## Refath Bari

Managing Editor, The Paper

Published on February 28, 2022

Have you ever wondered why the ocean is blue? Or why certain musical notes sound harmonious but others dissonant? Turns out they both boil down to one answer: harmonics.

As a kid, I always liked to jump on the playground swing and stare at the sky, wondering if the swing could ever take me to the moon. Although I’ve since been told this is not feasible, that has not stopped me from trying. If you’ve ever been pushed on a swing, you know that it takes just the right amount of force to do the job. Too light of a force will barely move you, but the goldilocks amount of force will have get you in the swing of things. That goldilocks’ amount of force is the “resonant frequency” of the swing. It’s just the same way that an opera singer, if he/she sings at just the right pitch (but no higher or lower) can shatter a wine glass. Or, if people walking on a bridge at just the right pace (at its resonant frequency, in fact!) can actually collapse the bridge! Where have you seen resonance in your life?

Just like a swing, a bridge or a wine glass, a water molecule (H2O) has resonance! In fact, water has three resonant frequencies because it has greater degrees of freedom than a swing, bridge, or glass, all of which can vibrate in only one way. Water, however, can vibrate in three ways — symmetrically, asymmetrically, and via bending. Similar to the systems above, water begins vibrating after receiving external energy. Similar to a kid jumping around after too much chocolate, water begins vibrating all over the place after receiving some energy in the form of a photon (you can think of this as a tiny bit of light energy, perhaps coming from the sun). Each of the three vibrational modes of water is its own “fundamental”, and further variations in these modes act as harmonic overtones to the fundamental.

It just so happens that the resonant frequencies of water lie in the red portion of the electromagnetic spectrum (the spectrum of light waves). So when a bit of sunlight hits a water molecule, it excites some of water’s resonant modes and the H2O absorbs red portion of sunlight, andi emits back a bluish tint. That’s why the oceans are blue.

What about music? What does music have to do with resonance and harmonics? It all starts (as it often does) with an ancient greek philosopher, Pythagoras [1]. The most likely apocryphal tale is as follows: Pythagoras, upon passing a blacksmith’s shop, heard the various notes rung by the blacksmith’s hammers. Some hammers produced consonant notes, while others produced dissonant ones [4]. Recognizing that the pitch of the emitted sound seemed to depend on where the hammer was struck, Pythagoras began experimenting with his own strings. Consider the following: a string of length L with either ends fixed to a wall. Plucking this string results in the fundamental harmonic. Holding the string down at the center to suppress the fundamental harmonic and elicit the second harmonic (i.e., first overtone) results in twice the pitch as the previous pluck, and is a common technique in stringed instruments, known as flageolet [2]. One can halve the string length once again to result in twice the pitch, and so on.

​​In fact, the pitch is given by f_n=n*f_1, where n is the number of harmonics. An interactive simulation is shown below in which one can create the nth harmonic and listen to the pitch of the resulting sound. After some experimentation, one will recognize that the ratio between the first and second harmonics is an octave. Between the second and third harmonics is a perfect fifth; third to fourth is a perfect fourth, then a major third, and finally a minor third. Thus, when a guitar string is plucked near the ends, the higher-pitched overtones are excited, whereas a string plucked nearer the middle pronounces the fundamental harmonic.

Most modern pianos and stringed instruments use equal temperament, which forces a note an octave above a previous note to be twice the pitch. Since the western music scale has 12 notes between octaves, this means that the frequency of each key is 2^{\frac{1}{12}}. This guarantees that a note 12 keys later will be an octave higher in pitch. As shown above, a linear combination of harmonic series can ideally produce any arbitrary musical note. In the simulation below, one can manipulate the number of harmonics and note how the resultant wave changes.

To illustrate the idea of building on the fundamental to create any musical note, we now consider a “square”-like wave which produces an interesting series of pitches. We begin with the fundamental. Note that the fundamental is normally half a wavelength, but we have extended the time domain for the sake of illustration. Each harmonic is represented as y += np.sin((2n+1)(2np.pi)(x))/(2*n+1) in the simulation. Combining the first harmonic with the first overtone, we note that at some points, the two waves constructively interfere to produce higher peaks and at other points destructively interfere to form little valleys. Indeed, we can continue this process of building onto the first harmonic to build a square wave. Note that as n\rightarrow\infty, where n is the number of harmonics, our note approaches a square wave. Indeed, one may be interested in hearing how adding progressively more harmonics influences the pitch of the final note.

Fascinating how Physics connects seemingly separate systems like water and music! It turns out that the fundamental ingredient for life and the latest musical hits both operate on the same physical principle: harmonics.

Have any burning math or science questions? Why is the sky blue? Why is fire hot? What do springs have to do with atoms? Send them to thepaper@gtest.ccny.cuny.edu and you might see your question in the next article!