spectrum (CGS) and the Irregularity of the Spectrum (IRR), defined by the two formulas: N CGS =k_ k = N-I A IRR = log 20log Ak k=2 IA-IA*A where, in both equations, k is the number of the harmonic and Ak is the amplitude of the k-th harmonic. Results on CGS were coherent with the ones obtained from time-frequency analysis on harmonics amplitude. CGS, which varies from 1.7 to 6.2, has a linear trend when the finger is moved along the string; while, when the finger or hand inclinations are changed, it has a symmetric trend, almost parabolic with the maximum (CGS=4.7) in the normal position and the minima at the highest inclinations. These two symmetric trends can be - 392 - ICMC Proceedings 1999
Page 00000393 explained considering that, if the nail is symmetric, the timbre nuances depend only on the angle absolute value, not on its sign. CGS parameter is also related to changes in finger degree of relaxation. Results are quoted in Figure 1. As it can be seen CGS decreases when the degree of relaxation increases. 5.5 5 4.5 4 3.5 0 u.0 I ~. minimum average maximum Figure 1: Values of CGS when the string is plucked with an increasing degree of relaxation of the finger Changes in IRR were found significantly relevant only when the finger is moved along the string. It has a maximum value-(IRR=2.44) when the string is plucked in the middle, that is with maximum distance from the bridge; then it decreases to its global minimum (IRR=1.78) with the finger close to the bridge. IRR is almost randomly distributed in the interval from 2.03 to 2.21 when other plucking techniques are changed. Principal Component Parametrization of Mel Frequency Cepstral Coefficients The last analysis technique was the extraction of the Mel Frequency Cepstral Components (MFCC), introduced in . Its basic idea is to perform the cepstral analysis in mel, rather than in hertz, scale. Mel scale approximates how the frequency is perceived, hence it is linear with hertz below 1 kHz and then logarithmic. So the signal was filtered by a bank of triangular filters equally spaced in mel scale, obtaining a set of log-energies Xk, one for each filter. The set of log-energies is then transformed with the formula: MFCC, = X, cos[i(k - k-l where N is the number of triangular filters. In this analysis the filters were spaced by 150 mel, hence centered on the frequency of the harmonics under 1 kH; they were evaluated 30 coefficients. It was chosen to reduce the redundancy of the information contained in these coefficients using a multivariate statistical technique  known as the Principal Component Algorithm (PCA). With this.technique it is possible to map the space of MFCC parameters in a space with a smaller dimension and with uncorrelated axes. PCA performs a projection of the coefficients in this space giving the percentage of the global variance in the parameters explained by each new axis. The samples, obtained by the plucking techniques, were separately mapped in four different spaces. Results were similar for all the techniques. In all the cases a single component was enough to explain more than 70% of the global variance; the components after the first were not statistically relevant. Results for each single technique are quoted in Table 2. Technique 1st Comp. 2nd Comp. Finger movement 72.8% 7.9% Finger inclination 78.6% 13.3% Hand inclination 87.4% 6.7% Finger Relaxation 77.3% 11.4% Table 2: Percentage of global variance explained by the first and the second principal components, separately calculated on the different plucking techniques The fact that a single component is enough to explain most of the global variance, is coherent with the fact that, in each case, it was varied only one performing parameter. Hence the first principal component can be, probably, related to the effect in spectrum of the variations in the plucking techniques. To test this hypothesis, the MFCC can be projected, through PCA, in the first axis of the new space. In most of the cases it was found a good correlation among the positions of the samples on this axis and the variations in the plucking technique. In Figure 2 the different samples obtained moving the finger along the string are plotted. How it can be seen the value of the first component decreases when the finger moves toward the bridge. ~^i~-~~~-:----~:~--;::-;.^;-----. 600 0...... 0.. s 20 ~...... *...... -.....e o,.. | 6..-.....................-...|.............|.........i.... i.............:.... ~....;......... L....;i............... 0 5 10 15 20 Figure 2: Projections along the First Principal Component of samples obtained changing the finger position along the string This correlation is maintained also when both finger and hand inclinations are varied. Results are ICMC Proceedings 1999 - 393 -
Page 00000394 respectively plotted in Figure 3 and in Figure 4. AA. 35 30 25 20........................................................................... 0 0!::! i i i~ * ~ ~0 ' i ___'_63 ~~: i!I -45* -30' -15' 0* 15* 30" 60* Figure 3: Projections along the First Principal Component of samples obtained changing the finger inclination In both cases the first principal component has values symmetrical around the normal inclination. Also in this case this can be explained considering that the effect of these two techniques in timbre nuances depends only on the absolute value of the angle, not on its sign. The analysis developed on samples obtained by changing the degree of relaxation gave a trend similar to the one previously shown in Figure 1. 30....................................... J.... 30 0 0 0 10 0 0 ___........................ -90* -60* -30' 0' 30* 600 90* Figure 4: Projections along the First Principal Component of samples obtained changing the hand inclination Finally the whole analysis process, MFCC and PCA, was developed on the whole group of samples, including the ones obtained mixing different techniques. Unfortunately results were not as expected. Also in this case the weight of the first component is predominant, it explains 55.6% of the global variance while the second and the third components explain, respectively, 14.8% and 9.2%. It is interesting to note that each technique maintains its trend, if mapped on the first axis. Samples obtained with mixed techniques map coherently with the techniques used, even if they are more affected by changes on the finger position along the string. In order to separate the different timbre nuances, the samples were mapped on the plane of the first two principal components. Results were affected by the low variance explained by the second componerit: different techniques map close in the plane, even if it is possible to separate samples obtained by moving the finger along the string and samples obtained by varying the degree of relaxation of the finger. Changes on inclination, in both direction, are hard to discriminate. 4. CONCLUSIONS The timbre space of a classical guitar was analyzed in order to investigate the relationship among timbre nuances and plucking techniques. To this aim, a group of guitar sounds, played changing in four different ways the plucking techniques, were analyzed using different techniques. The possible variations of guitar timbre were highlighted in relation to the different plucking techniques. From analyses it emerged: (i) the relationship among plucking technique and timbre characteristics; (ii) that it is possible to recognize the plucking technique from the sound analysis. The system needs some improvement when more plucking techniques are applied together. A future development of this work will be the extension of the analyses to different notes, with different pitch and loudness, in order to have a complete description of the timbre space of the classical guitar. REFERENCES  Grey, J.M., Moorer, J.A., "Perceptual Evaluations of Synthesized Musical Instruments Tones", Journal of Acoustic Society of America, Vol. 62(2), pp. 454-462, 1977.  Cosi, P., De Poli, G., Prandoni, P.,'"Timbre Characterization with Mel-Cepstrum and Neural Nets", in Proc. of International Computer Music Conference, Aarhus, Denmark, pp. 42-45, 1994.  Boatin, N., De Poli, G., Prandoni, P., "Timbre Characterization with Mel-Cepstrum: a Multivariate Analysis", in Proc. XI Colloquium on Musical Informatics, Bologna, Italy, pp. 145-148, 1995.  Krumhansl, C.L., Why is musical timbre so hard to understand?, in Structure and perception electroacoustic sound and music, in S. Nielsen and O. Olsson (ed), Amsterdam, pp. 43-53, 1989.  Davis, S.B., Mermelstein, P., "Comparison of Parametric Representations for Monosyllabic Word Recognition in Continuously Spoken Sentences", IEEE Trans. on Acoustic, Speech, and Signal Processing, Vol. 28(4), pp. 357-366, 1980.  Beyerbach, D., Nawab, H., "Principal Component Analysis of the Short Time Fourier Transform", in Proc. of International Conference on Acoustic, Speech, and Signal Processing, Vol. 3, pp. 1725-1728, 1991. - 394 - ICMC Proceedings 1999