Many of us have serious problems with Math. Most can’t even hear the world to abruptly burst into tears. But, behold! Math gave important contributions to music. Read this cool article by Ultimate Guitar.
It’s no secret – not many people like math, it conjures thoughts of the cliché math teacher, blandly reciting equations in monotone, looking like Droopy Dog in a plaid blazer. But math can be fun, and if not fun, at least it can be useful. I studied mathematics at the University of Minnesota, particularly cryptography and number theory. At the time, it was just something I had to do to complete my degree – the only way I was able to make it relatable was by trying to decipher the codes left behind by the Freemasons and the Zodiac Killer. Later on, when I got into working as a guitar tech, I found another way to relate math to the real world.
Math is a cornerstone of music theory and instrument design. Math can explain to us why certain notes sound very pleasing when we group them together as chords, thus it is at the very root of music theory. Math is also at the very root of studies in acoustics and mechanics as they pertain to musical instruments.
Acoustics of playing guitar can be explained very simply as a string vibrating, which vibrates the air, which vibrates your eardrum. Your brain interprets these sounds as different tones and rhythms. These vibrations can be manipulated by our hands to create different rhythms and patterns or tones, all of which can be explained by math. Anytime we’re talking about vibrating air, we’re talking about waves, waves are something we can measure. There are some very basic math principles that can explain why the twelfth fret and an open string on your guitar are the same note, but one octave higher. Noting these mathematical relationships can not only help us understand the fretboard better, it can help us understand guitar construction, and perhaps even help us understand the way we hear sounds in general.
In this article we will introduce a couple mathematical concepts and their contribution to the world of music. Please note that these are only a couple of the mathematical concepts – the world of music theory is full of them but for the sake of time, we will highlight a couple that pertain to instrument design.
Timbre, Wavelengths, and Variances
You may have noticed that an A note on a guitar sounds different than an A played on a violin or saxophone or harmonica. It even sounds different, albeit slightly, from one guitar to another. We hear more of a drastic difference sonically with acoustic guitars than electric guitars. Electric guitars work with a magnetic field created by the vibrating string, where acoustic guitars only work with the air that the string moves and the resonant properties of the body of the guitar. We know that when a note is played on a guitar, we are not only hearing that note, we re not only hearing one frequency, we also hear several variances within the note. A frequency is measured in hertz (Hz) and is the measurement for how many times something happens in once second. For example, a middle C has a frequency of about 262 (the actual measurement is 261.63, but we’ll round up). This means that a middle C vibrates 262 times a second. The lower the frequency (Hz) the lower the tone. Some of the lowest frequencies we can hear with our ears, you can almost count the vibrations – the range for humans is roughly 20Hz to 20000Hz. Understanding frequencies can take on a vital role when mixing music – ensuring that the snare drum or vocals are not occupying the same frequency as the guitar. For the snare, this usually happens somewhere around 500Hz, depending on the individual snare and guitar, of course.
When we hear a note made by a musical instrument, we are not just hearing that frequency, that would just sound like a steady tone or a “beep”, we know that a C when plucked on a guitar has a lot more character – this character, and the overall sound of a given guitar is determined by the variances in the frequency – these are often referred to as overtones, partials, or harmonics.
These variances (or an instrument’s timbre) can be determined by a number of factors ranging from instrument construction and design, to our hands and how we attack the string. When a string is plucked, it vibrates the air, it also vibrates everything around it, including the instrument itself. It’s the same concept that can allow an opera singer to shatter a wine glass or the theory of “the brown note” which, in theory, would make someone void their bowels (see the South Park episode “World Wide Recorder Concert” for a highly detailed scientific documentary on the subject). Designing an instrument to best exploit and convey these resonances has been the pursuit of every luthier that has ever lived.
Back to the acoustic guitar, if we want to build a guitar to sound a certain way, we first pick the frequencies we want to feature, then we find the wood that best conveys those resonant frequencies. Mahogany, for example, gives us a darker sound because it reflects those frequencies in a different way than maple does. This is what luthiers have to think about when deciding on what kind of wood to use on which parts of the guitar. Some companies will use certain types of wood, even thermally modified wood, on certain parts, to give a guitar the resonant properties desired. At the root of all this science is an understanding of wavelengths and resonance.
The Golden Ratio, Phi, and the Fibonacci Sequence
Phi is the symbol referencing something known as The Golden Ratio. The Golden Ratio is obtained when a line is divided into two parts such that the longer part divided by the smaller part is equal to entire length of the line divided by the longer part. The Golden Ratio is approximately 1.618. Most acoustic guitars are similar or very close to a phi proportioned guitar, the curves often match the “The Golden Spiral”. So if you’ve ever wondered why most acoustic guitars look alike, and why they are that shape, you can thank math for that.
Interestingly enough, the Golden Ratio, particularly The Golden Spiral are based on a sequence of numbers called the Fibonacci Sequence which we find occurring all over the place in nature. The Fibonacci Sequence is where we add each number to the one preceding it in the sequence to get the next number. So it would go, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on.
You might recognize these numbers if you are familiar with music theory. There are 13 notes in a span of any note through its octave. There are 8 notes in a scale and the 3rd and 5th notes create the foundation for all chords. So it’s no surprise that this sequence of numbers has been used by numerous classical composers, but here’s an example of the Fibonacci Sequence being put to work by Tool in “Lateralus.” First, we see the sequence being used in the number of syllables in each line:
 white are
 all I see
 in my infancy
 red and yellow then came to be
 reaching out to me
 lets me see
 there is
 more and
 beckons me
 to look through to these
 infinite possibilities
 as below so above and beyond I imagine
 drawn outside the lines of reason
 push the envelope
 watch it bend
Tool’s nod to math doesn’t stop there, in addition to the concept of spirals being used throughout the song, The Golden Ratio and Phi rear their heads as well. Bach used to use The Golden Ratio to balance his songs – using it to decide where to put key changes, or other alterations in tempo or texture. Tool adopted that concept as well. Maynard begins singing one minute and thirty-seven seconds into the song, if we do that math, that’s 1.617 minutes (The Golden Ratio or Phi = 1.618). The time signatures of the chorus change from 9/8 to 8/8 to 7/8, symbolizing a spiral. In addition, the number 987 is part of the Fibonacci sequence.
If you’ve seen Tool live, you will note many visual examples of fractals being used as well. A fractal is, generally put, an image that contains a smaller image of itself. There are musical fractals as well which is where a piece of music will harmonize with a slower version of itself. Bach was famous for using these in pieces of his compositions, usually to build tension. Often visual representations are represented as spiraling shapes.
Pythagoras and the Rule of 18
Have you ever wondered why the frets towards the headstock are further apart than those closer to the bridge? Some of you may remember a Greek mathematician named Pythagoras who famously gave us the Pythagorean Theorem (a²+b²=c²) which you probably studied at some point in high school. He also figured out fret spacing for us. Pythagoras did a lot of work with ratios. He discovered that if we take a string that is vibrating and we cut the string in half, it will produce the same note, except one octave higher. This is why on open string and its relative 12th fret are the same note – the 12th fret is half way between the bridge and the nut.
The spacing of the frets is based on something we like to call Pythagoras’ Rule of 18 although the actual value we use for the calculation is 17.817. To calculate the location of each fret, we take the length of the string and divide it by 17.817 – this will give us the location of our first fret. To get the second fret, we take (string length – distance to first fret) divided by 17.817. The answer will give us the distance from the first fret to the second fret. This is a common procedure to use to determine fret spacing.
Another way to space frets is using the equation (distance from the nut to the bridge) divided by (2^(x/12)) where x = the number of the fret we want to find. This will give us the positions for an “equal-tempered scale”.