Continuing with my aim to design a sound system for accurate reproduction of digital piano, I made some measurements to determine the required amplifier power and driver excursion capabilities. For output levels on par with a 6′ grand piano, an adequate system will need to reproduce transient peaks of at least 109dB SPL @ 1m and be capable of displacing at least 50 cm3. See below for details.
To get an estimate of the sound levels produced by a real piano, I made some SPL measurements in the reverberant field while playing loudly on a 6′ grand with the lid up. Playing as hard as I can, on big chords with the sustain pedal engaged, I can push the reverberant level in my living room up to 95dBC. But at the loudest I would actually use to play music the reverberant level peaks at 90dBC, so I’ll take this as the target level.
These measurements were made in a fairly lively 600 m3 room, with an RT60 at least 0.6s. With these figures we can estimate that to achieve the same (90dB SPL) reverberant level in the same room, a loudspeaker radiating as a monopole would need to produce a direct sound field of 95dB SPL @ 1m.
However, my recorded waveforms when playing at this level indicate a crest factor of about 14dB, so we actually need to reproduce transient peaks of 95+14=109dB SPL @ 1m. The amplifier power needed to achieve this will depend on the loudspeaker’s efficiency. Here are the calculated requirements for a range of driver sensitivies:
|Sensitivity [dB SPL @ 1W]||Amplifier Power [W]|
These requirements are demanding but certainly achievable: there are plenty of drivers with 90dB sensitivity and transient power handling of 60W.
Low Frequency Output
Although a piano can produce very low frequencies (the lowest note A0 has a 28Hz fundamental) my measurements indicate the level actually falls off rapidly below about 80Hz. For example, when playing the note A0 the sound pressure is almost all due to the 4th harmonic at 109Hz, which generates at least 30dB higher SPL than the fundamental or 2nd or 3rd harmonics.
I made spectral measurements while playing loud octaves at the bottom end of the piano (playing octaves doubles up some spectral components, so is likely to produce the highest possible SPL at any given spectral peak). Here are the levels of the first four harmonics, scaled to our targeted total SPL of 95dB @ 1m:
Note that below 70Hz the piano doesn’t produce any spectral component louder than about 70dB SPL. In this range of the instrument the sound pressure is strongly dominated by harmonics of the fundamental.
Taking the maximum measured SPL of each spectral component and converting to the volume displacement required for a monopole radiator to produce that tone at the required level, we get the following graph of required volume displacement vs. frequency:
Thus, in terms of driver excursion, the most demanding spectral component is at 92Hz (produced by the 2nd harmonic of the note F#1 sounding together with the fundamental of F#2). We can calculate the driver excursion (Xmax) required for a circular piston in an infinite baffle to produce this tone at 95dB SPL @ 1m for a range of driver sizes:
|Piston Diameter||Xmax (center-to-peak)|
Not surprisingly, a 4″ driver isn’t likely to move enough air. Many 6″ drivers are capable of the needed 2mm displacement, but likely with significant distortion. Drivers 8″ and up are good candidates. Note that these excursion requirements double in the absence of a substantial baffle.