Music and Mathematics
The Relationship between the Worlds of Harmony and of Numbers
Mathematicians have been exploring the mysteries of music for centuries. The questions being asked are those which seek to better understand the perception of beauty.
It is surely not difficult for anyone with experience in the subject known as “music theory” to realize that there is a very definite and precise relationship between music and mathematics. There is, in fact, practically no end to the number of areas where math rears its head in the world of harmony, melody, and audio aesthetics.
Math as a Foundation for Tonal Quality
The very nature of what constitutes the audible sensation of music itself – the creation of sound waves vibrating at distinct frequencies – can create an almost endless amount of enjoyment to a sufficiently motivated mathematician.
It is here that one can explore the ideas of harmony and dissonance – of certain notes “blending” with certain specific notes, but clashing harshly with others. Such early mathematicians as Pythagoras (and certainly early, for mathematical inquiries into music may have been in existence even in ancient Babylonia during the fourth millennium, B.C.) around the fifth century, B.C.
Pythagorian Tuning
Pythagoras, while perhaps not fully understanding the scientific nature of sound waves and their relation to the human ear, surely understood that there were mathematical answers to musical problems. As such, he is said to have developed a math-based system of tuning.
Pythagoras’ tuning was based on the principle of perfect fifths – intervals between the first and the fifth note of the proposed scale would be tuned to a ratio of 2:3. Alternately, the ratio between the first and eight notes (which would be an octave) would be 1:2. What these ratios signify is the ratio of the wavelength of a given note to another.
With the basic idea of wavelengths and musical notes understood, it becomes a fairly simple mathematical exploration to discover all the different ratios of notes that might sound good together (harmonize). By combining notes which have complementary wavelengths – that is, the peaks and troughs of the waves coincide with each other – a nice-sounding harmony is created. When the peaks and troughs diverge, however, “ugly” dissonance is created.
One can easily hear this effect in action by listening to two instruments attempting to tune to the same note. When dissonance occurs, the rhythmic difference in wavelengths can be easily heard. As the two notes come closer together, the dissonant rhythm becomes slower and slower, until finally smoothing out into an even pitch.
The Golden Ratio and Fibonacci Sequence
While the applications of mathematics are many, and have kept mathematicians and musicians alike occupied for centuries, covering such mathematical areas as set theory, algebra and prime number theory, one of the most interesting uses of mathematics might be the application of the golden ratio and the Fibonacci sequence to the aesthetic principles of music.
The golden ratio is a scientifically mysterious ratio that is supposedly pleasing to the human sense of aesthetic, while the Fibonacci sequence, likewise, is a seemingly random sequence of numbers (1,1,2,3,5,8 – where every number is the sum of the two preceding it) which has been shown to have countless applications toward understanding the patterns of nature itself.
These two interrelated mathematical concepts have been show to play roles in the human perception of musical aesthetics. For instance, it has been shown that certain groupings of notes in a major scale – such as the first, fifth, major and minor sixths are demonstrative of consecutive numbers in the Fibonacci sequence, while at the same time demonstrating the golden ratio.
Is this a part of what helps a human being appreciate a piece of beautiful music?
This is one of those questions that mathematicians love to explore. For embedded in the deep language of music, there are countless mysteries to unlock – a great many of them purely mathematical in nature.
In this sense, an understanding of mathematics may very well help to better understand the human mind itself.
As Gottfried Liebnitz once said:
“Music is the pleasure the human soul experiences from counting without being aware that it is counting.”
References:
Khramov, Mykhaylo. “Sonontometry.”
“Golden Ratio: Aesthetic Uses.”
