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[Mac Kerr]

When you listen-you hear nothing like a sinewave.  It sounds more like a "click or a pop".  It sounds like the freq is much lower than the actual tone.

Is that also the case if the turn on/off always happens at a zero crossing? If it is not a zero crossing you are actually generating a high frequency component related to the rise time from zero to full signal.

Mac

[George Friedman-Jimenez]

...In the old days, doing the math was a very long and tedious process. I suppose, using a waveform that was repeating was trying to minimize some of the work, but, today, it isn't a concern, the analyzers do the work.

It will give a representation, it is up to us to determine what is relevent information.

One can minimize some non critical information by adjusting the lines of resolution.

Hammer

Without the math in the "old days", specifically Fourier's derivation of the Fourier series and the assumption of a repeating waveform, there would be no "analyzers" and there would be no digital sound equipment. We sometimes take for granted the science foundations upon which our current technology is built.

The question of what the output of a computerized FFT analyzer means when the input is a single cycle or part of a cycle is not so much a question of what is relevant, but rather whether the concepts with which we understand and apply FFT results to our work are correct or even defined at all. Sure, our black box will give a representation, but what do we need to know and what do we need to assume to interpret that?

The assumption of a repeating waveform is not to make the math calculations easier, it is fundamentally necessary to even define concepts like "frequency" and "wavelength". Without clear definitions of frequency and wavelength, how do we understand what our analyzer is telling us after it "does the work"?

Luis, I am still curious why you asked the original question that led to this very interesting discussion!

[Nick Hickman]

Gosh!  A few thoughts to add to the melee...

First, in a theoretical sense, a pure tone containing energy at a single frequency can only exist if it lasts for ever.  If it starts, stops, or changes in any way, it's an indication that energy at other frequencies is present.  As to how long it would take a listener to identify it as a pure tone, I have no idea.  I guess it's in the ear of the behearer.

As far as absolute delays involved in a loudspeaker propagating sound are concerned, in a practical sense I don't think there are any, although there are a lot of phase shifts taking place.  The instantaneous current in the voice coil of a loudspeaker above resonance, for example, accelerates the cone (thanks to the laws invented by Messrs Lorenz and Newton), but the cone velocity is 90 degrees out of phase with its acceleration, and the displacement is another 90 degrees, so 180 degrees to the applied current.  In the extreme nearfield, I believe air particle displacement is held to be in phase with the cone displacement.  In the far field of a travelling wave, air particle velocity ends up in phase with air pressure.

Plus, there are more phase shifts in the detector (whether that's an ear or a microphone) with, for example, an extra 90 degree shift in a dynamic microphone compared to a capacitor microphone (because its output is dependent on diaphragm velocity rather than displacement).

I really don't have much clue about the detail of what happens in the extreme nearfield of a radiator.  Even in the nearfield more generally, behaviour can be unintuitive because of the radiation contributions from the various bits of the surface.  It's easy, for example, to find situations where sound level increases with increasing distance from the source.  The nice tidy models of curved soundwaves leaving a radiator are really only a fiction in the nearfield.  (Not to mention that, in some cases, we deliberately make the radiator large so that its nearfield extends into the audience for a significant range of frequencies!)  Certainly, any delays at an atomic level pale into insignificance compared to the glacially slow rate at which a pressure wave propagates through the air.

Second, you can't analyse sound in an instantaneous manner except to say what the absolute pressure is at that moment: there's no concept of frequency until you introduce a time interval.  To use a Fourier technique you have to select a time interval and to assume that the waveform within that interval repeats identically for ever.  That can easily confuse matters, though can be helped by applying a windowing function to the data in the interval before transforming it to the frequency domain.

What we hear indeed doesn't always match up well with the actual frequency content of the energy responsible.  For example, a theoretical Dirac pulse contains energy at all frequencies.  If you process it (for example, with a filter), that's how it behaves, but we only sense it as a click.  When tones start or stop, there are lots of frequencies involved, but we don't really sense it like that.

Errors and emissions accepted (!)

Nick

[John Roberts {JR}]

Edit:
JR, in my first post I specifically asked us not to discuss trees falling in forests.
Actually, the OP's question is really more about psychoacoustics than philosophy or math. Ivan's and Tom's empirical observations answer the question pretty well, and JR's explanation about the length of the sound and the predominance of either the steady tone or the beginning/end is very clear as well for most cases. But what if we specifically create a single cycle sine wave that starts and ends on the zero crossing? It would have no step function to mess up the single frequency. Anyone know what that would sound like, or how (or if) it would Fourier transform?

Actually stopping and starting at zero crossings is a subtly different sound issue, at least in the real world. Yes, starting a tone burst offset from the zero crossing, adds additional HF energy from the rapid slewing involved to get from 0V silence to some voltage instantaneously.  For a proper sine wave the slew rate at zero crossings is maximum (COS 0'=1) but finite. The step function to jump to any finite voltage instantaneously is theoretically infinite and has frequency content all the way up to your top practical measurement capability.

I actually did a lot of work with this stuff in connection to one of my early jobs in electronics. I worked for a company that shifted the pitch of speech samples electronically, but since the stretched or shrunk audio output would no longer fit perfectly into the original time window, subsequent samples needed to be spliced together. Stopping and starting the samples at zero crossings, eliminated the huge HF energy content from that mechanism, while there was still the energy content associated with the length of the samples (step function).

Later while designing sundry dynamics processor I made my own bench tone burst generator to use as a worst case stimulus for testing. I added zero crossing logic to my simple burst generator to eliminate the excessive clicking.

I spent way too many hours on the bench listening to tone bursts, so while i may not be proficient in doing a FFT calculation on the back of an envelope, I learned the sounds that sundry tone bursts make the old fashioned way. 

Your perfect one cycle 20Hz burst, even starting and stopping perfectly at zero crossings will sound more like the on/off step, that a 20 Hz pitch. Increasing the number of full cycles displayed will give it more 20 Hz energy and dilute the fixed amount of step/thump on/off energy.

Note: In addition to starting and stopping at zero crossings, you generally want to display whole cycles. Starting and stopping on an odd half cycle adds a DC component, that may or may not interfere with system behavior and measurements, while IIRC DC content did not appreciably change the "sound" of longer bursts.

JR   

[Langston Holland]

Something visual on the duration of a tone vs. frequency:

http://soundscapesweb.com/files/PSW/Other/500HzToneBursts.mov

Now listen to those (6) tone bursts and find out the duration required for you to hear the transition from a click to a tone:

http://soundscapesweb.com/files/PSW/Other/500HzToneBursts.wav

Why 500Hz? I like 500Hz. :)

[Tim McCulloch]

Why 500Hz? I like 500Hz.

There is help and treatment for your problem, Langston.  Trust the group, we care. 

[Mac Kerr]

Why 500Hz? I like 500Hz.

For me it starts to sound like a tone on the 3rd sample, and is clearly a tone on the 4th. That is 8ms and 16ms, or 4 cycles and 8 cycles. Have you tried this with a lower frequency to see if it is the 8ms/16ms length, or the 4cyc/8cyc number of cycles that is the determining factor?

Mac

[George Friedman-Jimenez]

...Yes, starting a tone burst offset from the zero crossing, adds additional HF energy from the rapid slewing involved to get from 0V silence to some voltage instantaneously.  For a proper sine wave the slew rate at zero crossings is maximum (COS 0'=1) but finite. The step function to jump to any finite voltage instantaneously is theoretically infinite and has frequency content all the way up to your top practical measurement capability.

...Your perfect one cycle 20Hz burst, even starting and stopping perfectly at zero crossings will sound more like the on/off step, that a 20 Hz pitch. Increasing the number of full cycles displayed will give it more 20 Hz energy and dilute the fixed amount of step/thump on/off energy.

JR

JR, your first statement is a great clear explanation of why we would expect to get a click if we start the waveform at some point that is not a zero (voltage) crossing. But now that we are convinced, it is hard to reconcile that logic, essentially based on Fourier analysis, with your observation in your second quote. Why would a single smooth cycle with no step function induced harmonics sound like an impulse rather than a tone? And Langston's brilliant custom-produced audio visual demonstration further makes the point that 1 cycle really does sound like a click with (to my old ears) no perceptible tone content, and 2 cycles sound like 2 barely distinguishable clicks, while 4 cycles start to have a perceptible tone. That is really really interesting, thank you Langston. Let's ignore the hugely wide distribution of frequencies of the single cycle for now, but the audio demonstration is extremely interesting.

Here is one possible explanation, what do you all think? (No Fourier math, no digital gobbledygook)

Thanks to the work of Hermann Helmhoz, a genius physician/physicist in the 19th century, we know that the ear perceives sound via "hair cells" in the inner ear that vibrate in resonance when a sound of a certain pitch hits the ear. In an (over)simplified model of the inner ear, different frequencies of sound stimulate resonance of different length hair cells. In an (over)simplified model of resonance, a system vibrates more and more strongly as energy is added each cycle in sync with the velocity and position of the system. So a 500 Hz tone will cause a hair cell with resonant frequency 500 Hz to vibrate sympathetically, but won't stimulate the 450 Hz hair cell at the right point in each cycle to add energy each time. The 450 Hz hair (and all the other hairs) will vibrate, but won't increase its energy each cycle as the resonating hair cell would. It likely takes several cycles of absorption of energy and increase in amplitude of vibration of the resonant hair cell until it reaches an amplitude that is relatively much stronger than all the other (nonresonating) hair cells and we perceive a pure tone. This is oversimplified, and based on long long term memory, so if any Helmholz groupies out there see anything grossly misrepresented, please correct me.

Now imagine a single cycle of sound, say one cycle of a 500 Hz sine wave starting and ending at the zero crossing for simplicity. This sine wave of air pressure is transmitted to the ear drum, and via the 3 little bones, to the fluid of the inner ear. The fluid transmits the pressure impulse to all the hair cells of all lengths. All vibrate at 500 Hz. There is no second cycle wave to make the 500 Hz hair cell pick up more energy from the repeated vibration. The ear and brain perceive this as a single click, that would not be all that different if the original stimulus was a single cycle at 1000 Hz, 5000 Hz, or 100 Hz. There will be perception of sound at each frequency, but the brain won't be able to identify a specific frequency. It would sound like a rim shot, or a firecracker, or some other rapid atonic impulse.

Now if we play 2 cycles of the 500 Hz sine wave, the 500 Hz hair cell will gain strength of vibration the second time and the others won't (ignore the 1000, 1500 etc hair cells at harmonic frequencies for now). So now we have the possibility of perception of a specific tone. My ears and brain don't hear a tone in Langston's 4 ms 2 cycle sample, but maybe someone else's would. I hear 2 very close clicks. With 4 cycles (8 ms), now we can really expect some differential resonating to occur, with the 500 Hz hair cell gaining energy each cycle and the other frequency hair cells not. I and probably most of you can perceive this as a tone, although don't ask me to sing it yet. For 16 and 32 cycles, tone perception gets better and better, most of us could probably sing the right tone for both of these (and maybe the 4 ms sound).

What do you think?

[George Friedman-Jimenez]

For me it starts to sound like a tone on the 3rd sample, and is clearly a tone on the 4th. That is 8ms and 16ms, or 4 cycles and 8 cycles. Have you tried this with a lower frequency to see if it is the 8ms/16ms length, or the 4cyc/8cyc number of cycles that is the determining factor?
Mac

I agree, both with hearing a tone in the 4 cycle sample and that you are asking the right question. Is our inability to hear a tone for the 2 cycle sound due to the shortness of the sound, or to the few cycles of the sound? If the former, we would predict no better results with a 4 ms long 4 cycle 1000 Hz tone or a 4 ms long 8 cycle 2000 Hz tone. If the latter, it would become easier and easier to identify a specific tone as the frequencies became higher.

[Per Sovik]

When tones start or stop, there are lots of frequencies involved

No, there are not! Same with a square waveform, a pulse, or whatever. How something might be mathematically represented doesn't mean that it is. You can cut a tree into lots of cubes if you want to, it is not made up of cubes. While our senses certainly detect sound as a combination of sine waves that causes our detecting hair cells to resonate, and a pulse causes all the hair cells to flex and respond, doesn't actually mean that the nature of sound is defined by how we detect it or describe it.

George, totally agree with the Heimholz based expanation, good!

As for the length of the tone, I believe my wife would be able to hear the pitch of the third one, while myself I cant make it out before no.4. Having said that, I can definitely hear that no. 3 is predominantly monotimbral, and upon hearing it repeatedly I think I would at some time make out the pitch. A bit like getting a picture flashed before your eyes for a short fraction of a second, repeating the excercise will eventually enable you to make out what's in the picture.

[ProSoundWeb Community]