Music and Science: Harmonics
Making music is an inherently subjective experience and the creative process is different for every musician. However, the fundamental concepts behind creating sounds that work well together are in fact based on universal physical laws and mathematical theory. The notions of octaves, chords, keys, and the basic reasons why certain combinations of notes sound pleasing to our ears are all fully explicable using what we know about the physical properties of a sound wave and some simple math. Recognizing the scientific processes behind harmonics allows for a deeper understanding of what music really is, and that knowledge can be applied to anything from classical composition to engineering a recording session.
Harmonics are the basis of music. When we hear a simple chord—for example a C chord—we can discern three different pitches, but they somehow work together harmoniously to form a cohesive sound. Many musicians can explain this phenomenon using music theory: if you play a root note stacked with a major 3rd stacked with a minor 3rd (in this case C, E, G), you form a major triad, which we know to sound consonant. However, there are ways to understand harmonies on a deeper level. Why do we like the sound of a major triad? What makes a group of notes sound consonant or dissonant? How can we predict what notes will work together? It turns out that these questions can be answered using math and science.
The connection between harmony and math was explored as early as 500 BC, when Pythagoras, a Greek philosopher and mathematician, made the connection between string length and pitch. Through a process of trial and error, Pythagoras was able to numerically define musical intervals using what are now called Pythagoras’ ratios.
First he found that, with the same tension, width, and material, two strings of the same length sounded exactly the same. Then he discovered that if he cut one string in half (giving them a ratio of 2:1) and plucked the strings together, they still sounded like they belonged together, but the shorter string sounded higher—we call this interval an octave. If he made one string 3/2 the length of the other, he found that they sounded consonant, but not as similar as an octave. This interval became known as a perfect 5th (P5). Through this process he also discovered that a perfect 4th has a ratio of 4/3, and a major 3rd has a ratio of 5/4. The table below describes all the various ratios that can be found within the notes of a major scale.
Interval, # of semitones, ratio
His discoveries, while not fully understood scientifically at the time, were an introduction to the idea that music is not random, or even entirely subjective. You can make music by playing random notes until you figure out what sounds you like based on trial and error, but there are in fact ways to predict the way notes will sound together. There are patterns ingrained in the fundamental make up of sound waves, so music can in fact be objectively quantified and, to an extent, reduced to simple mathematical ratios.
Frequencies and Intervals
Pythagoras’ experimental conclusions can be verified theoretically if you consider what we know about frequency. As mentioned in the previous section of this series, the frequency of a sound wave describes the rate at which compressed areas of the wave—the “crests” of the wave—hit your ear. The faster they hit, the higher the pitch of the sound.
Different combinations of these frequencies will sound different to our ears. Take an octave interval, for example. You’ve probably heard this interval when a woman or child and a man sing the same melody. One voice is noticeably higher than the other, yet they still sound like they’re singing the same note. This is because the frequency of a note one octave up is exactly double the lower note (a ratio of 1:2), so the two waves are in phase with each other.
For example, a middle C has a frequency of about 262 Hz. This means its wave crests, notated by X’s on the image below, hit your ear around once every .0038 (1/262) seconds.
Source: https://plus.maths.org/content/magical-mathematics-music
At an octave higher, C has a frequency of 524 Hz. This means its wave crests hit your ear once every .0019 seconds. Since this is exactly double the rate of a middle C, it lines up with middle C every other crest.
Source: https://plus.maths.org/content/magical-mathematics-music
A perfect 5th, which is still considered to sound very consonant but not quite as hollow as an octave, lines up its wave crests almost as frequently as an octave. You can see in the diagram for C and G below that the ratio 3/2 relates directly to the way the wave crests line up. Every 3rd crest for G lines up with every 2nd crest for C, thus giving us the simple ratio, 3/2. Another way of looking at the relationship is that the crests from G arrive 1.5 times as fast as the wave crests from C, and their crests line up around once every .0075 seconds.
Source: https://plus.maths.org/content/magical-mathematics-music
By contrast, dissonance is created when wave pockets line up less frequently. When two waves are out of phase with only brief moments of synchronization, we hear them as too different and thus dissonant.
Pythagoras’ ratios, while mathematically correct, when implemented musically can pose problems with key changes. Pythagoras’ ratios themselves are translatable between keys—that is, a perfect 5th is a ratio of 3/2 whether it is in the key of C or the key of E. However, in order to play all of these different ratios in various keys, we need to be able to play essentially all frequencies within hearing range. This causes problems for incremented instruments, such as guitar and piano. Unlike instruments like trombone, voice, and violin, incremented instruments do not have every frequency at their immediate disposal, so if you want to play in a certain key, you have to tune the instrument to accommodate for the different required frequencies.
In order to remedy this tuning problem and simplify the way we assign note names, the equal-tempering system was created. Within an octave, which we discussed has a ratio of 2:1, we can divide the pitches into 12 increments, called semitones, so that the distance between each of them is 1.059463. You can check this by setting up the equation: (x^12)=2, because we are looking for the number that when multiplied by itself 12 times, doubles the number we started with. Then solve for x to get 1.059463.
By dividing an octave into 12 equally separated semitones, the equal-tempering system only implements certain frequencies. For example, take C, which as we know has a frequency of about 262 Hz. The closest semitone up from C defined by the equal-tempering system is C#, which is has a frequency of about 277 Hz. So, the 15 Hz between these two notes are unaccounted for and essentially ignored by the system. This method of discounting some natural frequencies between semitones allows for all keys to be accounted for even on an incremented instrument without retuning, remedying the problem that arose from Pythagoras’ well-tempering system. While the system has intrinsic inaccuracy due to rounding, it turns out that the discrepancies between the approximated frequencies from the equal-tempering system and the frequencies defined by Pythagoras’ ratios are not actually prominent enough to be picked up by the human ear. The equal-tempering system is now universally accepted, and because of it, changing keys is as simple as moving your capo up or down on a guitar, or using/avoiding different keys on a piano.
Knowledge of the science behind harmonics and the physical properties of waves is the basis behind any musical creation. By understanding the fundamental physical and mathematical principles of harmony, you are able define what consonance and dissonance really are, why certain chords sound good or bad, and how that relates to what you personally define to be music.
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