Why does a series of pulses generate a pitch?

[Daniel Petka]

TL;DR Summary

A single (very short) pulse has all the frequencies, so it should excite all the little hair sensors in the cochlea and I don't understand why sending this pulse repeatedly creates a pitch.

Here's the thing: the ear detects a pitch by splitting a sound wave into it's frequency components in the cochlea, which is in a way a spatial Fourier transform (ish...) But I never liked this analogy because it doesn't explain why I hear a pitch when a series of pulses entern my ear.

A single (very short) pulse has all the frequencies, so it should excite all the little hair sensors in the cochlea and I don't understand why sending this pulse repeatedly creates a pitch. A soundwave can be decomposed into sine waves, but if we go in that direction, it's kinda tricky to talk about the duration of the transform interval.. because I can still hear the individual pulses. T

here is this thing called wavelet transform, which decompises the signal into wave packets instead. But I found one example that I don't think can be explained by wavelets: chirped pulse trains. Imaging sending a pulse, then another pulse 0.1s later, another one 0.25s later etc. The immediate "frequency" (the time between the pulses) changes constantly and yet I can still hear a changing pitch. The effect can be replicated by playing two sawtooth waves, one at 10Hz and the other one at 10.1 Hz.

 

[berkeman]

I'm not sure I understand what your question is, but with respect to multiple pulses and the frequency components, do you think that lining up the pulses in the time domain to try to align the phases of the harmonic sinusoidal components might be important?

 

[BillTre]

The sensory cells within the inner ear structure respond to different frequencies.

Its my understanding that:
Some cells at low frequencies can respond with a spike (action potential) per wave.
Others can't respond as fast as the frequency they are set-up to sense. They use a frequency coding.
In both cases the sensory cells respond in a signal path that defines the frequency that would be perceived.

Here's a Wikipedia article: https://en.wikipedia.org/wiki/Neural_encoding_of_sound

 

[DaveE]

The Fourier Transform of a pulse train is dominated at low frequencies by the fundamental frequency (pulse repetition frequency) and its harmonics. That is why you hear "tones".

https://sceweb.sce.uhcl.edu/harman/CENG3315_DSP_Spring2020/00_3315_2021/3315_web_2021/Fourier Series References_2_28_2021.pdf

BTW: It's not a spatial FT, it's temporal.

 

[Daniel Petka]

The Fourier Transform of a pulse train is dominated at low frequencies by the fundamental frequency (pulse repetition frequency) and its harmonics. That is why you hear "tones".

https://sceweb.sce.uhcl.edu/harman/CENG3315_DSP_Spring2020/00_3315_2021/3315_web_2021/Fourier Series References_2_28_2021.pdf

BTW: It's not a spatial FT, it's temporal.

Well first of all, the ear doesn't perform a Fourier transform as you would need to listen from minus infinity to infinity. There is short time FT and wavelet but then.. what is the window length? And yes, it's essentially temporal, but your ear doesn't view it that way. The cochlea only knows which hair cells get excited. The ones closer to the edge of the ear correspond to high frequencies and the ones further down to lower ones. Hope that clears thing up a bit

 

[Daniel Petka]

The sensory cells within the inner ear structure respond to different frequencies.

Its my understanding that:
Some cells at low frequencies can respond with a spike (action potential) per wave.
Others can't respond as fast as the frequency they are set-up to sense. They use a frequency coding.
In both cases the sensory cells respond in a signal path that defines the frequency that would be perceived.

Here's a Wikipedia article: https://en.wikipedia.org/wiki/Neural_encoding_of_sound

Thank you, I'll look into it.

 

[Daniel Petka]

I'm not sure I understand what your question is, but with respect to multiple pulses and the frequency components, do you think that lining up the pulses in the time domain to try to align the phases of the harmonic sinusoidal components might be important?

Phase doesn't matter in most cases, but sometimes it matters a lot. You'll see what I mean if you open any online tone generator and play a 10Hz sawtooth wave and another 10.1Hz sawtooth wave in another tab at the same time. If pitch was related to frequency, you would only hear the same pitch all the time. But as you hear, the pitch changes constantly. And the question is why. I don't think you can ignore the time domain in this case. The higher pitches correspond to pulses being closer together (which is kind of intuitive but I haven't heard anyone mention this before)

 

[NTL2009]

Phase doesn't matter in most cases, but sometimes it matters a lot. You'll see what I mean if you open any online tone generator and play a 10Hz sawtooth wave and another 10.1Hz sawtooth wave in another tab at the same time. If pitch was related to frequency, you would only hear the same pitch all the time. But as you hear, the pitch changes constantly. And the question is why. I don't think you can ignore the time domain in this case. The higher pitches correspond to pulses being closer together (which is kind of intuitive but I haven't heard anyone mention this before)

Old thread I know, but it came up under the recent saxophone/growl thread and caught my attention.

I think you are starting with an assumption that may or may not be accurate, and that is warping your thinking. When I mixed a 10Hz and 10.1Hz sawtooth in Audacity, I hear a series of tone bursts (dat-dat-dat-dat-dat) that include a perceived pitch that rises and falls. The term "perceived pitch" is a bit tricky here, as there are many tones/noise in that resultant sound, nothing is static. I'm going to guess that the pitch I perceive is rising/falling somewhere between maybe 100~200Hz ?

You stated " If pitch was related to frequency, you would only hear the same pitch all the time" - that is questionable. Since you start with such low frequencies, any perceived pitch would be the harmonics, and if my estimate of 100~200, we are talking 10x ~ 20x frequencies. What I think is happening is that the different harmonics cancel and add at the beat frequency (1/10th Hz), and I suspect we are hearing these different harmonics being accentuated and attenuated as a rise/fall in pitch. Somewhat analogous to the "Shepard's Tone" audio 'illusion'.

If our ears latched on to the 20th harmonic of the 10 Hz tone @ 200 Hz, then that faded out and the 20th harmonic of the 10.1 Hz tone @ 202 Hz faded in, we would hear that as a slight rise in pitch. And since the tones are 'gated' (not continuous) we don't pick up on the exact point where the frequency changed, so our mind hears it as a continuous tone sliding in pitch.

If we continue this to the 21st harmonic, we would get 210, then 212.1 Hz. You can kind of see this in Audacity on their Spectrogram, but I think you'd need to do some more work to validate my thinking here, but I'm pretty sure it would hold up.

So the frequencies are not changing, we just 'tune into' different harmonics, and perceive that as a rising/falling pitch, but they are discrete frequencies.

 

[Daniel Petka]

Hey, the point is that there is no frequency. I tried this with just 2 isolated pulses and there is still an audible pitch, which quite predictably depends on the delay between the pulses.

 

[NTL2009]

Hey, the point is that there is no frequency. I tried this with just 2 isolated pulses and there is still an audible pitch, which quite predictably depends on the delay between the pulses.

I don't follow you. My post was about the 10Hz and 10.1Hz sawtooth waves.
What do you mean by "no frequency"? What has "no frequency" (a single pulse?)? What do you mean by "2 isolated pulses"? Non-repeating, but a delay between, or repeating? Please submit a graphical representation. Or an audio file, or Audacity settings to replicate.

 

[NTL2009]

TL;DR Summary: A single (very short) pulse has all the frequencies, so it should excite all the little hair sensors in the cochlea and I don't understand why sending this pulse repeatedly creates a pitch.

OK, looking at your OP - you are mixing terms in the same sentence, and it is confusing you.

Yes, a single pulse theoretically has "all the frequencies". How the ear responds to this is, I'm sure, complicated - but skip that for a second.

"I don't understand why sending this pulse repeatedly creates a pitch." - That's different. if you send the pulse repeatedly, you now have frequencies related to that repetition rate.

Regardless, I explained the 10 and 10.1 Hz sawtooth - what do you think of that?

 

[Daniel Petka]

I don't follow you. My post was about the 10Hz and 10.1Hz sawtooth waves.
What do you mean by "no frequency"? What has "no frequency" (a single pulse?)? What do you mean by "2 isolated pulses"? Non-repeating, but a delay between, or repeating? Please submit a graphical representation. Or an audio file, or Audacity settings to replicate.

Yes, exactly - non repeating. That's what I mean that there is no frequency, so I think the 10/10.1Hz is an unnecessary complication. Sending 2 pulses after each other works just fine. Explaining this with frequency only creates confusion imo.
I'm at vacation right now with just my phone, but you can replicate it easily by turning a numpy array that contains two impulses [1,0,0,0,...0,1] into a wav file using for example scipy.io.wavfile

 

[NTL2009]

Yes, exactly - non repeating. That's what I mean that there is no frequency, so I think the 10/10.1Hz is an unnecessary complication. Sending 2 pulses after each other works just fine. Explaining this with frequency only creates confusion imo.
I'm at vacation right now with just my phone, but you can replicate it easily by turning a numpy array that contains two impulses [1,0,0,0,...0,1] into a wav file using for example scipy.io.wavfile

Take your time, we can catch up as time permits (the beauty of asynchronous communication, after all, this thread is many months old already!).

I have not used numpy, so don't feel like digging into that at the moment. But I did use Audacity to create a few pulses - start with 0.5 Sec silence, a pulse, 0.1 silence, a pulse, 0.25 silence, a pulse, 0.5 silence. Export to wav. I didn't see a basic impulse generator in Audacity, so I just created a short burst of 1KHz square wave, and trimmed out just the positive half-cycle to use as the pulse. So each was 0.5 msec in duration.

As I expected, I hear three distinct 'clicks/thumps'. Perhaps someone with a more trained musical ear than mine would attribute a pitch to these 'clicks/thumps', but I just perceive it as a 'clicks/thumps' over a wide frequency band.

I'll experiment later, but I think if I shorten/lengthen the pulse, the thump will sound generally higher/lower in pitch (even if I can't pick out a distinct pitch). I guess I'll also try one that does not return to zero, but I doubt that will make a difference in perceived sound.

 

[Daniel Petka]

Take your time, we can catch up as time permits (the beauty of asynchronous communication, after all, this thread is many months old already!).

I have not used numpy, so don't feel like digging into that at the moment. But I did use Audacity to create a few pulses - start with 0.5 Sec silence, a pulse, 0.1 silence, a pulse, 0.25 silence, a pulse, 0.5 silence. Export to wav. I didn't see a basic impulse generator in Audacity, so I just created a short burst of 1KHz square wave, and trimmed out just the positive half-cycle to use as the pulse. So each was 0.5 msec in duration.

As I expected, I hear three distinct 'clicks/thumps'. Perhaps someone with a more trained musical ear than mine would attribute a pitch to these 'clicks/thumps', but I just perceive it as a 'clicks/thumps' over a wide frequency band.

I'll experiment later, but I think if I shorten/lengthen the pulse, the thump will sound generally higher/lower in pitch (even if I can't pick out a distinct pitch). I guess I'll also try one that does not return to zero, but I doubt that will make a difference in perceived sound.

I think the brute force way to do this in Audacity would be to delete samples until you have just two separate samples. 0.5ms is way too long.
Btw I wouldn't expect that you heard anything for 0.1 seconds delay between the pulses because that would correspond to 10Hz infra sound. 1-10ms spacing between the samples is more reasonable

 

[NTL2009]

When you have time, can you spell out exactly what you want, and what you expect someone to hear?

I took a shot at this in Audacity and will share the files here. File m2p is the wav output of 0.5 sec silent, a pulse of 3 samples (44.1K sample rate), then a 2nd pulse ~ 1 msec later.

m3 is similar, with 3 sample pulses at approximately 0.5S, 0.501,0.503, 0.506, 0.510, 0.515, and 0.521.

When I play the wav files, I just get a short 'tick' (m2p), or short 'zipper' sort of noise(m3).

Pretty much what I expected. What is unexpected?

Looks like the forum SW embeds the links, but if you open them it takes to where you can download the file and open in Audacity to see for yourself. I'd rather just give the links, I'll try a bit later.

m3.wav
m2p.wav

OK, that's better?

 

[Daniel Petka]

The wav file is here, you should hear 6 tones with descending pitches.

 

[NTL2009]

I sort of hear a descending pitch, but these are rather 'pitch-less' noise bursts, so it's not clear to my not so well trained ear. I hear 5 equally spaced 'blips' and the 6th comes in half the time (like 4 pizzicato notes with quarter note rests between, followed by 2 pizzicato notes with an eighth note rest between).

But importing into Audacity, I see that each of the six "pizzicato notes" consist of 2 full scale samples. The 1st 'note' has those two ~1 msec apart (I see 50 samples and it imported at a 50,000 sample rate), the 2nd 'note' has those 60 samples apart, the 3rd is 70 samples apart, the 4th is 80 apart, the 5th 90, and the 6th 100 samples apart. So it appears you carefully constructed this, and I'm not surprised that the wider spaced samples would have some lower frequency components to them. Are you?

Here's the zipped Audacity file with data folder - download and click on the .aup to load into Audacity - the '50-60-70-80-90-100' labels are on the track below for clarity, scroll down if you don't see them.

Link to zip file

 

[Daniel Petka]

I sort of hear a descending pitch, but these are rather 'pitch-less' noise bursts, so it's not clear to my not so well trained ear. I hear 5 equally spaced 'blips' and the 6th comes in half the time (like 4 pizzicato notes with quarter note rests between, followed by 2 pizzicato notes with an eighth note rest between).

But importing into Audacity, I see that each of the six "pizzicato notes" consist of 2 full scale samples. The 1st 'note' has those two ~1 msec apart (I see 50 samples and it imported at a 50,000 sample rate), the 2nd 'note' has those 60 samples apart, the 3rd is 70 samples apart, the 4th is 80 apart, the 5th 90, and the 6th 100 samples apart. So it appears you carefully constructed this, and I'm not surprised that the wider spaced samples would have some lower frequency components to them. Are you?

Here's the zipped Audacity file with data folder - download and click on the .aup to load into Audacity - the '50-60-70-80-90-100' labels are on the track below for clarity, scroll down if you don't see them.

Link to zip file

Don't know if I carefully constructed this, making two full scale samples was just the easiest thing to do. Either there is a pitch or not and there definitely is a pitch. People that I've shown it to heard it and they're not musicians with well-trained ear.

And yes, to me it was quite surprising that there would be a pitch at all, but yeah it's an old thread and you're right, it is kind of obvious. Some hair cells will respond more than others if each hair cell has a different resonant frequency.

 

[NTL2009]

By 'carefully constructed', I just meant that the increasing time of 10 samples between each of the two pulses that make up each 'note'. When it comes to signals like this, I don't think 'pitch' is so well defined, it's a perception thing to hear a pitch in something that is maybe better described as a noise burst. And some will pick up on it more than others.I have trouble picking out just what a drummer is doing when they "tune" their drums. To me, that's a whole different thing from tuning a guitar.

I don't know if the mechanics of the ear's response matter that much. I think the math would show there are lower frequencies being emphasized as you spread the time between those two samples. They are there, so we hear them - no different from most other 'hearing'.

I guess that's about all there is to this.