# Do sine waves (pure tones) have overtones?

I was told in my music theory lessons that most sounds we hear in our everyday life contain partials which make up the timbral quality of that sound, whilst the sound musical instruments generate contain partials that occur at integer values to the fundamental frequency and are called overtones.

Now my confusion lies here:
if octave equivalence (perceiving a note at twice or 1/2 its frequency the same) and consonance (the result of overtones overlapping) is caused by the overtone series that exists in standing waves, why do I still feel the the effects of consonance and octave equivalence in sine waves?
I remember studying that pure tones are made up of just the fundamental and no partials or harmonics and hence the term, but then what’s the reason we feel these effects that are caused by the overtone series?

• The part you missed is this: mathematically it's easy (!!) to show that any repeating waveform, of any shape, can be expressed as a sum of sine waves of various frequencies (possible an infinite sum for,e.g., square wave). – Carl Witthoft Dec 9 '19 at 13:50
• (@CarlWitthoft, maybe not soooo easy. :) – paul garrett Dec 9 '19 at 23:51

One pure sine wave does not have an overtone or harmonic. The sine wave is a single unit of information in some version of signal processing theory. It is a mathematical function.

The overtones or harmonics that contribute to musical instrument tone or timbre come from solutions to the wave equation, which coincidentally is also, in many cases, a sine wave or some modified version of sine wave. When you play any instrument you need to "attack" it to produce sound and that attack will necessarily produce overtones, unless the attack is due to sympathetic resonance with another source but even in this case the initial turning on of the source WILL have harmonics in it due to the leading edge of the wave. This is a fact of life that will never be absent.

As for the "pure tone". No such thing exists in the sphere of human experience. The mechanisms in the human ear are non-linear and generate aural harmonics when excited by an otherwise pure tone. So the brain will not be fed a single frequency when provided a single frequency driving force. This can be tested using expensive function generators and listening to intervals produced by a series of them. It is not fair to say that the theory failed in practice since in practice you always excite harmonics in your ear.

You would have to provide more information on what you mean by "... but then what’s the reason we feel these effects that are caused by the overtone series?" relative to a pure tone. Under what conditions do you think you have "felt" the overtone effects when listening to a pure tone or set of tones?

Other causes of this could be related to other acoustic properties of the room, earphones, instruments etc. Stiff rods for example produce overtones that do NOT follow the harmonic series and are quite dissonant. A stiff rod with clamped boundary on one end can produce an overtone that is (if memory is not too faulty) about an octave and augmented 5th above the fundamental. When struck with enough force this harmonic is very loud and can be perceived as a distinct note. It can also be picked up using a mic and a AD data acquisition card. I've experienced this myself in a controlled setting. A less known factoid is that metal strings in a piano and on an electric guitar can exhibit this stiff behavior especially when very short and under high tension. Coupled with the resonances of the body it is not impossible to hear some of the dissonance when one plays even a single note on an instrument.

In short (1) your ear is never harmonic free, (2) there is more to acoustics than the harmonic series, (3) unless you are in a controlled environment it isn't really fair to say that these effects are "perceived" when you expect them not to be. It is an oversimplification of physical acoustics.

Don't confuse "pure tone" with "I only played one note".

A sine-wave, by definition, has no overtones. (Whether it's practically possible to experience a pure sine-wave is another matter!)

Defining consonance by coincidence of overtones is an attractive theory, but it falls apart in real life, when it's realized that (a) we very often don't use 'perfect' tuning systems and (b) the overtones of a real instrument very often deviate from the harmonic series. Yet we hear consonance and dissonance, tension and release.

When you develop a workable Universal Theory of the Perception of Harmony let us know!

• I don't think that either (a) or (b) is a critique of the theory based on overtones. Even when those do not follow the harmonic series the basic theory behind comparing overtones in a similar frequency band provides a reliable measure of consonance and dissonance. The fact that things are slightly out in real life explains dissonance but does not contradict the explanation of consonance. I am a little confused by your answer. – ggcg Dec 8 '19 at 23:35

Consonance and octave equivalence are effects that the hearing is tuned to detect since it is very important for separating sound sources in complex situations. Whether the consonance is part of a single oscillator with overtones or two coherently acting sound sources is a detail distinction: with quite a number of sound sources, overtones are not exactly phase-locked to the fundamental and are thus not "harmonics" in the strict mathematical sense.

The other answers are good, but all overlook something: you can't hear a sine wave. You have to provide that as input to an output device (i.e. a speaker), which will the introduce vibrations into the environment that correspond to the input as closely as the speaker is capable. The speaker is necessarily imperfect and, like your ears, the room, etc., could play a part in the creation overtones.

EDIT: elaborating as requested in the comments:

This answer was an intent to provide more clarity and precision to other answers.

A sine wave is just a visualization of a trigonometric function. It is not a physical vibration. Your ears hear physical vibrations.

• You can set up a system (e.g. a computer) that can compute the sine as a function of time and generate an electrical signal that matches the output of the function.
• You can feed that into an electromagnetic device (a speaker coil) which will vibrate.
• You can connect something (a speaker cone) to the vibrating speaker coil in order to efficiently transfer the vibration into the air.
• Those vibrations in the air will combine with other environmental vibrations (ambient noise) en route to your ear.
• If you are in the vicinity, your ears will pick up those vibrations in the air and translate them into nerve signals to your brain. It is only at this point that you have heard anything. ** I'm still glossing over details about the inner ear here, but I feel that I can make my point without these details.

Every step of this process introduces opportunity for loss of fidelity of the original output of the sine function. The vibrations you are hearing are not a precise match for the sine function (although they might be quite close, depending on the equipment you chose and the environment in which you perform this experiment).

Each one of these losses in fidelity introduces an opportunity for overtones to arise.

I do not know with certainty that there is no possible way (whether naturally occurring or human-made) to present the brain with something equivalent to a perfect, pure sine wave -- it is impossible to prove a negative. However, I find it extremely unlikely that such a thing will ever exist. We can get close, but perhaps never close enough to completely eliminate all losses in fidelity.

• Could you explain what you mean by "you can't hear a sine wave"? Are you saying that perfect sine waves don't exist in nature? Because if so, elaboration would be greatly appreciated. – user45266 Dec 10 '19 at 0:53
• Yeah, could you please elaborate? – user11845919 Dec 10 '19 at 4:41
• Please see edited answer. :) – JakeRobb Dec 10 '19 at 15:07