A History of Equal Temperament

Did you know that there is more than one way to tune a piano? Back in J.S. Bach’s time, the pianos and harpsichords and clavichords were not always tuned by the same method as we tune now.

The system of tuning the piano that we use today is called equal temperament. Maybe the method of equal temperament seems natural and obvious now, but years ago, the scale was tuned many different ways. Even today, some types of tuning exist which are peculiar to a particular culture.

Today, I’ll be discussing what temperament is, the different types of temperament, the history of equal temperament, and how it became the most common tuning system we use today. 


Equal Temperament and its Alternatives 

Firstly, what is equal temperament? Well, it’s a method of tuning. We tune the piano in octaves, and these octaves consist of 12 semitones.

 Octave.png

These semitones are all equally far apart from each other. As we’ll see, there are many methods of tuning in equal temperament; for example, you can tune a piano to have 19 equally distanced semitones in an octave (provided there are 19 keys in an octave, on that piano) – it’s called 19 equal temperament. Listen to 19 equal temperament here.

Now, what are the alternatives to equal temperament? What did musicians use before equal temperament? Well, there are many alternatives, but the two main ones are meantone temperament and well temperament.

To understand the differences between equal temperament and these other methods, you must first understand the interval called the “pure” interval. If you want to learn about the frequencies of a pure interval, please read this article. For a quicker explanation, listen to this video, up until about 8:17.

In the video, maybe you hear the beats Trevor talks about. In a pure interval, there are no beats, whereas in our equal tempered interval, there are 5 to 6 beats per second. Why? Because, for example, our 3rd on the piano is just a little higher than a “pure” third. The reason we have to tune that way is because, like he said, there is simply no way you can tune in pure 3rds and not come out with a wolf interval. A wolf interval is an ugly discord that arises when trying to tune with pure intervals. Listen to the wolf interval below.

Now, let’s apply this knowledge to meantone tempering. 

Meantone tempering is a general term – there are many types of meantone tempering. The most common one is called quarter-comma meantone and attempts to tune using pure 3rds throughout the keyboard. However, when tuning using pure intervals, there are wolf tones. If you’re playing in the key of C with few accidentals, there’s nothing wrong with a piano that has been meantone tempered to the key of C. But problems arise when there are accidentals or a change of key. You can’t play in the key of F# if the piano was tuned to the key of C – wolf tones will be everywhere, since the pitches have been specifically tuned to allow pure intervals in the key of C. One would have to retune the whole piano to be able to play in the key of F#.

So, meantone tempering, while it is pure, is a pain for tuners and a limitation for composers. 

Now, let’s take a look at well temperament. (This is, like meantone tempering, also a general term.) “Well temperament” or “good temperament” is some sort of irregular temperament in which the 5ths are not pure, but not too impure either. This tempering of 5ths means that unlike meantone tempering, with its inability to change keys, well tempering allows this. J.S. Bach, who was a proponent of well tempering, wrote The Well-Tempered Clavier to demonstrate the musical possibilities of well temperament.

Although well tempering has no wolf 5th, there is a downside to tuning the 5ths so irregularly; doing this affects the 3rds, which then sound a little out of tune. Even so, this downside was well worth it when compared to the meantone temperament. Well temperament became widely used in the Baroque period, persisted through the Classical period, and even survived into the 19th century.

So, musicians found a way to avoid the wolf 5th without overly contaminating pure intervals, but they then faced a new problem. While most keys were all very well, others with more sharps or flats sounded out of tune. The solution? Equal temperament. Equal temperament does not have pure intervals; the equal temperament 3rd is a little higher than a pure 3rd and the equal temperament 5th is a little lower than a pure 5th. On the other hand, because every interval is equal, there’s no more trouble about retuning the piano. On a perfectly equal-tempered piano, the distance between any two adjacent tones is exactly the same as the distance between two other adjacent tones. Every key works the same, so transpositions, chromatic notes, and modulations are all smooth.


A History of Equal Temperament

Two people are generally credited with the calculation of equal temperament: Zhu Zaiyu (in 1584) and Simon Stevin (in 1585). However, like Cristofori’s pianoforte, there were numerous predecessors to equal temperament; Ling Lun (allegedly the creator of the Chinese pentatonic scale), wrote about the idea of equal division of the scale in the 27th century BC. Also, a complete set of chime bells from the tomb of the Marquis Yi of Zeng covers five full 7-note octaves, including 12-note semitones. Besides that, He Chengtian, a mathematician, described basic equal temperament around 400 AD.

However, the fact still remains that Zhu Zaiyu was the first to mathematically solve the problem of equal temperament. Zaiyu, a prince of Ming, spent 30 years performing research for the equal temperament idea originally contemplated by his father.

Vincenzo Galilei, the father of Galileo Galilei, was one of the first advocates of this 12-tone equal temperament. He composed a set of dance suites for each of the 12 notes of the chromatic scale. However, Galilei was not alone; Giacomo Gorzanis also wrote music based on equal temperament by 1567 and Francesco Spinacino wrote a Recercare de tutti li Toni (Ricercar in all the Tones) as early as 1507.

Meanwhile, in the west, Simon Stevin was working on the 12-tone system of equal temperament. He had, mathematically, the right ideas – he could have correctly calculated equal temperament. But due to the inaccuracy of his calculations, many of his numbers were off by one or two units.

Among the 17th-century keyboard composers, Girolamo Frescobaldi advocated equal temperament. On the other hand, the theorist Giuseppe Tartini was opposed to equal temperament, feeling that degrading the purity of each chord degraded all music.

Despite the critical opinions of some purists, equal temperament became well-known in the Baroque period, was the tuning method of choice during the Classical era, and became the standard system by the early Romantic era. 12-tone equal temperament was a perfect fit for the existing keyboard design and permitted total harmonic freedom at the expense of little impurity in each interval.


Thanks for reading!

~ Maggie

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