Commuted Waveguide Synthesis of the Clavichord

Vesa Va¨lima¨ki,* Mikael Laurson,† Commuted Waveguide and Cumhur Erkut* *Laboratory of Acoustics and Audio Synthesis of the Signal Processing Helsinki University of Technology Clavichord P.O. Box 3000 FIN-02015 HUT Espoo, Finland {Vesa.Valimaki, Cumhur.Erkut}@hut.fi †Centre for Music and Technology Sibelius Academy P.O. Box 86 Helsinki, Finland Downloaded from http://direct.mit.edu/comj/article-pdf/27/1/71/1853827/01489260360613353.pdf by guest on 26 September 2021 laurson@siba.fi The clavichord is one of the oldest keyboard instru- instrument is an Anthony Sidey clavichord manu- ments, and it is still often used in performances factured by Heugel in Paris, France, in 1988. This and recordings of Renaissance and Baroque music. clavichord is an unfretted one, so any combination The sound of the instrument is pleasant and ex- of notes can be played. The range of a clavichord is pressive but quiet. Consequently, the instrument anywhere from three octaves to over five octaves. can only be used in intimate performances for Our clavichord has 51 keys ranging from C2 to D6. small audiences. This is the main reason why the The instrument was tuned about a whole tone clavichord was replaced by the harpsichord and fi- lower than the standard modern tuning Hz): the nominal frequency of A4 is 395 440 ס nally by the modern piano, both of which produce (A4 a considerably louder output. Attempts have been Hz. The Werkmeister tuning system was used. made to amplify the sound of the clavichord using For each key of the clavichord, a pair of strings is a piezoelectric pickup (Burhans 1973). tuned in unison, as sketched in Figure 2 (see, for One of our motivations in this research is to give example, Thwaites and Fletcher 1981; Campbell the clavichord a new life in the digital world, and Greated 1987). However, the two strings are al- where the faint sound level of the instrument can ways slightly detuned around the same note, be- be amplified by simply turning a volume knob. The cause exact tuning is impossible manually. Every suggested synthesis model is based on digital wave- key forms one end of a lever that has a tangent at- guide modeling of string instruments (Smith 1992, tached to its other end. When a key is depressed, 1998; Va¨lima¨ki et al. 1996; Karjalainen, Va¨lima¨ki, the tangent hits the string pair and initiates vibra- and Tolonen 1998) and uses the principle of com- tion. One end of the strings has been damped with muted waveguide synthesis where the soundbox’s felt, and the other end goes over a bridge to the response is incorporated in the excitation signal tuning mechanism. Thus, the strings are freely vi- (Smith 1993; Karjalainen and Va¨lima¨ki 1993; Karja- brating between the bridge and the tangent, which lainen, Va¨lima¨ki, and Jańosy 1993). Special sam- works as both a hammer and a termination. The pling techniques are also employed. Musical tangent mechanism is rather noisy, as it excites modes of the soundboard but also itself causes examples produced using the proposed synthesizer sound from its moving parts owing to friction. will be included on a forthcoming Computer Music When the key is released, the tangent falls back Journal CD. with the aid of gravity, and the string vibration is allowed to propagate to the felt-covered end of the Acoustics of the Clavichord string, which efficiently damps the vibration. At the end of each note, another knock is heard as the A photograph of the clavichord used for the mea- tangent returns to its resting position. The some- surements in this study is shown in Figure 1. The what mistuned strings of each pair are coupled via Computer Music Journal, 27:1, pp. 71–82, Spring 2003 a non-rigid bridge, and thus both beats and a two- ᭧ 2003 Massachusetts Institute of Technology. stage decay result (Weinreich 1977). Va¨ lima¨ ki, Laurson, and Erkut 71 Figure 1. Clavichord used Figure 2. Tangent mecha- in this study. nism of one key of the clavichord and a string pair associated with it. Downloaded from http://direct.mit.edu/comj/article-pdf/27/1/71/1853827/01489260360613353.pdf by guest on 26 September 2021 The first thing that people usually notice about Felt the clavichord is that the sound level is very low. Tangent The maximum sound pressure level at 1 meter is only about 50 dB or 60 dB, depending on the indi- String vidual construction of the instrument. This makes pair this ancient instrument a sound source less effi- cient than a human speaker. There are many rea- sons for the weak output (see Campbell and Lever Bridge Key Greated 1987; Fletcher and Rossing 1991). The Tuning strings are thin and their tension is low, and they pins radiate sound inefficiently. The soundboard is small and light, and it cannot amplify the sound much. A particularly interesting feature in the clavichord is the mechanical aftertouch known as Be- bung. When the player increases pressure on a key, phonic aftertouch, which would really be needed, is the tangent is raised more, which in turn increases only available in high-end keyboard controllers. the tension of the string pair, resulting in a raise of As the key pressure is changed, the tangent pitch. This enables continuous control of vibrato, causes a slightly different change in the tension of which is a used performance style. Aftertouch has the two strings of a pair. This feature affects the been part of keyboard controllers since the 1980s, timbre of clavichord tones by introducing a flang- and it is thus easy to include this control in the ing effect. It is heard during and after the attack, clavichord synthesizer. However, a fully poly- particularly in the bass register. 72 Computer Music Journal Figure 3. (Top) Waveform tone at 1.1 sec, and (bot- Figure 4. Envelopes of the (dashed line), 4th (dash- of a clavichord tone (A3) tom) the time history of its lowest partials of the tone dot line), and 5th (dotted showing the irregular over- fundamental frequency shown in Figure 1: 1st line). all decay pattern and the (nominal fundamental fre- (solid line with circles), thump at the end of the quency is 197.5 Hz). 2nd (solid line), 3rd 1 Ϫ80 Ϫ100 0 Level Ϫ120 Magnitude (dB) Ϫ1 Downloaded from http://direct.mit.edu/comj/article-pdf/27/1/71/1853827/01489260360613353.pdf by guest on 26 September 2021 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 Time (sec) 200 199 lower than the threshold of audibility for this ef- F0 (Hz) fect. According to Ja¨rvelaïnen and Va¨lima¨ki (2001), 198 the threshold at 196 Hz (the nearest frequency 197 tested) is 4.4 Hz. In the case of the clavichord used 0 0.2 0.4 0.6 0.8 1 1.2 Time (sec) in our measurements, during normal playing the pitch glide can be inaudible for most listeners. Au- dible pitch glides can be generated easily, however. Figure 5 gives an example of a tone that contains Properties of Single Clavichord Tones an exaggerated pitch vibrato, which should be obvi- ous for all listeners. Note that the vibrato width is Figure 3 shows the envelope of a recorded clavi- typically about 2 Hz or less, but can occasionally chord tone. The irregular, non-exponential decay of peak higher. The vibrato rate is about 7 Hz in this the tone can be observed. The key mechanism of case, which is slightly faster than what would be the clavichord generates a loud knock at the begin- usual in a musical context. The above fundamental ning and end of a tone, which is characteristic to frequency measurements reflect the effective pitch, the sound of the instrument. In Figure 3, a burst lo- where the contributions of both strings are present. cated at 1.1 sec corresponds to the thump caused Impulsively driven string instrument sounds are by release of the key, at the same time the signal always at least weakly inharmonic (Fletcher and starts to decay quickly. The envelope curves of the Rossing 1991). This is caused by the stiffness of the first five harmonics are presented in Figure 4. Note string material. It is well known that piano tones, that regular exponential decay (i.e., linear decay on particularly in the bass range, are highly inhar- a dB scale) is rare: pronounced beating and other ir- monic, which affects both the timbre and the tun- regularities are observed in many harmonics (e.g., ing. Measurements of inharmonicity have not been the beating of the fourth partial). published previously for the clavichord, so we con- The fundamental frequency of clavichord tones ducted them ourselves. This was needed to decide varies over time, as illustrated by an example in whether we had to implement the inharmonicity the lower part of Figure 3. This may be partly using an allpass filter, as is customary to do in caused by tension modulation (Tolonen, Va¨lima¨ki, waveguide synthesis (Jaffe and Smith 1983; Bank and Karjalainen 2000), which is a purely physical 2000). We used our parameter estimation software phenomenon in vibrating strings, but also by the to extract partial frequencies of clavichord tones as pressure of the player’s finger (i.e., the mechanical a function of time. Using a least-squares fit, we es- aftertouch), which directly controls string tension timated from the data the inharmonicity coeffi- during playing. A time-domain finite-difference cient.

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