Since the medieval ages, music has evolved tremendously. Today, music is split into a variety of genres, including pop, hip hop, rock, and jazz. However, they all have one common feature: they are out of tune. In fact, almost all modern music is out of tune. But why?
Deriving Pure Intonation Using the Harmonic Series:
Just Intonation is a tuning system that consists of intervals that use simple ratios. It is the purest way to tune any interval in music. The mathematical concept that derives pure or just intonation is the harmonic series. The harmonic series is defined as the summation of all the positive unit fractions: 1+ ½ +⅓ +¼+⅕+⅙…
In music, suppose you have a string that vibrates at A4=440 Hz, the standard concert pitch used in music today. Now, when vibrating the string, it will produce the 440 Hz frequency first (the fundamental frequency) followed by a series of quieter pitches known as overtones that are produced by dividing the next term of the harmonic series. So if you have a fundamental frequency of 440Hz, the second overtone will be 880 Hz, followed by 1320 Hz, 1760 Hz, 2200 Hz and so on. Why is this so? Going back to the string that vibrates at 440 Hz, if you were to shorten the string to ½ the length, the original string would vibrate only ½ the speed of the shortened string. Hence, the harmonic series is used to calculate how the fundamental frequency is ½ of that of the second overtone, ⅓ of that of the third overtone, and so on.
However, when we analyze the ratios between each overtone’s frequency, we start to notice something interesting. The second overtone has a ratio of 2:1 to the fundamental frequency which is the ratio of an octave. When playing these two notes together, the sound is so consonant that you feel like they are the same notes but one is higher, hence the name of the octave. But the ratio between the third overtone and the second overtone is 3:2. This is the ratio of a justly tuned perfect fifth, the second most consonant interval. If we follow this pattern, the next ratios would be 4:3, 5:4, and 6:5; they are the ratios of a justly tuned perfect fourth, major third, and minor third respectively. However, as we continue to move on, the intervals get more and more dissonant. Some ratios, 12:11 and 8:7 for example, do not divide into numbers with simple decimals so they are too dissonant to represent any of the intervals that we recognize in the most commonly used tuning system today. We eventually derive 9:8 and 16:15 to be the major second and minor second respectively, the next two intervals that are recognizable.
But how do we derive the other intervals such as the minor sixth or the major seventh? These intervals cannot be derived using the harmonic series alone. Going back to how the octave is a ratio of 2:1, we simply have to divide that by the ratio of the inversion. The inversion of a minor sixth is a major third. So to find the ratio of a justly tuned minor sixth, we divide 2:1 by the ratio of a major third, 5:4, to get 8:5. We apply this same method to derive the remaining intervals to get 5:3, 16:9 (sometimes substituted for 9:5), and 15:8, which correspond to the major sixth, minor seventh, and major seventh respectively. However, there is still one interval that is missing: the tritone. The tritone is defined as an interval of six semitones, also known as a diminished fifth or augmented fourth. One way to derive the ratio is to start from the ratio of a major third (four semitones) and multiply it by the ratio of major second (two semitones) to get the ratio of a tritone (six semitones). So the ratio would be 5:4 * 9:8 to get 45:32 or 1.40625 in decimals. However, this ratio was considered too precise and complex for a justly tuned interval so it was universally accepted to settle with 7:5 or 1.4 as a simpler ratio for the tritone. Another way to derive the ratio of a tritone is to divide the octave. The tritone is six semitones, exactly half of that of the octave so we can derive the ratio of the tritone simply by taking the square root of the ratio of an octave (2:1) to get a decimal of approximately 1.41. However, irrational numbers cannot be used for just intonation so again, we end up with the ratio of 7:5 from before. Due to the convoluted process of deriving the tritone, many in the Medieval Ages considered this interval to be evil, giving the name of the Devil's Tone, resulting in its banning from any use in music at that time.
So now we have all that we need to form a chromatic scale. If we start at A=440Hz, using this tuning system, we get the notes to be A4=440 Hz, A#4=469.33 Hz, B4=495 Hz, C4=528 Hz, C#4=550 Hz, D4=586.67 Hz, D#4=616 Hz, E4=660 Hz, F4=704 Hz, F#4=733.33 Hz, G4=792 Hz, G#4=825 Hz, A5=880 Hz.
However, this tuning system--just intonation--has several major limitations, which is why it is not used in modern music. The first limitation is that not all semitones on a scale are created equal, which results in different ratios for the same intervals when modulated. For example, the perfect interval of A4 to E4 (3:2) does not have an equal ratio to that of D#4 to A#4 (approx. 1.52…). In fact, the D#4 to A#4 interval would sound quite dissonant. Even though the ratio 1.52 seems very close to the ratio of 3:2 or 1.5, it will sound noticeably out of tune due to the number of Hz adding up. Consequently, modulating to a new tonal center would result in the music being out of tune. To fix this issue, a temperament—the adjustment of intervals in tuning to fit the scale for different keys—would need to be applied.
The Most Commonly Used Tuning System Today: 12-Tone Equal Temperament:
Throughout history, various tuning systems were experimented to fix the issues that came with just intonation. Those tuning systems include Meantone Temperament, Equal Temperament, and Well Temperament. Eventually, Equal Temperament became the standard tuning system that is most commonly used in today’s music. Equal Temperament is a system that divides the octave equally on a logarithmic scale. There are various equal temperaments, but the most commonly used today is 12-Tone Equal Temperament (12-TET), which divides the octave into twelve equal divisions (semitones). Since there are twelve semitones in an octave, each note’s frequency will have a ratio of 2^(1/12) to the note that is a semitone lower. Since equal temperament divides an octave equally on a logarithmic scale, these ratios result in irrational numbers. Therefore, other than the octave, all the intervals are going to be slightly out of tune, but in tune enough so that our ears barely notice anything. The equally tempered perfect fifth for example, now has a ratio of 2^(7/12) or roughly 1.498, still close to its 3:2 justly tuned counterpart so the difference is barely audible. The most noticeably out of tune interval is the major third, which now has a ratio of roughly 1.259, which is 13.49 cents sharper than its justly tuned counterpart. Although some intervals are farther from their justly tuned counterparts than the major third is, they are not as noticeable due to their already dissonant nature. Even though these intervals are out of tune, this fixes the issue from modulation in just intonation. All twelve keys will sound out of tune but equally out of tune now.
However, other equal temperament tuning systems have been experimented, including 19-TET, 29-TET, 31-TET, 34-TET, 41-TET, and 53-TET, which offer even more in tune intervals than 12-TET does. (24-TET is not included because it is a common multiple of 12, resulting in the same tunings). But they do not synchronize well with the piano (a common 12-TET instrument and they are difficult to notate, so they are not used as commonly as 12-TET is.
Equal Temperament Applications
With modulation being easy, new types of music have emerged including atonality, serialism, pop and jazz. Jazz, for example, uses many improvisational techniques such as tritone substitutions, which would not have been possible without the adoption of Equal Temperament. Most tuners also default to equal temperament, which make tuning the piano or the guitar easier. However, they should not be used to tune certain string instruments such as the violin, viola, and cello, which are designed to be tuned in just intonation. The violin’s strings are designed to be 1.5x longer than the one above it to maintain the ratio of a perfect fifth so you should use your ears or manually calculate the frequencies from your tuner to tune it. This presents a difficulty when playing with the piano or an equally tempered instrument since the violinist has to adjust their intonation to align with the piano.
Nonetheless, Equal Temperament is an essential concept in music theory due its versatility, standardization, and worldwide influence. It is after all what makes modern music out of tune but frankly, it is better that way.
Works Cited:
Amir, R., & Evstigneev, I. V. (2016). On Zermelo’s theorem. ArXiv [Math.CO]. https://doi.org/10.48550/ARXIV.1610.07160
Brams, S. J., & Davis, M. D. (2023). game theory. In Encyclopedia Britannica.
Klosky, D. (n.d.). Chess - chessprogramming wiki. Chessprogramming.org. Retrieved March 17, 2023, from https://www.chessprogramming.org/Chess