Music: An Imperfect System

Everybody knows about Pythagoras and his theorem. But did you know his studies on music are the origins for Western Music?

Pythagoras was the first to note that mathematics applied to noises created pleasing sounds. He noticed that an anvil being struck in unison with another anvil half its size (2:1), or an anvil 2/3 its size (3:2), produced sounds he enjoyed. The ratio of 2:1 is what we now call an octave (P8) and the 3:2 ratio is what we call a perfect 5th (P5). The simpler the ratio is between two pitches, the more their sound waves overlap, making them more consonant. The P8, P5, and P4 (4:3) are the most consonant intervals (hence the word “perfect” associated with them) and were the main focus for writing music during the Medieval period. Using simple ratios to represent intervals is known as “Just Intonation.” Singers and string players find these simple ratios by using their ears to produce the most consonant sounds. Since other instruments have less control and keyboard instruments can’t be retuned in the middle of a piece of music, tuning systems developed to balance the relationship between just intonation intervals and convenience.

Once again, we have Pythagoras to thank for the first tuning system which we fittingly call “Pythagorean tuning.” This tuning system and the 12-tone scale we all know and love is the result of continually finding the P5 above a given pitch:

C-G-D-A-E-B-F#-C#-G#-D#-A#-F- and then one P5 more and we’re back to C right? Well, not quite (if that were the case this entry would be titled “Music: A Perfect System” instead). The problem is that if you use the 3:2 ratio to get from C back to C, you don’t get exactly the same note. The C you end up with is 23.46 cents sharper than the original, a discrepancy known as the “Pythagorean comma.” In order to fit all 12 pitches into an octave using this Pythagorean tuning system, one of the perfect fifths is lowered by the Pythagorean comma and is known as a “wolf” fifth because of its noticeably out of tune sound which resembles a howl. This wolf fifth can be placed anywhere the player wishes and is placed on a P5 they are unlikely to use.

Starting in the Renaissance period, the major third (M3) became a more desired interval, but using the Pythagorean tuning system its ratio is 81:64. As a result, the tuning system known as “meantone temperament” was developed using justly tuned M3rds (5:4). Since an octave is divided evenly by M3rds (C-E-G#-C) and a M3 is divided evenly into two major 2nds (M2), each M2 serves as the exact halfway point (the mean) between a note and the M3 above it. The P5ths are no longer justly tuned, but 11 of the 12 are only 5.38 cents flat compared to their justly tuned counterparts. This discrepancy is suitable but the remaining P5 is a wolf fifth that is 35.68 cents sharp. In addition to this wolf fifth, meantone temperament also results in 4 wolf M3rds which are 41.06 cents sharp and 3 wolf minor thirds (m3) which are 46.44 cents flat. Meantone temperament allowed M3rds to be an extra consonant (though still considered a lesser consonant than P8, P5, and P4) and allowed keyboard instruments to modulate to other keys for the first time; but only keys with 3 accidentals or less were feasible to use.

During the Baroque period, writing music transitioned from using the various “Church Modes” (Lydian, Dorian, etc), which determine the pattern of ascending intervals for the octave, to the major and minor keys we still frequently use today. This made the keys more important than the intervals themselves. For the first time, being able to use all 12 major keys and all 12 minor keys became important. Any temperament that is irregular and allows this is known as a type of “Well Temperament.” These involve altering the Pythagorean tuning, but instead of having the wolf fifth, the Pythagorean comma is displaced by lowering some of the justly tuned P5 by the same fraction. Organist, composer, and theorist Andreas Werckmeister wrote examples of well temperament. In the one he recommends as being the most efficient, he suggests lowering the P5ths C-G, G-D, D-A and B-F# by 1/4 the Pythagorean comma. Like meantone temperament, the keys with more accidentals are less in tune, but unlike the other tuning systems, these keys still sound in tune. As a result, with well temperament each key has its own distinct sound (key color) which composers liked to explore.

Once all 24 keys became usable, music in the Classical and Romantic periods focused on using, and modulating to, any key. This, combined with the popularity of the piano as an ensemble instrument led to the creation of the tuning system we still use today, “Equal Temperament.” In equal temperament, the octave is divided into exactly 12 equal pitches. This means every interval (M2, M3, P5, etc.) is always the exact same distance regardless of the key. Before equal temperament, a note and its enharmonic equivalent (G#-Ab, D#-Eb, etc.) were considered two different pitches with different frequencies. In fact, enharmonic used to mean “two notes less than a semitone apart” instead of its current meaning of “two equivalent notes (or keys) that are spelled differently.” Equal temperament allows composers to modulate to other keys through enharmonic modulations seamlessly. However, using equal temperament means every interval except the P8 is out of tune in comparison to its just intonation counterpart, with the P4 and P5 only 1.96 cents flat and sharp respectively and the M3 and minor 6th (m6) 13.69 cents flat and sharp respectively. Also, the key colors created from well temperament no longer exist in equal temperament.

Music is an imperfect system, but that’s what has made music so interesting and appealing. As tuning systems developed and changed, composers were able to write music in new and different ways; and as composers aspired to write music in new and different ways, the tuning systems developed and changed. Each tuning system has its own advantages as well as disadvantages. The real problem is not that music is an imperfect system but that we perpetuate the belief that equal temperament is the final (and only) solution in determining what frequencies we use. It is incredibly reliable and should be considered the default system. Without equal temperament, many 20th century music techniques (atonality, serialism, etc.) would have never surfaced. But just like there are an infinite amount of numbers between 1 and 2, there are potentially an infinite number of frequencies to use within an octave. If we only use equal temperament, we limit pitches to precise frequencies that are set in stone; and that would be imperfect.

Note: If you want to read more about tuning systems, I recommend the book “How Equal Temperament Ruined Harmony (And Why You Should Care)” by Russ W. Duffin.