Can we verify this mathematically? Simple fraction small denominator , as expected. And the interval between B and C? There is a sound clash between B and C notes separated by a semitone apart do not sound well together. Complex fraction, as expected. If you understand a bit of mathematics you noticed that, when doing the calculations of frequencies and roots, we work intrinsically with the logarithm to the base 2.
For this reason, piano builders placed the shape of a logarithm chart on the body of the piano, to make a reference to this mathematical-musical discovery. There are many other mathematical explanations for various questions in music, and to show them here it will be necessary to address more advanced subjects, such as the Fourier series. A Fourier series can be used to describe the behavior of a wave in physics.
Basically, this series is formed by a main harmonic added to other secondary harmonics. The equation can be described as follows:. When a string vibrates, what we hear is not a single pure sound, but a superposition of several sounds , whose frequencies are all multiples of the fundamental frequency.
The fundamental frequency is the main one, which contributes the most, and the other multiple frequencies are called harmonics. The sound of a tuning fork, for example, does not have the highest harmonics, basically containing only the fundamental frequency.
This is great as a reference for this reason the tuning fork is usually used to tune instruments , but in contrast, the tuning fork has a poor sound, without much beauty or richness of sound.
The proportion with which harmonics are added to a fundamental frequency contributes to the production of timbre characteristic of each instrument. This is why the harmonica has a different sound than the flute, for example.
And obviously, different materials produce different harmonics. This means that the production of a quality instrument takes into account each characteristic of each material, such as the type of wood used in the body of a guitar. Going a step further, when a musician plugs an instrument through a cable to the amplifier, each element in this circuit may end up filtering some harmonics, which reduces the sound quality.
This is why it is very important to invest not only in an instrument, but in each particular equipment. The sound engineering process, which develops analog-digital devices to capture sound waves and store them digitally, always seeks to preserve the original wave shape as much as possible.
Studio editing such as noise removal also use these concepts, trying to identify which harmonics are polluting the original wave. Our goal was to show you how music works mathematically and how logical relationships are understood by our brain. Obviously, we did everything here using approximations rounded numbers , as a more accurate analysis would be tedious for most readers and would also require more rigorous mathematical and physical testing. It is not necessary to memorize everything we teach in this topic, just keep in mind that music did not come out of nowhere, it is the result of a numerical organization.
All of this interpretation is done by our brains. The bottom line is that if you are a musician, then you are in one way or another a mathematician, as the feelings of pleasure you feel when listening to music hide subliminal calculations. The more you practice, study and know music, the more this skill will develop. You will probably start to enjoy listening to music that did not bring you great feelings before.
We can compare this with a physics student from the 1st semester. If he reads a modern physics book, it will look like Greek to him; it will not bring him any pleasure. But a few years later, when he has reached a solid mathematical foundation and comes across this same book, perhaps he will come to love this subject and want to dedicate his life to it.
Toggle navigation. Mathematics and Music. When we play this string, it vibrates see the drawing below : Pythagoras decided to divide this string into two parts and touched each end again. The sound produced was exactly the same, only higher since it was the same note an octave higher : Pythagoras did not stop there. He decided to try what the sound would look like if the string was divided into 3 parts: He noticed that a new sound appeared, different from the previous one.
Over time, the notes were given the names we know today. The mathematics of musical scales Many peoples and cultures have created their own musical scales. And we have already seen that a note multiplied by 2 is itself an octave higher.
By finding notes from frequencies Everything we discussed will become clearer by looking at the notes on a piano: If we consider that the first C leftmost has an f frequency, the second C one octave higher will have a 2 f frequency.
Follow the logic: But what about frequencies 3 f , 5 f , 6 f and 7 f , where they are? By performing the same procedure, we can find note 5 f , which is halfway between frequencies 4 f and 6 f : Now we can also think backwards to find the frequencies of the first octave.
It tends to be that people are good at math and science or art and music, as if the two elements could not be placed together logically. In actuality, math and music are indeed related and we commonly use numbers and math to describe and teach music. Musical pieces are read much like you would read math symbols. The symbols represent some bit of information about the piece. Musical pieces are divided into sections called measures or bars. Each measure embodies an equal amount of time.
Furthermore, each measure is divided into equal portions called beats. These are all mathematical divisions of time. Fractions are used in music to indicate lengths of notes.
In a musical piece, the time signature tells the musician information about the rhythm of the piece. A time signature is generally written as two integers, one above the other. The number on the bottom tells the musician which note in the piece gets a single beat count. The organization Roberts chose has the advantage of allowing students, particularly those with a musical background, to start in familiar territory, the piano, and probably prevents some confusion.
A person who has played an instrument or sung in a choir will feel right at home with notation they understand, and a person without much musical experience has probably at least seen a piano keyboard. Starting with pitch and interval perception instead and then moving to temperament could have the effect of pulling the rug out from under people. Twelve half-steps in an octave is practically an axiom in Western music, and it could be confusing to have to justify it at the outset.
The next four chapters feel more like a collection of case studies. Since each topic is largely self-contained, instructors would have ample opportunity to pick and choose the sections that will work the best for their particular classes. Chapter 5, on musical symmetries, explores Bach fugues and the more literally symmetric music of modern composers such as Bartok and Hindemith.
This chapter also contains a gentle introduction to group theory. Chapter 6, on change ringing, delves more deeply into group theory. Change ringing is an unusual musical practice that originated in 17th century England.
It takes place in bell towers with large bells that are mounted so they can swing in a full circle. One person stands under each bell holding a rope, and the bells are rung in sequence.
Experienced change ringers learn to carefully control the timing of the bells, so bells can change positions in the sequence. Change ringing is not about artistic or emotional expression.
Instead, melodies are basically combinatorial. Ringers will start by playing a descending scale, and in subsequent rounds, neighboring bells will swap positions. My illustrious career as a change ringer lasted about three weeks at the beginning of graduate school, but in that brief time I was floored by the fact that we were actually performing permutation groups.
Chapter 7 is on tone music, another topic that gets a fair amount of press as a place where mathematics and music intersect. The final chapter, on modern music created with mathematics, has three case studies of modern composers who use mathematics in their music in some way.
Throughout the book, in another contrast to other textbooks about music and mathematics, Roberts often uses music as a motivation for introducing a mathematical topic and takes a detour into the mathematics itself.
In the chapter about rhythm, he ventures into infinite geometric series using dotted rhythms for motivation. Although I have a lot of experience in mathematics, music, and their intersection, I learned a few things myself. I had not known much about the way Indian classical music led to the discovery of the Hemachandra—Fibonacci sequence 1, 1, 2, 3, 5, 8, In the introduction, Roberts makes some suggestions about how to use the text for either a year-long or one-semester class on math and music.
I have not taught such a course, but it seems like the text would be appropriate for a freshman seminar or liberal arts math class with a group of students that is diverse with regard to their mathematical and musical backgrounds. The text has plenty of worked-out examples, exercises, and two interesting projects for instructors to choose from.
The project in the final chapter is particularly compelling: write music that is influenced by mathematics in some way. I only regret that I did not get to listen to any of the pieces his students wrote! The views expressed are those of the author s and are not necessarily those of Scientific American. Follow Evelyn Lamb on Twitter. Already a subscriber? Sign in.
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