The mathematics of amplifiers

[HolmeBass]

Fair enough. Let's agree to disagree.

I wonder what your bonafides are w/re to bass amplifier and bass speaker cabinet designs.

It seems like the kind of thing that probably does have a right answer, doesn't it?

 

[HolmeBass]

This is a great way to demonstrate the math, but the thread was totally lost at the conclusion. 2dB is definitively not noticeable. It takes at least 3dB to even register as an audible difference to the human ear.

Ugh, no it does not. 1 dB is just above the JND. It is plainly audible, especially at high SPL.

This is a falsehood, a canard, misinformation.

Ok I replied before reading past, I see this has been sorted.

It seems misinformation travels faster than a wild fire, while good information is stuck in slow motion sometimes.

 

[el murdoque]

I'm admittedly a bit out of my area of expertise here but I'm not aware of the IEEE providing a standard breakdown for the audio range, which IIRC they generally refer to in documents setting standards for electroacoustic laboratory measurement as the 20 Hz to 20 kHz frequency range. This is in contrast to the way they do formally divide up the frequency spectrum for RF energy (e.g., HF, VHF, UHF, etc.).

However, I do think most audio engineers would agree that subwoofer systems are not designed to operate in the same frequency range as bass guitar cabinets, and therefore the design goals and designs trades would be different. Obviously there are physics/acoustics and electrical/audio engineering similarities being considered, but IME a good subwoofer would make a terrible bass cab (for most people) and vice versa.

The reason is because while many of the fundamental frequencies of the notes on the bass guitar do fall in a subwoofer range (in my mind for live sound approximately 100 Hz and below...and that is admittedly subjective), a lot of what we enjoy in a bass guitar note is in the second and third harmonic range and often higher, especially for playing up the neck and/or getting good articulation from some higher frequency definition to the notes.

I'm not sure very many engineers would accept that a subwoofer is designed (or in any way optimized) for frequencies above 200 Hz in any design circumstance I'm aware of. In most live performance systems I have used, subs are assigned to handle frequencies below around 50 Hz. Interestingly enough this is where many traditional bass cabs start to attenuate the output more significantly. There are sealed bass cabs that can be quite a few dB down by 50 or 60 Hz and they still sound great, and are not at all similar to subwoofer designs IME...actually they are kind of the opposite.

I know some of the bass cab design engineers on TB can add more detail and will likely correct a few things I said, but I believe the above thoughts are at least in the ballpark.

Getting in on this:
As far as I know, the human hearing has problems locating low frequencies. That's why the whole Idea of 2.1 systems came up. Toss the subwoofer ... wherever and only position the satellites properly. Since low frequencies are hard to locate and high frequencies are easy to locate, my guess is that someone did all the testing and came up with the exact frequency range that works for subs ...

 

[SunByrne]

Getting in on this:
As far as I know, the human hearing has problems locating low frequencies. That's why the whole Idea of 2.1 systems came up. Toss the subwoofer ... wherever and only position the satellites properly. Since low frequencies are hard to locate and high frequencies are easy to locate, my guess is that someone did all the testing and came up with the exact frequency range that works for subs ...

To a first order of approximation (meaning: this is not universally true, there are many edge cases), human hearing can't localize very well below about 100Hz. This is one reason why most HT systems cross over at 80Hz.

 

[Luther]

I listen with my eyes. 115 for the win.

 

[agedhorse]

To a first order of approximation (meaning: this is not universally true, there are many edge cases), human hearing can't localize very well below about 100Hz. This is one reason why most HT systems cross over at 80Hz.

The key to this is that the stage and surround channels (cinema term, I worked in the cinema industry a career or two ago) help focus the location of the sound so that the subwoofer location doesn't really matter much. Combine with the more omni-directional aspect of the LF source and it can be quite convincing.

 

[MechaSparky]

I listen with my eyes. 115 for the win.

I listen with my back. 115 for me too.

 

[toughluck]

To a first order of approximation (meaning: this is not universally true, there are many edge cases), human hearing can't localize very well below about 100Hz. This is one reason why most HT systems cross over at 80Hz.

This is one of the most persistent myths.

You can absolutely localize the origin of low frequency sound, with the caveat that the location gets somewhat less precise the lower the frequency gets, but you can localize even 20 Hz sound just fine if you focus on it.

What cannot be localized is sound that is propagated not as a sound wave, but as a pressure mode. This occurs when the wavelength of a frequency below a certain threshold is too long to fit within an enclosed space. The entire volume of air in the room is alternately pressurized and rarefied and as such, the entire volume of air acts as the source of sound, which is why it is not possible to localize the exact source.

That threshold frequency below which pressure mode dominates as the sound propagation method is the Schroeder frequency. The smaller the room, the higher this frequency is, but in large rooms, concert halls, let alone during outdoor concerts, low frequency sound can be clearly located.

 

[agedhorse]

This is one of the most persistent myths.

You can absolutely localize the origin of low frequency sound, with the caveat that the location gets somewhat less precise the lower the frequency gets, but you can localize even 20 Hz sound just fine if you focus on it.

What cannot be localized is sound that is propagated not as a sound wave, but as a pressure mode. This occurs when the wavelength of a frequency below a certain threshold is too long to fit within an enclosed space. The entire volume of air in the room is alternately pressurized and rarefied and as such, the entire volume of air acts as the source of sound, which is why it is not possible to localize the exact source.

That threshold frequency below which pressure mode dominates as the sound propagation method is the Schroeder frequency. The smaller the room, the higher this frequency is, but in large rooms, concert halls, let alone during outdoor concerts, low frequency sound can be clearly located.

This too is mostly mythology and falsehoods, especially once you add boundary reflections into the equation. It's also backed up by empirical evidence.

 

[toughluck]

Sure, but boundary reflections in a room will confuse direction cues for higher frequencies as well.

In free space this is not a consideration and you can locate low frequency sound sources without much of a problem, and I have my own experience of this, but the plural of anecdote is not data, so I'm unsure whether this counts (it does insofar as a single exception means that the rule has to be revisited).

Within dense buildup, dealing with multiple boundaries, where direct sound is too attenuated to make out and you mostly hear reflections, you will have just as much of a problem locating higher frequency sound — in a busy street, you will often be unable to locate emergency vehicles from their sirens alone.

Interestingly, this affects low frequency sounds less because longer wavelengths mean that directivity cues will only be smeared, not repositioned to a completely different source.

 

[agedhorse]

Again, we will have to agree to disagree.

 

[SunByrne]

You can absolutely localize the origin of low frequency sound, with the caveat that the location gets somewhat less precise the lower the frequency gets, but you can localize even 20 Hz sound just fine if you focus on it.

[citation needed]

Actually, no, I don't need a citation, because I know there isn't one. How do I know this?

<professor mode on>

Humans localize sound by differences between what is perceived at the two ears, essentially by taking the between-ear differential in terms of both intensity and time: Interaural Time Difference (ITD) and Interaural Level Differnces (ILD). This is why you can't locate a sound that is directly behind you vs. directly in front of you without moving your head a little; there's no difference between what each ear gets. ILD doesn't work at all below about 1kHZ so we're talking about ITD for localizing low-frequency sounds.

ITD is based on the phase mismatch between the two ears. The phase difference is a function of wavelength, and yes, the usefulness of this cue decreases as frequency decreases. The average difference between the two ears is on the order of 8 inches (depending of course how big your head is). The wavelength of a 1 kHz sound is about 1.13 feet, meaning there's a large phase difference between the two ears, making the differential a useful predictor. The wavelength of a 100Hz sound, however, is of course a little over 11 feet. Your 8-inch spaced ears won't see much of a phase difference here, so there's almost nothing upon which to base localization.

There's a great graph of this in a 2014 PLoS One paper by Smith and Price, though it only goes down to 250 Hz. Azimuth angle on the x-axis is the angular displacement from "straight ahead" so 0 is straight ahead and 90 is directly off to one side. Y-axis is the angular phase difference between what arrives at each ear. Lines are model predictions assuming a spherical head (not 100% accurate for most people but quite close) and data points are empirical. Fit is obviously very good:

It doesn't take much mathematical intuition to see that the curve for 100Hz will be awfully flat, meaning that at that frequency, there's almost no spatially useful information in the signal. At 20Hz it is almost perfectly flat and there's nothing there at all.

This is just the physics of the situation; you can't localize low frequencies because of geometric limits on the validity of the cue that your auditory system uses to solve the problem.

<professor mode off>

So how is this a myth, exactly?

I mean, most people can barely even hear a 20Hz pure tone at all, much less localize it. If it's high-intensity enough you might be able to "feel" it more than hear it (which I believe is why HT bass-heads care about sub-20Hz extension), and maybe with the help of your vestibular system you might have some idea about direction, but that's not likely to be particularly accurate.

If someone says they can localize something below 100Hz, first, I'm dubious. Second, it's almost certainly being done by hearing overtones and/or some kind of distortion that actually occurs at higher frequencies such that they can be localized. This is why I said "to a first order of approximation" in my original post, because the presence of pure tones without overtones/distortion is rare, so in practice people can sometimes do it—but they're doing it because of higher-frequency content in the signal, not because this is actually possible for pure tones.

 

[agedhorse]

[citation needed]

Actually, no, I don't need a citation, because I know there isn't one. How do I know this?

<professor mode on>

Human localize sound by differences between what is perceived at the two ears, essentially by taking the between-ear differential in terms of both intensity and time: Interaural Time Difference (ITD) and Interaural Level Differnces (ILD). This is why you can't locate a sound that is directly behind you vs. directly in front of you without moving your head a little; there's no difference between what each ear gets. ILD doesn't work at all below about 1kHZ so we're talking about ITD for localizing low-frequency sounds.

ITD is based on the phase mismatch between the two ears. The phase difference is a function of wavelength, and yes, the usefulness of this cue decreases as frequency decreases. The average difference between the two ears is on the order of 8 inches (depending of course how big your head is). The wavelength of a 1 kHz sound is about 1.13 feet, meaning there's a large phase difference between the two ears, making the differential a useful predictor. The wavelength of a 100Hz sound, however, is of course a little over 11 feet. Your 8-inch spaced ears won't see much of a phase difference here, so there's almost nothing upon which to base localization.

There's a great graph of this in a 2014 PLoS One paper by Smith and Price, though it only goes down to 250 Hz. Azimuth angle on the x-axis is the angular displacement from "straight ahead" so 0 is straight ahead and 90 is directly off to one side. Y-axis is the angular phase difference between what arrives at each ear. Lines are model predictions assuming a spherical head (not 100% accurate for most people but quite close) and data points are empirical. Fit is obviously very good:

View attachment 7318912

It doesn't take much mathematical intuition to see that the curve for 100Hz will be awfully flat, meaning that at that frequency, there's almost no spatially useful information in the signal. At 20Hz it is perfectly flat and there's nothing there at all.

This is just the physics of the situation; you can't localize low frequencies because of geometric limits on the validity of the cue that your auditory system uses to solve the problem.

<professor mode off>

So how is this a myth, exactly?

I mean, most people can barely even hear a 20Hz pure tone at all, much less localize it. If it's high-intensity enough you might be able to "feel" it more than hear it (which I believe is why HT bass-heads care about sub-20Hz extension), and maybe with the help of your vestibular system you might have some idea about direction, but that's not likely to be particularly accurate.

If someone says they can localize something below 100Hz, first, I'm dubious. Second, it's almost certainly being done by hearing overtones and/or some kind of distortion that actually occurs at higher frequencies such that they can be localized. This is why I said "to a first order of approximation" in my original post, because the presence of pure tones without overtones/distortion is rare, so in practice people can sometimes do it—but they're doing it because of higher-frequency content in the signal, not because this is actually possible for pure tones.

Very well explained, thank you!!!

 

[SunByrne]

Very well explained, thank you!!!

You are most welcome. There are several TBers whose explanations have consistently benefitted me. You are most definitely on that list, so I'm pleased to hear you got something out of it.

 

[toughluck]

This is why you can locate low frequency sound only approximately and you may have to listen in for a while, turning your head around, to find the source.
The average distance between ears is ca. 200 mm.
Wavelength of a 100 Hz sound wave is ~3430 mm; maximum phase difference between ears is ~21°
Wavelength of a 50 Hz sound wave is ~6860 mm; maximum phase difference between ears is ~10.5°
Wavelength of a 20 Hz sound wave is ~17150 mm; maximum phase difference between ears is ~4°

Wavelength of a 1 kHz sound wave is ~343 mm; maximum phase difference between ears is ~210°
At azimuth angle of 1°, the phase difference is ~3.7°. If you can localize the difference in position of a 1 kHz sound source that shifted by about 8.7 cm at a distance of 5 meters, you can localize the position of a 20 Hz sound, let alone any higher frequency. It will take longer, it will take listening to it for a while for the brain to process the data, but you will eventually tell where it's located.

This assumes that you can turn, nod and tilt your head to capture as many cues as possible.

 

[agedhorse]

Sorry, the absurdity here is simply stunning.

@Sun Byrne brought up an important point that can impact the appearance of low frequency localization… the impact of harmonics on the process.

Since there can be as much as 50% total harmonics in a bass guitar signal, the harmonics being higher frequencies essentially mask the impact of the fundamental.

 

[blr2]

Since there can be as much as 50% total harmonics in a bass guitar signal, the harmonics being higher frequencies essentially mask the impact of the fundamental.

Even more actually, especially in the lower octave and if roundwound strings are used.

 

[agedhorse]

Even more actually, especially in the lower octave and if roundwound strings are used.

Agreed, I was attempting to avoid the argument of exaggerating…

 

[SunByrne]

This is why you can locate low frequency sound only approximately and you may have to listen in for a while, turning your head around, to find the source.
The average distance between ears is ca. 200 mm.
Wavelength of a 100 Hz sound wave is ~3430 mm; maximum phase difference between ears is ~21°
Wavelength of a 50 Hz sound wave is ~6860 mm; maximum phase difference between ears is ~10.5°
Wavelength of a 20 Hz sound wave is ~17150 mm; maximum phase difference between ears is ~4°

Wavelength of a 1 kHz sound wave is ~343 mm; maximum phase difference between ears is ~210°
At azimuth angle of 1°, the phase difference is ~3.7°. If you can localize the difference in position of a 1 kHz sound source that shifted by about 8.7 cm at a distance of 5 meters, you can localize the position of a 20 Hz sound, let alone any higher frequency.

No, actually your math—which is at best approximate but basically does what I need—shows the opposite.

Essentially, the problem your brain is trying to solve is mapping a phase differential to a an azimuth. That is f(phase angle, freq) -> azimuth. This is essentially drawing a line from phase angle to the mapping function in the figure I provided above (with the particular mapping function identified by frequency), and then drawing a line from that function down to the azimuth.

But your perceptual system has limited resolution of the "phase angle" input. It's not drawing a line, but a smear. That input smear is, by your own math, about 4 degrees wide. Since the entire input domain to the relevant function at 20Hz is, again by your own math, also only 4 degrees wide, it means the output is a smear that's 90 degrees wide. You can't localize anything if you can't tell the difference between 0 and 90.

It is, in fact, even more complicated than this, but this alone should be enough.

It will take longer, it will take listening to it for a while for the brain to process the data, but you will eventually tell where it's located.

I'm totally willing to be proved wrong here, but given what the math says, I'm going to need data. If that's actually true, then it shouldn't be hard for you to show me empirical data that shows accurate localization for a 20Hz pure tone.

This assumes that you can turn, nod and tilt your head to capture as many cues as possible.

No, this is a mistake. THERE IS ONLY ONE CUE. The only thing your brain has to work with is the phase differential. Since the math shows that cue is non-predictive at 20Hz, you can sample it as many times as you want—it still won't actually tell you anything.

 

[bucephylus]

Glad we settled that.

Anyhow, then the Sound Engineer asks you to turn down, because your stage volume is interfering with the FoH mix.

Which is faster at 25 mph; a Ferrari or a Ford?

Oh, right; we are discussing making music. Hmmm.