Some believe that the most fundamental aspects of reality -- everything that we are and everything that we perceive -- ultimately come down to mathematics. It is the language of science and perhaps the only surviving bastion of irrefutable truths in the aftermath of the Age of Enlightenment. It is this irrefutable quality, this perfection of sorts, that also lends mathematics a certain beauty; in fact, the origin of all beauty may come down to simple mathematical relationships.
Our brains perceive mathematical relationships in countless ways, but perhaps none are so direct as the way in which we process sound. Suppose I were given two tuning forks, one designed for a frequency of 440 Hz and the other for 880 Hz. When I strike the first, the metal begins to vibrate, moving back and forth at a rate of 440 times per second. This vibration, in turn, causes the surrounding air molecules to oscillate at the same frequency, an oscillation that travels outwards from the tuning fork and reaches my ear. Assuming that the tuning fork continues to vibrate at this frequency, my brain will interpret the oscillation of air molecules as a steady and constant "pitch."
Now suppose I strike the second tuning fork, which vibrates 880 times per second. This new pitch corresponds to a frequency twice that of the first, and if I strike the second tuning fork while the first is sounding, they together produce a sound something like this. Notice how smoothly the pitches blend together. The interval heard here is called an "octave," a musical term reserved for any pair of pitches with a frequency ratio of 2:1. Our ear easily identifies the relationship between these two pitches because their frequencies are in a small, natural-number ratio to one another. By contrast, listen to the major seventh, an interval that corresponds to a frequency ratio of about 15:8. The blend is not nearly so pleasing to the ear.
When two pitches blend together well, like the octave, they are referred to as "stable," or a "consonance." Those that don't blend so well, such as the major seventh, are referred to as a "dissonance." In medieval music, the consonances were the octave (2:1), the perfect fifth (3:2, listen), and the perfect fourth (4:3, listen). In the early 15th century, starting with the Burgundian School, intervals of a major third (5:4, listen) and a major sixth (5:3, listen) began to be treated as consonances, allowing for developments such as triadic harmony.
Our brains perceive mathematical relationships in countless ways, but perhaps none are so direct as the way in which we process sound. Suppose I were given two tuning forks, one designed for a frequency of 440 Hz and the other for 880 Hz. When I strike the first, the metal begins to vibrate, moving back and forth at a rate of 440 times per second. This vibration, in turn, causes the surrounding air molecules to oscillate at the same frequency, an oscillation that travels outwards from the tuning fork and reaches my ear. Assuming that the tuning fork continues to vibrate at this frequency, my brain will interpret the oscillation of air molecules as a steady and constant "pitch."
Now suppose I strike the second tuning fork, which vibrates 880 times per second. This new pitch corresponds to a frequency twice that of the first, and if I strike the second tuning fork while the first is sounding, they together produce a sound something like this. Notice how smoothly the pitches blend together. The interval heard here is called an "octave," a musical term reserved for any pair of pitches with a frequency ratio of 2:1. Our ear easily identifies the relationship between these two pitches because their frequencies are in a small, natural-number ratio to one another. By contrast, listen to the major seventh, an interval that corresponds to a frequency ratio of about 15:8. The blend is not nearly so pleasing to the ear.
When two pitches blend together well, like the octave, they are referred to as "stable," or a "consonance." Those that don't blend so well, such as the major seventh, are referred to as a "dissonance." In medieval music, the consonances were the octave (2:1), the perfect fifth (3:2, listen), and the perfect fourth (4:3, listen). In the early 15th century, starting with the Burgundian School, intervals of a major third (5:4, listen) and a major sixth (5:3, listen) began to be treated as consonances, allowing for developments such as triadic harmony.
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