Tuning and Temperament(Pietro Aron - "Toscanello in Musica", Venice, 1523).
Aron's scheme was refined by a myriad of musical theorists and scholars in the following centuries. There are countless different varieties of Meantone intonation, with the comma divided among the rest of the intervals in ever more esoteric ways, but all follow the same basic philosophy.
Despite its drawbacks, meantone was the dominant European keyboard tuning system for over 400 years, from its introduction in the late 15th century until well into the 19th century.
Werkmeister - Kirnberger - Thomas Young
Well temper was a further development of Meantone intonation, designed to allow use of all the key signatures. It was pioneered by Andreas Werckmeister (1645 - 1706), among others, in the late 17th century. "Well Temper" tamed the wolf, but left varying degrees of "out-of-tuneness" in the more distant keys. This gave rise to the concept of "key colour" - as different keys are out of tune in differing amounts, and in different places, they have different characters.
Opponents of equal temper argue that much is lost when Bach or Mozart is performed in equal temperament, as these composers were well acquainted with the different "key colours", and used them deliberately for effect. "Playing Bach's Well-Tempered Clavier in today's equal temperament is like exhibiting Rembrandt paintings with wax paper taped over them", in the words of one scholar. Mozart's music also gains another dimension when played in its original temperament.(Equal temperament has no "key colours". As all its intervals are equal, the blend of intervals is the same in every key, so all keys sound alike. "Key colour" is a feature of irregular temperaments like Meantone and Well Temper.)
Well temper and equal temper are not the same thing. It is true that Well Temper was often called equal temper when it first came into vogue but this was not because all its intervals were equal, but because it made it all keys equally usable. Each key, however, had a slightly different character: Bach wrote "The Well-Tempered Clavier" in all 24 key signatures to exploit those differences, not because there were no differences.
It is documented that both Bach and Werckmeister were opposed to the rigid mathematical treatment implied by the term "gleichschwebende" - which translates literally as "like-beating". "Like-beating" implies that parallel intervals beat equally in all keys - which is only true of equal temper.
Well temperament was a kind of halfway house between meantone and our modern equal temperament - and came in many flavours. Well temperament has been researched by many music theorists and there are many different schemes. Some keyboard tuners use different schemes depending on what music will be played on the instrument (or customer preference). Here for example are three of the most widely-accepted well-tempered schemes, compared to the equal-tempered scale, with the differences tabulated in cents:
The Well-Tempered Clavier
"The exact tuning of his instruments as well as of the whole orchestra had his greatest attention. No one could tune and quill his instruments to please him. He did everything himself." (1774, New Bach Reader, #394) Forkel then presented it thus:
"Nobody could install the quill-plectrums of his harpsichord to his satisfaction; he always did it himself. He also tuned both his harpsichord and his clavichord himself, and was so practised in the operation that it never cost him above a quarter of an hour. But then, when he played from his fancy, all the 24 keys were in his power; he did with them what he pleased. He connected the most remote as easily and as naturally together as the nearest; the hearer believed he had only modulated within the compass of a single key. He knew nothing of harshness in modulation; even his transitions in the chromatic style were as soft and flowing as if he had wholly confined himself to the diatonic scale. His Chromatic Fantasy, which is now published, may prove what I here state. All his extempore fantasies are said to have been of a similar description, but frequently even much more free, brilliant, and expressive." (1802, New Bach Reader, p436. English translation by Kollmann, 1820.) (CPE Bach and Forkel quotes borrowed from www.larips.com, with thanks.) Unfortunately neither Bach nor his sons ever saw fit to write down explicit instructions on how to tune this temperament, and it has thus sadly been lost to us for almost 250 years. However, according to Dr. Bradley Lehman, A.Mus.D. (harpsichord), of Goshen College, Indiana, the key to the mystery has been hidden in plain sight since 1722 - in Bach's own hand, on the title page of the original manuscript of "Das Wohltemperirte Clavier". Dr. Lehman's intimate knowledge of Bach's music, and extensive hands-on experience of harpsichord tuning, enabled him to crack the code at last. His breakthrough came in April 2004 and is making huge waves internationally in the world of the harpsichord and the pipe organ. A number of newly-built pipe organs using Lehman's "New" Bach Temperament have already been installed in music colleges and churches by one of America's leading organ builders, and work is in hand on more instruments. A newly-built "Bach/Lehman" organ was recently dedicated in Laajasalo Church in Helsinki, Finland. Harpsichord concerts in this temperament are being performed around the world. For further information, see:
Dr. Lehman's fascinating website. Back to top
There is little question but that the majority of fretted instrument makers and players had adopted equal temperament - or as close an approximation to it as they could achieve - by the 16th century. Nicola Vicentino wrote in 1555 about the difficulties arising from the parallel usage of two systems - meantone for keyboard instruments, and equal temperament for the fretted instruments. ("L'antica musica ridotta alla moderna prattica", Rome, 1555.) Orchestras of the period became quite large, to compensate for the fact that their different sections could not all play at the same time.
Alongside Pietro Aron, two other major contributors to the tempering revolution of the 16th century were Gioseffo Zarlino and Francisco Salinas. While both Zarlino and Salinas propounded similar irregular tunings for keyboards, both men also recognised the necessity for equal temperament on fretted instruments. Indeed Salinas, a blind professor of music in Naples, gave the first precise mathematical definition of equal temperament when in 1577 he wrote:
"We judge this one thing must be observed by makers of viols, so that the placing of the frets may be made regular, namely that the octave must be divided into twelve equal parts equally proportional, which twelve will be the equal semitones." (De musica libri VIII, Salamanca, 1577.)
Before the 16th century, very little was known about the generation and transmission of sound. The first scientifically valid experiments were made by Vincenzo Galilei (1520 -1591), a professional musician, lutenist, composer, theorist, and teacher - and father of Galileo Galilei (1564 - 1642). Using much the same methodology as Pythagoras (suspending weights from strings), he gathered data which enabled him to establish the relationship between pitch and frequency, and state the laws of harmony and dissonance, essentially as they are understood today. He also demonstrated that the natural frequency of a vibrating string is dependent on its length, weight, and tension.
Musicians had undoubtedly become quite skilled in the art of adjusting their frets (which in instruments of the time usually consisted of lengths of gut, tied around the neck) by trial and error, or by simple geometrical methods. But the first appearance in writing of a practical approximation for placing frets in equal temperament - the 18:17 ratio, or "Rule of 18" - was given by Vincenzo Galilei in his "Dialogo della musica antica e moderna", published in Florence in 1581.
Galilei explained the need for equal semitones logically and correctly - since the frets are placed straight across the six strings, the order of diatonic and chromatic semitones is the same on all strings. In chords, therefore, a C# might be sounded on one string, and a Db on another - this will be a very false octave unless the instrument is in equal temperament.
The Rule of 18
The "Rule of 18" puts the first fret at 1/18th of the distance from the nut to the bridge, the second fret at 1/18th of the distance from the first fret to the bridge, and so on. This system made it simple to mark out a fret scale on a piece of wood using just a ruler and a compass, and it remained the basis of practical fretboard construction for over 300 years.The Rule of 18 results (theoretically) in intervals of 99 cents:
This almost certainly gave acceptable intonation with the natural gut strings of the period, which were far more elastic than modern-day steel or nylon strings. The instruments of the day seldom had more than 8 or 9 frets to the body, so a minor adjustment to the string length at the bridge would suffice to tame any tendency to "out-of-tuneness" up the neck.
The emerging technology of the 1800's had a profound impact on keyboard tuning. The issue was forced by advances in piano design. In the search for more power, high-tension, cast-iron-framed pianos, using newly-developed, much stronger wire, came into use. As the tension increased, so did the number of audible overtones. Dissonances that could be tolerated on the much more lightly strung harpsichords and fortepianos became unbearable on the new high-tension instruments. Musicians and composers - the Romantics in particular - wanted a temperament that would let them modulate freely without encountering the wolf. The first pianos to be tuned to equal temperament were produced by Broadwoods in the middle of the 19th century.
The 19th century was a time of tremendous advances in materials technology and engineering. Science was becoming synonymous with progress in all walks of life, and musical instrument design was not immune to its inroads. The "mathematically correct" equal temperament held great appeal in this atmosphere of trust in science, particularly since vast improvements in the accuracy and sophistication of machine tools were making it possible to mass-produce musical instruments to a degree of precision undreamed of a few generations earlier.
Equal temperament is the ultimate compromise. Tonal purity is sacrificed for freedom of modulation. Depending on your viewpoint, equal temperament either a) makes every key equally in tune, or b) makes every key equally out of tune... The idea is to make it possible to play all intervals and chords, in all keys, with the same relative accuracy. Although every key is very slightly out of tune, every key is also useable. No key sounds worse than any other key. The same applies to all chords.The final countdown The key to the equal tempered scale is the number 1.0594631, the twelfth root of two. This number has a bunch of tricks up its sleeve! If you multiply one by it twelve times you get two (and if you divide two by it twelve times you get one...).
1 x 1.0594631 = 1.0594631
The right-hand column of figures defines the semitones of the equal tempered scale, which gives us 12 semitones of 100 cents each and thus an octave of exactly 1200 cents. The old Rule of 18 was not so far off - extrapolation of the twelfth root of two ratio gives us a divisor of 17.817152.
The equal-tempered fifth is fairly "sweet" at 700 cents compared to the just fifth at 702 cents, and the equal-tempered fourth works well at 500 cents compared to 498 cents for the just fourth - but thirds, sixths and sevenths fare less well. Thirds are especially troublesome, as the equal-tempered minor third is 16 cents flat to the just minor third, and the equal-tempered major third is 14 cents sharp of just. The equal-tempered major sixth is 16 cents sharp of just, and the equal tempered major seventh is 12 cents sharp of just. The only interval which is identical in the two scales is the octave.© Paul Guy 1999 - 2006