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How Keyboard Instruments Were Tuned:  A Brief History

1. Bach Did Not Recommend Equal Temperament.

Did your physics teacher tell you that Bach invented equal tempera­ment?  That Bach was first to get this brilliant idea:  Compute the 12th root of two, then tune all semitones to that frequency ratio?

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 Clavier,

That tho’ the Octave may be divided into 12 equal Semitones by geometrical Methods, that is, 13 Lines may be constructed, which shall be in continued gometrical Proportion, and the greatest to the least be as 2 to 1, yet none of these Terms can be exprest by rational Numbers, and so ’tis impossible that such a Scale could express any true Musick.
(Alexander Malcolm, Treatise of Musick, 1721, page 304)

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.

2. Why Is Temperament Necessary?

Look at the keys on your piano. Start at middle C. Go twelve fifths up and seven octaves down (not necessarily in that order – you are allowed to intermix octaves and fifths as to keep near the middle of the keyboard). You will end at the key where you started.

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.

3. Meantone Temperament

Let us try another exercise on the keyboard: Four fifths up, two octaves down, and then one major third down – and we are back at the key where we started.

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.

4. Well Temperament

To flatten all the fifths is not necessary. It is possible to flatten only some of them, and arrive at a tempera­ment in which all keys are playable.

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.

Young Bach. Portrait from commons.wikimedia.org.
Source: Wikimedia.  Public Domain
Taciturn about tuning?
Johann Sebastian Bach (1685-1750) as young.
Portrait attributed to J. E. Rentsch, the Elder

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 more content-rich 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.)

5. That’s All

Bach never published details of how he tuned. I once read an anecdote saying that Bach avoided discussing the subject. I went to the library again some time later in order to document that anecdote – just to find that most tuning books had been taken off the shelf to make room for literature of ever­lasting value such as “Get Started With Windows Vista.” (The good news is that the physics textbook mentioned above had disappeared too.)

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)


NOTES AND REFERENCES

[1] Equal temperament was described well before Bach’s time by Prince Zhu Zhaiyu (1536-1611), Simon Stevin (1548-1620) and Martin Mersenne (1558-1648). Its history can be traced back to ancient times.

[2] 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

[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.

[4] Bradley Lehman, “Bach’s extraordinary tempera­ment: our Rosetta Stone - 2, ”
Early Music Vol. 33, No. 2, May 2005, pp. 211-231.

[5] Stephen Bicknell, “Temperament: A Beginner’s Guide.” Online at
www.albany.edu/piporg-l/tmprment.html and at www.stephenbicknell.org/3.6.04.php.


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