Pythagorean, Equal, Meantone, 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. This is 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".
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 he wrote music for various Well Temperaments (WT). These were temperaments that struck a compromise between meantone and Pythagorean. This concept worked because Pythagorean tuning ended up with notes that were too sharp, while meantone is too 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 were devised by Werkmeister and by Young. If we broadly classify tunings as Meantone, WT, or Pythagorean, then ET is a WT because ET is neither sharp nor flat. There is no record of the temperaments Bach used. We can only guess at the temperaments from the harmonies in his compositions, especially his “Well Tempered Clavier”, and these studies indicate that essentially all the details of tempering were already worked out by Bach’s time (before 1700) and that Bach used a temperament not very different from Werkmeister.
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.
In the last 100 years or so, ET had been almost universally accepted because of its musical freedom and the trend towards increasing dissonance. Piano tuners liked it because it can hide minor changes in tuning that can occur just a few days after tuning. All the other temperaments are generically classified as "historical temperaments", which is clearly a misnomer. The 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. This greatly complicated issues because now musicians were dealing not only with pure chords versus wolves, but with colors that were not easily defined. The extent to which the colors can be brought out depends on the piano, the pianist, the listener, and the tuner. Note that the tuner can blend stretch (see "What is stretch?" near the end of section 5) with temperament to control color. After listening to music played on pianos tuned to WT, ET tends to sound more muddy and bland. Thus key color does matter. More important are the wonderful sounds of pure (stretched) intervals in WT. 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 chords for the dissonant ninths in the first movement of his Moonlight Sonata that are least dissonant in WT, and are much worse in ET. These great composers were acutely aware of temperament. Most works from Chopin's and Liszt's time were composed with ET in mind and key color is not an issue. Although these compositions sound different in ET and WT to the trained ear, it is not clear that WT is objectionable (for Chopin, etc.) because pure intervals always sound better than detuned ones. The conclusion is that Bach was right: WT should be used for everything, although some musicians might complain that Chopin sounds too bright in WT.
My personal view for the piano is that we should get away from ET because it deprives us of one of the most enjoyable aspects of music -- pure intervals, that was the motivation for creating the chromatic scale. You will see a dramatic demonstration of this if you listen to the last movement of Beethoven's Waldstein played in ET and WT. Meantone can be somewhat extreme unless you are playing music of that period (before Bach), so that we are left with the WTs. For simplicity and ease of tuning, you cannot beat Kirnberger. I believe that once you get used to WT, ET will not sound as good even for Chopin, once you get used to it. Therefore, the world should standardize to the WTs. Which one you choose (Kirnberger, Werckmeister, Valloti, Young) does not make a big difference for most people because those not educated in the temperaments will generally not notice a big difference even among the major temperaments, let alone among the different WTs. This is not to say that we should all use Kirnberger but that we should be educated in the temperaments and have a choice instead of being straight-jacketed into the bland ET. This is not just a matter of taste or even whether the music sounds better. We are talking about developing our musical sensitivity and knowing how to use those really pure intervals.
The biggest disadvantage of WT is that if the piano is out of tune by even a small amount, the dissonance becomes audible, whereas it is much less audible in ET. In fact, most acoustic pianos today will require more frequent tunings if tuned to WT. Therefore, WT will become more practical when the self-tuning pianos become available. There are no such problems with the electronic pianos, and in addition, you can change temperament with the flick of a switch. Another problem with WT is that transposition can change the key color. Of course, WT does not produce all pure intervals – every WT is a compromise just as ET is a compromise.
I believe that these WT drawbacks are minor compared to the advantages; I would be happy if all piano students developed their sensitivity to the point at which they can notice that a piano is very slightly out of tune. And music teachers should be even happier if their students start arguing about which WT is the best. It is about time we listened to Bach, who knew all about ET, but has been trying to tell us to use WT for the last 200 years.