And yes, for the purests who insist that there is harmonic information above 6k in a bass, there certainly is. And it certainly colors the instrument's tone.
However. Time for a giant pile of arithmetic and assorted acoustic physics.
(Exceptionally long post warning)
Let's look at a basic 4-string, 24-fret bass. The highest fundamental frequency it'll produce is about 390Hz (G-string, 24th fret). 6KHz is another four octaves above that. From a raw mechanical perspective (-6/octave), that's an overtone that's about -24db from the highest regular fundamental before the filter ever kicks in. So, again leaving out the filter, the next octave point (12KHz) is about -30db, and the last octave point (24KHz) is about -36db.
So, even though there is string energy present above 6k, it's generally more than 24db down from the highest fundamental a 4-string can produce, leaving aside the pickup response.
We don't really need to worry about this rolloff, though, as it represents what the string is doing. Which is, after all, what we're trying to accurately reproduce. It merely addresses the question of what's present above 6KHz before adding in any additional losses the instrument system may impose.
(Also, what's above 6KHz includes some incredibly nasty crap such as thermal noise and RF interference. If we're going to play in that space, we'll need to be able to separate signal from shit. More on that later.)
Now, let's factor in the pickups. There are three factors involved in determining a pickup's frequency response. First, its electrical characteristics as coupled to the amplifier chain. Coil inductance, capacitance, impedence, loading, etc. Let's ignore that for the moment, since that varies from pup to pup. Second, the pickup's magnetic aperature. Again, this varies from pup to pup, so we'll leave that alone for a sec as well.
Finally: pickup placement, relative to the vibrating string. This may actually have the greatest impact on an instrument's tone, beyond its mechanical construction. A pickup is only looking at a small section of the string, and so only a particular subset of the overtone sequence being generated by the instrument. That subset represents a collection of energy resources that differ in frequency, amplitude and phase. It's the phase/amplitude relationship that's of concern here. Basically, the pickup's aperature and placement act as elements of a mechanical filter that rejects any part of the string's harmonic structure that it can't see. So the phase and amplitude of the individual harmonics create sum-and-difference resonances that ultimately result in a comb-filter.
Looking at our G24 note at 392 Hz, with a pair of pickups each with a 1 inch aperature, sitting 3.75 and 7.5 from the bridge, respectively. Both pickups are at full volume. The resulting response curve shows a +6db peak at about 1.1KHz, followed by a fairly sharp notch (-60db) an octave higher at about 2.2KHz. So, just in the pickup aperature and location, there's already a 60db/octave rolloff about 2/5 octaves above the fundamental! Of course, as a notch, there's energy above that node point as well. The next peak occurs at about 3.3KHz, which sits about -9db from the fundamental. There's another -60db notch at about 3.85KHz, followed by another -9db peak at about 4.4KHz. Finally, there's another peak at just about 6KHz that's of equal amplitude to the fundamental.
The general effect is to create a response hole between 2.2KHz and 4.4KHz, where there's very little support for whatever string information is actually there, and which creates an upper hinge point of about 6.6KHz, after which the peaks within the comb filter roll off at about 6db.
So, _mechanically_, this instrument shows a 6db/octave rolloff low-pass response with a 6.6KHz pole, when reproducing a 392Hz fundamental. Again, this is _before_ adding in the pickup's electrical characteristics; this is purely the pup's location and aperature.
Let's look at the other end of the spectrum: an open E string at 41Hz. Same pickup setup as above. The resonant peak hits at about 125Hz; first comb notch hits at about 250Hz; etc. The mechanical filter here has its low-pass hinge at about 2KHz, again with a 6db/octave rolloff.
So, mechanically, the instrument displays a self-variable low-pass response, with the filter opening as the fundamental note increases in pitch.
NOW, add in the pickup's electrical response. A pickup will generally have a resonant frequency, above which it naturally rolls off. A classic passive Fender pickup will have a resonance (when installed and loaded with volume and tone pots and used with a cable) of about 2.5KHz (give or take .5KHz), above which it'll roll off at about 6db/octave.
A low-impedence active pickup, such as Alembic builds, will have a resonant peak somewhere between 4KHz and about 7.5KHz, depending on how low-Z is really is, and how much gain the designers are making up in the preamps. (Again, we're leaving out any preamp filtering, so far, and are merely looking at the total harmonic energy available in the system.)
Putting all the above stuff together:
The string rolls off at 6db/octave from the fundamental.
The instrument rolls off at 6db/octave from the pickup's position/aperature low-pass hinge point.
The pickup rolls off at 6db/octave from its resonant peak.
So, let's take a bass, playing an open G string (roughly 100Hz). Low-Z pups with aperature and location specs as described above, and a 5KHz resonant peak. What do we have?
The string rolls off at 6db/octave above 100Hz.
The instrument rolls off at 6db/octave above 3KHz.
The pickup rolls off at 6db/octave above 5KHz.
That gives us an aggregate signal of (-36 plus -6 plus -3) equals -45db down at 6KHz. Go up another octave and you're down -63db at 12KHz.
That's a real long way down, for the next to last octave, let alone trying to squeeze 20KHz out of the system. That's all there is. To get anything else out of the system, (i.e., to get back to the raw string's response) you'd need to provide boost to offset the instrument's mechanical rolloff and the pickup's electrical rolloff, or about 9db at 6KHz (remember that number, btw.) and about 21db at 12KHz.
That'd take roughly a four-pole 15KHz high-pass filter. Take half a second and think about what else would be getting amplified by 21db or more, and then remember how much effort RonW has put into keeping noise and radio frequency interference OUT of Alembic's instruments...
So, let's add in Alembic's active filters. Rather than trying to boost the last two octaves, where there's very little information to begin with, and which will give us at least +21db of noise gain, let's see what we can do to at least try to recover some of that string response that's being eaten by the electro-mechanical structure of the bass itself.
Remember that 9db number I mentioned above? It's not a coincidence that Q-switch's maximum gain provides a 9db peak at 6KHz. Or that the filter sweep tracks right through the sweet spot where that aperature/phase cancellation comb filter hole is. The goal is to recover the raw string information that disappears into the first few notches in the comb filter (and to boost those first few peaks that are -9db down from the fundamental), and to extend the instrument's electromechanical response up to where the strings themselves finally start to give up at 6KHz.
Trying to push the filter by increasing the sweep range from 6KHz up to, say, 12KHz, would require a Q-switch with 21db of gain, followed by a six-pole low-pass filter, simply to push the amplified noise and RF back down by 36db or more. And the easiest way to create a six-pole filter is to use an elliptical filter, which happens to be exactly what the comb filter is, in the first place. Remmeber that the comb filter has peaks and valleys spaced all along the overtone chain from the fundamental signal. Put two of them in a circuit, and you'll get places where the peaks coincide. The technical term for that phenomenon is oscillator.
So, let's not go that way, and see what we can do to tame some of the peak-and-valley chaos that's already built into the mechanical comb filter. How about we simply squash is into oblivion? What do we lose?
We lose the upper two octaves of a string that're already down -24db from the highest fundamental the bass will produce, and we lose any circuit noise generated in the pickups themselves or by the first preamp stages.
We gain a measure of immunity from RF interference in the preamps, but more importantly, we gain that really neat 9db boost that exactly offsets the electromecanical filtering effect of the bass, AND, we get user-controllable support for the more significant nasties from the first few nodes of the electromecanical comb filter.
So, we trade a hatfull of major headaches that occur above 6KHz for an instrument that gives you as close to a 100% accurate picture of what the string is actually doing as you're ever going to get with magnetic pickups.
(For a look at the electromechanical comb filter in action, take a look at
this really cool demonstration page by J. Donald Tillman. Also, take a run over to
for a nice table of the Frequency of Musical Notes, from MichiganTech's Physics department.)
Sorry to be so long-winded, but there's a huge amount of background to cover whenever you start trying to back-engineer Ron Wickersham. :-)
nic
