is the introduction to this writeup
Meantone tuning: Mending the rift within the lute
In order to play music with a full range of major and minor chords within a single key, a subtle form of detuning has to be used, called temperament. The commonest form of this is meantone tuning. This means that each perfect fifth is made equal in size: i.e. each has the same ratio of frequencies. (Strictly speaking, Pythagorean tuning also fits this definition, but it is not usually called meantone.) The name comes from the fact that the tones C-D and D-E are now the same size, thus the note D is in the exact middle (mean) of the third C-E. The alert reader will immediately see that the meantone method can be extended to all the other degrees of the chromatic scale by taking fifths of the same size as E-B, B-F#, F#-C#, ... and C-F, F-Bb, Bb-Eb, ... at least up to a point.
To discuss meantone tunings we need a way of describing exactly the size of intervals and dividing them into N equal parts. This means taking the Nth root of the ratio describing any given interval. This is where cents come in: a cent is the 1200th root of 2, which means that if you divide an octave into 1200 equal intervals, each one is a cent. (The number 1200 was chosen because there are 12 semitones in an octave, so each is 100 cents if they are made equal.) The syntonic comma is about 22 cents. For notes near middle C, the number of beats per second in a perfect fifth is about half of the number of cents it is out of tune. (This is very useful in tuning by ear.)
The principle of constructing meantone temperaments is very simple. Each fifth is out of tune by some fraction F of a syntonic comma, and the major third is out of tune by one syntonic comma minus 4 × F. The simplest meantone tuning is quarter-comma (F=1/4), where each perfect fifth is out of tune by a ratio of the 4th root of 80/81, which is about 5.5 cents. Now when you pile up four quarter-comma tempered fifths, you get a major third which is exactly in tune. The quarter-comma tuning was used extensively in the Renaissance and early Baroque periods. It has the disadvantage that the distonation of each fifth is noticeable, if mild. The reason why quarter-comma meantone went out of fashion is that it has a very large wolf (the subject of another node) which becomes evident in music with a wider range of modulation into different keys. Without going into detail, the basic problem is that G# and Ab (or any enharmonic equivalent notes) would have to be quite different pitches, whereas there is only one key or fret with which to play them. As a result either the key of A has to have a very sharp leading-note, or C minor a very flat minor sixth, and one fifth (usually G#-Eb) is very much out of tune, being much sharper than the others.
A few instrument-makers did build keyboards with split keys and supernumerary strings or pipes to allow for this; but this became a futile exercise since composers in the 18th century began to write in keys remote from C major and also increasingly used enharmonic modulation, in which one single pitch had to serve as both G# and Ab (etc.) at the same time.
Fifth-comma meantone is very similar, except that now the major third is out of tune by the same amount (namely 1/5 syntonic comma, about 4.4 cents) as the perfect fifth. In this case the fifth and the third beat at an equal frequency when the major triad is sounded, which supposedly produces a particularly satisfying sound. Fifth-comma also has a big wolf, but not so big as quarter-comma. Sixth-comma has slightly wider fifths again, and the thirds are out of tune by one-third syntonic comma; the wolf is still quite noticeable. Sixth-comma meantone was in fact the system by which intonation was taught to vocalists and string players in the mid-eighteenth century -- for example Mozart. Theoretically, 1/6-comma meantone is easy to deal with because it results from a division of the octave into 55 equal parts, each of which is 21.8 cents, or (to a very good approximation) 1 syntonic comma. Then each tone is 9 of these parts, except for G#-Bb which is 10 parts.
All keys are equal, but some are more equal than others
One can go on making F smaller in the hope of eliminating the wolf, and this actually occurs when F is 1/11, that is about 2 cents (or 1/12 of a Pythagorean comma). The major thirds are out by 7/11 syntonic comma, so they are moderately discordant. This is equal temperament. It allows you to play in all keys and use any type of enharmonic modulation, and every key will sound equally out of tune. The fifths are almost exactly in tune, but the thirds are not very good. The problem of wide major thirds corresponds to another mathematical discrepancy called the lesser diesis, which is the difference between three major thirds (Ab-C-e-g#) and an octave (Ab-ab).
Equal temperament (ET) has been the standard tuning now for more than a century, so it is not surprising that most listeners do not notice the out-of-tune thirds. However when it was being introduced, for example in Victorian Britain, there were many protests by experienced musicians who found that it made organs sound worse, particularly for stops tuned a tenth or a seventeenth above the fundamental (so-called quint stops or mutations). Many organs actually had these stops removed because they sounded so bad in ET. Previously the organs had been in some variety of irregular temperament in which the fifths are of different sizes (some being wider than ET) and the most commonly-used major thirds have better intonation. On a piano you can easily hear the out-of-tune thirds by playing, say, C and e', which should produce a strong beating effect.
In music with a restricted range of keys -- Renaissance, early Classical, folk, rock-n-roll, R'n'B, country, gospel... -- there is no reason to accept the sub-optimal distribution of the syntonic comma which ET offers. On the guitar it is easy to experiment with tuning the fourths a couple of cents wider, and there are many recordings of harpsichords and organs with 1/4-comma, 1/5-comma, 1/6-comma (etc.) meantone tuning. You might wonder what happens musically when you meet the wolf note - a very flat Ab or a very sharp D#, for example. If the music was intended to be played in meantone, these notes will be used in such a way that they don't disrupt the piece and can actually add to its expressive range. I did read somewhere that the pianos in a famous Nashville studio were tuned in an unequal temperament, simply because they were always being used to play the same three chords. Now you know why.