- Email: [email protected]
Separation of piano keyboard vibrations into tonal and broadband components
Applied Acoustics 68 (2007) 1104–1117 www.elsevier.com/locate/apacoust
Separation of piano keyboard vibrations into tonal and broadband components M. Keane
*
Acoustics Research Centre, University of Auckland, Private Bag 92019, Auckland, New Zealand Received 19 April 2006; received in revised form 8 June 2006; accepted 14 June 2006 Available online 14 August 2006
Abstract A new method is presented for splitting musical notes into tonal and broadband components by removing the harmonic peaks from the spectra. This method was used to analyse the key vibrations of eight pianos (four grand and four upright). The broadband component of vibrations was found to be more intense in the upright pianos than in the grand pianos, while no such difference was found in the tonal component. As grands are considered to be superior to uprights, this suggests that it may be possible to improve upright pianos by reducing the strength of the broadband component of vibrations. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Piano; Vibration; Tonal; Broadband
1. Introduction When examining recordings of pianos, it is convenient to divide the recorded signals into tonal and broadband components. The tonal component comes from the vibration of the strings, and contains harmonics that are approximately integer multiples of the fundamental. Each harmonic is slightly raised due to the bending stiffness of the string; this effect is known as inharmonicity [1]. The admittance of the soundboard can cause small modifications to the lowest harmonics [2].
*
Tel.: +64 963 085 00. E-mail addresses: [email protected], [email protected].
0003-682X/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2006.06.004
M. Keane / Applied Acoustics 68 (2007) 1104–1117
1105
The broadband component, also known as the ‘knock’ or ‘thump’, is all the vibration and sound that does not come from the strings. It is generated by impacts in the action and keys at the start of each note, and sounds something like a wooden box being struck. Spectrograms showing the knock may be found in Bork [3] and Galembo [4]. It is readily audible in the extreme treble of the piano, and grows less so with lower notes as it becomes more heavily masked by the tonal component (a result of the upward spread of masking in the human hearing system [5]). Blackham [1] found that the knock is necessary to make piano tones sound natural. Marshall and Nielsen [6] showed that under certain circumstances the knock noise can be too strong, leading to an unpleasant tone. The knock is seen in both the radiated sound and in vibrations of the body of the piano. In this paper, it is the vibration of the keys that is discussed. For convenience, the components of acceleration will be referred to as tonal acceleration component (TAC) and broadband acceleration component (BAC), and likewise for pressure giving tonal pressure component (TPC) and broadband pressure component (BPC). The unqualified tonal component (TC) or broadband component (BC) refer to both pressure and acceleration. By examining spectrograms of sample notes, it is found that there are large differences in the ratio of BC to TC between radiated sound and key vibrations. In recorded vibrations, the BAC is 5–10 dB louder than the TAC for the first 50–100 ms of the note, and it is only once the BAC has decayed sufficiently that the TAC is seen. In radiated sounds, the TPC dominates the BPC, even at note onset. The harmonics of the TPC are always at least 20 dB louder than the BPC, thus the BPC is partially masked by the TPC. 1.1. Quality of pianos Upright pianos are universally considered to be of lower quality than grands. There are several factors adversely affecting the upright that are unavoidable due to the nature of the instrument, for example the strings in uprights are shorter than in grands. The strings must be thicker to compensate, and this increases the bending stiffness. Higher stiffness increases inharmonicity, giving a ‘metallic’ tone [1]. Another reason is that the return action of the keys is governed by springs rather than gravity, which affects the feel and repetition of the instrument. For the purposes of this paper it will be assumed (although it may not be true in every conceivable case) that grand pianos are superior to uprights, and making changes to uprights that bring them closer in performance to grands will always be an improvement. Apart from these well known considerations, few authors have proposed measures of the quality of piano tones, and those that have focus on the treble range. Galembo [4] gives spectrograms of C8 notes from three different pianos. For these notes the fundamental of the TPC is above 4 kHz, so the BPC is especially prevalent. The judged note quality in listening tests increased when the BPC was less intense, of narrow spectrum, and decayed rapidly, while the TPC was more intense and decayed slowly. The upright presented in the paper was clearly deficient by these measures when compared with the Steinway grand piano. This led Galembo to propose that tone quality of pianos is related, in the high treble, to how quickly the tonal component becomes audible over the broadband component. Galembo initially removed the TPC by damping the strings with felt. This method gives the BPC but the TPC cannot be recovered. The TPC was also removed by comb filtering recorded piano tones, in which case the TPC may be recovered. However, this method is
1106
M. Keane / Applied Acoustics 68 (2007) 1104–1117
problematic, as the filter will remove any BPC that is within the filter stopbands. Ideally some of the energy in the stopbands should be retained in the BPC. In addition to Galembo, Smurzynski [7] suggests ‘‘the squared ratio between sound level of the tone with the harmonics removed and the sound level of the original tone’’ and Revvo [8] suggests ‘‘the ratio between the levels of the original tones and the same tone modified by mechanical damping of the strings’’. All three measures are similar in that for tones to be assessed as high quality, they require low BPC levels. Those studies were concerned with radiated sound of treble tones. This paper intends to consider vibration of the keys, and a larger range of notes, extending as low as C3. 1.2. Aural and tactile feedback When playing, the pianist receives aural and tactile feedback, both of which are important in monitoring performance [9–12]. To date, vibration feedback in pianos has not received extensive study, however Galembo [11] found that pianists’ subjective assessments of a piano were affected more by vibration and touch/feel than by sound. Tests were carried out where either the aural or tactile feedback (via a shaker) to the pianist was delayed by 250 ms, as in the study by Finney [10]. The experimental setup is shown in Fig. 1. The two channels were independent, so it was possible to delay the aural and tactile feedback separately. Delayed aural feedback was found to be extremely disruptive to smooth playing, as in other studies [9,10]. Delayed tactile feedback was reported by the participants to be disconcerting, but they were still able to play as smoothly as they would without the feedback present. These tests suggest that tactile feedback contributes to the subjective feel of an instrument but the pianist does not use it to the same degree as aural feedback for monitoring and controlling their own performance.
Headphones Keyboard 250 ms delay
250 ms delay
Amplifier
Shaker
Fig. 1. Experimental setup for delayed aural or tactile feedback. The output from an electronic keyboard (Yamaha DX100) was fed to the pianist’s headphones (Sennheiser HD 414) and a shaker (Bru¨el and Kjær 4809) which vibrated the keyboard. Feedback via either channel could be delayed by 250 ms by an ART DXR delay, which could be bypassed to provided undelayed feedback.
M. Keane / Applied Acoustics 68 (2007) 1104–1117
1107
The audience by contrast are interested only in the radiated sound. For them, the vibrations transmitted to the pianist are only significant insofar as they enable the pianist to perform well. Thus to investigate tactile feedback is to focus on the experience of the pianist, and the manner in which the pianist subjectively evaluates the instrument. The evaluation of the radiated sound by the audience is therefore a topic which must be excluded from the current discussion. 2. Recordings and separation Recordings were made of four uprights: a Kawai K50E, a Kawai NS20A, a Kessels and a Yamaha YUX, and four grands: a Kawai GS70, a Kawai RX2 and two Steinway model Ds which will be referred to as Steinway D1 and D2. All belonged to the School of Music at the University of Auckland and were in good condition, being regularly tuned and maintained by a registered piano tuner. The Kessels upright, manufactured in 1955, provided a useful comparison with the more recent models, which were all between 5 and 15 years old. While it was not practicable to perform a subjective comparison of the eight pianos, an indication of relative quality can be gained from the knowledge that the four upright pianos are used as student practice instruments, while the two Steinway grand pianos are used for concert performances, and the two Kawai grands are used for teaching purposes. Approximate resale values of the Kawai grands were four to six times higher than the uprights, and those of the Steinway grands were approximately ten times higher. Thus the grand pianos were significantly superior to the uprights. Recordings of both radiated sound and key vibrations were made, however there were problems with the sound recordings. The first was that due to the differences in geometry between uprights and grands, it was difficult to choose a point at which to make the sound recordings that was equivalent. As a compromise, the microphone (Bru¨el and Kjær 4165) was placed at the point where the pianist’s head would be, so the recording was essentially of the sounds the pianist would hear. The second problem was that, due to external constraints, it was not possible to record all the pianos in the same room, so the pianos were recorded in widely differing acoustic environments. Thus it was not possible to draw any strong conclusions from the sound results; instead it is the vibration results on which this paper will focus. Some references to sound spectra are included where useful for comparison, but they should be regarded as tentative. For the vibration recordings, the accelerometer (PCB 352 C68) was attached with wax to the piano on the F4 key. This key was held down by a small weight (160 g) so that it was in contact with the keybed. During the course of this investigation measurements were made on various keys, and it was found that there was relatively little variation between measurement points. For the results shown the F4 key was chosen because it was found, in the analysis of a number of pieces of classical music, to be the mean pitch used on the piano [13], thus the measurement represents the average point at which the pianist feels vibrations while playing. As this key could not be struck, it was skipped in the analysis. To strike each note, a weight of 340 g was held so that it touched the key. It was then released to fall and depress the key. Under the weight a small strip of foam prevented clicks. This was repeated three times for each note, and a total of 48 notes were recorded, chromatically from C2 to C7.
1108
M. Keane / Applied Acoustics 68 (2007) 1104–1117
A comparison of the recordings for each key showed a high degree of repeatability. Of the 324 levels eventually used (acceleration, BAC and TAC), the difference of the note level from the mean level for that note was less than 0.2 dB for 171 notes, 0.2–0.4 dB for 89 notes, 0.4–0.6 dB for 51 notes, 0.6–0.8 dB for 6 notes and 0.8–1.0 dB for 7 notes. The microphone and accelerometer signals were recorded on DAT at 48 kHz, then transferred digitally (via SPDIF) to a PC and encoded as WAV files for analysis in Matlab. Each note was held for at least 1.5 s, but in the analysis all were truncated to 1 s in length, to avoid the sound of the note let–off. By this time the BC has decayed to below the noise floor, but the TC was often much longer and was therefore truncated. The effect of the truncation was not significant, due to the low amplitude and stable harmonic structure at this stage of the note development. 2.1. Basis of separation algorithm Fig. 2 shows a comparison of the sound spectra of two notes, C5 and C8, recorded on a grand piano. The time window was 1 s long, thus the decay of the C5 note was truncated. The harmonics of the C5 note can be clearly seen, at 524 Hz and integer multiples. The fundamental of the C8 note can be seen at 4285 Hz; the relatively large bandwidth reflects the rapid decay of very high treble notes. Between the harmonics of the C5 note and below the fundamental of the C8 note, the BC spectrum can be clearly seen. In contrast to the TC which has sharp peaks and is zero elsewhere, the BC varies slowly with frequency, decreasing at a rate of around 20 dB per octave. This allows the design of an algorithm to separate a recorded note into TC and BC, as described below.
100 C5 (524 Hz)
Level (dB, arb. 0)
50
0
–50
50 C8 (4285 Hz) 0
–50
–100 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
f (Hz)
Fig. 2. Spectra of C5 (524 Hz) and C8 (4285 Hz) notes recorded on a grand piano (Kawai GS70). Below the fundamental of the C8 note there is only BC.
M. Keane / Applied Acoustics 68 (2007) 1104–1117
1109
2.2. Detection of harmonics Detection of individual harmonics is a difficult task, and one for which elaborate methods have been devised [2,14–16]. For the purposes of this study, two methods were developed which were not as robust as some of those used by others, but were suitable for the task at hand. Both used the power spectrum of the signal after the BC has died away, leaving the relatively slowly decaying TC. Since both methods gave some false positives, the results from the two were compared, and a harmonic was confirmed if found by both methods. In the first method, the spectrum was lowpass filtered (the raw data of the spectrum were treated as if they formed a signal), giving an approximate spectrum, without the sharp peaks at the harmonics. The filter cutoff used was proportional to the pitch of the note. A threshold was created by adding 5 dB to this lowpassed spectrum, and the local peaks where the spectrum was greater than this threshold were identified. Peaks below the note fundamental were ignored. In the second method, the spectrum was again treated like a signal, and its Hilbert transform was found. The Hilbert transform may be thought of as a complex envelope follower, so in this case the magnitude of the Hilbert transform followed approximately the envelope of the spectrum [17]. It gave a relatively smooth function that retained peaks due to the harmonics in the original spectrum, but ignored the smaller peaks associated with noise. Any peaks found in the Hilbert transform that matched those from the first method were identified as the harmonics of the note. Both methods are illustrated in Fig. 3, which demonstrates the detection of the harmonics of a C5 note. Circles show any local peaks in the spectrum that are above the threshold, while stars show the locations of Hilbert transform peaks. Asterisks mark the confirmed
40
Spectrum Lowpass spectrum + 5dB Spectrum peaks Hilbert transform peaks Confirmed harmonics
Level (dB, arb. 0)
30
20
10
0
–10
–20
–30 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
f (Hz) Fig. 3. Example of harmonic detection method, for a C5 note. (s) Peaks in the spectrum, (q) the locations of Hilbert transform peaks, and ( ) the confirmed harmonics.
*
1110
M. Keane / Applied Acoustics 68 (2007) 1104–1117
harmonics, where both methods agree. The method does not identify harmonics that are too weak, as shown by the false negative at 4.3 kHz, which was identified by the Hilbert transform but was below the 5 dB offset. 2.3. Removing harmonic content The technique used to reconstruct the BC was to create an approximate BC spectrum from the available data, and then use the inverse Fourier transform to return to the time domain. The majority of the BC spectrum may be simply taken from the spectrum of the original signal, however near each harmonic it is the TC spectrum which dominates. Consequently, in a small frequency range around each harmonic the BC spectrum must be inferred, based on the assumption that it varies linearly over such a small range. Therefore, the separation proceeded as follows. The spectrogram of the note was taken, using a 4096 sample Hamming window. For each time window in the spectrogram, for each harmonic, there is a peak in the spectrum that must be replaced with an approximation of the BC. Two endpoints were chosen, 50 Hz to either side of the peak (although individual harmonics varied, this was found to be a good estimate of where the peaks died back to the level of the BC), and a linear interpolation was made between them. This was used as the approximate BC for that portion of the spectrum, replacing the peak which was originally there. The remainder of the BC spectrum, away from the harmonics, was taken directly from the original note, as in those regions the TC makes no contribution. Fig. 4 demonstrates the process for two harmonics of a C5 note. For each harmonic, the BC spectrum is created by interpolating between the points chosen on 80 Original spectrum BC spectrum approximation Interpolation endpoints Harmonic content removed
70
Level (dB, arb. 0)
60
50
40
30
20
10 0
500
1000
1500
f (Hz)
Fig. 4. Example of removal of spectral peaks, for a C5 note. Asterisks mark the endpoints chosen for the interpolation, and the dashed line shows the interpolated spectrum. The shaded area represents the harmonic content which is removed.
M. Keane / Applied Acoustics 68 (2007) 1104–1117
1111
either side of the peak. The shaded area therefore represents the harmonic content that is removed. The inverse spectrogram (since the spectrogram is calculated via the Fourier transform, it is reversible) was then taken to reconstruct the BC signal. Within each window the signal was divided by the Hamming window, and multiplied by a triangular window (because the windows used overlapped in time by 50%) so that for the reconstructed signal the envelope response was flat. Thus the BC was found, and by subtracting it from the original signal the TC was also recovered. This algorithm works equally well whether the input is vibration or sound, but it is limited to notes in the middle range of the piano. With low notes, the harmonics are closer together in frequency, so the time window should be lengthened to increase the frequency resolution. However, there is time variation in the amplitude and harmonic structure (shimmer and jitter), so the assumption of an unchanging signal within each window is no longer valid. With high frequency notes, the harmonics decay very rapidly, at around the same rate as the BC. Therefore the assumption that the BC spectrum varies linearly over the range of each TC harmonic is no longer valid, as each harmonic is spread over too large a frequency range. Due to these limitations, the algorithm can only be used between approximately F2 and C7. For all results given in this paper, only notes from C3 to C6 were used, to give a margin of safety. Fig. 5 shows a typical acceleration signal separated into TAC and BAC by the algorithm. The BAC is large initially, but it quickly dies away to below the level of the TAC. The separated signals may be tested by any of the myriad measures that have been proposed for examining musical timbre, such as spectral centroid, harmonic onset synchrony, inharmonicity, shimmer, jitter and methods utilising cepstral analysis [18–20]. The absence of the BC makes the note harmonics easier to investigate by these methods.
1.5 1 Combined 0.5
Acceleration (m/s2)
0 –0.5 0.5 Tonal 0 1 Broadband
0.5 0 –0.5 –1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
Fig. 5. Example total acceleration, TAC and BAC recorded for an F#4 note recorded on a grand piano (Kawai RX2).
1112
M. Keane / Applied Acoustics 68 (2007) 1104–1117
This study focused on the peak level and spectrum of each signal. Since the measurements were of time-varying quantities, difficulty was encountered in describing the relative recorded levels. However, all the notes decayed approximately exponentially with decay rates that increased from the bass to the treble. Thus the results found would be unchanged regardless of the method used to describe levels. For this paper, the level given is the peak absolute acceleration attained (this will be during note onset), converted to decibels relative to 1 m/s2. It would be interesting to examine the time variation of the recordings, though unfortunately there is not space to go into detail here. For all the notes, the decays of the tonal and broadband components were approximately exponential. For the broadband components, there was no variation in decay rates across the compass. Similar rates were found in the uprights and grands, between 75 and 105 dB/s. Measuring tonal component decay rates presented more difficulty, as there was interaction between individual harmonics, and variation across the compass. For the uprights, decay rates were approximately 15 dB/s at C3, 25 dB/s at C4, and 40 dB/s at C6. For the grands the decay rates were around 10 dB/s at C3, 20 dB/s at C4, and 40 dB/s at C6. These represent average values however, and the rates for individual notes could vary by ±10 dB/s from those given. 3. Results Figs. 6–8 are plotted as box and whisker plots: the boxes have lines at the mean and the upper and lower quartiles, and the whiskers show the extent of the remainder of the data. The results showed significant differences between the uprights and grands measured, as well as a large degree of variation between the pianos within those divisions. 18
2
Peak acceleration (dB re. 1 m/s )
16
14
12
10
8
6
4
2
K50E
NS20A Kessels Uprights
YUX
GS70
RX2
D1
D2
Grands
Fig. 6. Total acceleration levels of notes from C3 to C6 in four upright pianos (left) and four grand pianos (right).
M. Keane / Applied Acoustics 68 (2007) 1104–1117
1113
16
12
2
Peak acceleration (dB re. 1 m/s )
14
10
8
6
4
2
0
–2
K50E
NS20A Kessels Uprights
YUX
GS70
RX2
D1
D2
Grands
Fig. 7. Broadband acceleration component levels of notes from C3 to C6 in four upright pianos (left) and four grand pianos (right).
The mean acceleration level before separation was 4 dB higher in the upright pianos than in the grands. Fig. 6 shows the distribution of total acceleration for the eight pianos studied. The piano with the lowest mean acceleration was the GS70, followed by D2 and D1. The piano with the highest acceleration was the Kessels. The RX2 was unusual, as it was the only one of the grand pianos with a large mean acceleration. The NS20a had a lower mean acceleration than the other uprights. Fig. 7 shows that the BAC levels were 5 dB higher in the uprights than in the grands. Again, the RX2 had the largest mean acceleration amongst the grand pianos. Fig. 8 shows that the TAC levels were approximately the same between the two groups. The Kessels had lower TAC levels than the other pianos, it is not known whether this is related to the age of that piano. The BAC levels were much higher than the TAC in all the pianos, thus the differences seen in Fig. 6 were caused mostly by the BAC and not the TAC. 4. Broadband spectra Fig. 9 shows the mean BAC and BPC spectra for the four upright and four grand pianos. For each piano there was very little difference in broadband spectra between notes. The largest peak in the BAC was found at 60 Hz. Above this there was a secondary peak near 1.5 kHz, around 25 dB lower than the first. The BPC spectra were approximately flat up to 800 Hz, then dropped at 12 dB/octave. It can be seen that the BAC and BPC spectra were not strongly correlated, and the differences found between uprights and grands in the BAC were not replicated in the BPC.
1114
M. Keane / Applied Acoustics 68 (2007) 1104–1117 10
Peak acceleration (dB re. 1 m/s2)
5
0
–5
–10
–15
–20
K50E
NS20A Kessels
YUX
GS70
Uprights
RX2
D1
D2
Grands
Fig. 8. Tonal acceleration component levels of notes from C3 to C6 in four upright pianos (left) and four grand pianos (right).
40 Grand broadband spectrum Upright broadband spectrum
30 20 10
Level (dB, arb. 0)
0 –10
Acceleration
–20 40 30 20 10 0
Sound pressure
–10 –20
2
3
10
10 f (Hz)
Fig. 9. Mean BAC and BPC spectra from C3 to C6 averaged over four upright and four grand pianos.
M. Keane / Applied Acoustics 68 (2007) 1104–1117
1115
5. Discussion As discussed above, for this paper it is assumed that grands are superior to uprights, in agreement with the general experience (and perhaps prejudices) of most pianists. Thus, bringing the uprights closer in performance to the grands would be a valuable improvement. The BAC levels in the grand pianos were around 5 dB lower than the BAC in the uprights, while the TAC levels were on average 1 dB higher. Thus the BAC/TAC ratio, as given in Table 1, was on average 6 dB higher in the uprights (although the Kawai RX2 had a BAC/TAC ratio of 10.1 dB, which was higher than both the NS20A and the YUX). The difference limen for detection of differences of tactile vibration intensity is in the range of 1–2 dB [21], so a difference of 6 dB is large enough to be detected by the pianist. It is not surprising, given the difference in geometry, that differences exist between upright and grand pianos in terms of vibration levels. The results in this paper, on their own, are not to be regarded as evidence that a lower BAC/TAC ratio is desirable. What remains to be proven is whether the differences seen are significant to a majority of pianists. This was further investigated by means of a subjective experiment, presented in a paper which has been submitted to Music Perception. In that work, one of a pair of upright pianos of the same make and model was modified to reduce the BAC level by 6 dB, without changes to the BPC or the radiated sound. A group of experienced pianists played on both pianos, and showed a statistically significant preference for the piano with the lower BAC level. Another group of pianists (the control) showed no preference for either piano before the modifications were made. Thus the study showed that reducing the BAC level of that upright piano improved the subjective evaluation. Along with the results given in this paper, this strongly suggests that it is desirable for upright pianos to have lower BAC/ TAC ratios. It is important to note that only the pianist feels the key vibrations. For the audience, it is the radiated sound which is central to the assessment of instrument quality. It is uncertain whether the BPC/TPC ratio, by analogy with the BAC/TAC ratio, is important in evaluating piano tones. The BPC spectra show that this is unlikely to be the case, as significant changes in the spectra relating to piano quality were not found.
Table 1 Grand and upright mean acceleration levels, in dB relative to 1 m/s2, and the BAC/TAC ratio, in dB Piano
Type
BAC mean
Kawai K50E Kawai NS20A Kessels Yamaha YUX Kawai GS70 Kawai RX2 Steinway D1 Steinway D2
Upright Upright Upright Upright Grand Grand Grand Grand
10.3 8.5 11.6 10.9 3.4 8.2 5.1 3.3
TAC mean 0.8 0.8 3.5 1.7 0.3 1.9 0.2 1.9
BAC/TAC 11.1 9.3 15.1 9.3 3.7 10.1 4.9 1.4
1116
M. Keane / Applied Acoustics 68 (2007) 1104–1117
6. Conclusions A new method was presented for splitting recorded piano notes into tonal and broadband components, with several advantages over previously proposed methods. The results from the eight pianos tested show that there are large differences in key vibrations between upright and grand pianos. The principal differences found were the upright pianos tested had total acceleration levels 4 dB higher than the grands; the BAC levels were 5 dB higher in the uprights, this accounted for the majority of the measured difference in total acceleration; the BAC/TAC ratios were twice as large in the uprights as in the grands; there were only small differences in broadband spectrum between the eight pianos, and across the compass of each piano. These results suggest that lowering the BAC/TAC ratios in upright pianos to bring them closer to the ratios found in grands is desirable. Acknowledgements The author is very grateful to the anonymous reviewers whose suggestions have significantly improved the quality of this paper. References [1] Blackham E. The physics of the piano. Sci Am 1965;213:88–96. [2] Ortiz-Berenguer L, Casaju´s-Quiro´s F, Torres-Guijarro S. Multiple piano note identification using a spectral matching method with derived patterns. J Audio Eng Soc 2005;53:32–43. [3] Bork I. Sound radiation from a grand piano. In: Proceedings 100th convention of the audio engineering society, Copenhagen, 1996. [4] Galembo A. Quality of treble piano tones. In: Proceedings Stockholm musical acoustics conference, Stockholm, 2003. p. 147–150. [5] Zwicker E, Fastl H. Psychoacoustics: Facts and models. New York: Springer-Verlag; 1999. [6] Marshall H, Nielsen J. On the transmission of ‘impact noises’ from a grand piano. University of Auckland, 1997. [7] Smurzynski J, Objective and subjective measurements on treble piano sounds. Czechoslovakian Scientific and Technological Society, Hradec Kra´love´, 1983. [8] Revvo A, Criteria for objective quality evaluation of piano sounds. In: Novelties of musical instrument industry. Scientific Research and Design Technological Institute of Musical Industry, Moscow, 1988. [9] Banton L. The role of visual and auditory feedback during the sight-reading of music. Psychol Music 1995;23:3–16. [10] Finney S. Auditory feedback and musical keyboard performance. Music Percept 1997;15:153–74. [11] Galembo A. Perception of musical instrument by performer and listener (with application to the piano). In: Proceedings international workshop on human supervision and control in engineering and music, Kassel, 2001. p. 257–66. [12] Askenfelt A, Jansson E. On vibration sensation and finger touch in stringed instrument playing. Music Percept 1992;9:311–50. [13] Keane M. Statistical analysis of classical piano recordings in MIDI format. In: Proceedings New Zealand acoustical society conference, Wellington, 2004. [14] Guillemain P, Kronland-Martinet R. Characterization of acoustic signals through continuous linear time– frequency representations. Proc IEEE 1996;84:561–85. [15] Godsill S, Davy M. Bayesian harmonic models for musical pitch estimation and analysis. Bayesian Statistics VII. Oxford: Oxford University Press; 2003.
M. Keane / Applied Acoustics 68 (2007) 1104–1117 [16] [17] [18] [19] [20] [21]
1117
Kay S. A fast and accurate single frequency estimator. IEEE Trans Acoust 1989;37:1987–90. Hahn L. Hilbert transforms in signal processing. Boston: Artech House; 1996. Grey J. Multidimensional perceptual scaling of musical timbres. J Acoust Soc Am 1977;61:1270–7. Pollard H, Jansson E. A tristimulus method for the specification of musical timbre. Acustica 1982;51:162–71. Fletcher H, Blackham E, Stratton R. Quality of piano tones. J Acoust Soc Am 1962;34:749–61. Gescheider G, Bolanowski S, Verillo R, Arpajian D, Ryan T. Vibrotactile intensity discrimination measured by three methods. J Acoust Soc Am 1990;87:330–8.
Separation of piano keyboard vibrations into tonal and broadband components M. Keane
*
Acoustics Research Centre, University of Auckland, Private Bag 92019, Auckland, New Zealand Received 19 April 2006; received in revised form 8 June 2006; accepted 14 June 2006 Available online 14 August 2006
Abstract A new method is presented for splitting musical notes into tonal and broadband components by removing the harmonic peaks from the spectra. This method was used to analyse the key vibrations of eight pianos (four grand and four upright). The broadband component of vibrations was found to be more intense in the upright pianos than in the grand pianos, while no such difference was found in the tonal component. As grands are considered to be superior to uprights, this suggests that it may be possible to improve upright pianos by reducing the strength of the broadband component of vibrations. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Piano; Vibration; Tonal; Broadband
1. Introduction When examining recordings of pianos, it is convenient to divide the recorded signals into tonal and broadband components. The tonal component comes from the vibration of the strings, and contains harmonics that are approximately integer multiples of the fundamental. Each harmonic is slightly raised due to the bending stiffness of the string; this effect is known as inharmonicity [1]. The admittance of the soundboard can cause small modifications to the lowest harmonics [2].
*
Tel.: +64 963 085 00. E-mail addresses: [email protected], [email protected].
0003-682X/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2006.06.004
M. Keane / Applied Acoustics 68 (2007) 1104–1117
1105
The broadband component, also known as the ‘knock’ or ‘thump’, is all the vibration and sound that does not come from the strings. It is generated by impacts in the action and keys at the start of each note, and sounds something like a wooden box being struck. Spectrograms showing the knock may be found in Bork [3] and Galembo [4]. It is readily audible in the extreme treble of the piano, and grows less so with lower notes as it becomes more heavily masked by the tonal component (a result of the upward spread of masking in the human hearing system [5]). Blackham [1] found that the knock is necessary to make piano tones sound natural. Marshall and Nielsen [6] showed that under certain circumstances the knock noise can be too strong, leading to an unpleasant tone. The knock is seen in both the radiated sound and in vibrations of the body of the piano. In this paper, it is the vibration of the keys that is discussed. For convenience, the components of acceleration will be referred to as tonal acceleration component (TAC) and broadband acceleration component (BAC), and likewise for pressure giving tonal pressure component (TPC) and broadband pressure component (BPC). The unqualified tonal component (TC) or broadband component (BC) refer to both pressure and acceleration. By examining spectrograms of sample notes, it is found that there are large differences in the ratio of BC to TC between radiated sound and key vibrations. In recorded vibrations, the BAC is 5–10 dB louder than the TAC for the first 50–100 ms of the note, and it is only once the BAC has decayed sufficiently that the TAC is seen. In radiated sounds, the TPC dominates the BPC, even at note onset. The harmonics of the TPC are always at least 20 dB louder than the BPC, thus the BPC is partially masked by the TPC. 1.1. Quality of pianos Upright pianos are universally considered to be of lower quality than grands. There are several factors adversely affecting the upright that are unavoidable due to the nature of the instrument, for example the strings in uprights are shorter than in grands. The strings must be thicker to compensate, and this increases the bending stiffness. Higher stiffness increases inharmonicity, giving a ‘metallic’ tone [1]. Another reason is that the return action of the keys is governed by springs rather than gravity, which affects the feel and repetition of the instrument. For the purposes of this paper it will be assumed (although it may not be true in every conceivable case) that grand pianos are superior to uprights, and making changes to uprights that bring them closer in performance to grands will always be an improvement. Apart from these well known considerations, few authors have proposed measures of the quality of piano tones, and those that have focus on the treble range. Galembo [4] gives spectrograms of C8 notes from three different pianos. For these notes the fundamental of the TPC is above 4 kHz, so the BPC is especially prevalent. The judged note quality in listening tests increased when the BPC was less intense, of narrow spectrum, and decayed rapidly, while the TPC was more intense and decayed slowly. The upright presented in the paper was clearly deficient by these measures when compared with the Steinway grand piano. This led Galembo to propose that tone quality of pianos is related, in the high treble, to how quickly the tonal component becomes audible over the broadband component. Galembo initially removed the TPC by damping the strings with felt. This method gives the BPC but the TPC cannot be recovered. The TPC was also removed by comb filtering recorded piano tones, in which case the TPC may be recovered. However, this method is
1106
M. Keane / Applied Acoustics 68 (2007) 1104–1117
problematic, as the filter will remove any BPC that is within the filter stopbands. Ideally some of the energy in the stopbands should be retained in the BPC. In addition to Galembo, Smurzynski [7] suggests ‘‘the squared ratio between sound level of the tone with the harmonics removed and the sound level of the original tone’’ and Revvo [8] suggests ‘‘the ratio between the levels of the original tones and the same tone modified by mechanical damping of the strings’’. All three measures are similar in that for tones to be assessed as high quality, they require low BPC levels. Those studies were concerned with radiated sound of treble tones. This paper intends to consider vibration of the keys, and a larger range of notes, extending as low as C3. 1.2. Aural and tactile feedback When playing, the pianist receives aural and tactile feedback, both of which are important in monitoring performance [9–12]. To date, vibration feedback in pianos has not received extensive study, however Galembo [11] found that pianists’ subjective assessments of a piano were affected more by vibration and touch/feel than by sound. Tests were carried out where either the aural or tactile feedback (via a shaker) to the pianist was delayed by 250 ms, as in the study by Finney [10]. The experimental setup is shown in Fig. 1. The two channels were independent, so it was possible to delay the aural and tactile feedback separately. Delayed aural feedback was found to be extremely disruptive to smooth playing, as in other studies [9,10]. Delayed tactile feedback was reported by the participants to be disconcerting, but they were still able to play as smoothly as they would without the feedback present. These tests suggest that tactile feedback contributes to the subjective feel of an instrument but the pianist does not use it to the same degree as aural feedback for monitoring and controlling their own performance.
Headphones Keyboard 250 ms delay
250 ms delay
Amplifier
Shaker
Fig. 1. Experimental setup for delayed aural or tactile feedback. The output from an electronic keyboard (Yamaha DX100) was fed to the pianist’s headphones (Sennheiser HD 414) and a shaker (Bru¨el and Kjær 4809) which vibrated the keyboard. Feedback via either channel could be delayed by 250 ms by an ART DXR delay, which could be bypassed to provided undelayed feedback.
M. Keane / Applied Acoustics 68 (2007) 1104–1117
1107
The audience by contrast are interested only in the radiated sound. For them, the vibrations transmitted to the pianist are only significant insofar as they enable the pianist to perform well. Thus to investigate tactile feedback is to focus on the experience of the pianist, and the manner in which the pianist subjectively evaluates the instrument. The evaluation of the radiated sound by the audience is therefore a topic which must be excluded from the current discussion. 2. Recordings and separation Recordings were made of four uprights: a Kawai K50E, a Kawai NS20A, a Kessels and a Yamaha YUX, and four grands: a Kawai GS70, a Kawai RX2 and two Steinway model Ds which will be referred to as Steinway D1 and D2. All belonged to the School of Music at the University of Auckland and were in good condition, being regularly tuned and maintained by a registered piano tuner. The Kessels upright, manufactured in 1955, provided a useful comparison with the more recent models, which were all between 5 and 15 years old. While it was not practicable to perform a subjective comparison of the eight pianos, an indication of relative quality can be gained from the knowledge that the four upright pianos are used as student practice instruments, while the two Steinway grand pianos are used for concert performances, and the two Kawai grands are used for teaching purposes. Approximate resale values of the Kawai grands were four to six times higher than the uprights, and those of the Steinway grands were approximately ten times higher. Thus the grand pianos were significantly superior to the uprights. Recordings of both radiated sound and key vibrations were made, however there were problems with the sound recordings. The first was that due to the differences in geometry between uprights and grands, it was difficult to choose a point at which to make the sound recordings that was equivalent. As a compromise, the microphone (Bru¨el and Kjær 4165) was placed at the point where the pianist’s head would be, so the recording was essentially of the sounds the pianist would hear. The second problem was that, due to external constraints, it was not possible to record all the pianos in the same room, so the pianos were recorded in widely differing acoustic environments. Thus it was not possible to draw any strong conclusions from the sound results; instead it is the vibration results on which this paper will focus. Some references to sound spectra are included where useful for comparison, but they should be regarded as tentative. For the vibration recordings, the accelerometer (PCB 352 C68) was attached with wax to the piano on the F4 key. This key was held down by a small weight (160 g) so that it was in contact with the keybed. During the course of this investigation measurements were made on various keys, and it was found that there was relatively little variation between measurement points. For the results shown the F4 key was chosen because it was found, in the analysis of a number of pieces of classical music, to be the mean pitch used on the piano [13], thus the measurement represents the average point at which the pianist feels vibrations while playing. As this key could not be struck, it was skipped in the analysis. To strike each note, a weight of 340 g was held so that it touched the key. It was then released to fall and depress the key. Under the weight a small strip of foam prevented clicks. This was repeated three times for each note, and a total of 48 notes were recorded, chromatically from C2 to C7.
1108
M. Keane / Applied Acoustics 68 (2007) 1104–1117
A comparison of the recordings for each key showed a high degree of repeatability. Of the 324 levels eventually used (acceleration, BAC and TAC), the difference of the note level from the mean level for that note was less than 0.2 dB for 171 notes, 0.2–0.4 dB for 89 notes, 0.4–0.6 dB for 51 notes, 0.6–0.8 dB for 6 notes and 0.8–1.0 dB for 7 notes. The microphone and accelerometer signals were recorded on DAT at 48 kHz, then transferred digitally (via SPDIF) to a PC and encoded as WAV files for analysis in Matlab. Each note was held for at least 1.5 s, but in the analysis all were truncated to 1 s in length, to avoid the sound of the note let–off. By this time the BC has decayed to below the noise floor, but the TC was often much longer and was therefore truncated. The effect of the truncation was not significant, due to the low amplitude and stable harmonic structure at this stage of the note development. 2.1. Basis of separation algorithm Fig. 2 shows a comparison of the sound spectra of two notes, C5 and C8, recorded on a grand piano. The time window was 1 s long, thus the decay of the C5 note was truncated. The harmonics of the C5 note can be clearly seen, at 524 Hz and integer multiples. The fundamental of the C8 note can be seen at 4285 Hz; the relatively large bandwidth reflects the rapid decay of very high treble notes. Between the harmonics of the C5 note and below the fundamental of the C8 note, the BC spectrum can be clearly seen. In contrast to the TC which has sharp peaks and is zero elsewhere, the BC varies slowly with frequency, decreasing at a rate of around 20 dB per octave. This allows the design of an algorithm to separate a recorded note into TC and BC, as described below.
100 C5 (524 Hz)
Level (dB, arb. 0)
50
0
–50
50 C8 (4285 Hz) 0
–50
–100 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
f (Hz)
Fig. 2. Spectra of C5 (524 Hz) and C8 (4285 Hz) notes recorded on a grand piano (Kawai GS70). Below the fundamental of the C8 note there is only BC.
M. Keane / Applied Acoustics 68 (2007) 1104–1117
1109
2.2. Detection of harmonics Detection of individual harmonics is a difficult task, and one for which elaborate methods have been devised [2,14–16]. For the purposes of this study, two methods were developed which were not as robust as some of those used by others, but were suitable for the task at hand. Both used the power spectrum of the signal after the BC has died away, leaving the relatively slowly decaying TC. Since both methods gave some false positives, the results from the two were compared, and a harmonic was confirmed if found by both methods. In the first method, the spectrum was lowpass filtered (the raw data of the spectrum were treated as if they formed a signal), giving an approximate spectrum, without the sharp peaks at the harmonics. The filter cutoff used was proportional to the pitch of the note. A threshold was created by adding 5 dB to this lowpassed spectrum, and the local peaks where the spectrum was greater than this threshold were identified. Peaks below the note fundamental were ignored. In the second method, the spectrum was again treated like a signal, and its Hilbert transform was found. The Hilbert transform may be thought of as a complex envelope follower, so in this case the magnitude of the Hilbert transform followed approximately the envelope of the spectrum [17]. It gave a relatively smooth function that retained peaks due to the harmonics in the original spectrum, but ignored the smaller peaks associated with noise. Any peaks found in the Hilbert transform that matched those from the first method were identified as the harmonics of the note. Both methods are illustrated in Fig. 3, which demonstrates the detection of the harmonics of a C5 note. Circles show any local peaks in the spectrum that are above the threshold, while stars show the locations of Hilbert transform peaks. Asterisks mark the confirmed
40
Spectrum Lowpass spectrum + 5dB Spectrum peaks Hilbert transform peaks Confirmed harmonics
Level (dB, arb. 0)
30
20
10
0
–10
–20
–30 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
f (Hz) Fig. 3. Example of harmonic detection method, for a C5 note. (s) Peaks in the spectrum, (q) the locations of Hilbert transform peaks, and ( ) the confirmed harmonics.
*
1110
M. Keane / Applied Acoustics 68 (2007) 1104–1117
harmonics, where both methods agree. The method does not identify harmonics that are too weak, as shown by the false negative at 4.3 kHz, which was identified by the Hilbert transform but was below the 5 dB offset. 2.3. Removing harmonic content The technique used to reconstruct the BC was to create an approximate BC spectrum from the available data, and then use the inverse Fourier transform to return to the time domain. The majority of the BC spectrum may be simply taken from the spectrum of the original signal, however near each harmonic it is the TC spectrum which dominates. Consequently, in a small frequency range around each harmonic the BC spectrum must be inferred, based on the assumption that it varies linearly over such a small range. Therefore, the separation proceeded as follows. The spectrogram of the note was taken, using a 4096 sample Hamming window. For each time window in the spectrogram, for each harmonic, there is a peak in the spectrum that must be replaced with an approximation of the BC. Two endpoints were chosen, 50 Hz to either side of the peak (although individual harmonics varied, this was found to be a good estimate of where the peaks died back to the level of the BC), and a linear interpolation was made between them. This was used as the approximate BC for that portion of the spectrum, replacing the peak which was originally there. The remainder of the BC spectrum, away from the harmonics, was taken directly from the original note, as in those regions the TC makes no contribution. Fig. 4 demonstrates the process for two harmonics of a C5 note. For each harmonic, the BC spectrum is created by interpolating between the points chosen on 80 Original spectrum BC spectrum approximation Interpolation endpoints Harmonic content removed
70
Level (dB, arb. 0)
60
50
40
30
20
10 0
500
1000
1500
f (Hz)
Fig. 4. Example of removal of spectral peaks, for a C5 note. Asterisks mark the endpoints chosen for the interpolation, and the dashed line shows the interpolated spectrum. The shaded area represents the harmonic content which is removed.
M. Keane / Applied Acoustics 68 (2007) 1104–1117
1111
either side of the peak. The shaded area therefore represents the harmonic content that is removed. The inverse spectrogram (since the spectrogram is calculated via the Fourier transform, it is reversible) was then taken to reconstruct the BC signal. Within each window the signal was divided by the Hamming window, and multiplied by a triangular window (because the windows used overlapped in time by 50%) so that for the reconstructed signal the envelope response was flat. Thus the BC was found, and by subtracting it from the original signal the TC was also recovered. This algorithm works equally well whether the input is vibration or sound, but it is limited to notes in the middle range of the piano. With low notes, the harmonics are closer together in frequency, so the time window should be lengthened to increase the frequency resolution. However, there is time variation in the amplitude and harmonic structure (shimmer and jitter), so the assumption of an unchanging signal within each window is no longer valid. With high frequency notes, the harmonics decay very rapidly, at around the same rate as the BC. Therefore the assumption that the BC spectrum varies linearly over the range of each TC harmonic is no longer valid, as each harmonic is spread over too large a frequency range. Due to these limitations, the algorithm can only be used between approximately F2 and C7. For all results given in this paper, only notes from C3 to C6 were used, to give a margin of safety. Fig. 5 shows a typical acceleration signal separated into TAC and BAC by the algorithm. The BAC is large initially, but it quickly dies away to below the level of the TAC. The separated signals may be tested by any of the myriad measures that have been proposed for examining musical timbre, such as spectral centroid, harmonic onset synchrony, inharmonicity, shimmer, jitter and methods utilising cepstral analysis [18–20]. The absence of the BC makes the note harmonics easier to investigate by these methods.
1.5 1 Combined 0.5
Acceleration (m/s2)
0 –0.5 0.5 Tonal 0 1 Broadband
0.5 0 –0.5 –1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
Fig. 5. Example total acceleration, TAC and BAC recorded for an F#4 note recorded on a grand piano (Kawai RX2).
1112
M. Keane / Applied Acoustics 68 (2007) 1104–1117
This study focused on the peak level and spectrum of each signal. Since the measurements were of time-varying quantities, difficulty was encountered in describing the relative recorded levels. However, all the notes decayed approximately exponentially with decay rates that increased from the bass to the treble. Thus the results found would be unchanged regardless of the method used to describe levels. For this paper, the level given is the peak absolute acceleration attained (this will be during note onset), converted to decibels relative to 1 m/s2. It would be interesting to examine the time variation of the recordings, though unfortunately there is not space to go into detail here. For all the notes, the decays of the tonal and broadband components were approximately exponential. For the broadband components, there was no variation in decay rates across the compass. Similar rates were found in the uprights and grands, between 75 and 105 dB/s. Measuring tonal component decay rates presented more difficulty, as there was interaction between individual harmonics, and variation across the compass. For the uprights, decay rates were approximately 15 dB/s at C3, 25 dB/s at C4, and 40 dB/s at C6. For the grands the decay rates were around 10 dB/s at C3, 20 dB/s at C4, and 40 dB/s at C6. These represent average values however, and the rates for individual notes could vary by ±10 dB/s from those given. 3. Results Figs. 6–8 are plotted as box and whisker plots: the boxes have lines at the mean and the upper and lower quartiles, and the whiskers show the extent of the remainder of the data. The results showed significant differences between the uprights and grands measured, as well as a large degree of variation between the pianos within those divisions. 18
2
Peak acceleration (dB re. 1 m/s )
16
14
12
10
8
6
4
2
K50E
NS20A Kessels Uprights
YUX
GS70
RX2
D1
D2
Grands
Fig. 6. Total acceleration levels of notes from C3 to C6 in four upright pianos (left) and four grand pianos (right).
M. Keane / Applied Acoustics 68 (2007) 1104–1117
1113
16
12
2
Peak acceleration (dB re. 1 m/s )
14
10
8
6
4
2
0
–2
K50E
NS20A Kessels Uprights
YUX
GS70
RX2
D1
D2
Grands
Fig. 7. Broadband acceleration component levels of notes from C3 to C6 in four upright pianos (left) and four grand pianos (right).
The mean acceleration level before separation was 4 dB higher in the upright pianos than in the grands. Fig. 6 shows the distribution of total acceleration for the eight pianos studied. The piano with the lowest mean acceleration was the GS70, followed by D2 and D1. The piano with the highest acceleration was the Kessels. The RX2 was unusual, as it was the only one of the grand pianos with a large mean acceleration. The NS20a had a lower mean acceleration than the other uprights. Fig. 7 shows that the BAC levels were 5 dB higher in the uprights than in the grands. Again, the RX2 had the largest mean acceleration amongst the grand pianos. Fig. 8 shows that the TAC levels were approximately the same between the two groups. The Kessels had lower TAC levels than the other pianos, it is not known whether this is related to the age of that piano. The BAC levels were much higher than the TAC in all the pianos, thus the differences seen in Fig. 6 were caused mostly by the BAC and not the TAC. 4. Broadband spectra Fig. 9 shows the mean BAC and BPC spectra for the four upright and four grand pianos. For each piano there was very little difference in broadband spectra between notes. The largest peak in the BAC was found at 60 Hz. Above this there was a secondary peak near 1.5 kHz, around 25 dB lower than the first. The BPC spectra were approximately flat up to 800 Hz, then dropped at 12 dB/octave. It can be seen that the BAC and BPC spectra were not strongly correlated, and the differences found between uprights and grands in the BAC were not replicated in the BPC.
1114
M. Keane / Applied Acoustics 68 (2007) 1104–1117 10
Peak acceleration (dB re. 1 m/s2)
5
0
–5
–10
–15
–20
K50E
NS20A Kessels
YUX
GS70
Uprights
RX2
D1
D2
Grands
Fig. 8. Tonal acceleration component levels of notes from C3 to C6 in four upright pianos (left) and four grand pianos (right).
40 Grand broadband spectrum Upright broadband spectrum
30 20 10
Level (dB, arb. 0)
0 –10
Acceleration
–20 40 30 20 10 0
Sound pressure
–10 –20
2
3
10
10 f (Hz)
Fig. 9. Mean BAC and BPC spectra from C3 to C6 averaged over four upright and four grand pianos.
M. Keane / Applied Acoustics 68 (2007) 1104–1117
1115
5. Discussion As discussed above, for this paper it is assumed that grands are superior to uprights, in agreement with the general experience (and perhaps prejudices) of most pianists. Thus, bringing the uprights closer in performance to the grands would be a valuable improvement. The BAC levels in the grand pianos were around 5 dB lower than the BAC in the uprights, while the TAC levels were on average 1 dB higher. Thus the BAC/TAC ratio, as given in Table 1, was on average 6 dB higher in the uprights (although the Kawai RX2 had a BAC/TAC ratio of 10.1 dB, which was higher than both the NS20A and the YUX). The difference limen for detection of differences of tactile vibration intensity is in the range of 1–2 dB [21], so a difference of 6 dB is large enough to be detected by the pianist. It is not surprising, given the difference in geometry, that differences exist between upright and grand pianos in terms of vibration levels. The results in this paper, on their own, are not to be regarded as evidence that a lower BAC/TAC ratio is desirable. What remains to be proven is whether the differences seen are significant to a majority of pianists. This was further investigated by means of a subjective experiment, presented in a paper which has been submitted to Music Perception. In that work, one of a pair of upright pianos of the same make and model was modified to reduce the BAC level by 6 dB, without changes to the BPC or the radiated sound. A group of experienced pianists played on both pianos, and showed a statistically significant preference for the piano with the lower BAC level. Another group of pianists (the control) showed no preference for either piano before the modifications were made. Thus the study showed that reducing the BAC level of that upright piano improved the subjective evaluation. Along with the results given in this paper, this strongly suggests that it is desirable for upright pianos to have lower BAC/ TAC ratios. It is important to note that only the pianist feels the key vibrations. For the audience, it is the radiated sound which is central to the assessment of instrument quality. It is uncertain whether the BPC/TPC ratio, by analogy with the BAC/TAC ratio, is important in evaluating piano tones. The BPC spectra show that this is unlikely to be the case, as significant changes in the spectra relating to piano quality were not found.
Table 1 Grand and upright mean acceleration levels, in dB relative to 1 m/s2, and the BAC/TAC ratio, in dB Piano
Type
BAC mean
Kawai K50E Kawai NS20A Kessels Yamaha YUX Kawai GS70 Kawai RX2 Steinway D1 Steinway D2
Upright Upright Upright Upright Grand Grand Grand Grand
10.3 8.5 11.6 10.9 3.4 8.2 5.1 3.3
TAC mean 0.8 0.8 3.5 1.7 0.3 1.9 0.2 1.9
BAC/TAC 11.1 9.3 15.1 9.3 3.7 10.1 4.9 1.4
1116
M. Keane / Applied Acoustics 68 (2007) 1104–1117
6. Conclusions A new method was presented for splitting recorded piano notes into tonal and broadband components, with several advantages over previously proposed methods. The results from the eight pianos tested show that there are large differences in key vibrations between upright and grand pianos. The principal differences found were the upright pianos tested had total acceleration levels 4 dB higher than the grands; the BAC levels were 5 dB higher in the uprights, this accounted for the majority of the measured difference in total acceleration; the BAC/TAC ratios were twice as large in the uprights as in the grands; there were only small differences in broadband spectrum between the eight pianos, and across the compass of each piano. These results suggest that lowering the BAC/TAC ratios in upright pianos to bring them closer to the ratios found in grands is desirable. Acknowledgements The author is very grateful to the anonymous reviewers whose suggestions have significantly improved the quality of this paper. References [1] Blackham E. The physics of the piano. Sci Am 1965;213:88–96. [2] Ortiz-Berenguer L, Casaju´s-Quiro´s F, Torres-Guijarro S. Multiple piano note identification using a spectral matching method with derived patterns. J Audio Eng Soc 2005;53:32–43. [3] Bork I. Sound radiation from a grand piano. In: Proceedings 100th convention of the audio engineering society, Copenhagen, 1996. [4] Galembo A. Quality of treble piano tones. In: Proceedings Stockholm musical acoustics conference, Stockholm, 2003. p. 147–150. [5] Zwicker E, Fastl H. Psychoacoustics: Facts and models. New York: Springer-Verlag; 1999. [6] Marshall H, Nielsen J. On the transmission of ‘impact noises’ from a grand piano. University of Auckland, 1997. [7] Smurzynski J, Objective and subjective measurements on treble piano sounds. Czechoslovakian Scientific and Technological Society, Hradec Kra´love´, 1983. [8] Revvo A, Criteria for objective quality evaluation of piano sounds. In: Novelties of musical instrument industry. Scientific Research and Design Technological Institute of Musical Industry, Moscow, 1988. [9] Banton L. The role of visual and auditory feedback during the sight-reading of music. Psychol Music 1995;23:3–16. [10] Finney S. Auditory feedback and musical keyboard performance. Music Percept 1997;15:153–74. [11] Galembo A. Perception of musical instrument by performer and listener (with application to the piano). In: Proceedings international workshop on human supervision and control in engineering and music, Kassel, 2001. p. 257–66. [12] Askenfelt A, Jansson E. On vibration sensation and finger touch in stringed instrument playing. Music Percept 1992;9:311–50. [13] Keane M. Statistical analysis of classical piano recordings in MIDI format. In: Proceedings New Zealand acoustical society conference, Wellington, 2004. [14] Guillemain P, Kronland-Martinet R. Characterization of acoustic signals through continuous linear time– frequency representations. Proc IEEE 1996;84:561–85. [15] Godsill S, Davy M. Bayesian harmonic models for musical pitch estimation and analysis. Bayesian Statistics VII. Oxford: Oxford University Press; 2003.
M. Keane / Applied Acoustics 68 (2007) 1104–1117 [16] [17] [18] [19] [20] [21]
1117
Kay S. A fast and accurate single frequency estimator. IEEE Trans Acoust 1989;37:1987–90. Hahn L. Hilbert transforms in signal processing. Boston: Artech House; 1996. Grey J. Multidimensional perceptual scaling of musical timbres. J Acoust Soc Am 1977;61:1270–7. Pollard H, Jansson E. A tristimulus method for the specification of musical timbre. Acustica 1982;51:162–71. Fletcher H, Blackham E, Stratton R. Quality of piano tones. J Acoust Soc Am 1962;34:749–61. Gescheider G, Bolanowski S, Verillo R, Arpajian D, Ryan T. Vibrotactile intensity discrimination measured by three methods. J Acoust Soc Am 1990;87:330–8.