Harmonic Orders

[Bastinium]

I'm looking for a mathematical representation of a fundamental wave and it's harmonic modes.
For example a wave of 590 nm what are its 'octaves'?

Can the resulting formula be represented in polar coordinate 'vector' space?

 

[Ibix]

I'm not sure what you are asking.

A single monochromatic wave is described by a sine wave, $A\sin(\omega t-kx)$, where $\omega=ck$, $k=2\pi/\lambda$, $\lambda$ is the 590nm you specified, and $A$ is the amplitude (how bright the light is, roughly).

Octaves are double the frequency (i.e. half the wavelength), so would be the same thing but with $\lambda$ in nm being 590/2, 590/4, 590/8 etc.

Does that help?

 

[Bastinium]

I'm not sure what you are asking.

A single monochromatic wave is described by a sine wave, $A\sin(\omega t-kx)$, where $\omega=ck$, $k=2\pi/\lambda$, $\lambda$ is the 590nm you specified, and $A$ is the amplitude (how bright the light is, roughly).

Octaves are double the frequency (i.e. half the wavelength), so would be the same thing but with $\lambda$ in nm being 590/2, 590/4, 590/8 etc.

Does that help?

Resonant conjuncts, like in music...

 

[Ibix]

Resonant conjuncts, like in music...

Yes - those are integer or half-integer frequency multiples as I described. They're less interesting in optics than in music because our visual range is a bit less than one octave so we can't see any (at least, not any fundamentals).

We can exploit resonant cavities for various applications - anti-reflection coatings on lenses and optical filters used in optical fibre communications work that way, as do the colours in an oil film on water. And interference effects are related phenomena that lead to the rainbow colours on CDs as well as the colours on some bird feathers and butterflies.

 

[Bastinium]

Yes - those are integer or half-integer frequency multiples as I described. They're less interesting in optics than in music because our visual range is a bit less than one octave so we can't see any (at least, not any fundamentals).

We can exploit resonant cavities for various applications - anti-reflection coatings on lenses and optical filters used in optical fibre communications work that way, as do the colours in an oil film on water. And interference effects are related phenomena that lead to the rainbow colours on CDs as well as the colours on some bird feathers and butterflies.

For multiplexing discreet digital data packets with exceptional resolution and quality without inferential degeneracy degradation

 

[berkeman]

For multiplexing discreet digital data packets with exceptional resolution and quality without inferential degeneracy degradation

I may be misunderstanding your thread start and reply, but no. You cannot encode lots more information in harmonics. To encode more information you need to choose orthogonal and non-harmonic frequencies to be able to use unique data decoding.

https://en.wikipedia.org/wiki/Spread_spectrum

https://en.wikipedia.org/wiki/Orthogonal_frequency-division_multiplexing

 

[Bastinium]

I may be misunderstanding your thread start and reply, but no. You cannot encode lots more information in harmonics. To encode more information you need to choose orthogonal and non-harmonic frequencies to be able to use unique data decoding.

https://en.wikipedia.org/wiki/Spread_spectrum

https://en.wikipedia.org/wiki/Orthogonal_frequency-division_multiplexing

Octavial Frequency Integration (OFI) multiplexing was just born today. LoL

 

[Ibix]

Octavial Frequency Integration (OFI) multiplexing was just born today. LoL

Not sure if you're joking or missing the point. Precisely because resonant frequencies are resonant, it's harder to use resonant devices to distinguish them. Hence you want your carrier frequencies to be as far from resonant with each other as you can arrange.

 

[Bastinium]

Not sure if you're joking or missing the point. Precisely because resonant frequencies are resonant, it's harder to use resonant devices to distinguish them. Hence you want your carrier frequencies to be as far from resonant with each other as you can arrange.

But it can be distinguished

 

[Baluncore]

Octavial Frequency Integration (OFI) multiplexing was just born today. LoL

This seems like a good place to dump some fundamental notes:

Consider a note played on an instrument, that note has a fundamental frequency. The second harmonic is twice that frequency, one octave above the fundamental. The third harmonic is at three times the fundamental. Then the fourth harmonic is at four times the fundamental, that is, two octaves above the fundamental. The harmonic numbers rise linearly, while the octave numbers rise logarithmically.

When the fundamental is a pure sinewave, it will have no harmonic energy. If the fundamental is distorted in amplitude, it will have harmonics with odd numbers, even numbers, or both. The harmonic content is dependent on the form of the amplitude distortion. Different instruments colour the harmonics differently.

The rise and the fall of the fundamental's envelope in time, introduces modulation energy close to the fundamental. Those energy sidebands, or "skirts", are dependent on the way the note rises and falls, how it is played, plucked, struck, or blown. The skirts are naturally present, duplicated about each harmonic of that fundamental.

All the modulation information is present within the skirts of the fundamental, without any need to know which harmonics are present. The information content of any harmonic cannot be changed without influencing the shape of the fundamental. The information encoded on the fundamental cannot differ from that encoded on the second harmonic, one octave above, nor indeed, on any harmonic.

Music sounds good to us because our ears include the tapered cochlea structure. That is a logarithmic mechanical frequency analyser, that stimulates the hair cells, driving the nerves to the brain. The nerve fibres near the middle of the auditory nerve carry the high frequency audio information, while those nearer the outside carry the low frequency information. The relative phase of the energy in the skirts and harmonics are not identified by the cochlea, so are lost.

That explains how our ears can cover many octaves, typically from 20 Hz to 20 kHz. Since 2¹⁰=1024, the factor of 1000 in frequency covers the nine octaves we hear. Although those nine octaves include the first 1000 harmonics of a 20 Hz fundamental, the harmonic energy falls off rapidly over very few octaves. It follows that we can hear only the second harmonic of 10 kHz.

 

[DaveE]

Octavial Frequency Integration (OFI) multiplexing was just born today. LoL

Still born?

 

[Bastinium]

This seems like a good place to dump some fundamental notes:

Consider a note played on an instrument, that note has a fundamental frequency. The second harmonic is twice that frequency, one octave above the fundamental. The third harmonic is at three times the fundamental. Then the fourth harmonic is at four times the fundamental, that is, two octaves above the fundamental. The harmonic numbers rise linearly, while the octave numbers rise logarithmically.

When the fundamental is a pure sinewave, it will have no harmonic energy. If the fundamental is distorted in amplitude, it will have harmonics with odd numbers, even numbers, or both. The harmonic content is dependent on the form of the amplitude distortion. Different instruments colour the harmonics differently.

The rise and the fall of the fundamental's envelope in time, introduces modulation energy close to the fundamental. Those energy sidebands, or "skirts", are dependent on the way the note rises and falls, how it is played, plucked, struck, or blown. The skirts are naturally present, duplicated about each harmonic of that fundamental.

All the modulation information is present within the skirts of the fundamental, without any need to know which harmonics are present. The information content of any harmonic cannot be changed without influencing the shape of the fundamental. The information encoded on the fundamental cannot differ from that encoded on the second harmonic, one octave above, nor indeed, on any harmonic.

Music sounds good to us because our ears include the tapered cochlea structure. That is a logarithmic mechanical frequency analyser, that stimulates the hair cells, driving the nerves to the brain. The nerve fibres near the middle of the auditory nerve carry the high frequency audio information, while those nearer the outside carry the low frequency information. The relative phase of the energy in the skirts and harmonics are not identified by the cochlea, so are lost.

That explains how our ears can cover many octaves, typically from 20 Hz to 20 kHz. Since 2¹⁰=1024, the factor of 1000 in frequency covers the nine octaves we hear. Although those nine octaves include the first 1000 harmonics of a 20 Hz fundamental, the harmonic energy falls off rapidly over very few octaves. It follows that we can hear only the second harmonic of 10 kHz.

Could you elaborate more about this "logarithmic rising"?

 

[Baluncore]

Could you elaborate more about this "logarithmic rising"?

The notes on a piano are not spaced linearly in frequency. Octaves are based on the log to the base 2 of the frequency, each octave covers twice the frequency range of the previous octave. A4 is now defined as 440 Hz, so A5 will be 880, and A6 will be 1760 Hz. Likewise, A3 is 220 Hz, A2 is 110 Hz, and A1 is 55 Hz.

Within each octave there are 12 notes that are not equally spaced in frequency, each note is higher than the previous note by a factor of ¹²√2 = 1.059463
https://en.wikipedia.org/wiki/Piano_key_frequencies

https://en.wikipedia.org/wiki/Music_and_mathematics

 

[Bastinium]

The notes on a piano are not spaced linearly in frequency. Octaves are based on the log to the base 2 of the frequency, each octave covers twice the frequency range of the previous octave. A4 is now defined as 440 Hz, so A5 will be 880, and A6 will be 1760 Hz. Likewise, A3 is 220 Hz, A2 is 110 Hz, and A1 is 55 Hz.

Within each octave there are 12 notes that are not equally spaced in frequency, each note is higher than the previous note by a factor of ¹²√2 = 1.059463
https://en.wikipedia.org/wiki/Piano_key_frequencies

https://en.wikipedia.org/wiki/Music_and_mathematics

✔This is going in the right direction, now to devise a packet series representation of the octave! ️

 

[berkeman]

✔This is going in the right direction, now to devise a packet series representation of the octave! ️

Please keep in mind that we don't support personal research at PF. We are happy to help answer your questions about science and correct misunderstandings (as addressed so far in this thread), but we cannot help you try to develop new science that should be published first before being discussed here. That is pretty clear in the PF rules (see INFO at the top of the thread).