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Harmonics Explorer

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Re: Harmonics Explorer

#21

Post by roelschroeven »

laszlokorte wrote: post #11
Prcmaker wrote: post #9

I like the thing, but it misses what I enjoyed teaching about these most: phase. Many people know white noise is nominally 'all frequencies at the same intenisty', yet those taught Fourier mathematics are also taught that the same recipe makes a pulse. The difference is all in the phase information, and why I maintain to this day the Nyquist-Shannon sampling theorem, as typically applied, is incorrect.

Yes phase information is really important. Try this: 1. Fourier Transform an Image 2. Set all magnitues the the spectrum to 1.0, but do not change the phase 3. Inverse Transform and look at the result 4. Now try the same, but this time keep the maginutes unchanged but change all phases to 0° Spoiler: When changing all amplitudes the image is still regocognizable, when changing all phases, it is not. See example: [1]…

Phase information is important, but as far as I understand it, it is not important in audio: our ears are insensitive to phase.

(Phase is important when combining different sinuses of the same frequency, because the sum of those will be different depending on their relative phase, but that's a different matter and not relevant here.)

Changing the phases of the different frequencies will result in a waveform that looks different, but it will sound the same. Our ears are like a spectrum analyzer that only records the volume of each frequency, and is unable to record the phase.

Re: Harmonics Explorer

#22

Post by xeonmc »

laszlokorte wrote: post #11
Prcmaker wrote: post #9

I like the thing, but it misses what I enjoyed teaching about these most: phase. Many people know white noise is nominally 'all frequencies at the same intenisty', yet those taught Fourier mathematics are also taught that the same recipe makes a pulse. The difference is all in the phase information, and why I maintain to this day the Nyquist-Shannon sampling theorem, as typically applied, is incorrect.

Yes phase information is really important. Try this: 1. Fourier Transform an Image 2. Set all magnitues the the spectrum to 1.0, but do not change the phase 3. Inverse Transform and look at the result 4. Now try the same, but this time keep the maginutes unchanged but change all phases to 0° Spoiler: When changing all amplitudes the image is still regocognizable, when changing all phases, it is not. See example: [1]…

Explaining concretely: a uniform spatial displacement of the image corresponds to a ramping phase shift across the frequency spectrum.

i.e. if you shift an image by 1cm, then the 1 rad/cm frequency component gets its phase "fast forwarded" by 1rad, the 1.5 rad/cm component forwarded by 1.5rad, and 2 rad/cm by 2rad and so on.

By subtracting each frequency's phase from their original distribution, you are basically displacing them each by a different distance from one another, decohering the image entirely.

Re: Harmonics Explorer

#24

Post by xeonmc »

roelschroeven wrote: post #21
laszlokorte wrote: post #11

Earlier quoted context omitted.

Yes phase information is really important. Try this: 1. Fourier Transform an Image 2. Set all magnitues the the spectrum to 1.0, but do not change the phase 3. Inverse Transform and look at the result 4. Now try the same, but this time keep the maginutes unchanged but change all phases to 0° Spoiler: When changing all amplitudes the image is still regocognizable, when changing all phases, it is not. See example: [1]…

Phase information is important, but as far as I understand it, it is not important in audio: our ears are insensitive to phase. (Phase is important when combining different sinuses of the same frequency, because the sum of those will be different depending on their relative phase, but that's a different matter and not relevant here.) Changing the phases of the different frequencies will result in a waveform that look…

It's insensitive to individual phase of audible frequencies, but the concept of phase itself is very important in audio in the form of time delays/echoes, if you are dealing with long sample lengths.

See my sibling comment explaining how translation corresponds to ramping phase shift/"fast-forwarding" each frequencies such that the shifted distance are the same across the spectrum.

Re: Harmonics Explorer

#27

Post by wholinator2 »

roelschroeven wrote: post #21
laszlokorte wrote: post #11

Earlier quoted context omitted.

Yes phase information is really important. Try this: 1. Fourier Transform an Image 2. Set all magnitues the the spectrum to 1.0, but do not change the phase 3. Inverse Transform and look at the result 4. Now try the same, but this time keep the maginutes unchanged but change all phases to 0° Spoiler: When changing all amplitudes the image is still regocognizable, when changing all phases, it is not. See example: [1]…

Phase information is important, but as far as I understand it, it is not important in audio: our ears are insensitive to phase. (Phase is important when combining different sinuses of the same frequency, because the sum of those will be different depending on their relative phase, but that's a different matter and not relevant here.) Changing the phases of the different frequencies will result in a waveform that look…

To expand on the other comment. You wouldn't be able to tell the difference in a single sine wave with phase set to either 0 or 180 degrees. But if you add in another sine wave at 0 degrees, the two 0's will add up and the 0 and 180 will completely cancel.

Phase makes a huuuuge difference in audio engineering. There isn't a single song that gets mixed without intense consideration of phase interactions between the different tracks. Getting it wrong can result in catastrophic damage to the audio signal that reaches your ears. If you have a speaker capable, try switching the leads that feed the signal on one of the speakers and see how it sounds! Everything that's exactly the same between the two speakers will sound hollow and tinny, the frequency balance will completely degrade

Re: Harmonics Explorer

#28

Post by klysm »

mcqueenjordan wrote: post #3

This is really cool. I used to do a lot of overtone ("harmonic") throat singing and playing with this tool reminded me of those days. For anyone curious, vowels are mostly just how we perceive different harmonic distributions. Put differently, harmonics are the basis of what it means to pronounce a different vowel. The human voice is basically just a harmonic chord, with different distributions of the 2nd, 3rd, 4th,…

Is this related to formants? I don’t actually know what that term means but I’ve heard it used in this context

Re: Harmonics Explorer

#29

Post by ubavic »

laszlokorte wrote: post #11
Prcmaker wrote: post #9

I like the thing, but it misses what I enjoyed teaching about these most: phase. Many people know white noise is nominally 'all frequencies at the same intenisty', yet those taught Fourier mathematics are also taught that the same recipe makes a pulse. The difference is all in the phase information, and why I maintain to this day the Nyquist-Shannon sampling theorem, as typically applied, is incorrect.

Yes phase information is really important. Try this: 1. Fourier Transform an Image 2. Set all magnitues the the spectrum to 1.0, but do not change the phase 3. Inverse Transform and look at the result 4. Now try the same, but this time keep the maginutes unchanged but change all phases to 0° Spoiler: When changing all amplitudes the image is still regocognizable, when changing all phases, it is not. See example: [1]…

Tangential: Your Fourier Cuboid is very cool project. I have added it to awesome-interactive-math [1] list.

[1]: https://github.com/ubavic/awesome-interactive-math/

Re: Harmonics Explorer

#30

Post by elevaet »

This is very cool.

If the creator is reading these comments, my one piece of feedback would be that I think it would be more interesting/useful if the harmonics were expressed as multiples or ratios of the fundamental.

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