Tuning and Temperament
Imagine a world in which the units used for linear measurement were not quite commensurate—one in which, by some quirky royal decree, let us say, twelve official inches did not quite make an official foot, or three feet exactly a yard. Most citizens, presumably, would be aware of a problem only rarely, but anyone whose profession depended upon precise measurement would long since have become expert at making fine distinctions; we can be sure that architects and carpenters, for instance, would have come to distinguish unabashedly between “inches” and “twelfths of a foot.” The units of our musical world—those we call “intervals”—are, in fact, of a similarly incommensurate nature, although unlike the units of our metaphorical example, their size is not determined arbitrarily; their mathematical ratios reflect basic acoustical phenomena. And unlike the professionals in our metaphor, musicians—those who must deal constantly with the problem—are for the most part unaccustomed to discussing it intellectually, generally preferring an intuitive approach. In fact, so out of favor is a “scientific” approach to intonation that to mention it may arouse suspicion among other musicians as to one's musical sensibilities.
It was not always so; in centuries before the nineteenth, a firm grasp of the mathematical foundations of music was considered to be one of the highest attainments of a good musician. But perhaps more relevant to the present discussion is the fact that different musical priorities in earlier times led to solutions different from the usual modern ones. (In our linear analogy above, our experts made a provisional redefinition of the inch in terms of the official foot; they might just as well have found reason to redefine the foot in terms of the official inch instead.) In order to appreciate these earlier solutions—and certainly in order to put them into practice ourselves—we have to have an understanding of both the underlying theory and terminology. Neither the conceptual basis nor the attendant math is really all that complicated, although the full ramifications of some intonation schemes can appear rather threatening. Fortunately, all the hard work, both theoretical and practical, has been done—over and over, in fact—and we are in a position to reap the benefit. It is the purpose of this short chapter to introduce some of the basic concepts and terms, provide some historical context, and serve as a guide to the copious resources already available to the performer.
Central to tuning theory is the idea of ratio or proportion. Before the nineteenth century, the intervallic ratios were understood in terms of string lengths on the monochord; more recently they have come to be understood in terms of vibration frequencies. Fortunately, the ratios themselves are the same, only inverted. Thus the octave, produced by a 2:1 ratio of string lengths, is also produced by a 1:2 ratio of frequencies; the fifth can be thought of as either 3:2 or 2:3, and the fourth as 4:3 or 3:4. All that really matters is consistency: choose one form or the other and stay with it, at least in any one calculation. Remember, too, that in adding ratios, one multiplies; in subtracting, one divides. Thus, adding a fifth (3:2) to a fourth (4:3), we get a ratio of 12:6 (3 × 4: 2 × 3), which reduces to 2:1—the ratio of an octave. Subtracting a fourth from a fifth should give us a major second; as with fractions, dividing by a ratio is the same as multiplying by its reciprocal (i.e., inversion), so that 3:2 multiplied by 3:4 (the reciprocal of the fourth, 4:3) gives us 9:8 as the ratio of a major second. (There is, incidentally, some research to suggest that this subtraction of a fourth from a fifth is pretty much what our brains are doing naturally and subconsciously to determine the size of a second.)
It was recognized from ancient times that a stack of six major seconds—a whole-tone scale, if you will—exceeds an octave by a fractional amount, equaling about an eighth of a tone. (This discrepancy is called a “comma”—a “ditonic” or “Pythagorean” comma, to be exact.) This small interval can be divided up and distributed equally along the chain of ascending fifths and descending fourths comprising the octave, all without most listeners being any the wiser; this, of course, is exactly what is done to achieve our standard modern system, equal temperament. (It is customary in discussions like this, by the way, to treat of both fourths and fifths as “fifths,” ignoring the octave displacements thus implied. For convenience, this convention will be followed from here on.) One serious problem remains, however, which is a lot harder to eliminate or hide—that concerning the major third. To back up just a little: on our journey up the whole-tone scale, long before reaching the problem of the octave, we find another discrepancy—another comma—this time between two 9:8 tones and a pure or “just” major third (with a ratio of 5:4). This “syntonic comma” is only slightly smaller than the ditonic comma we met above; it is still large enough to turn an otherwise sweet consonance into a comparative dissonance. (The ratio of the syntonic comma—that is, two tones less a pure major third—is 81:80; the math runs as follows: 9:8 × 9:8 = 81:64; multiplying by 4:5—the reciprocal of 5:4—results in 324:320, which reduces to 81:80.) If we substitute the slightly compressed major seconds of equal temperament for 9:8 (just) tones, we are only a little better of; the major third of equal temperament is only slightly less dissonant than the Pythagorean ditone—the true technical name for the interval made up of two 9:8 tones. (Why musicians since about 1800 have been less disturbed than Renaissance and Baroque ones about the impurity of major thirds is a complex question. The answer has a lot to do with tone color—the impurities are more noticeable with certain timbres than others—as well as the changing role of the major third itself; it has come to be prized for its dynamic quality, which is even intensified when a “leading tone” or “tendency tone” is infected toward its resolution.1)
If we want to maintain the purity of the major third, we have only a few choices. One, called “just intonation,” is to leave all but one of the intervening fifths pure, making that one bear the full brunt of the problem (and thus rendering it hopelessly dissonant, as well as producing a tone significantly smaller than 9:8—10:9, to be exact). On an instrument of fixed pitch (like a keyboard), no matter which of the fifths making up the third is made to be impure, it will be a bad choice for some chords, even within a single tonality. Thus just intonation is really feasible only on flexible-pitched instruments (and the voice, of course), where the decisions can be continually renegotiated. Much more practical on keyboards is to distribute the comma equally over all four fifths, subtracting a quarter of it from each. The result, when extended to the complete octave, is the system called “quarter-comma meantone” temperament. (“Meantone” itself refers to the size of the major seconds, which represent a mean or average between the unequal-sized tones of just intonation.) Meantone temperament is far less restricted than just intonation, although it does impose some of its own limitations, particularly on a keyboard of normal design. Its limitations are due to its strong differentiation between enharmonically related notes, which is due in turn to the failure of three pure major thirds to add up to an octave. Thus, for instance, the note 
—the octave of 
—is higher than the 
three pure thirds above 
(i.e., top of the series 
) by about a fifth of a tone. (The discrepancy this time is known as a “diesis.”) Obviously both 
and 
cannot be obtained by the same key. On a normal keyboard with twelve notes to the octave, choices must be made; the usual “black-key” selections are 
, and 
. (In England—for the keyboard music of Henry Purcell, for instance—the standard choice was 
instead of 
) Another possibility is to provide “split” keys. Two usually suffice, giving an additional 
and 
(available from the raised rear halves of the 
and 
keys, respectively). Although meantone, like any temperament, is in the strictest sense achievable only on instruments of fixed pitch, its pure thirds and compressed fifths are basic to the tuning of many early woodwinds. Being somewhat flexible, however, the latter can in practice “untemper” the fifths and achieve something closer to just intonation.
Also often classed as forms of meantone temperament are those regular temperaments in which the major thirds are allowed to be somewhat larger than pure. (A “regular” temperament is one in which all usable fifths are of the same size; these include all twelve in equal temperament, but in meantone—in which the circle of fifths does not close—there is always one dissonant or “wolf” fifth.) These forms of meantone are named according to the amount by which the fifths are tempered: two-ninths comma, one-fifth comma, one-sixth comma, and so forth. There are also many “irregular” temperaments—ones that mix different sizes of fifth. These range from informal amendments to meantone (such as Michael Praetorius's recommendation that one slightly untemper some of the fifths of quarter-comma meantone in order to make 
usable as 
in a pinch2) to various so-called well-tempered systems—”circulating” temperaments in which all keys are playable, but in which those nearer C major (those more often used) are better in tune at the expense of the more distant. In discussing these systems, some English-speaking theorists have painted themselves into something of a linguistic corner. Having made a rigorous distinction between “tunings” (in which the relationships can be described by ratios) and “temperaments” (which involve irrational numbers), they find the expression “well-tempered tuning” to be an oxymoron. (The distinction itself is ultimately more significant to theory than practice, by the way, since the irrational intervals can be approximated well enough by ratios that no one can tell the difference by ear.) Some resort to the somewhat odd-sounding expression “well temperaments,” using “well” as an adjective; others call them “good” temperaments, keeping “good” in quotes to remind us of its special technical meaning. In any case, one of the most influential of these well-tempered systems is known as “Werkmeister III,” being the third system offered by Andreas Werkmeister in his Musicalische Temperatur of 1691. (He had actually first published it a decade earlier in his Orgel-Probe.) Here the fifths C-G-D-A and 
are each made a quarter comma—ditonic, in this case—small; all the rest are pure.
These are the main types of tuning/temperament available to the seventeenth-century musician. There is no question as to the dominance of quarter-comma meantone temperament for keyboards throughout Europe during the first half of the century, and indeed in many places through the second half, as well. This is not to say, however, that alternatives were unthinkable. Even some of the simplest transpositions strained at the limits of the system and were an acknowledged source of frustration for many. (For instance, when dorian on G is transposed down a tone to 
are required as notes of the basic scale, and 
may be needed as ficta; when instead it is transposed down a minor third to 
will certainly be needed as ficta. None of these accidentals is available on a normal, twelve-key keyboard tuned in the usual meantone configuration.) Retuning to suit the transposition (as detailed by Gian Paulo Cima in 16063) is possible for string keyboards, although the practice seems to have been unusual; it is impractical, of course, for organs. Split keys were quite a common solution (except, apparently, in France), but they are awkward at best and were unacceptable to many players. However, both the theory of equal temperament and its common use on fretted instruments had been recognized since the sixteenth century, and its use on keyboard instruments found some strong proponents in the seventeenth—albeit perhaps more among theorists (such as Marin Mersenne) than performers. Notable among the latter, however, was Girolamo Frescobaldi, famous for some daring modulations. But, as an indication of the prevailing climate, his recommendation in 1640 to have the organs of Bernini's new apse in the Basilica of San Lorenzo in Domaso tuned to equal temperament was subverted; conservative attitudes prevailed, and the instruments were tuned to meantone.
The chief rivals to quarter-comma meantone in the latter part of the century were the less extreme versions of meantone (i.e., ones with thirds larger than pure) and certain irregular systems. Étienne Loulié claimed that fifth-comma meantone was “better and more in use” than any other temperament,4 and his statement is echoed by some other writers. At the same time, however, the system known as tempérament ordinaire was an irregular one, in which the fifths 
were made larger than pure. (It seems to have resulted from a fortuitous misunderstanding of Mersenne's unclear directions5 for tuning ordinary quarter-comma meantone!) The history of the acceptance of irregular temperaments in Germany is still a controversial topic. It was once suggested (from circumstantial evidence) that Dieterich Buxtehude had the organs of St. Mary's, Lübeck, tuned to Werkmeister III in 1683; however, the feasibility of such a retuning in the documented timeframe has since been chal-lenged.6 (How—or even whether—a number of his more adventurous organ compositions could have been accommodated on an instrument limited to quarter-comma meantone remains a matter of discussion.) Irregular circulating temperaments of these kinds were praised in the early eighteenth century for the distinctive quality they brought to different keys, which are, of course, indistinguishable in equal temperament (as well as in meantone, for that matter, as long as its bounds are not overstepped). It might be argued, in fact, that well-tempered systems restored to tonality some of the variety of color lost with the passing of modality.
The seventeenth-century tuner would have had the choice between tuning completely by ear (counting beats or judging the “favors” of slightly mistuned intervals) or by mechanical means—matching pitches with a monochord. The modern tuner has the same choice, except that the monochord has been superseded by the electronic tuning “box.” There are numerous written sources giving instructions for tuning by ear.7
Electronic aids also vary in thoroughness; the fanciest have several built-in tunings and can be programmed for a few more. But one can manage very well with the simplest ones that have a meter reading in “cents” (hundredths of an equal-tempered semitone). All one needs is a chart of the deviations (in cents) of each note from its equal-tempered value; these deviations can be extrapolated from sources such as J. M. Barbour's classic Tuning and Temperament. For instance, in Table 22, p. 26, specifying values for quarter-comma meantone, we find that E is 386 cents above C, or 14 cents shy of 400—its value in equal temperament. The meantone value for F is 503 cents, or 3 cents higher than in equal temperament. The only problem is that Barbour has centered his calculations on C; we need to center ours a little farther to the right on the chain of fifths (on D or A, say) in order not to have too many of our notes come out fat. (This is easily accomplished by adding a constant positive number of cents to each deviation.) With A as our “ground zero,” the deviations for quarter-comma meantone are as follows: 
, 
; if needed, 
. The deviations for fifth-comma meantone (also centered on A) are instead 
, 
. Those for Werkmeister III (centered this time on D) are 
,
.
One of the “hot issues” among early music practitioners nowadays concerns the tuning of fretted instruments. These were assumed by most writers of the period to have been universally and inexorably in equal temperament. The reason has certainly been explained often enough: in order to produce the unequal semitones of other systems and at the same time offer the standard choice of accidentals, at least some frets would have to run “zigzag”—clearly impossible for the tied-on frets of normal lutes and gambas. However, just as some keyboardists sought alternatives to the prevailing meantone standard, some lutenists and gambists took pains to mollify the egregious major thirds of equal temperament, through both playing technique and adjustment of frets. Speaking directly to this question is Lindley, Lutes. As in much of his writing on tuning, Lindley here transcends the merely descriptive by making astute qualitative judgments about the musical effects of different systems on different parts of the repertory.8
The importance of pure vertical relationships—and particularly the pure major third—to seventeenth-century musicians cannot be overemphasized. Clearly, any departures from purity would have been viewed, even by a Mersenne or a Frescobaldi, as a necessary evil, not the ideal. For modern musicians, used as we are to equal temperament—not to mention “tendency-tone” infections—singing and playing using pure thirds definitely represents learned behavior. Of immense help here are practicing and performing to the accompaniment of a keyboard (particularly an organ) in quarter-comma meantone. Also helpful is listening to recordings of such instruments, concentrating on the serene consonances—the real raison d'être of this temperament—at least as much as on the melodic aspects that may frst command one's attention. Woodwind players should be encouraged to obtain meantone versions of their instruments; rather than increasing one's burden (the usual fear), these actually predispose the instrument to better intonation. (One must, of course, help out a little bit by remembering to use the right fingerings—specifically those that differentiate between enharmonic pairs.) In the end, intonation is at least as important as timbre, if not more so, in recapturing the favor of early music.
NOTES
1. For a discussion of this issue, as well as many others, see Duffin, How Equal.
2. Praetorius, Syntagma II: 155; note that only parts 1 and 2 have been translated into English (see Bibliography); parts 3 and 4, pp. 81–236, have yet to be translated.
3. Rayner, “Enigmatic”: 23–34; see also the reference to Cima in Apel, History: 417–418.
4. Loulié, Nouveau: 28.
5. Mersenne, Harmonie universelle, vol. 3:108–109 (concerning tuning the spinet) and 364–365 (concerning tuning the organ); the relevant page numbers in the Chapman translation are 161–162 and 447–449, respectively.
6. See Snyder, Dieterich Buxtehude: 82–86 and 228–232 (2nd ed.), for a discussion of this issue.
7. An excellent introduction is in Lindley, “Instructions”: 13–23. Much more detailed and thorough, dauntingly so, are the books by Jorgensen, Tuning the Historical and Tuning. The latter give exact beat rates for all intervals; these have been calculated for a pitch standard of a' = 440 and have to be adjusted slightly for other standards. Also useful is Tittle, A Performer's Guide, and Donahue, A Guide to.
8. His article “Temperaments” in the New Grove remains one of the best introductions to the whole subject. See also Duffin, “Tuning,” for many practical suggestions concerning this issue and others.
BIBLIOGRAPHY
Barbour, Tuning and Temperament; Cima, Partito; Donahue, A Guide to; Duffin, How Equal; Duffin, “Tuning”; Jorgensen, Tuning; Jorgensen, Tuning the Historical; Lindley, “Instructions”; Lindley, Lutes; Lindley, “Temperaments”; Loulié, Nouveau; Praetorius, Syntagma II; Rayner, “Enigmatic”; Snyder, Dieterich Buxtehude; Tittle, A Performer's Guide.