The term “meantone” is a relatively modern invention; it seems to have first been used by Robert Smith in his book Harmonics (1749), in which he spoke of the “the temperament of mean tones”. For most of the period in which these temperaments were used, however, they were never referred to by this name. The term refers to the fact that the major third is divided into two “tones” of equal size, each of which are the geometric “mean” of the major third, or in other words, the square root of the ratio of the major third (whatever size it may be). This is the natural outcome of using four tempered fifths of equal size between the two notes which make the third. In the natural state of affairs, using a tuning according to the overtone series, these two tones are of slightly different sizes, C – D (9/8) being slightly larger than D – E (10/9).

The structure of a regular meantone temperament is very simple; eleven fifths are all tempered by the same amount, creating eight good major thirds and nine good minor thirds. The only variation among the different versions is the degree of tempering, or in vector terms, the steepness of the downward slope according to which the position of each successive note in the chain of fifths is determined. The less steep the slope, the less pure the major third, and the less dissonant the bad fifth and thirds which span the gap. Historical variations range from 1/3 comma, which produces pure minor thirds, to 1/6 comma, which produces major thirds noticeably wider than pure though not as bad as in ET. The only rational for using major thirds wider than pure is to reduce the tempering of the fifths. None of the regular meantones reduce the slope to the point where the wolf fifth becomes a good fifth, or where the bad major thirds (or the “false fourths”, as Prinz called them in 1696) to become good thirds. Thus we can but wonder as to why anyone would abandon the wonderfully-delicious sound of the pure major thirds of 1/4 comma meantone, as the organ builder Silbermann was reported to have done in favor of 1/6th comma meantone.

Some modern writers favor 2/9ths comma meantone because when a chord is played in root position/close spacing, the major third and the fifth beat at almost exactly the same rate. However, this acoustic curiosity is of little musical value, since the occurrence of adjacent root-third-fifth chord spacing is rare. In a similar fashion, others praise 2/7ths coma meantone, in which the downward slope is even steeper than that required for making pure major thirds, meaning that the major thirds are about as much too narrow as the minor thirds are too wide. In this case, in both the minor and major thirds beat at a nearly-identical speed with the fifth in a close-spaced triad, but once again the value of this so-called “ornamental beating” is highly questionable.

Such “equal beating” considerations are projections of the modern piano tuner’s bag of tricks onto the musical context. In reality, though, they are two completely different acoustical settings which most likely have little to do with one another; we simply don’t listen to music like the tuner listens to intervals when tuning, nor are the harmonies in music realized in the simplistic close spacing used by the tuner. I no longer believe that the perception of beats per se has anything to do with interval/chord quality in a musical environment. In any event, the clear historical preference is for 1/4 comma meantone, based on the overwhelming number of times it is described in the extant literature (see the list of reference notes).