The previous blog post analyzed the physics of sound, namely the sound waves which music is comprised of and how they function, particularly in intonation and volume as it is perceived by the human ear. This then leads into the topic of today’s blog post: the overtone series, which is largely due to the properties of standing waves, and which can also be explained mathematically by the harmonic series. In order to understand the overtone series, one must first understand what an overtone is.
The notes that we perceive as music and specific pitches are caused by longitudinal standing waves travelling at specific frequencies. For example, C4, also known as middle C on the piano, has a frequency of 261.63 Hz and a wavelength of 131.87 cm (Source). In the Crash Course video of the previous blog post, it was established that standing waves are produced by reflecting a wave off of the end of a tube, and the properties of the standing wave were greatly affected by whether the tube was closed or open, as it affected the interference of the wave with itself. For review, click here and view the crash course video.
Within these specific frequencies, or pitches,there are more pitches than the dominant one (the fundamental) that people hear.. These are known as overtones. In short, overtones are the natural parts of any pitch heard when it is sounded (Source). The fundamental is the longest wave that can fit into the tube (the instrument), while the other waves in the tube, multiples of the fundamental, are overtones.
There is a distinct order the overtones, which are dictated by physics and described mathematically. In theory terms, however, the first overtone is an octave above the fundamental, second is an octave and a fifth above the fundamental, the third is two octaves, the fourth is two octaves and a third, and continues in a similar fashion (Source). An example of the series, based on C, can be seen below.
(Source)
Of course, as aforementioned, music is described mathematically and is bound by physics, which can be seen in the ratios of the frequencies. Music is described mathematically by an infinite sum of sines and cosines multiplied by appropriately valued coefficients — infinite mathematically, but in practice only a handful of overtones really matter. (Source) This is largely the reason that the fundamental pitch of A, at 440 Hz, has a series of overtones mathematically related to to the fundamental pitch. When the note A at 440 Hz is played, there is an audible overtone ratio of 1:2, or 880 Hz, and 1:4, 1760 Hz (Source). Either way one analyzes the overtone series, each successive pitch is weaker in strength and importance than the one before; furthermore, overtones are usually felt more than they are heard (Source).
This Crash Course video further explains the physics, and some of the math, of the overtone series quite well.
The previous video clearly demonstrates the reasons why different instruments have different sounds, even when producing the same pitches travelling at the same frequencies. Different standing wave properties–caused by two fixed ends (a piano), two loose ends (flute), and one fixed end and one loose end (pan flute)– produce very distinct sounds. Furthermore, the differing lengths of these instruments affect the wavelengths that they produce. Namely, the distinct overtones of each instrument form the characteristic sound of the instrument.
In an example given by The Method Behind the Music (Click here to visit their page), all the waves are of the same frequency; however, the overtones produced by each instrument are different. This produces distinct waveforms for each instrument, as is seen in the image below. The violin has a jagged waveform that produces a sharper sound; the smooth waveform of the piano produces a purer, more even tone throughout the registers, which is closer to that of the sine wave.
In all this, we see once again that math, science, and music are all incredibly entwined with one another. Science and math explain why music works, while also dictating the constraints which music must function in; conversely, music is often a means through which people understand the math and science concepts. Music is vital to us all, no matter what path we take in life. Perhaps this is why so many of the greatest minds of humanity also played instruments and studied music. Albert Einstein was an accomplished pianist and violinist, and he even once stated, “If I were not a physicist, I would probably be a musician. I often think in music. I live my daydreams in music. I see my life in terms of music” (Source). Thomas Edison played the piano; Jean-Jacques Rousseau composed an opera; Leonardo DaVinci not only created masterful artistic works, but he was also an accomplished musician and invented musical instruments (Source); Pythagoras created a system of tuning that was used for decades and believed that the planets and stars moved according to mathematical equations that corresponded to musical notes, creating the “Musical Universalis” (Source). Perhaps Plato summed it up best of all, “Music… gives soul to the universe, wings to the mind, flight to the imagination, and charm and gaiety to life and everything” (Source).
