Author: Ellie Wang
Editors: Misha Wichita and Emily Yu
Artist: Olivia Yuan
In the world of music, the Western music system is by far the most prevalent. This standardized system of 12 notes grouped in octaves may seem like an arbitrary one. Yet, the reason why it immediately sounds good or bad when playing certain notes together is because of math.
Sound is the result of vibrating objects bumping into air molecules, which hit other air molecules, and so on, through either a solid, liquid, or gas. This movement ends up as a wave of vibrations that the ears receive and the brain perceives as sound. Like all wave-motion phenomena, sound waves are measured in terms of frequency, the number of times a wave cycle repeats in one second. This is represented by the unit, 1/second or Hertz (Hz).
The frequencies of the Western music system were chosen for how they lined up with each other—in other words, for their ratios. For example, take a common tuning note: A4 or the A note of the 4th octave. This particular pitch has a frequency of 440 Hz. A5—an octave above—has a frequency of 880 Hz. 440 Hz to 880 Hz simplifies to a 1:2 ratio and is the simplest ratio besides playing the same note in unison, which yields a 1:1 ratio. Pythagoras’s theory of the consonance, “pleasantness,” or dissonance, “unpleasantness,” of each interval between two notes is based upon how simplified the ratios are. The simpler the ratio, the more consonant or pleasant the interval, and vice versa: the more complex the ratio, the more dissonant.
After an octave, the second-most simplified ratio would be a fifth, which has a ratio of 2:3. If this pattern is continued above and below, it spans 7 octaves, intervals of 8 notes, with just fifths, or intervals 5 notes apart, touching upon each of the 12 types of notes that Western music has. Then, fill in the spaces with more intervals, and the Western music system is created with Pythagorean tuning, where all fifths are perfectly consonant. Despite creating intervals of uniform size, this tuning of pitches within the octave creates intervals of major thirds and major sixths that are too sharp or too far apart for the ear to accept as musical tones by 22 musical cents (a cent being equal to 1/1200 of an octave). This specific way of tuning pitches then results in extremely dissonant ratios of 81:80.
In other attempts to produce a music system with both uniform and pleasant-sounding intervals, several more tunings were made. Yet, tuning itself is based upon the simpler ratios of the overtone series— a series of notes with frequencies matching the overtones produced by musical instruments. These are the less dominant frequencies simultaneously playing at double, triple, quadruple, and so on from the fundamental frequency, the dominant frequency of note played. While this tuning produces pleasant intervals, it also creates uneven intervals. Meantone temperament, a system centered around major thirds by positioning the major third perfectly in the middle of fifths, suffers from the same problems as Pythagorean tuning.
Currently, the most widely accepted tuning system is equal temperament, which spaces out the 12 notes of the octave into equal semi-tones or half-steps. Equal temperament creates a music system that sounds agreeable to the listener by compromising on creating uniform intervals that allow for harmonious use of all 12 notes in all combinations, even with intervals that don’t match the perfectly euphonious overtone series. Since the 19th century, most, if not all, of music created adheres to this exact system, standardizing music produced across the world.
Citations:
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The Editors of Encyclopaedia Britannica. “Meantone Temperament | Music Theory, Tuning
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