[Argo Duck]

I haven't read the whole thread but intuitively, and in fourier terms, the OP makes a lot of sense to me.
 
Need to think about this more though. Subscribed.

 

[Tilpo]

I think I get ac500 point a bit better now. But rather than saying that treble frequencies are part of a bass note, I think the important factor is that the quick changes introduced by treble are harder to reproduce. This means that adding treble to the mix makes the whole waveform harder to reproduce, therefore if the driver is good at reproducing the treble accurately the details in the lower tones aren't lost when the driver messes up when playing high pitched tones.
However this explanation of the phenomena would not allow for ac500's proposition that completely removing the treble would reduce the detail in the bass. It would remain the same.
In the example you gave us you said that removing the treble would result in a plain ol' boring sine wave. Maybe that sine wave is what is there in real life, what we would measure with Fourier transformations. The extra perceived detail in the bass is completely imaginary. That is not to say that such imaginary detail can give a nice effect, but it isn't necessarily true that the bass reproduction is better.

 

[ac500]

I think you're kind of getting it but as hard as it is to believe, you have to realize that these "quick changes" are no different than treble from a mathematical point of view. There is no "illusion" here... that waveform will very tangibly change into a simple sinewave if you remove frequencies above ~200hz. There's no illusion to it, you can actually observe it as you remove them if you view this kind of effect on an oscilloscope or equivalent software.
 
I'm almost tempted to write a program that lets you draw your own waveform and then mess around adjusting certain frequencies to see how it modifies the resulting output wave. Sadly I don't have time to devote to a project like that currently. 

 

[Tilpo]

I think you're kind of getting it but as hard as it is to believe, you have to realize that these "quick changes" are no different than treble from a mathematical point of view. There is no "illusion" here... that waveform will very tangibly change into a simple sinewave if you remove frequencies above ~200hz. There's no illusion to it, you can actually observe it as you remove them if you view this kind of effect on an oscilloscope or equivalent software.

I agree to all of that, but what I think is that the bass tone was a simple sine wave to begin with. It's the treble that's creating the illusion of detail to the bass. This illusion is either created by the ear, or by non-linearities in the sound reproduction of the driver.

I'm almost tempted to write a program that lets you draw your own waveform and then mess around adjusting certain frequencies to see how it modifies the resulting output wave. Sadly I don't have time to devote to a project like that currently.

You can. It's as simple as plotting a couple sine functions in a plotting program. If you know your way around Linux, try to experiment a bit with GNUPlot.

 

[Argo Duck]

Hmm, hope I'm not adding confusion here as I could be way off track, but it strikes me something like ac500's complex 40Hz waveform is a realistic scenario.
 
To offer a simple thought experiment, suppose two 40Hz tones occurred, with a time delay (phase-shift) of 1/400 s (equivalent to 400Hz). Once summed, and with appropriate amplitudes, a more complex waveform roughly like the example results.
 
Note that - in effect - a '400Hz component' has to be playable in order for the wave to be accurately reproduced, even though there is no 400Hz fundamental present.

 

[J0nny]

OK, I couldn't really be bothered reading the last two pages, but... Do we even know if a complex sinusoidal wave at any given instant is actually the sum of an infinite series of simple sine waves in real life, or is this simply a mathematical way to analyse the complex sine wave? Maybe I shouldn't have said complex. Lets not get started on complex analysis.

 

[ac500]

> Note that - in effect - a '400Hz component' has to be playable in order for the wave to be accurately reproduced, even though there is no 400Hz fundamental present.

Exactly. There is a 400hz component present however (mathematically), just not in the same way we typically intuitively think of it.
 
However just a slight correction - two 40Hz tones summed cannot form a waveform with higher frequency components. You'll just get a sinewave with a different amplitude (volume) and phase. So you cannot actually increase the complexity of a signal no matter how many frequencies you sum at the given frequency.
 
Proof: http://www.wolframalpha.com/input/?i=sin%28x%29%2Bsin%28x%2Bc%29

 

 

[ac500]

> Do we even know if a complex sinusoidal wave at any given instant is actually the sum of an infinite series of simple sine waves in real life, or is this simply a mathematical way to analyse the complex sine wave?
 
In real life there are always different ways to mathematically represent the same thing. Just because it may be intuitive to think of sound as a time-domain waveform doesn't mean it's not "really" a sum of sine waves, because it's just as much sine waves as it is anything else. The two representations are absolutely equivalent in every way.
 
Since we're talking about frequency issues (treble, bass, etc.) it makes a lot of sense to talk about frequency domain (so, sums of sinewaves = fourier series).
 
But further than that, it actually makes a lot of sense physically to talk about sums of sinewaves. When you learn about simple harmonic motion (basically the resulting path of a spring 0 oscillatory acceleration), you'll see that what results is a simple sinewave. In other words, oscillatory acceleration (the exact function of a headphone driver) produces output naturally as a function of sinewaves, if you do the calculus. So it's even more natural to think of headphone driver output in the frequency domain.
 
But ultimately it doesn't matter - mathematically and physically time domain and frequency domain represent the exact same signal.

 

[Argo Duck]

Aargh, of course. School math. Been a bit too long.
Quote:

> Note that - in effect - a '400Hz component' has to be playable in order for the wave to be accurately reproduced, even though there is no 400Hz fundamental present.

Exactly. There is a 400hz component present however (mathematically), just not in the same way we typically intuitively think of it.
 
However just a slight correction - two 40Hz tones summed cannot form a waveform with higher frequency components. You'll just get a sinewave with a different amplitude (volume) and phase. So you cannot actually increase the complexity of a signal no matter how many frequencies you sum at the given frequency.
 
Proof: http://www.wolframalpha.com/input/?i=sin%28x%29%2Bsin%28x%2Bc%29

 

 

 

[J0nny]

Quote:

> Do we even know if a complex sinusoidal wave at any given instant is actually the sum of an infinite series of simple sine waves in real life, or is this simply a mathematical way to analyse the complex sine wave?
 
In real life there are always different ways to mathematically represent the same thing. Just because it may be intuitive to think of sound as a time-domain waveform doesn't mean it's not "really" a sum of sine waves, because it's just as much sine waves as it is anything else. The two representations are absolutely equivalent in every way.
 
Since we're talking about frequency issues (treble, bass, etc.) it makes a lot of sense to talk about frequency domain (so, sums of sinewaves = fourier series).
 
But further than that, it actually makes a lot of sense physically to talk about sums of sinewaves. When you learn about simple harmonic motion (basically the resulting path of a spring 0 oscillatory acceleration), you'll see that what results is a simple sinewave. In other words, oscillatory acceleration (the exact function of a headphone driver) produces output naturally as a function of sinewaves, if you do the calculus. So it's even more natural to think of headphone driver output in the frequency domain.
 
But ultimately it doesn't matter - mathematically and physically time domain and frequency domain represent the exact same signal.

ac500, what I'm saying is this: say I play my bass guitar at low E, i.e. the open E string. I record the sound through a nice condenser mic. placed next to my guitar amp. I play back the sound through my headphones and my PC. I can hear the note but I can also hear the character of my guitar and the amp. i.e. the tone of my bass and the amp, the slight distortion etc. So we know that I haven't recorded a perfect sine wave of, for arguments sake, y = f(t) = sin(41.204t) (low E is a 41.204 Hz tone) but there are little "errors" in it. So we obviously need to express this as a more complicated function. So this is the scenario, the driver of my headphone is tracing out the function I just talked about. But if we record it we see that it doesn't adhere it exactly.
 
Does it reproduce an imperfect signal because what it is doing is reproducing an infinite sum of sine and cosine waves, who's sum converges to the function above, or is it simply because the headphone driver has a less than ideal transient response, and cannot "turn around" quick enough to trace the function correctly?
 
Is there any proof that a driver is actually reproducing an input signal as the Fourier decomposition of the said signal?

 

[Tilpo]

ac500, what I'm saying is this: say I play my bass guitar at low E, i.e. the open E string. I record the sound through a nice condenser mic. placed next to my guitar amp. I play back the sound through my headphones and my PC. I can hear the note but I can also hear the character of my guitar and the amp. i.e. the tone of my bass and the amp, the slight distortion etc. So we know that I haven't recorded a perfect sine wave of, for arguments sake, y = f(t) = sin(41.204t) (low E is a 41.204 Hz tone) but there are little "errors" in it. So we obviously need to express this as a more complicated function. So this is the scenario, the driver of my headphone is tracing out the function I just talked about. But if we record it we see that it doesn't adhere it exactly.
 
Does it reproduce an imperfect signal because what it is doing is reproducing an infinite sum of sine and cosine waves, who's sum converges to the function above, or is it simply because the headphone driver has a less than ideal transient response, and cannot "turn around" quick enough to trace the function correctly?
 
Is there any proof that a driver is actually reproducing an input signal as the Fourier decomposition of the said signal?

Just to clarify, those 'little errors' are mostly harmonics and form what we call the timbre of the sound. This is actually not that difficult to measure.
Say the function of the fundamental harmonic (the low E tone in this case) is of function sin(x), then the function of the wave produced by the guitar is more or less sin(x)+a1*sin(2x)+a2*sin(3x)+a3*sin(4x)... where aN denotes some decreasing constant of amplitude.
This does make it a little more complicated, but not by much.

 

[ac500]

> Does it reproduce an imperfect signal because what it is doing is reproducing an infinite sum of sine and cosine waves, who's sum converges to the function above, or is it simply because the headphone driver has a less than ideal transient response, and cannot "turn around" quick enough to trace the function correctly?
 
> Is there any proof that a driver is actually reproducing an input signal as the Fourier decomposition of the said signal?
 
This ability to "turn around" quickly -- the transient response -- is by definition "good treble".
 
Any signal is equivalent to its Fourier "decomposition", so there's no difference at all - they're just like two mathematical languages to refer to the same thing. The thing about a fourier series is it's easier to talk about the effects of treble, that's all.
 
Probably the easiest way to prove this would be to take a sample input signal with the claimed "fast turn around signal that isn't treble", pass it through a low-pass filter (which can be done without any sort of fourier series btw - this can be done in the time domain since as I said, these are mathematically identical ways of referring to the same thing), and then show that this "fast turn around" signal has now become a undefined blob.
 
To prove it rigorously though, we'd have to define a mathematical understanding of this detail (the fast changes to the signal), then prove that any given signal sent through a low pass with cutoff frequency x will remove such detail.
 
I'd like to do this, but we'd have to agree on what detail looks like in a waveform - we need a mathematical description of what it is. I propose a simple definition: given an arbitrary waveform of a base tone of period T, we can choose two points within a cycle of this waveform of very short distance (delta t << T) such that the difference of their amplitudes is considerably more than what is possible under a simple sinewave (lim0(sin (2pi/T)t)=(2pi/T)t, so delta amplitude >> 2pi*t/T ). In other words, there is at least one point on this wave in which it deviates from a simple sinewave in such a way that requires a fast response.

 

[Tilpo]

Probably the easiest way to prove this would be to take a sample input signal with the claimed "fast turn around signal that isn't treble", pass it through a low-pass filter (which can be done without any sort of fourier series btw - this can be done in the time domain since as I said, these are mathematically identical ways of referring to the same thing), and then show that this "fast turn around" signal has now become a undefined blob.

Isn't that obvious in the sense that you might be removing harmonics? The harmonics of bass note can easily extend into mid range. For example a cello can easily have 10 harmonics.
Say the cello plays a 200Hz fundamental tone. If you put it through a low-pass filter of ~400Hz, then the last 8 harmonics will be attenuated, or even removed completely! That's like turning it into a completely different instrument, since timbre is exactly what our brain uses to identify instruments.

 

[Alondite]

I could be way off on this (and I probably am), but the OP seems to make sense to me. Transient response is simply a combination of impulse and step response correct? If that's the case, then it seems to me that good treble certainly plays a part in detail over the spectrum.

Take a look at the impulse+step response graphs over at innerfidelity. Take the HD800 for example. It is widely considered to have some of, if not the best transient response of any headphone. The graph clearly shows the presence of high frequencies (the sharp rise and fall, and overall detail of the graph), while something that sounds slow and muddy (the Beats Solo) has a very gentile, rolling graph with very little detail. It lacks sharp peaks and thus clearly high frequencies.
 
Also take a look at the square wave graphs. The leading edge (higher frequencies) of the graphs for the HD800 have a very clear and defined edge well above "zero", indicating strong performance in the higher frequencies of both the 30hz and 300hz square waves, while the Beats are quite the opposite.
 
Again, I could be way off here, but that's how I understand it.

 

 

[Armaegis]

Here's a complex waveform for you guys, pulled out of Bach's Tocatta & Fugue (the haunted house song) about 20 seconds in, a moderately low organ note: