[2.2.3] Pythagorean, Meantone, Equal, and "Well" Temperaments

Historical developments are central to discussions of temperament because the music of the time is tied to the temperament of the time. Pythagoras is credited with inventing the Pythagorean Temperament at around 550 BC, in which the chromatic scale is generated by tuning in perfect 5ths, using the circle of 5ths. The twelve perfect fifths in the circle of fifths do not make an exact factor of two. Therefore, the final note you get is not exactly the octave note but is too high in frequency by what is called the "Pythagorean comma", which is about 23 cents (a cent is one hundredths of a semitone). Since a 4th plus a 5th make up an octave, the Pythagorean temperament results in a scale with perfect 4ths and 5ths, except at the end where you get a very bad dissonance. It turns out that tuning in perfect 5ths leaves the 3rds in bad shape, another disadvantage of the Pythagorean temperament. Now if we were to tune by contracting each 5th by 23/12 cents, we would end up with exactly one octave and that is one way of tuning an Equal Temperament (ET) scale. In fact, we shall use just such a method in the section on tuning. The ET scale was already known within a hundred years or so after invention of the Pythagorean temperament. Thus ET is not a "modern temperament" (a frequent misconception).

Following the introduction of the Pythagorean temperament, all newer temperaments were efforts at improving on it. The first method was to halve the Pythagorean comma by distributing it among two final 5ths. One major development was Meantone Temperament, in which the 3rds were made just instead of the 5ths. Musically, 3rds play more prominent roles than 5ths, so that meantone made sense, especially during an age when music made greater use of 3rds. Unfortunately, meantone has a wolf worse than Pythagorean.

The next milestone is represented by Bach's Well Tempered Clavier in which music was written with "key color" in mind, which required various Well Temperaments (WT). These were non-ET temperaments that struck a compromise between meantone and Pythagorean. This concept worked because Pythagorean tuning ended up sharp, while meantone is flat. In addition, WT presented the possibility of not only good 3rds, but also good 5ths. The simplest WT was devised by Kirnberger, a student of Bach. Its biggest advantage is its simplicity. "Better" WTs (all temperaments are compromises and they all have advantages and disadvantages) were devised by Werckmeister and by Young (which is almost the same as Valotti). If we broadly classify tunings as Meantone, WT, or Pythagorean, then ET is a WT because ET is neither sharp nor flat.

The violin takes advantage of its unique design to circumvent these temperament problems. The open strings make intervals of a 5th with each other, so that the violin naturally tunes Pythagorean. Since the 3rds can always be fingered just (meaning exact), it has all the advantages of the Pythagorean, meantone, and WT, with no wolf in sight! In addition, it has a complete set of frequencies (infinite) within its frequency range. Little wonder that the violin is held in such high esteem by musicians.

Sine about 1850, ET had been almost universally accepted because of its musical freedom and the trend towards increasing dissonance. All the other temperaments are generically classified as "historical temperaments", which is clearly a misnomer. Most WTs are relatively easy to learn, and most harpsichord owners had to tune their own instruments, which meant that they used WT. This historical use of WT gave rise to the concept of key color in which each key, depending on the temperament, endowed specific colors to the music, mainly through the small de-tunings that create "tension" and other effects. After listening to music played on pianos tuned to WT, ET tends to sound more muddy and bland. Thus key color does matter. On the other hand, there is always some kind of a wolf in the WTs which is reduced in ET.

For playing most of the music composed around the times of Bach, Mozart, and Beethoven, WT works best. As an example, Beethoven chose intervals for the dissonant ninths in the first movement of his Moonlight Sonata that are less dissonant in WT. These great composers were acutely aware of temperament. You will see a dramatic demonstration of WT if you listen to the last movement of Beethoven's Waldstein played in ET and WT. This movement is heavily pedaled, making harmony a major issue.

From Bach's time to about Chopin's time, tuners and composers seldom documented their tunings and we have precious little information on those tunings. At one time, in the early 1900s, it was believed that Bach used ET because, how else would he be able to write music in all the keys unless you could freely transpose from one to the other? Some writers even made the preposterous statement that Bach invented ET! Such arguments, and the fact that there was no "standard WT" to choose from, led to the acceptance of ET as the universal tuning used by tuners, to this day. Standardization to ET also assured tuners of a good career because ET was too difficult for anyone but well trained tuners to accurately tune.

As pianists became better informed and investigated the WTs, they re-discovered key color. In 1975, Herbert Anton Kellner concluded that Bach had written his music with key color in mind, and that Bach used a WT, not ET. But which WT? Kellner guessed at a WT which most tuners rejected. Subsequent search concentrated on well known WTs such as Kirnberger, Werckmeister, and Young. They all produced key color but still left open the question of what Bach used. In 2004, Bradley Lehman proposed that the strange spirals at the top of the cover page of Bach's "Well Tempered Clavier" manuscript represented a tuning diagram (see Larips.com), and used the diagram to produce a WT that is fairly close to Valloti. Bach's tunings were mainly for harpsichord and organ, since pianos as we know them today didn't exist at that time. One requirement of harpsichord tuning is that it be simple enough so that it can be done in about 10 minutes on a familiar instrument, and Lehman's Bach tuning met that criterion. Thus we now have a pretty good idea of what temperament Bach used.

If we decide to adopt WT instead of ET, which WT should we standardize to? Firstly, the differences between the "good" WTs are not as large as the differences between ET and most WTs, so practically any WT you pick would be an improvement. We do not need to pick a specific WT - we can specify the best WT for each piece we play; this option is practical only for electronic and self-tuning pianos that can switch temperaments easily. In order to intelligently pick the "best" WT, we must know what we are seeking in a WT. We seek one of two properties: (1) pure harmonies or (2) key color. We can not have both because they tend to be mutually exclusive. Pure harmony is an improvement over ET, but is not as good as key color. We already encountered this type of phenomenon in "stretch" (see section 5.10 below) whereby the music sounds better if the octave is tuned slightly sharp. Unlike stretch, however, key color in our chromatic scale is accompanied by dissonances in certain situations because the Pythagorean comma is too big. With this caveat, therefore, we should pick a WT with the best key color and least dissonance, which is Young. If you want to hear what a clear harmony sounds like, try Kirnberger, which has the largest number of just intervals.

We now see that picking a WT is not only a matter of solving the Pythagorean comma, but also of gaining key color to enhance music – in a way, we are creating something good from something bad. The price we pay is that composers must learn key color, but they have naturally done so in the past. It is certainly a joy to listen to music played in WT, but it is even more fascinating to play music in WT. Chopin is somewhat of an enigma in this regard because he loved the black keys and used keys far from "home" (home means near C major, with few accidentals, as normally tuned). He probably considered the black keys easier to play (once you learn FFP, section III.4.2, Ch. One), so that the fears many students feel when they see all those sharps and flats in Chopin's music is not justified. Chopin used one tuner who later committed suicide, and there is no record of how he tuned. Who knows? Could it be that he tuned Chopin's piano to favor the black keys? Because of the "far out" keys he tended to use, Chopin's music does not benefit much from WT, as normally tuned. Conclusions: We should get away from ET because of the joy of playing on WT; if we must pick one WT, it should be Young; otherwise, it is best to have a choice of WTs (as in electronic pianos); if you want to hear pure harmonies, try Kirnberger. The WTs will teach us key color which not only enhances the music, but also sharpens our sense of musicality.