The Well-tempered tuning, page 1 
(The history of the piano and its regulation starts on page 3)

The Invention of Temperament, a historic background

Equal Temperament is probably the single most important development in Western European music in the last 400 years. Equal Temperament is the tuning system by which all the notes in our Western music are organised and structured.

Pythagoras (born about 569 BC in Samos, Ionia – died about 475 BC), is the starting point in tuning theory. According to the legend, his discovery happened when he was passing a blacksmith. He reacted to the sound of the hammers on the metal. After some investigation he discovered that the weights and lengths of metal that produced harmonious notes were simple ratios: 1, 1/2, 2/3, etc. Then he started experimenting with his monochord. By halving the length of the string, then halving it again, it produced a new note, the same but an octave higher. Similarly, striking a string two-thirds of the original length gave him another natural tone, one fifth above the note he started on.

The tone derived from a 2/3 fraction is called a dominant, precisely because it has such a powerful influence on the harmonic scheme. This dominant is also ever-present in all natural sounds in the so-called harmonic series.

 

Now the problems start

If you divide a string by 2/3 and continue to divide each part of the string that is left over 12 times, you will have a short string left and most importantly, you will be back with the same tone you started with, only at a much higher pitch. This is known as the circle of fifth’s, which in turn will give us the twelve notes on the keyboard.

 

The circle of fifths, containing all the 24 tones and keys.

Now imagine the same string, dividing it by splitting it halfway seven times, you will end up with a short string that should be identical with the above mentioned string. The problem is that it’s not! The difference is called the “Pythagorean comma” and the difference is 23,46001038465 cent (or percent) of a half tone.

This flaw made theorists and musicians spend the next two thousand years trying to find a pleasing solution to fit all 12 tones into the scale. That is the story of the search for “Equal Temperament”.

This difference (or natural fault in the scale) is contained in the Well-Tempered tuning system but evenly distributed among all of the 12 tones in the scale. But how did they solve this problem in Pythagoras' time? The answer is that they didn’t. Music in his day had much simpler harmonic features compared to ours; the average instrument of that time could only cope with a scale of 6 pitches, A B C D E and F. The tones were derived from the circle of fifths, and arranged into a scale ladder within a single octave span. The “black notes” weren’t invented until later, but they had always been there. It’s just that they weren’t used.

Later on, from the first of the millennium after Christ, European music mainly used 8 of the 12 notes – A B C D E F G A – In other words, the medieval musician used only the “white keys” on a modern keyboard.

 

 

The development of harmony

The development of harmony was an enormous change in the history of music, a change that derived its impetus from the invention of the music notation system by Guido de Arezzo around AD1000. Now composers could write down much more complicated music. This is the invention that separated the European tradition irreversibly from all its counterparts elsewhere in the world!

But, the problems with tuning continued. If you start using harmony, the Pythagorean comma will come into play. The medieval musicians solved the problem by using only pure intervals such as 5ths, 4ths, and octaves. As perfect natural dominants, or 5ths, means imperfect 3rds, the latter where not used at all. The reason was simple: if you used a perfect natural 3rd, it would mean imperfect ratios for the 5ths.

Around the year 1000, musicians working for the Church began combining notes to achieve a richer sound and create new effects. This was considered radical, and it took another two hundred years for them to try three notes together at the same time. The Church also slowed this process by condemning the use and combinations of notes that derived from impure ratios.

But, you cannot keep anyone away from trying out something new...  At the beginning of the 15th century, composers started to experiment boldly with previously forbidden intervals. Intervals formerly considered unstable, 3rds and 6ths, now came in use. John Dunstable’s motet, Veni Sancte Spiritus, written for a special service of thanksgiving held in Canterbury Cathedral in 1416, is an important milestone. The many instances of 3rds and 6ths made it well known and imitated throughout Europe.

The Pythagorean tuning system, which had managed pretty well with 5ths, 4ths and octaves, was now in a serious crisis.

During the early Renaissance, with the increased use of these “new” intervals and combinations, musicians increasingly allowed 3rds and 6ths to prevail harmoniously. To do this they had to tune their instruments differently. The Renaissance temperaments that favour 3rds and 6ths over 5ths and 4ths, are called “meantone”. In order for the notes to fit neatly together inside an octave they tried many different ways of tuning and came up with many different solutions. This is what is meant by “tempering” the tuning.

At the beginning of the 15th century the situation was quite unlike that of previous generations. A whole new set of chords and intervals became unusable. There were certain keys you couldn’t use at all. The basic problems of the Pythagorean comma were still present!

 

The problem solver or J. S. Bach’s Well-Tempered Clavier    Page 2