Mathematics is often viewed by humans as being abstract and ultimately unable to be understood by anyone unwilling to devote large amounts of time and effort to its study. This is an unfortunate and erroneous view because in reality mathematics is the systemized study of what we find in the world around us. Mathematics is defined by the world. And consequently the world is made of mathematics. One major example of a concept and practice with which many, if not most, humans are familiar and that is deeply rooted in mathematics is music. The mechanics of music, music theory, and many actual compositions all demonstrate mathematical aspects.
briefing on Pythagoras
It is unexpected and unfortunate to "math in music" enthusiasts that few people recognize the connection between mathematical concepts and music. This was not always the case. One of the first documented instances of this recognition comes from the great Pythagoras, philosopher and prodigious mathematician of ancient Greece. He and his followers devoted their lives to mathematics and saw divinity in number theory. They considered all of their contributions to math to be forms of worship - which encouraged them to push the boundaries of math. On the subject of math's involvement in music, Boethius,1 a later philosopher who followed Pythagorean ideals, wrote of "four mathematical disciplines, of which music is one..." The other three are concerned with investigation of rational truth, but music concerns not only speculation, but also human behavior...As an analysis of the Pythagorean point of view, another2 author wrote "it is a fulcrum between the material world and the meta-reality of number, contributing to the dialogue of correspondence between the two."
Pythagoras and his followers are forever remembered for their (sometimes misled)contribution to the field of mathematics, but their contribution to the field of music has also been very rich. Several of these will be outlined later.
Every aspect of music is concerned with mathematics, from patterns of the rhythm to the reason why certain notes are dissonant. The most tangible aspects of mathematical involvement include the mechanics of music and melody, the modes- a name for the groupings of notes, such as scales-, and instances of famous or important aspects of math integrated into actual compositions.
"mechanics of music"
In order to understand the more complex involvement of math in music, such as configurations of notes or even full compositions, it helps if one knows the very basics. The phrase "mechanics of music" refers to how individual notes are heard and differentiated from each other, as well as why certain notes sound harmonious or discordant when played together. The notes one hears when music is played or sung are universally defined despite the medium used to make them. How can this be, when the notes have different consistencies?
Notes are specified pitches on a scale. Pitch is determined by the frequency of the sound wave. Frequency is the measure of the number of times a single wave passes a given point. Therefore, a note is determined by the length of the wave produced. For instance, Middle C has a frequency of 262 Hz. Anything that can produce a sound wave with this frequency can produce a C.
Assonance and dissonance between notes are determined by wavelength as well. When two or three notes are played together, their waves are traveling at different rates.
If the crests of the waves are overlapping every few waves and at a regular pattern, the sound is considered to be consonant and is generally regarded as more esthetically pleasing. If the crests of the sound waves overlap only seldom or the pattern s repetition takes many waves to repeat, the sound is considered dissonant. Perhaps an easy way to express this would be by comparing the frequencies. Following is a list of notes, frequencies of these notes, and ratios of said notes to Middle C:
note: frequency approximate ratio.
This is intended to show that certain notes sound more assonant because of the comparative frequencies. For example, the frequencies of the notes C, E, and G match up very well because the the ratio of E to C is 5:4, the ratio of G to E is 6:5, and the ratio of G to C is 3:2. These three notes are a type of combination of notes found often in music called chords. In fact, they exemplify the definition of chords: chords are groups of notes whose frequencies form simple ratios to each other. When played together, the frequencies form simple patterns. This concept has been explained more thoroughly in Sol-Fa - The Key to Temperament.
The next mechanism of music that can be described is the making of actual notes. Pythagoras had much to do with discoveries in this area. Pythagoras realized that if a blacksmith, hit two pieces of metal and one of them is twice the mass of the other, the sounds produced will be exactly one octave apart. This caused him to experiment further and ultimately to discover that pitch is proportional to mass. He illustrated this with bells, glasses of water, and string.
The use of strings is the easiest way to illustrate Pythagoras’s findings because they afford a unique ability to experiment with ratios. In one of his most famous experiments, a stretched string was divided by simple arithmetical ratios (1:2, 3:4, 2:3, .....) and plucked. In this way he demonstrated that the intervals, or distances between tones, that the string sounded before and after it was divided are the most fundamental intervals the ear perceives. These intervals, which occur in the music of nearly all cultures, either in melody or in harmony, are the octave, the fifth and the fourth, in the scale with which we, as westerners, are familiar. (They are named this because they involve a cross over eight notes, five notes, and four notes in our scale, respectively.)
With the newfound knowledge that Pythagoras and his followers had, they began to make patterns out of the notes. This pattern making is a tendency among every culture and the configurations of notes that result are known in music terminology as modes.
Pythagoras and his followers, for example, considered the numbers two, three, and five to be sacred, representing heaven, earth, and the joining of the two, respectively. They developed a mode that involved these three numbers.
All other Greek music was centered on the intervals mentioned above, the octave, the fifth, and the fourth. English music that came about in about the 14th Century, however, was based on a unique system of thirds and sixths. Our Western heritage derives almost entirely from these two systems.
Arabic speaking countries have a system that contains several differant intervals between the frequencies of the notes, including one that is 1.5 times further apart in frequency than the western system. This organization means that musicians from one background have a very hard time learning music from the other because the ratios that are subconsciously learned while playing are not found in the same way in the different types of music. This is also why Arabic music has such a distinct and easily recognized sound to our ears: we have learned the ratios of our western scales to the point where anything else sounds discordant.
In fact, even when the step size is the same, different modes lead to characteristics by which we can distinguish music. This began with ancient cultures, and has progressed with them. Therefore we can distinguish between genres of music based on modes. For instance Blues progressions are easily discernable. This makes little sense unless modes are taken into consideration; after all, Blues progressions are just series of notes, as is all music. However, at the heart of most Blues music is the minor hexatonic scale, which has a pattern of 3-2-1-1-3-2 where each digit indicates the number of half steps up (or down) the scale. Therefore the hexatonic scale for A would be A-C-D-Db-E-G-A. Because this pattern is so prominent in Blues, whenever pieces of it are stylized and played, starting from any note, the sound is very "bluesy." This happens because we begin to associate certain ratios to blues music after hearing it only a few times, and then can recognize the ratio when recreated.
Most rock music and folk music is written in major scales and major keys, and therefore it is hard to pinpoint certain scales that sound like one of these specific genres (as they are differentiated by style instead of ratio).
math in music : some examples
When music is based upon naturally developed scales, it becomes easily recognizable because of the ratios of the notes to one another. This is not because the makers of the music were purposefully trying to achieve mathematical ratios, but because of the contrary: the ratios were what made the sound they wanted. There are many instances of artists including in their music other ratios that are important to mathematics. Some seem to simply find these important mathematical ratios to be more aesthetically pleasing. Others purposefully include them as a tribute to the beauty of mathematics, proportions, and therefore nature. With many more it is unknown whether the artist intended the ratios to be included in the work, and if so for what purpose. All that is known of these often prodigious and talented musicians is that their music did involve mathematical ratios, often complexly and intricately woven into the music.
One well-known artist that falls into the latter category is Mozart. It is said that Mozart was intrigued by mathematics, and one characteristic of all of his music is his "elegant proportions." Modern analyses of his works have revealed that the golden ratio appears in many of his compositions. The golden ratio is described starting with a line that is one unit long. The line is divided into two unequal segments, so that the shorter one equals x and the longer one equals (1 - x). The ratio of the shorter segment to the longer one is set equal to the ratio of the longer segment to the overall line; that is,
(1-x)/x = x/1
That equality leads to a quadratic equation that can be used to solve for x, and substituting that value back into the equality yields a common ratio of approximately 0.618. Mozart s works are often divided into two sections, the Exposition in which the musical theme is introduced, and the Development and Recapitulation in which the theme is developed and revisited. The numbers of measures in each were compared, and it was found that in nearly all of his works, Mozart divided them using the golden ratio, with the Exposition as the shorter of the parts and the Development and Recapitulation as the longer. Because some of the compositions deviate substantially from the ratio, however, while most others seem to be as close as they can get in whole numbers, there is now a long-standing debate among experts. Did Mozart, a man who seemed to respect mathematics, purposefully integrate the ratio into his music, or did he simply find that his pieces sounded better when he used ratios he had no knowledge of? Once again, we will never know definitively what the answer is to this question; we only know that the ratio is there.
math in music : contemporary examples
Because the Fibonacci sequence occurs so often in nature, icluding in the spirals of pinecones, the splitting of trees, the number of petals on a flower, et cetera, nature is thought to be the source of a surprising connection between the series and ragtime music.
A Fibonacci series is defined by the expression:
Fn+1 = Fn-1 + Fn , if n>1 where F0 = 0 and F1 = 1
Ragtime music was analyzed and broken down bar by bar by mathematicians. They found that the number of notes in a bar was a Fibonacci number surprisingly often. This occurance of a mathematical sequence in music is thought to be a natural progression caused by an either conscious or unconscious immitation of nature.
While the motives of some composers are ambiguous, and while the inclusion of mathematics in ragtime is not thought to be influenced by human purposefullness, there are artists who center their works around mathematical concepts as a tribute to the beauty of nature, of order, or just of math. Examples of these people can be found throughout history, from the Pythagoreans who played almost all of their music in certain ratios, through the middle ages with composers who followed a stringent plan for their music, to the present day where "math music" is becoming a well know term and "fractal music" is becoming more and more popular.
One band that has apparently included mathematical ratios for the pure love of knowledge and as a tribute to the beauty found in it is Tool, a modern progressive rock band. The Fibonacci sequence is referenced when lyrics of one song, "Lateralice," are sung in syllables corresponding to the first six Fibonacci numbers, then cycling down and up again. In other words, the first line has one syllable, the second has one syllable, the third has two, the fourth 3, the fifth 5, and so on, forming a repeating pattern of 1, 1, 2, 3, 5, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8. They have included this sequence perhaps in and attempt to bring the beauty of nature into their art, perhaps as little treat for the knowledgeable. The pattern is quite noticeable in the lyrics because of the halting rhythm when there are several syllables with pauses between them, and while there is never an explanation made by the band, it is obvious that the configuration is no accident.
The presence of mathematics permeates music.
As one delves further into the idea of mathematics and music intertwining, it is only natural to begin to wonder how it is possible that there is so much interconnectedness between the two. Is there perhaps something in our music theory that creates certain configurations? Are the certain configurations simply a product of the human mind, an entity that requires order and can create it even out of absolute chaos? Or is there just something appealing to us about certain ratios, some sort of innate magnetism toward unnamed configurations and patterns of numbers, expressed through music and other mediums? Perhaps at the root of all of these questions is the human search for order in the unknown, peace in chaos.... mysticism, and the divine. Such a situation would not really be surprising: both make references to the divine on a regular basis.
Humans have included music in their lives since the dawn of civilization and as we have progressed as a race, so, too, has our ability to recognize and define patterns progressed. As soon as mathematics arose as a discipline, as an abstract idea to be contemplated, learned from, and supplemented, the subject was recognized in music. From the way notes are produced with a string to the actual frequency of individual sound waves, much has been learned about music, and more is continuously being discovered. But at the same time, over all of these eras, as music has created new maths, so, too, has math created new music. This endless dichotomy creates a relationship that is as beautiful as it is intriguing.