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Johann Sebastian Bach
(1685 - 1750)
Preferred well temperament over equal temperament.
By the beginning of the 18th century, musicians everywhere were using a variety of tuning systems and keyboards.
In fixed tone instruments, the Pythagorean comma, one-quarter of a semitione, was distributed amongst the 12 semitones of each octave in different ways:
- Meantone temperament allowed modulation into certain keys, which worked as long as certain "wolf" tones were avoided. They were called wolf tones because they sounded like howling wolves.
- Well temperament allowed for full modulation and eliminated wolf tones at the price of sacrificing a little pure harmony here and there. Since the intervals between the pure semitones still varied in size, when a scale began on a different note or key, its tuning characteristics were unique.
- Equal temperament, where each of the 12 notes are evenly distributed within the octave by, starting at one, multiplying the frequency of the previous note by 1.059463094 until two is reached. Pure harmony is sacrificed for the convenience of not having to retune when there is a change of key.
J. S. Bach clearly demonstrated his tuning preference. The Well Tempered Clavier* is the title he gave to the study books he released in 1722 and 1744, each consisting of 24 preludes and 24 fugues composed in each of the 12 major and 12 minor keys. Composed "for the profit and use of musical youth desirous of learning, and especially for the pastime of those already skilled in this study", it is obvious that Bach's main purpose was to demonstrate the feasibility of a tuning system that would allow for composition in every key.
Many manuscript copies were made by his pupils and copies spread steadily over Europe with his fame. Both books were published in 1801. Equal temperament had been introduced in Germany about 100 years earlier.
Bach made a diagram at the top of the title page of the Well Tempered Clavier visually describing how the clavier should be tuned for the the well tempered tuning required. When the diagram is turned upside down, the loops represent the distribution of the Pythagorean comma between the 12 fifths of the octaves, in the order of the cycle of fifths, starting with the F - C interval on the left.
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The first five large loops (C, G, D, A, E) enclose three smaller loops indicating that for these five intervals, a perfect fifth should be flattened by a factor of 3.
- The next three large loops without inner loops (B, F#, C#) indicate three pure intervals of a fifth, without any beats, perfectly in tune.
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The three large loops at the right (G#, D#, A#) indicate three intervals enclosing one smaller loop indicating that these fifths should only be flattened by a factor of 1.
Dividing the Pythagorean comma into these proportions and applying them mathematically calls for the following:
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Starting with the F to C interval tuned to a perfect fifth, flatten the fifth by 1/6 of the Pythagorean comma for the first five intervals, keep the next three fifths pure and then flatten the remaining three fifths by 1/18 of the comma.
For electronic instruments, the tuning variance in cents from equal temperament is as follows, based on A440, starting with C:
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+6, +4, +2, +5, -2, +8, +2, +4, +5, 0, +6, 0, +6. cents
* Clavier means any keyboard instrument.
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