Then you are not alone. It is wrong. But I heard something similar at school, and encountered it again some years ago in a middle-school physics textbook on the shelf at a public library in Copenhagen, Denmark.
Equal temperament1 is what you get when all semitones are tuned to the same size. (Or very nearly the same size – in practice, the intervals must be adapted to certain physical properties of the strings).
Owen Jorgensen, former professor of piano tuning at Michigan State University, observes in his big Tuning book2 that methods for tuning equal temperament accurately on keyboard instruments were not developed until the 20th century. According to a quote on page 63 in the same book, Alexander Malcolm commented on equal temperament as follows in 1721, the year before Bach finished the first part of his Well-Tempered Claviricher,
This and various other references to equal temperament tell us that the concept was discussed in the 18th century. Not all comments were equally negative; but there are no signs in the old literature that Bach invented, used, or recommended equal temperament. There are signs that Bach did not like mathematics. Maybe a few physics teachers should be taught to check the facts before teaching outside their area of competence!
Equal temperament makes the music sound the same way in all keys except for pitch level. However, the tradition at and before Bach’s time was that different keys should sound differently in a way that suited the music. This is sometimes called key color.
As late as in the 1950’es, a tuner in my neighborhood in rural Denmark made my piano to sound one way in E major, well suited for Handel’s Harmonious Blacksmith, and with a different sound in A flat major, perfect for Schubert’s Impromptu in that key. Obviously, that tuner did not tune equal temperament. The instrument sounded most of all as if The Art of 19th-Century Temperament 2 was not entirely lost.
If you do the math on the back of an envelope, and use pure intervals throughout, something different is going to happen. The frequency ratio is 3:2 for a pure fifth, 2:1 for a pure octave. A few multiplications will show you that your trip will end at a frequency ratio 531441:524288 above the tone where it started. That quantity is called the ditonic comma or the Pythagorean comma. It equals 23.460 cents, where a cent is a logarithmic unit defined such that a 2:1 pure octave is 1200 cents. An equal-tempered semitone is 100 cents, as observed by Alexander John Ellis who introduced these units around 1885. Cents were not used at Bach’s time.
So, twelve pure fifth do not fit the keyboard. Each of them is 1.955 cents too big (that is, 1/12 of 23.460 cents), What can we do?
In equal temperament, we simply tune all fifths 1.955 cents smaller than pure. If we do that accurately, we get the same result as in the mathematical exercise mentioned in the beginning of this document: All semitones will get the same size – and all keys will sound identical apart for pitch level.
(Note how well equal temperament fits into the philosophy of the industrial society and its quest for product uniformity. Is it a coincidence that equal temperament became widespread during the same historical period as industry? Could that be one of the reasons why no one questioned the musical merits of equal temperament until the second half of the 20th century?)
Tuning was done very differently in the 17th and 18th centuries.
In frequencies, if we first go four 3:2 pure fifths up and two 2:1 octaves down, that will bring us to a frequency ratio 81:64 above the point where we started.
So, in order to get back to the frequency where we started, we will have to go downwards by an interval of frequency ratio 81:64. Such an interval is called a Pythagorean third. It sounds like a colorful or even somewhat dissonating major third. It exceeds a 5:4 pure major third by an interval of frequency ratio 81:80.
During much of music history, a Pythagorean third was regarded as too wide to be useful. Again, the solution was to flatten some or all of the fifths.
The quantity 81:80, which is an interval of size 21.506 cents, is called the syntonic comma. If we flatten eleven of the keyboard’s twelve fifths by 1/4 of that amount, the major thirds become pure. The 12th fifth becomes a diminished sixth, usually chosen to be located between G sharp and E flat. It dissonates strongly and is often called the wolf. Such a tuning is called one-quarter syntonic comma meantone temperament.
This and other kinds of meantone temperament were used for centuries. When an intrument is tuned in meantone temperament, you can only play in keys with up to about three accidentals – but those keys will make the music sound in its full beauty. (This is how my music teacher explained it to the listeners when he played renaissance music on the historical Compenius Organ in Frederiksborg Slotskirke, Hillerød, Denmark, in a radio brodcast around 1962.)
The range of playable keys in meantone temperament can be extended if the instrument is built with sub-semitones, also known as split keys. Historically, some instruments had, for instance, separate keys for d sharp and e flat, and for g sharp and a flat. That is how the new organ in Örgryte Nya Kyrka near Göteborg, Sweden, inaugurated in 2000, is built. It is tuned in one-quarter syntonic comma meantone temperament.
The German tuning theorist Andreas Werckmeister published a number of temperaments of that kind in 1681 and 1691. The most widely known one is called Werckmeister III Correct Temperament No. 1, or simply Werckmeister III. (Werckmeister himself called it “Num. 3.”) In that temperament, four of the fifths, namely C–G, G–D, D–A, and B–F#, are flattened. The rest of the fifths are pure.
In Werckmeister’s terminology, a keyboard instrument tuned along such lines was said to be well-tempered (wol temperirt or wohl temperiret). The similarity with Bach’s title Das wohltemperirte Clavier – that is the spelling Bach used on his hand-written title page – can hardly be a coincidence. Most scholars tend to agree that Bach must have used some kind of well-tempered tuning, but not necessarily one of the tunings published by Werckmeister.
In a properly designed well temperament, the various keys sound differently in a manner that suits the music.
Numerous players have found that Bach’s music sounds richer in content when played in a well temperament than it does in equal temperament. Melodies sound more melodious, harmonies more harmonious. For a person who is used to equal temperament, the experience of playing a well-tempered instrument for the first time can be strong. One player used seven exclamation marks on his website to express his feelings about having spent part of his life with a distorted impression of Bach’s keyboard music. Others have compared it with seeing Rembrandt van Rijn’s paintings in colors vs. black-white reproduction.
Analysis of the Well-Tempered Clavier has shown that Bach composed differently in different keys. The British physicist and instrument maker John Barnes went so far as to reverse-engineer Bach’s usage of major thirds in the preludes of the WTC, in order to compute the well temperament that would suit them best. The result, known as the Bach/Barnes temperament, came to be widely known.
More recently, an ornament on the old hand-written title page of the Well-Tempered Clavier has been identified as a possible source of information about Bach’s tuning. Bradley Lehman3–4 analyzed it in a relatively straightforward way and arrived at a temperament that suits Bach’s keyboard music (not only the WTC) to a high degree of perfection.
I have tuned my piano in Lehman’s proposed Bach temperament since 2005 and like it very much. There are only two points where I think more historical information would be nice to have if it could be found: The widest major third if E–G#, which is historically unusual but not impossible; and, if the ornament is a tuning instruction for educational purpose, we ought to be able to understand why none of Bach’s pupils or colleagues mentioned it in their writings.
Could it be that they did but it has gone unnoticed because modern scholars did not have enough information to understand the context? That is a far-fetched idea of my own.
(A year’s time after the above was written, a simpler idea is coming to my mind:
Bach designed his ornament as an allusion to his temperament, or to one of his temperaments, in part for fun, in part so that it could be used for educational purpose. But maybe he never actually used the ornament for educational purpose. Hence the absence of historical references to it.)
Well – accurate and reliable information about historical keyboard temperaments can be found in numerous places on the Internet, and to repeat it here would serve no purpose.
If you are a beginner and would like to know more, Stephen Bicknell’s guide5 is one possible place to start.
If you have access to the New Grove in a library, either a paper edition or the electronic edition at Oxford Music Online, be sure to read Mark Lindley’s articles about Temperaments and Well-Tempered Clavier. Buy Jorgensen’s book if you can get hold of a copy. Look for information everywhere and read all you can find. Do not uncritically accept everything – and be aware that some authors, even in respected encyclopedias from as late as the 20th century, did not distinguish between well temperament and equal temperament. When someday you feel you know enough, read Bradley Lehman’s twin Rosetta papers3–4 from beginning to end, and study his site.
WARNING. If you have not tuned a modern piano before, take lessons with a professional tuner before you try. To damage the piano is easy.
(Author: Bjarne Pagh Byrnak)
 Owen Jorgensen, “Tuning. Containing: The Perfection of Eighteenth-Century Temperament, The Lost Art of Nineteenth-Century Temperament, and The Science of Equal Temperament, Complete with Instructions for Aural and Electronic Tuning.” Michigan State University Press, East Lansing, 1991. ISBN 0-87013-290-3
 Bradley Lehman, “Bach’s extraordinary temperament:
our Rosetta Stone – 1, ”
Early Music Vol. 33, No. 1, February 2005, pp. 3-23.
Supplementary material available electronically.
 Bradley Lehman, “Bach’s extraordinary temperament:
our Rosetta Stone - 2, ”
Early Music Vol. 33, No. 2, May 2005, pp. 211-231.