The implications of this finding are staggering.
       This finding means that the whole model we have been taught of the sound from musical instruments, as consisting of a fundamental sine wave plus harmonic overtone sine waves at a few discrete frequencies, is, though crudely useful for some primitive purposes, fundamentally and wholly wrong for the sophisticated purpose of high fidelity reproduction. That's especially true for sharp signal changes or transitions, such as attack transients, which we of course want accurately reproduced by our equipment. Virtually all musical attack transients actually contain an infinite spread of frequencies, extending all the way down to DC and as high as the risetime of the attack warrants. Indeed, the only musical sounds that obey the common sine wave model are legato notes which begin slowly and gradually, without any attack at the beginning transition, for example many notes played by woodwinds or organ pipes.

C.1. Transients of Midrange and Treble Notes

       This finding also means that most attack transients, even for musical notes whose sustained pitch is at some discrete frequency, actually contain an infinitely dense and very wide spread of frequencies. And this in turn means that they will excite and trigger any and all misbehaviors in all your system components, even though those misbehaviors might nominally be at specific frequencies far removed from the discrete frequency representing the sustained pitch of the musical note after the initial transition and transient attack.
       For example, even though the sustained pitch of a musical note might be in the lower midrange, its transient attack at the beginning contains energy at an infinite spread of frequencies, so it will excite and trigger a misbehaving resonance in the upper midrange of your main loudspeaker's midrange driver, and this misbehaving upper midrange resonance will then continue to ring and artificially color the sustained portion of the musical note, artificially coloring it both by the foreign tonal energy in the upper midrange, and also by the foreign material sound of this ongoing resonance (say a papery mechanical coloration from the paper midrange cone breaking up). Or, the high frequency energy contained this attack transient could trigger tweeter misbehavior, even though the sustained later portion of the musical note lies in the lower midrange.
       As another example, more to the point here, this attack transient, for a nominally lower midrange musical note, also contains energy at frequencies down to DC, so it would also trigger all the misbehaviors discussed above in conventional subwoofers. In other words, it is a fallacy to assume that the many misbehaviors discussed above for conventional subwoofers, and the many sonic degradations therefrom, occur only when deep bass musical notes come along. These conventional subwoofer misbehaviors are in fact triggered by virtually all attack transients. Indeed, these conventional subwoofer misbehaviors stick out more like a sore thumb, and are more obvious and obnoxious, when the musical note with the triggering attack transient is not nominally a bass note, but instead has its sustained pitch in the midrange or treble, and is therefore nominally a midrange or treble note. That's because the sustained portion of this midrange or treble note does not have much or any bass energy that might usefully mask or hide the ongoing, ringing misbehavior of a conventional subwoofer.
       This finding also means that transient attacks actually require far more extended treble capability by your system, to be reproduced correctly, than the traditional musical model, of musical instruments emitting only sine wave fundamentals plus overtones, would suggest is adequate. For example, we measured the frequency content of a gentle cymbal kiss (such as just before the final coda in Rachmaninov's Second Piano Concerto), focusing in on the attack transient that starts this gentle, quiet cymbal kiss. Its actual frequency spectral content was, as predicted, a very wide, infinitely dense spread of frequencies. The shape of the energy content, as a function of frequency, was a broad hump, with skirts extending far above and far below the center frequency of the main peak of this hump. And, what was most interesting, for the point of our discussion here, was the frequency at which this center peak of the hump was located. This frequency could in a sense be called the fundamental frequency of this gentle cymbal kiss, since the maximum energy was at this frequency. What was the fundamental frequency of this gentle, quiet musical transient? It was 40,000 hz!! And the spectral content showed large amounts of energy extending far beyond 100,000 Hz, the limit of our measuring microphone and FFT analyzer.
       Incidentally, as a technical aside, if a musical transient is identically repeated many times, and at identical temporal intervals, then its characteristics fall between a single transient and a sine wave that identically repeats, so its actual spectral content does not range as wide in frequency as does the actual spectral content of a single transient, nor is its spectral content infinitely dense.  But in practice, even repeated musical transients are not performed identically, and not at identical intervals, so they become individually different transients, similar to truly singular transients that are never repeated. Thus, for practical purposes, we can regard virtually all music as consisting of singular transients, actually containing infinitely dense spectral energy, and containing a very wide frequency spread of energy, down to (but not including) DC.
       We've just discussed the staggering implications of our finding for musical notes that are nominally midrange and treble musical notes. Similar implications naturally also apply to all vocal sounds, dialogue, and sound effects, which of course are richly laden with attack transients.

C.2. Bass Content of Transients

       Now, what about notes that are nominally bass notes? Well, which musical instruments actually produce the deepest, lowest frequency bass?
       Let's take a moment for a pop quiz. Consider the mighty pipe organ, often cited as the musical instrument with bass reach to the lowest frequency of interest for high fidelity reproduction. And let's pit this pipe organ against the most absurd possible competition, musical instruments that are nominally considered to be the highest frequency instruments in the orchestra, such as the violin or triangle.
       Take a guess as to which real musical sound has the deepest bass, and therefore requires the deepest bass frequency extension from a subwoofer to be correctly reproduced by your system, the biggest pipe of a pipe organ, or the attack transient of a triangle ting - or, even more absurdly, a pluck on the highest frequency violin string, stopped down to its shortest length to yield the highest frequency (this is a real musical sound, such as you'll find a little over a minute into the fourth movement of Bartok's Fourth String Quartet).
       The pipe organ's biggest pipe (found on only a few organs) plays at 16 Hz, and, because an organ note starts slowly, it does not have any significant attack transient, so its frequency content does not extend below 16 Hz. In contrast, the attack transient from the highest frequency violin string, stopped down to its highest frequency, actually contains energy all the way down to zero Hz (DC)!!
       This shocking result is predicted by the fundamental theory of frequency (for those who truly understand the concept of frequency). And we verified this result by actually measuring the frequency content of a real violin pluck, wherein the FFT analyzer showed energy all the way down to (but not including) DC.

C.2.i. Mathematical Corroboration

       We then further corroborated this result by asking our friend Glenn Rankin to do a mathematical analysis. Glenn used as a model a single cycle of a sine wave, which, like the FFT analyzer showed, would not have any energy content at DC itself, and which would be a decent model for a string being plucked in one direction, then spontaneously going back in the other direction, and then quickly being damped to zero. So, what do you think is the actual frequency content of a sine wave, but only a single complete cycle thereof? In particular, how low in frequency does the energy content of this signal actually extend? Well, you might say, that depends on the frequency (or periodicity) of this sine wave cycle.
       No, it doesn't. And this serves as a good illustration of the fundamental misunderstanding of sine waves and music that we have all been taught, and that most design engineers still mistakenly believe. You see, this signal is not a sine wave. For it to be a sine wave, it would have to continue cycling, unchanged, forever, with no beginning and no end. But this signal does change, just like all our program material does. This signal first is nothing, then starts, then has a waveform shape that deceptively is the same as a single sine wave cycle, and then stops. So this signal, rather than being a sine wave, is instead a transient, which appears and then disappears over time. The start and stop of this single sine wave cycle might be relatively gradual, rather than a sudden, steep attack, but it is still a transient. Note also that this means that even clarinet and pipe organ notes, which begin and end gradually, and which also have a simple sine wave shape while they are playing, are nevertheless still transients, because they too are temporary, and therefore they too actually contain a spectral frequency spread that extends beyond the single frequency represented by their pitch.
       What then about the actual frequency content of this single sine wave cycle? Glenn Rankin's mathematical analysis, transforming this time domain signal into the frequency domain, showed that its actual spectral frequency content extends all the way down to DC. It does not happen to include DC, because this model signal is symmetrical about the zero axis. But this signal still requires a subwoofer that extends all the way down to (but in this case not including) zero Hz, in order to be reproduced accurately by your system. And this is true regardless of the nominal frequency or periodicity of the sine wave, whose single cycle we excerpted to make this model signal.
       Incidentally, the full spectral frequency spread, of energy contained in this single cycle of a sine wave, has a broad hump shape, with energy spectrally extending far above, as well as far below (down to DC) the frequency of the hump's peak. And the frequency of the hump's peak is at the frequency or periodicity represented by the frequency of the sine wave whose single cycle we have excerpted. Thus, it makes good sense to speak of this peak, of the broad hump's spectral energy distribution, as representing the fundamental frequency of the transient (just as we did above for the cymbal kiss, whose spectral energy distribution hump was centered about, and peaked at, 40,000 Hz).

D. Summary of Actual Program Content at Bass Frequencies

       We have now demonstrated that musical transients, even ones at nominally high frequency, actually have spectral energy content down to DC, and therefore require a subwoofer that can play accurately down to DC, if your system is to reproduce these musical transients accurately. We demonstrated this by analyzing the true fundamental concept of frequency, and by actual measurement, and by mathematical analysis. The only remaining step, to prove the pudding and close the case, is to corroborate this by listening comparisons, which we will discuss below.
       Since musical transients, including nominally high frequency ones, actually contain spectral energy down to DC, then obviously so also do all vocal utterances, which a laden with starting and stopping transients, and nominally occur in the midrange. And film sound effects obviously can contain very low frequency energy, including sound effect transients that nominally have a very high frequency.
       Today's recording chains can have very good, very extended response to very low bass frequencies. In fact, with today's solid state circuitry and digital recording, both of which can be perfect down to DC, the only filter acting to reduce very low frequency spectral energy is the recording microphone capturing the original sound. Microphones can extend very low in frequency (especially omnidirectional ones), and a microphone's low frequency rolloff slope is not that steep, so microphones can still put a reasonable amount of spectral energy onto a recording of a live sound, say down to .1 Hz or so. Thus, it clearly becomes important to have a subwoofer that extends accurately down to say .1 Hz, in order to correctly reproduce live sounds as they actually exist on our program material.
       Then, consider that many film sound effects never pass through a microphone at all, but instead are synthesized electronically and put directly onto the soundtrack. Thus, many film soundtracks contain abundant energy at extremely low frequencies. For example, the massive thuds of a dinosaur's foot might be synthesized by gating a sine wave oscillator set at say 3 to 5 Hz, so proper reproduction of this sound effect requires a subwoofer that is comfortable putting out huge amounts of bass energy at 3 to 5 Hz, plus a heck of a lot lower in frequency, in order to accurately reproduce the transient start and stop of each footfall (as we just saw above, a gated signal comprising a burst of a single or a few cycles of sine wave actually contains spectral energy all the way down to DC).

E. Human Limits for Low Bass Perception

       Let's turn now to the low frequency limits of human perception. Scientists tell us that humans cannot hear below below 20 Hz, which implies that there is no need for subwoofers to extend below 20 Hz, regardless of what we found above about program material actually extending far below 20 Hz.
       But again, what scientists have taught us is woefully fallacious. And again, the reason for their mistaken belief is that they don't understand the fundamental concept of frequency, and erroneously rely upon a sine wave model.

E.1. Sine Waves vs. Transients; Frequency vs. Time Domain

       Scientists test this human hearing limit only with continuous, unchanging sine wave tones. The standard test simply plays a sequence of sine wave tones, at ever lower frequencies, until the human subject reports that he can no longer hear the sine wave tone.
       But this test is doubly irrelevant. It is firstly irrelevant because it does not represent the type of bass that is to be heard from our program material. As discussed above, the vast majority of low bass from our program material consists of bass transients, not steady bass tones (the only exceptions being the pipe organ and the contrabassoon). It is secondly irrelevant because the human mechanism for perceiving transient bass events, in the actual time domain in which they really occur, might well be very different from the human hearing mechanism they are testing, which involves hearing continuous tones in the frequency domain.
       These scientists fail to comprehend that real program material is composed of changing transients, not unchanging tones, and they fail to see that the human mechanism for perceiving and evaluating changing transients in the time domain might well be totally different from our mechanism for hearing unchanging sine wave tones in the frequency domain.
       What, then, would be a more relevant, more meaningful way to test human low frequency perception capability? Let's take a lesson from above, where we found that it was more revealing and relevant to evaluate subwoofer performance in the time domain, rather than the frequency domain, and where we found that a transient test signal was a more revealing and relevant probe of subwoofer performance than a sine wave was. So, let's evaluate human performance via the same principles we found so helpful in evaluating subwoofer performance. Let's test human performance with a transient test signal, a signal that probes and exercises human perception capabilities in the time domain.
       This plan also fits in perfectly with the real task that we are testing humans for. Since program material contains far more occurrences of low bass information arising from transients rather than from steady tones, it's obviously far more relevant to test human low frequency perception ability using transient signals rather than steady sine wave tones. And, in case humans do employ a different detection or analysis mechanism for transients than they employ for steady tones, perhaps a mechanism that operates in the time domain rather than the frequency domain, our use of a transient test signal would correctly and relevantly engage this human mechanism -- whereas the scientists' use of steady sine wave tones would utterly fail to engage this human mechanism, thus totally failing to reveal the true low frequency perception abilities of humans, especially for precisely the type of bass information (transient rather than steady tone) that humans will be receiving most from their program material.

E.2. Experiment to Test Human Bass Sensing Limit

       As we saw above, the simplest possible transient test signal is a step signal, which has merely one transition or change. The step signal as a test contains all frequencies, so in one instant it presents a human subject with all frequencies he could possibly perceive. The basic idea of the experiment is to play a step signal repeatedly for the human subject, while the lowest frequencies of the step signal are being electrically filtered at various low frequency points, in order to discover where the human can detect a qualitative difference in the sound of the step, as the low frequency rolloff point is changed, and where he cannot.
       This experiment could be designed in various specific ways, and we'll just describe one protocol as an example. First, play the step signal (from a signal generator) through a bass (high pass) filter that filters out frequencies below say 40 Hz, and play this step repeatedly, so the human subject gets acquainted with its sonic qualities. Then, change the filter to filter out frequencies only

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