Fundamental frequencies and subharmonics

[Doctor X]

Hi,

 

Fundamental frequency is the lowest partial in signal analysis, and the harmonics of the fundamental are multiple integrals of that.

 

But I've heard of 'subharmonics' where people are claiming that you can have integral fractions of the fundamental frequency in music. don't understand this, as I thought the fundamental was the lowest, by definition.

 

Anyone who is clued up in Acoustics that could perhaps explain this in more depth? Is there such a thing as subharmonics and how does work in relation to the fundamental?

[Strange]

In this context, it may be better to think of the fundamental as the "natural" frequency of the system; i.e. the frequency it would oscillate at if unforced. Under certain conditions, it can then be forced to vibrate at frequencies that are integer fractions of the fundamental (as well as the fundamental and harmonics).

[studiot]

Subharmonics are indeed integral fractions of the fundamental frequency in acoustics.

There is no magic in this.

 

Acoustic sources are intended to produce a particular frequency (or several frequencies).

We call this the fundamental frequency.

Each musical note has one fundamental frequency which is the same for all instruments that produce it.

 

However

 

All sources also create distortion due to their particular method of producing the sound.

 

For scientific purposes we want that distortion to be as little as possible

 

For musical purposes it is that 'distortion' or colouring of the sound that leads to the individual characteristic of instruments.

 

The most common form of distortion is harmonic distortion where integral (whole number) and or reciprocal integral multiples of the fundamental frequenciy are mixed in with the fundamental sound.

Integral multiples are called harmonics and reciprocal integral multiples are called sub harmonics.

 

Normally it is only the multiples or harmonics that are wanted.

These can appear in significant proportions of the overall output sound, although for most instruments the fundamental is still the largest component.

Some wind instruments can be 'overblown' and the output sound is largely a harmonic.

 

A physics way of looking at the fundamental is to consider a stringed instrument.

Sound sources vibrate as stationary or standing waves, so they have nodes and antinodes.

If you think about it the ends of the string are fixed so they must be nodes.

You cannot fit part of a wave between the fixing points there must be a whole number of half cycles between the fixing nodes.

(half cycles because a node is a zero crossing point and it is half a cycle betwen these in a wave).

 

The smallest number of half cycles is obviously 1 This is the fundamental.

You can have as many larger integral numbers of 'wiggles' as you like, jus so long as you have a node at each end. These are the harmonics.

 

but

 

You cannot produce a wave with a lower frequency on its own because there would not be a node at each end of the string.

 

So sub harmonics can only appear as a distortion of the fundamental or higher (harmonic) frequency.

I gather you have posted this question in a proper scientific forum, in search of better information than provided elsewhere.

 

That is sensible and you have come to the right place. But the spirit of this (and many other) forums is that the discussion and information is for all to share so should be put into the thread, not some other way.

 

Technically harmonics or harmonic functions are solutions of Laplaces Equation, which does not contain a forcing term.

This includes sub harmonics, the fundamental and harmonics.

The theory of this is called potential theory.

 

Harmonics are specified in terms of frequency.

Physically the lower the frequency the longer the wavelength, which is the amount of distance you need to accomodate (fit in) the wave.

The largest physical dimension (eg the length of my string) provides the largest wavelength and thus the lower limit of pure tone frequency that can be generated by a system (rather as Strange said).

 

So, although they may exist mathematically, you cannot generate pure tones lower than the fundamental.

However as I noted in post#3 they may be present as distortion to a higher frequency, fundamental or overtone.

 

However loudspeakers are considerably more complicatated than this.

 

A further complication, often ignored in heated loudspeaker discussions, is "Will the soundwave at the frequency considered fit into the room or is it too large?"

All too often 'experts' argue about a wave that could never be developed in the listening room.

This is of course why concert halls always sound different from your lounge at home.

They are much larger so can accomodate longer wavelength, lower frequencies sound waves.

 

Please address further questions within the thread.

Edited February 26, 2014 by studiot

[Doctor X]

Thank you very much for your informative replies studiot.

 

So, although they may exist mathematically, you cannot generate pure tones lower than the fundamental.

However as I noted in post#3 they may be present as distortion to a higher frequency, fundamental or overtone.

 

 

 

In the case of helmholtz resonance, where you blow/whistle across the top of the surface to determine the fundamental frequency, and the note you hear from the bottle is lower than the whistle. That is supposedly the subharmonic?

[studiot]

Yes and No it is not normally called a sub harmonic, but it could be.

 

Like the louspeaker, the Helmholtz resonator is complicated (although not as much as the loudspeaker since there is no electric to mechanical energy conversion stage).

 

The key word is resonator.

 

You can excite an oscillator with any of the harmonics of its fundamental frequency.

 

So with respect to the resonant (fundamental) frequency of oscillation of the air inside the bottle the effect works best (is at its most efficient) when the rush of passing air contains harmonics of the resonant frequency of the bottle.

This is easy to achieve since the passing air approximates to white noise which contains all frequencies.

So the bottle 'picks out' or selects energy at is resonant frequency.

 

Incidentally, I don't know how much you understand about the mechanical mechanism of the action but the whole of the air in the bottle does not resonate.

A plug of air in the neck is bounced up and down between the pressure of the passing air stream and the restoring force due to the springiness of the air in the large volume of the body of the bottle.

This plug of air thus acts as a piston like a loudspeaker cone.

It is this that generates the tone you can here, transferred to the general air in the room.

[Enthalpy]

Fundamental frequency is the lowest partial in signal analysis, and the harmonics of the fundamental are multiple integrals of that.

 

But I've heard of 'subharmonics' where people are claiming that you can have integral fractions of the fundamental frequency in music. don't understand this, as I thought the fundamental was the lowest, by definition.

 

Only periodic signals are made of a fundamental and harmonics.

Musical sounds are not periodic, hence contain other frequencies. These can be non-integral partials, subharmonics...

 

Non-periodicity is a fundamental quality of musical sounds, essential to our perception. Imitating a music instrument through a periodic synthesis fails in most cases. This knowledge is already 20-30 years old but still not spread enough, as most books and professors still claim that sound quality results from harmonics.

 

So much that the spectrum, even non-harmonic, isn't necessarily the best approach to characterize or synthesize a musical sound. Researchers have had more success using simple tricks in the time domain. The frequency domain would contain the same information, but not as a simple description.