EXCEPCIÓN DE CADUCIDAD DE LA ACCIÓN

SEUNGHUN KIM1

([email protected])

,MOONSEOK KIM

([email protected])

,AND WOON SEUNG YEO1

([email protected])

1

Graduate School of Culture Technology, KAIST, Daejeon, Korea

This article proposes a commuted waveguide synthesis model for the geomungo, a Ko- rean traditional plucked-string instrument featuring unusually wide vibrato. To model pitch fluctuation as well as the decay characteristics of its harmonic partials, a time-varying loss filter with a sinusoidal loop gain is used. Filter parameters are estimated by approximating the gains, and excitation signals are generated by inverse filtering the original sound. Real-time implementation of the algorithm for the development of the virtual instrument is also discussed.

0 INTRODUCTION

Physical modeling synthesis is based on the use of the mathematical models of physical acoustics for computer simulation of sound [24]. By manipulating control param- eters, users can modify the physical structure of the sound source (usually a musical instrument) or performer’s inter- action with it, thereby avoiding the need to record every possible sound from the source.

Since the introduction of the Karplus-Strong (KS) al- gorithm for plucked string instruments [13], many studies on physical modeling synthesis have been proposed. Jaffe et al. suggested the use of the KS algorithm for the synthesis of stringed instrument tones [11]. Smith proposed a digital waveguide theory that simulates traveling waves with digi- tal delay lines [28] [29] [31]. Based on the assumption that the commutativity of the linear time-invariant (LTI) system is valid, damping and dispersion are also lumped at specific points to reduce the computational cost and allow real-time sound synthesis.

Karjalainen et al. showed that bi-directional digital waveguides could be reduced to a single-delay loop (SDL), a generalized form of the KS algorithm [12]. Here, the over- all transfer function from the input (the excitation signal in the digital waveguide model) to the output on the bridge is obtained from the partial transfer functions. Since this results in several filters forming a cascade structure, the SDL model is conceptually simpler and easier to imple- ment. Considering this, a plucked string instrument model contains three components—excitation signal, string, and instrument body—each of which can be considered as a fil- ter. However, the order of the filter for modeling the body is usually very high and requires too much computational cost for real-time sound synthesis.

To resolve this, Smith proposed commuted waveguide synthesis [30], i.e., rather than forming a distinct filter, the body part is combined with the original excitation signal as a new input, which is stored in the excitation table and used to generate sound through the string model. Excitation samples can be extracted from the digital recordings of the instrument by inverse filtering of the string model, which is a cascade form of several filters. Parameters of the model can be estimated by analyzing the decays of harmonics from recorded samples.

Examples of instrument sound synthesis based on the digital waveguide theory include wind [14] [25] [26], bowed string [27] [37], and plucked string. V¨alim¨aki et al. presented a real-time digital waveguide model for several plucked string instruments including the guitar, banjo, man- dolin, and kantele [33]. They adopted the commuted waveg- uide synthesis model and Lagrange interpolation for frac- tional delay, and proposed a real-time synthesis model on a digital signal processor. This was followed by an improved model to implement a guitar synthesizer [36]. Synthesis models for the piano [2] [32] and harpsichord [35] have also been investigated. In addition, spring reverberation ef- fect was simulated based on similar filter components used in digital waveguide theory [34].

While the majority of digital waveguide synthesis ex- amples feature western musical instruments, some studies have dealt with old Asian plucked-string instruments. Erkut et al. proposed a synthesis model for the ud and the Renais- sance lute, which originated in the Middle East [8], and implemented the glissando (a glide from one pitch to an- other) on fretted/fretless instruments. A model-based sound synthesis algorithm of the guqin, a Chinese plucked string instrument, was proposed by Penttinen et al. based on the commuted waveguide method [23]. The whole synthesis

system includes a body model filter, a ripple filter for fla- geolet tones, multiple SDL string models for inharmonic partials (called phantom partials), and a friction model. Re- cently, model-based sound synthesis of the dan tranh, a Vietnamese plucked string instrument, has been reported [5]. Here, estimated loop gain values and loop coefficient values of the dan tranh were compared with those of the gayageum [4], a Korean traditional plucked string instru- ment with 12 strings. The fundamental sound production mechanism of these instruments however, is basically simi- lar to those of their western counterparts and does not prop- erly reflect either the differences in physical structure of the instrument (or materials used) or the playing techniques, which can play a crucial role in generating distinctive tones. This paper deals with the digital waveguide synthesis of the geomungo, a Korean traditional plucked string instru- ment. Due to vigorous playing techniques, extreme vibrato (usually more than 100 cents) with a noticeably fluctu- ating decay of the harmonics is characteristic of typical geomungo sound. While Mellody et al. found a similar am- plitude variation in their analysis of the vibrato tones of the violin [20], research on digital waveguide synthesis of (not only Asian, but also western) stringed instruments con- cerning this phenomenon is rare. When modeling vibrato, most of the previous examples considered fundamental fre- quency as the main control parameter but did not pay at- tention to fluctuations in the amplitudes of the additional partials, thereby failing to resynthesize the target vibrato tone properly.

In this work we propose a new commuted waveguide synthesis algorithm that can handle changes in the decay of vibrato tones. Instead of a constant value, the algorithm uses a sinusoidal function to control its loop gain and hence models the time-varying decay pattern of the vibrato partials more naturally.

1 THE GEOMUNGO

Known as a remolded guqin instrument imported from China before the fifth century, the geomungo is one of three major traditional string instruments from Silla, an ancient Korean kingdom. The name came from geomun (meaning “black”) and go (meaning “zither”). The pitch range of the geomungo is approximately three octaves—the widest of all the traditional Korean musical instruments, and its sound is generally described as low and resonant. Although the instrument was usually played solo by scholars before the modern era, it is also used widely to play bass in Buddhist music and modern court music nowadays. Since the 1980s, new songs have been composed for the geomungo, and consequently the playing technique and structure of the instrument have been improved. Composer and geomungo virtuoso Jin Hi Kim introduced the instrument to the world, and developed an electric geomungo [15].

1.1 Structure

Fig. 1 shows a picture of the geomungo and the names of its strings. The dimension of the instrument is approx-

Fig. 1. The geomungo [10][21].

imately 160 cm long, 22 cm wide, and 10 cm tall. The body is hollow inside to amplify the vibration from the strings. The curved top of the body is made of paulow- nia wood and the back is crafted from chestnut wood. Six strings made of twisted silks are fastened to the body. The names of each string are moon-hyun, yu-hyun, dae-hyun,

gwae-sang-cheong, gwae-ha-cheong, and moo-hyun (hyun

and cheong mean “string” and “sound,” respectively). Yu- hyun, dae-hyun, and gwae-sang-cheong are stretched over sixteen frets called gwae, while each of the rest is sup- ported by a movable bridge called the anjok (meaning “a seagull’s foot”). The geomungo does not have stan- dard frequencies for tuning, but its six strings are typi- cally tuned to E2 (moon-hyun), A2 (yu-hyun), D2 (dae- hyun), B1 (gwae-sang-cheong), B1 (gwae-ha-cheong), and B2 (moo-hyun). Yu-hyun is the thinnest string with clear and crisp sonic characteristics, while the thickest, dae- hyun, produces a low and rich sound. These two strings are used most frequently in playing and cover a wide range of pitch.

1.2 Techniques

To play the geomungo, the performer places his/her left foot under right thigh and puts the instrument on it to prop up one edge. The string is then plucked with his/her right hand—both downward and upward—using a 20-cm long plectrum (called a sooldae) while controlling the tension of the string with his/her left hand (Fig. 2).

There are three plucking techniques (dae-jeom, joong-

jeom, and so-jeom, where jeom means “point” and dae, joong, and so mean “large,” “medium,” and “small,” re-

spectively) [18] as well as four left-hand techniques (nong-

hyun, choo-sung, toh-sung and jeon-sung) [16] [17], many

of which cause vibrato. Tables 1 and 2 summarize the tech- niques.

Table 1. Right-hand plucking techniques of the geomungo.

dae-jeom To hit the string very strongly, thereby including the sound of the body being hit joong-jeom To hang the plectrum on the string and then plucking the string

so-jeom To pluck the string weakly

Fig. 2. Picture of typical geomungo playing.

2 ANALYSIS

The sounds of the geomungo were recorded for analysis in an anechoic chamber. A microphone (Rode Classic II) was placed toward the back of the instrument body, 50 cm away. Sounds generated with different playing techniques, e.g., nong-hyun, choo-sung, and jeon-sung, and/or plucking styles were recorded at 12 positions on yu-hyun and dae- hyun. Plectrum-hit body sounds were also recorded for the simulation of the dae-jeom technique.

Fig. 3 shows the waveforms of three geomungo tones from the same string (yu-hyun, pressed on the fourth fret). Fig. 3a is a tone without any left hand techniques, while Figs. 3b and 3c are sounds of the nong-hyun technique, with the latter featuring greater intensity and faster vibrato.

2.1 Frequency Response

Changes in the frequency response of vibrato sounds along time were measured by short-time Fourier transform (STFT) [1], which is defined as follows:

Ym(k)= N₋₁  n=0 w(n)y(n + m H)e−2πjkn/N_, m= 0, 1, 2, ..., k = 0, 1, 2, ..., N − 1

where N is FFT size, w(n) is a window function, and H is the number of overlapping samples. Fig. 4 shows the STFT results of the geomungo tones in Fig. 3. Pitch mod- ulations are apparent in Figs. 4b and 4c. In addition, the seventh and higher partials are negligible in loudness, i.e., about 40 dB less than the fundamental, and decay instantly. This phenomenon is common to all the geomungo tones

0.2 0.4 0.6 0.8 1 1.2 1.4 −0.2 −0.1 0 0.1 0.2 Time (s) Amplit u de (a) 0.2 0.4 0.6 0.8 1 1.2 1.4 −0.2 −0.1 0 0.1 0.2 Time (s) Amplit u de (b) 0.2 0.4 0.6 0.8 1 1.2 1.4 −0.2 −0.1 0 0.1 0.2 Time (s) Amplit u de (c)

Fig. 3. Waveforms of recorded geomungo tones of yu-hyun (pressed on the fourth fret): (a) With no left hand technique, (b) nong-hyun, (c) nong-hyun, with greater intensity and faster vibrato.

Table 2. Playing techniques with the left hand of the geomungo.

choo-sung To press on the string to produce slightly higher pitch

toh-sung To pull the string down

jeon-sung To vibrate the plectrum around one position on the string nong-hyun Similar to vibrato including jeon-sung and toh-sung

Fig. 4. Short-Time Fourier Transform (STFT) results of the geo- mungo tones in Fig. 3. Here N is the next power of 2 greater than the length of each tone, and H is half the window size (2048 sam- ples). Blackman-Harris window is used for analysis. Compared with (a), pitch modulations are apparent in (b) and (c).

we recorded and is considered for the development of the synthesis algorithm (as discussed later).

2.2 Fundamental Frequency Curve

Information on the fundamental frequency of a tone is required for its digital waveguide synthesis. In the case of a vibrato tone, its fundamental frequency can be considered as a function of time (hence denoted as f0(n)). To estimate

the fundamental frequency, we used the YIN algorithm developed by Cheveign´e et al. [7]

dt(τ) =  1, ifτ = 0 dt(τ)/[(1/τ) _τ j=1dt( j )] otherwise, 0.2 0.4 0.6 0.8 1 1.2 140 150 160 170 180 Time (s) F u ndamental Freq u ency (Hz) (a) 0.2 0.4 0.6 0.8 1 1.2 140 150 160 170 180 Time (s) F u ndamental Freq u ency (Hz) (b) 0.2 0.4 0.6 0.8 1 1.2 140 150 160 170 180 Time (s) F u ndamental Freq u ency (Hz) (c)

Fig. 5. Fundamental frequency curves of geomungo tones on Fig. 3 based on the cumulative mean normalized difference function. Compared with (a), (b), and (c) show pitch modulations of nearly 10 ∼20 Hz, which is an unique feature of vibrato tones of the geomungo. where dt(τ) is dt(τ) = t+W−1_ j=t (xj− xj+r)2.

This allowed us to avoid the selection of zero-lag. The fundamental frequency at timeτ was then estimated as

f0(τ) = fs/dt(τ).

where fsis the sampling rate.

Fig. 5 shows the fundamental frequency curves of the same geomungo tones in Fig 3. Compared with Fig. 5a, Figs. 5b and 5c show larger pitch modulations reaching up to nearly 10 Hz, i.e., 100 cents. This feature is distinguish- able from the case of the guitar [9], whose vibrato does not usually exceed 1 Hz.

H(z)

Z

Output

F(z)

fractional delay filter

Excitation