- Equal Temperament is the standard tuning system in Western music
- Meantone Temperament and Well Temperament were developed as alternatives to Pythagorean Tuning
- Equal Temperament allows for playing in every key without retuning instruments
Equal Temperament has been the standard tuning system in Classical music and really western music in general since the 18th century. It is a system that tunes octaves perfectly and breaks each octave into twelve equally-spaced semitones.
The result is a tuning system that works equally well in every key but does not quite reach perfectly just tuning in any key.
This is, of course, only the short answer: as with every development in Western Art Music, there is a long history leading up to the invention of equal temperament and a long list of innovations since. Read on for the full story.
What Is A Tuning System?
Let’s start with the basics: a tuning system defines musical pitches’ number, spacing, and exact frequencies. There are many tuning systems, and they vary across time and space.
We will attend to tuning systems in order of time and start as close to the beginning as possible with what is known as Pythagorean tuning.
Pythagorean Tuning
Understanding Pythagorean tuning first requires understanding ratios. If you were to pluck a taut string, it would vibrate at a certain frequency, and that frequency would sound like a certain pitch to you.
If you then plucked a string half as long but with the same thickness and tautness, it would vibrate twice as fast and thus sound one octave above the pitch of the first string.
Thus, the 2:1 ratio defines the perfect octave. Let us now turn to 3:2, the second-simplest ratio, which defines the fifth. Pythagorean tuning is constructed by stacking perfect fifths on top of each other (and reducing to octave equivalents) until all twelve notes of the octave are accounted for. Because Pythagorean tuning utilizes these simple numerical ratios, it is a form of just tuning.
Pythagorean tuning is misattributed to Pythagoras: it originated in Ancient Mesopotamia and was only later adopted by the Ancient Greeks. Pythagoras, a mathematician obsessed with ratios living around 500 BCE in Greece, was credited by modern music theory writers.
It is worth noting that the Chinese Shí-èr-lǜ scale is built from the same intervals and was developed independently between 600 and 240 BCE.
The Pythagorean method for finding pitch has a certain elegant simplicity to it, especially as it uses the two simplest ratios (which correspond to the two purest intervals) to find every pitch. Still, the result is more complex and far from perfect. Because each interval comes from stacking fifths on top of fifths, the ratios eventually grow into more complex numbers.
The perfect fourth is simple enough, as it is an octave (2:1) minus a fifth (3:2), so you end up with a ratio of 4:3. The major third, on the other hand, is the octave equivalent of four perfect fifths stacked on top of one another, so it is the equivalent of 3:2 raised to the fourth power and then reduced down the octave twice, or 81:64.
As a result, the major third above the tonic in a given scale tuned to the Pythagorean tuning system ends up being nearly 8 cents higher than via the equal temperament system with which you are likely more familiar.
Things get especially problematic when considering enharmonic equivalents, which do not really exist in Pythagorean tuning: in using the perfect fifth to establish just tuning, there is about a quarter-tone that ought to exist between two enharmonically equivalent pitches, known as the Pythagorean comma.
Yet, if you look at instruments that pre-date equal temperament, you do not find keyboards with two black keys directly adjacent to one another.
The solution is to use perfect fifths to construct the scale, but to leave one interval, also called the wolf interval, badly out of tune to absorb all of the Pythagorean commas.
This move is similar to the concept of a leap year: 365.25 days is an awkward length for a year, so rather than deal with a quarter day each year, we just count an extra day every four years.
The problem with leaving a wolf interval in a scale is it limits a composer’s or player’s access to pitch. One of the pitches of the wolf interval must be excluded from an entire piece of music, or else it will reveal how badly out of tune the interval is.
This is fine if a piece of music sticks to a single key and does not need to use that forbidden pitch, but music that modulates and uses the full set of pitches cannot be played in this tuning system. Fortunately, other tuning systems later broadened the scope of harmonic possibilities in Western art music.
Meantone Temperament
There was some writing that might possibly have been about meantone temperaments in the late 15th century. But by the 16th century, there were unmistakable instructions available for tuning instruments using various forms of meantone temperament.
Essentially, meantone temperament involves “tempering” a just interval, like the perfect fifth, so that it can be used to find a just smaller interval, like the major third.
Meantone temperaments are measured by the syntonic comma by which the original just interval is tempered. Quarter-comma meantone, in that case, is a meantone temperament in which the perfect fifth is lowered by ¼ of the syntonic comma so that four fifths (reduced by two octaves) results in a just major third one syntonic comma lower than a Pythagorean major third.
Essentially meantone temperaments trade perfectly tuned fifths for perfectly tuned thirds, which to a violinist’s ear is just plain wrong, as bowed string instruments are typically tuned in perfect fifths.
Though various meantone temperaments can be used to get closer to greater harmonic freedom, they still do not work well as they extend beyond a set of closely related keys. For modulations all over the full range of tonal harmony, one has to go to Well temperament.
Well Temperament
The goal of well temperament was to get some intervals perfectly in tune and to temper others so that music could be played without excluding one or both of the pitches associated with a wolf interval. Essentially, well temperaments eliminate wolf intervals by tempering intervals throughout the tuning system.
Well temperament is historically associated with J.S. Bach’s “The Well-Tempered Clavier,” two sets of preludes and fugues in all 24 major and minor keys. Well temperament is implicated because the full range of keys would not be playable on a clavichord or harpsichord that was not tuned accordingly.
Well temperament, like meantone temperament, comes in many forms, with distinct types of well temperament being named after their inventors. Bach is reported to have tuned his own harpsichords to modulate to distant keys without audiences noticing the difference, suggesting that his preferred well temperament might have been close to equal temperament.
Equal Temperament
Finally, we arrive at equal temperament, which tunes all octaves perfectly but then tempers every other interval in dividing the octave into 12 equally-sized semitones.
All intervals are then measured in terms of semitones so that the octaves are perfect, but every other interval is just a little bigger or a little smaller than it would be in just tuning. In tempering every interval to a small degree, it is possible to play in every key without re-tuning the instrument.
Equal temperament has been the most common tuning system since the 18th century and likely the only system most people in the present day have heard.
However, the origin of equal temperament stretches back a few centuries earlier as a practical means of tuning lutes. Fretted string instruments would require a means of tuning to equally sized semitones, as that is how they are set up.
Vincenzo Galilei (father of Galileo) wrote in Florence in 1581 about a system composed of these equivalent half-steps. Zhu Zaiyu wrote something similar in China in 1596, and Marin Mersenne did so in France in 1636.
Though equal temperament made a lot of sense theoretically, tuning forks were not invented by John Shore until 1711. Thus, systems using more just intervals were easier to utilize in practical situations.
It is likely that some well temperaments approached equal temperament in de facto application, as might have been the case with J.S. Bach’s claviers.
Andreas Werckmeister and other German music theorists advocated successfully for the widespread adoption of equal temperament, and it was generally accepted in Germany and France by the end of the 18th century and in England in the 19th century.
Equal Temperament vs. Just Intonation
When it comes down to it, equal temperament has had the staying power it has had because it allows instrumentalists playing fixed-pitch instruments to play in every key with decent intonation. A fifth in equal temperament is only about 2 cents (or two-hundredths of a semitone) off from just intonation, which is imperceptible unless you focus very hard on hearing the difference.
That said, only some instruments are fixed in pitch. A violinist, for example, tunes their strings in fifths based on just intonation once their A is tuned to 440 Hz.
Micro-adjustments in finger placement determine the exact tuning of intervals once the strings are tuned, which frees the violinist up to tune their thirds just how they want them in the context and key in which they are playing.
A fretted instrument is different in that frets determine the tuning, but fretless string instruments provide the instrumentalist more freedom in terms of intonation.
Wrapping Up
Tuning and temperament are one of the many topics in music that could be more cut and dry. Temperament exists due to the imperfections of “perfect intervals” in determining a complete set of intervals.
Balancing an octave into twelve equal semitones and building intervals based on them (also known as equal temperament) seems a simple and elegant solution. But, it also introduces small imperfections throughout the system.
These imperfections make it possible to drape the system of equal temperament across all of pitch and govern every key equally, though, so the result is a “good enough” system, but one that is far from perfect.
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