Difference between Notes and Harmonics

You may have noticed that on every instrument (almost every …) there are several Cs, As and so on … ; so what’s the difference between a C4 and a C5 for example ?

Why the sound of a C5 is “higher” then the sound of a C4 ?

Why the sound of a A4 is higher then C4 but lower then C5 ?

The fact that notes are “higher” or “lower” with respect to each other is taken into account by a “feature” called “pitch”.

The pitch is not a scientific measure of the notes but it is directly connected with the “frequency” which instead it is, so in a way the concept of pitch can indeed be used to define the relationship of sounds in terms of “how much high or low they are”. So we can safely say that the pitch of a note A4 is higher then the pitch of C4 and lower then the pitch of C5.

Also with non-instrumental sounds we can use the same concept of pitch, for example the chirp of a little bird has a higher pitch then then a distant thunder. We can apply the concept of pitch with every sound even the ones (which are a lot) that don’t have a name or a representation on the score.

So why we connect the concept of “pitch” (non scientific) with the concept of “frequency” (scientific) ? And why we use the measurement unit Hertz (Hz) to give a value to these features ?

A very looooong time ago “Pythagoras” conducted studies on strings, he discovered that a string with a longitude “L” was doing a certain number of oscillation within a certain unit of time, let’s say for simplicity that a string with length “L” was doing “N” oscillations within 1 second.

Pythagoras discovered (and measured) that the same string, made of the same material and blocked with the same tension but with half of the length (so L/2) was doing the double of oscillation (so 2N) in the same unit of time (1 second).

So he discovered (among other marvelous things) that in 1 second:

- a string with length L was making N oscillations

- a string with length L/2 was making 2*N oscillation

He also noticed that the sound of the shortest string was “higher” then the sound of the longest string ! And this was incredible because changing the length of the strings one could have generated sounds at different pitches, thus music !

But let’s stick to the science; the number of oscillations within a second is called frequency and it is measured in Hz (Hertz).

So in the above observation the length L can be measured in “meters” (m) and the number of oscillations N in one second can be measured in “Hertz” (Hz). And thus we know now that halving the length we double the frequency; this is exactly what happens today in our stringed musical instruments! But in addition to that we also discovered that changing the thickness of the strings has an impact on the oscillation frequency and also the material.

But now you might be wondering, how do we measure the frequency ? I mean the string of the note A4 in the piano (or a guitar) oscillates at 440 Hz meaning 440 oscillations every second ! How do I count them ? How do I measure them ?

A looong time ago (but less long then Pythagoras :) ), there was a cool guy called Jean-Baptiste Joseph Fourier, this guy was a mathematician and he discovered (among other things) a feature of mathematical functions that could be extended and applied also to musical signals (and not only musical’s).

Let’s see how Fourier’s work applies to musical signals (sounds).

Notice that until now we never talked about “harmonics”, we only talked about notes and oscillations (vibrations) which justifies the usage of a measurement called “Frequency” and also the general extension to the concept of “pitch”.

Let’s consider a sound of a musical instrument, for example D2 note played by a piano. (In case you are interested this note has MIDI number 38 and it’s found on the piano at key number 18), so it’s a low pitched note.

If we record that note with a proper software using a normal microphone we obtain a digital copy of that sound which looks like this:

The maximum peaks of this ideal sound are “spaced” (in time) by the same amount of the maximum peaks of the real sound but you can see here that everything is more clear and the signal (blue line) oscillates between the maximum and the minimum in a smooth and continuous way.

The time between two consecutive maximum peaks is called “period” (T) and the inverse of the period (1/T) is the frequency ! The distance between the maximum peak and the minimum (vertically) is called instead “Amplitude”.

So what Fourier said (he did not say this applied to music) is that (in music) whatever piece of sound we have, it is made of a sum of numerous “pure” sounds (like the ones above) at different frequencies and different amplitudes. The great thing about this is that in music all those “pure” sounds are not only theoretical, they exist for real inside the real sound ! The problem is that it’s not possible to isolate them properly, but they exist, they are there, they are real vibrations, or better, they are components of the main vibration. So in practice a string does not perfectly move up and down at a specific frequency (number of oscillations in a second) in a smooth and continuous way, in the middle of an oscillation, it moves in a complex way because this is the result of its physical nature ! It is a consequence of the existence of all those other “pure” components that affect the overall movement of the string (we talk always about strings but it can be whatever, also air in wind instruments for example).

In the figure above one complete oscillation (one period) is highlighted and the real signal (down) is compared with the mathematical signal (the perfect and ideal string) (up); see how the real signal contains inside other “smaller” pure waves with the same shape of the one above (upper math wave) but smaller and shorter !

So indeed the real signal is made of a complex overlap of pure waves, at different frequencies and different amplitudes, thus in the figure above only the largest pure component is represented, the one with the lowest frequency and maximum amplitude, many other components sum to this one, and together they generate the real signal in the end. Each of these pure (bigger or smaller) oscillations are the results of the internal vibration of the physical matter which is complex and heterogeneous.

Nowadays, thanks to Fourier, all of these components (including the biggest overall oscillation) can be calculated (and today visualized) using the Fast Fourier Transform (FFT).

The figure below is the “famous” FFT of the real signal we have been talking about until now and the peaks in that plot are indeed all those pure waves that summing up all together, literally compose the real signal !

On the horizontal axis of that plot there is the frequency and on the vertical axis the amplitude (let’s say the “energy” of the component). So each peak is a little “pure” mathematical wave that is present for real inside the real sound.

All of these components (represented by peaks in this graph) are called “harmonics”.

The first harmonic (marked in red), which is the lowest in frequency and the biggest in energy is also called “fundamental”, this first harmonic is exactly the pure mathematical wave used for comparison in the previous image !! This is just the first of many harmonics (peaks with higher frequency) but it’s the most important because we conventionally attribute its frequency to the whole signal, thus in this case we say that what we are analyzing is the piano note D2 at 73 Hz, but you know now that this is just a convention and scientifically this is wrong because the D2 piano note does not have only the first harmonic (the one at 73 Hz) it has maaaaany other harmonics at higher frequency, for example the second harmonic (the one at 146 Hz) is also very strong and it’s there, it’s in the real signal, same thing applies to the third harmonic, the fourth and so on ….

So the fact that we attribute the frequency of the first harmonic to the whole signal is just a convention, it’s not scientific, it’s a way for us to distinguish between notes !

Clearly this does not disrupt the concept of “pitch”, in fact, given two different notes I will always perceive “higher” (with higher pitch) the one whose fundamental (first) harmonic has higher frequency.

So going back to the central C (C4), it is conventionally said that this note is the one at frequency 261 Hz, but you know now that this is just the frequency of its first harmonic ! And that inside the C4 there are many other harmonics with higher frequency ! The one at 261 is just the first harmonic, the one at lowest frequency. But still I will perceive the C4 higher (with higher pitch) then the D2 (which has the first harmonic at 73 Hz).

As a last example we can consider the A4 (as before), this note is the one used for tuning the instruments of an orchestra and we refer to it as the note at 440 Hz, again this is just the frequency of its first harmonic and it has inside many other harmonics at higher frequencies; and again I will perceive this note as higher then the central C (261 Hz).

Since the harmonics are strictly correlated with the physics of the musical instrument, the “pattern” they create can also be considered as a sort of “fingerprint” of the musical instrument, this concept is extensively used in many pitch detection algorithms including the one I developed.

For the purposes of this article the information provided should be enough for you to understand the difference between notes and their harmonics, but if you still have doubts or questions you can always write to me using this form.