REFLECTION
Being a lover of both music and math, I wanted to bring my two passions together. So, for my history of math class I wrote and presented on a small portion of the connection between math and music. I learned a lot while conducting my research, and truly enjoyed presenting my findings to the class. The following is the paper that I wrote.
For many people, mathematics is an enigma. Characterized by the impression of numbers and calculations taught at school, it is often accompanied by the feelings of rejection and disinterest, and it is believed to be strictly rational, abstract, cold and soulless.
Music, on the other hand, has something to do with emotion, with feelings, and with life. It is present in all daily routines. Everyone has sung a song, pressed a key on a piano, blown into a flute, and therefore made music. It is something people can interact with; it is a way of expression and a part of everyone’s existence.
The motivation for investigating the connections between these two apparent opposites therefore is not very obvious, and it is unclear in what aspects of both topics such a relationship could be sought. Moreover, if one accepts some mathematical aspects in music such as rhythm and pitch, it is far more difficult to imagine any musicality in mathematics. The countability and the strong order of mathematics do not seem to coincide with an artistic pattern.
However, there are different aspects, which indicate this sort of relationship. Firstly, research has proved that children playing piano often show improved reasoning skills like those applied in solving jigsaw puzzles, playing chess or conducting mathematical deductions (Mann 17). Secondly, it has been noticed in a particular investigation that the percentage of undergraduate students having taken a music course was about eleven percent above average amongst mathematics majors (Mann 18).
This paper examines the relationship between mathematics and music from two different points of view. The first describes some ideas about harmony; tones and tunings generated by the ancient Greeks and the second illuminates artistic attributes of mathematics.
In the time of the ancient Greeks, mathematics and music were strongly connected. Music was considered as a strictly mathematical discipline, dealing with number relationships, ratios, and proportions. In the quadrivium (the curriculum of the Pythagorean School) music was placed on the same level as arithmetic, geometry and astronomy. This interpretation totally neglected the creative aspects of musical performance. Music was the science of sound and harmony (Grout 37).
The basic notions in this context were those of consonance and dissonance. People had noticed very early that two different notes do not always sound pleasant (consonant) when played together. Moreover, the ancient Greeks discovered that to a note with a given frequency only those other notes whose frequencies were integer multiples of the first could be properly combined. If, for example, a note of the frequency 220 Hz was given, the notes of frequencies 440 Hz, 660 Hz, 880 Hz, 1100Hz and so on sounded best when played together with the first (Grout 44).
The most important frequency ratio is 1:2, which is called an octave in the Western system of music notation. Two different notes in such a relation are often considered as principally the same (and are therefore given the same name), only varying in their pitch but not in their character (Grout 77). The Greeks saw in the octave a ‘cyclic identity’. The following ratios build the musical fifth (2:3), fourth (3:4), major third (4:5) and minor third (5:6), which all have their importance in the creation of chords (Grout 75).
The difference between a fifth and a fourth was defined as a ‘whole’ tone, which results in a ratio of 8:9. These ratios correspond not only to the sounding frequencies but also to the relative string lengths, which made it easy to find consonant notes starting from a base frequency. Shortening a string to two thirds of its length creates the musical interval of a fifth for example (Grout 78).
All these studies of ‘harmonic’ ratios and proportions were the essence of music during Pythagorean times. This perception, however, got its importance at the end of the Middle Ages, when more complex music was developed. Despite the ‘perfect’ ratios, there occurred new dissonances when particular chords, different keys or a greater scale of notes were used (Grout 81).
The explanation for this phenomenon was the incommensurability of thirds, fifths and octaves when defined by integer ratios. By adding several intervals of these types to a base note, one never reaches an octave of the base note again (Grout 82). In other words, an octave (1:2) cannot be subdivided into a finite number of equal intervals of the Pythagorean type (x: x +1 | x being and integer). Adding whole tones defined by the ratio 9:8 to a base not with the frequency f, for example, never creates a new note with the frequency 2f, 3f, 4f or similar. However, adding six whole tones to a note almost creates its first octave defined by the double frequency (Grout 83).
Considering these characteristics of the Pythagorean intervals, the need of another tuning system developed. Several attempts were made, buy only one has survived until nowadays: the system of dividing an octave into twelve equal (‘even-tempered’) semi-tones introduced by Johann Sebastian Bach. Founding on the ratio 1:2 for octaves, all the other Pythagorean intervals were slightly tempered (adjusted) in order to fit into this new pattern. A whole tone no longer was defined by the ratio 9:8, but by two semi-tones (each expressed by 12*(2^1/2)) obtaining the numerical value 12*(2^1/2) * 12*(2^1/2) = 1.1225. The even-tempered fifth then was defined by seven semi-tones and therefore slightly bigger than the Pythagorean fourth (Grout 86).
The controversy within this tempering process is that the human ear still prefers the ‘pure’ Pythagorean intervals, whereas a tempered scale is necessary for complex chordal music. Musicians nowadays have to cope with these slight dissonances in order to tune an instrument in a way that it fits into this even-tempered pattern.
With the evolution of this more complicated mathematical model for tuning an instrument, and with the increased importance of musicality and performance, music and mathematics in this aspect have lost the close relationship known in ancient Greek times. As an even-tempered interval could no longer be expressed as a ratio (12*(2^1/2) is an irrational number), the musicians learned to tune an instrument by training their ear rather than by applying mathematical principles. Music from this point of view released itself from mathematical domination (Grout 88).
All these aspects of mathematical patterns in sound, harmony and composition do not convincingly explain the outstanding affinity of mathematicians for music. Being a mathematician does not mean discovering numbers everywhere and enjoying only issues with strong mathematical connotations. The essential relation is therefore presumed to be found on another level.
It is noticeable that the above-mentioned affinity is not reciprocated. Musicians do not usually show the same interest for mathematics as mathematicians for music. One therefore must suppose that the decisive aspect cannot lie in arithmetic, the part of mathematics people sometimes consider to be in fact the whole subject. It is probably more the area of mathematical thinking, mind-setting and problem-solving which creates these connections (Bernstein 33).
In mathematical research, the omnipresence of words such as beauty, harmony and elegance is found. Whereas musicians sometimes develop a particularly well-formed melody or apply an outstanding harmony, mathematicians often seek ‘simple’ and elegant proofs. Moreover, the sensations in solving a mathematical problem seem to be similar to those appearing when performing a musical work. Most important is the creative aspect, which lies within both of these disciplines (Bernstein 49).
Interesting evidence for this idea has been presented by Bernstein, who compared the history of music with the history of mathematics based on the following three main arguments (59):
1. “Mathematics has many of the characteristics of an art.
2. Viewed as an art, it is possible to identify artistic periods in mathematics: Renaissance, Baroque, Classical, and Romantic.
3. These periods coincide nicely and share many characteristics with the corresponding musical epochs, but are significantly different from those of painting and literature.”
Relating to concepts such as dualism (Baroque), universality (Classical) and eternity (Romantic), he draws out surprising similarities between the evolution of mathematics and music.
Moreover, he outlines the necessity of a change in mathematical education towards a more musical style. “Students should make mathematics together (as in fact professional mathematicians do), not alone. […] And finally, students should perform mathematics; they should sing mathematics and dance mathematics.” This would probably help people understand what mathematics really is, namely not divine, but mortal, and not law, but taste.
In spite of the highly speculative aspect within such ideas, this is probably the fundamental point of view when seeking connections between mathematics and music. It is the musicality in the mathematical way of thinking that attracts mathematicians to music. This, however, is difficult for people, who are not familiar with this particular pattern of mind, to comprehend. It is therefore probable – as has been stated by Bernstein – that the degree of understanding such relationships is proportional to the observer’s understanding of both mathematics and music.
This paper has outlined two different approaches to the question of how mathematics and music relate to each other. The first showed the particular perception of music by the ancient Greeks putting less importance on melody and movement than on tone, tuning and static harmony. The most fundamental approach, however, was the second, in which connections were revealed concerning the artistic aspect of the mathematical way of thinking.
It is obvious that these are only examples for investigating such a relationship and that other comparisons could be attempted. However, these two represent probably the most often discussed concepts and ideas and are particularly suitable for providing a first impression of this topic. Whatever links between music and mathematics exist, both of them are obviously still very different disciplines, and one should not try to impose one on the other. It would be wrong to attempt explaining all the shapes of music by mathematical means as well as there would be no sense in studying mathematics only from a musicological point of view. However, it would be enriching if these relationships were introduced into mathematical education in order to release mathematics from its often too serious connotations.
It is important to show people that mathematics, in one way, is as much an art as it is a science. This probably would alter its common perception, and people would understand better its essence and its universality. This task, however, could take several years.
Bibliography:
Bernstein, Leonard. The Infinite Variety of Music. Simon and Schuster, New York: 1966.
Grout, Donald Jay. A History of Western Music. Norton & Company, New York: 1960.
Mann, William. Music in Time. Harry Abrahms Inc., New York: 1982.