Ross W. Duffin • Case Western Reserve University

T uning and temperament in music from the Middle Ages to the Baroque Era is a

fascinating subject. It can have an enormous effect on the sound of a musical performance but, ironically, is simultaneously complex and easy to ignore. In recent years, it has become a "given" that performers wishing to follow principles of historical performance will employ some temperament with an historical pedigree, particularly if they are performing with a keyboard instrument such as a harpsichord. Out of curiosity, I regularly ask performers of Baroque music what temperament they are using and increasingly, to my dismay, the answer is "Vallotti" or "Vallotti/Young". Exactly why that should cause me any dismay is the subject of this article.

It could be argued that what is needed is a full exposition of problems and solutions in using historical temperaments, not a specialized study. I have chosen to focus on what I perceive to be a current problem, however, hoping that the explanations will give readers an understanding of the basic issues along with the specifics of this topic. Words that are highlighted have pop–up definitions available, so help should be only a click away. Some basic explanation is needed first anyway.

Notes that are higher in pitch vibrate faster than lower notes, and the ratio between those vibrations determines the interval between the notes. Thus, a note vibrating twice as fast as another will sound an octave higher. Without the technology to measure vibrations, even medieval theorists noticed that this pitch relationship applies to string lengths as well: a string half as long as another of the same material and tension sounds an octave higher. We therefore say that the ratio of the octave is 2:1. This is the simplest, least problemmatic interval, and the easiest to tune as well.

When notes are perfectly in tune with one another according to the harmonic series (the overtones of a musical sound that are present in varying strengths and help to determine its tone color), the interval produced is clean and stable. When the notes are close but not in perfect agreement, there is interference between the respective harmonic series of the two sounds, and that creates a noticeable "beat" or wavering of the interval, most prominently at the lowest conjunction or shared harmonic in both series. (The octave is the easiest interval to tune because its beats are so easy to hear and match to the notes of the interval: the higher note itself belongs to the harmonic series of the lower note so the beats are in unison with the higher note.) The more out of tune the notes are, the faster the beat. As the beats get slower and the sound suddenly becomes tranquil, it is clear that the notes are in tune. Other intervals have beats that are more difficult to relate to the pitches of the interval, so those are harder to tune. But all intervals with notes in some kind of harmonic relationship to one another create beats unless the interval is acoustically pure, or else so far out of tune that we can’t perceive the beats.

After the octave, the next simple interval is the fifth. Its ratio is 3:2, which means that a fifth is created when one note is vibrating 1 1/2 times as fast as the other (or with string lengths in a 3:2 proportion). Fifths occur in every triad and therefore in most harmonies from the period in question. It would be nice to be able to use acoustically pure, beatless fifths in performance, but it’s not that simple. The problem is that tuning a series (commonly though in this case inaccurately called a "circle") of pure fifths, say, up from C to G to D and so on, eventually, you’d expect to arrive back at C again.

Unfortunately, when you expect to be arriving back on C with the twelfth fifth in the series, the note you tune with that last pure fifth is actually quite a bit higher than the original C or any octave multiple of it. That difference–about 1/4 of a semitone–is something we call the Ditonic comma or, perhaps more familiarly, the Pythagorean comma in deference to the Greek theorist’s "discovery" of the harmonic series and the 3:2 proportion of the fifth.

Thus, pure fifths, which would seem to be desirable, are incompatible with our twelve–note octave. The solution generally used today is to narrow (temper) the fifths by a tiny amount–1/12 of the Pythagorean comma or about 1/50 of a semitone–so that the comma discrepancy is spread out over the whole series and the fifths really do make a circle. Thus, the C of arrival is in tune with the original one. A byproduct of this system is that all of the semitones in the octave are equal, but the name Equal Temperament (ET) actually comes from the uniform narrowing of all of the fifths. The virtue of ET is that it can be used in any key from C to C# or Cb major with no change in the quality of the chords and no limitation on the use of enharmonic notes like G# and Ab. There is a price to pay, however.

After the fifth (3:2) and its partner the fourth (4:3)–like Ginger Rogers, doing everything the fifth does only backwards (I don’t know about the high heels)–the next harmonic interval of critical importance is the major third (5:4). In the harmonic series, the major third is a much narrower interval than we’re used to, but its beatless, tranquil quality can be appreciated after a few hearings. Unfortunately, the pure major third (PM3rd) cannot be easily reconciled to a twelve–note octave either: tuning a series of PM3rds up from C (C—E, E—G#, G#—C) produces an octave that is excruciatingly narrow. Neither can the PM3rd and the pure fifth be reconciled in a twelve–note octave. Tuning four pure fifths in series (C—G, G—D, D—A, A—E) produces a C—E third that is much wider than an acoustically pure third, a discrepancy known as the syntonic comma. Only by tempering each of the intervening fifths by 1/4 of the syntonic comma can the resultant third be made pure. Systematically tempering the fifths by this amount is a solution–quarter–comma meantone–that most Renaissance musicians chose, preferring the sweetness of its eight PM3rds, even though the fifths are almost 2 1/2 times narrower and more dissonant than fifths in ET. This is the cost of a system that favors PM3rds:

1. Fifths so narrow that, especially without the mitigation of the perfectly euphonious thirds, they sound starkly dissonant.

2. A limitation on the number of usable chords to those involving only two or three sharps or flats–enough for most Renaissance music but increasingly inadequate for Baroque music.

On the other hand, the cost of the flexible, universally usable system of ET is M3rds that are all equally bad. Quarter–comma fifths are just that: 1/4 of a syntonic comma narrower than pure fifths. ET M3rds, however, are 2/3 of a syntonic comma wider than pure! It’s a heavy price to pay for utility, and even though some theorists discussed ET at least from the early 17th century, most musicians elected not to use it, largely because the thirds were so awful and because it was so difficult to tune by ear.

So what tuning system did Baroque musicians use?

S ome continued to use quarter comma meantone in spite of its limitations: it’s

arguably still the best choice for much music of the early– to mid–17th century and it continued to be used for organs long after that. Others chose to compromise the purity of the thirds slightly, tempering all the fifths equally but not as much as in quarter comma meantone. These regular meantone systems–1/5, 1/6 comma, etc.–feature good chords in the same places, but although the thirds are not as good as in quarter comma, the fifths are better, and complex harmonies are better. All of these regular meantone systems share one problem interval, however: a "wolf"–one fifth that isn’t even really a fifth because it is so far out of tune.

Thus, theorists were motivated to explore other tuning solutions. Many of these involved a core or "backbone" of tempered fifths, along with some fifths tempered not at all (e.g., Kirnberger), not so much (e.g., Neidhardt), or even tuned wider than pure (e.g, tempérament ordinaire). These are all termed "irregular" systems because they use different sizes of fifths. Those that eliminate the wolf and are serviceable in a variety of keys are called "well" temperaments, or even "circulating" temperaments if they can be used–for better or worse–in all keys without adjustment. This is the category to which belong the temperaments devised by Francesc’ Antonio Vallotti and Thomas Young.

Vallotti (1697–1780) was a theorist and composer who worked in Padua. His Trattato della scienza teorica e pratica della moderna musica wasn’t published until 1779, but he claimed his theories had all been worked out by 1728. Actually, only Book 1 out of four was published in 1779–the rest languished in manuscript until 1950–and his temperament discussion appeared in Book 2, chapter 4. Vallotti’s temperament was not unknown, however, since it was cited in England by William Jones in 1781 (Physiological Disquisitions) with a reference to endorsement by Tartini. This is perhaps understandable since Tartini was concertmaster of the standing orchestra under Vallotti’s direction at Padua. Thomas Young (1773–1829) was a polymath whose temperament was given as part of an address to the Royal Society in January, 1800. (Philosophical Transactions, pp. 143–47).

The temperaments advocated by these two men are very similar. Vallotti’s calls for a series of six fifths (F—C—G—D—A—E—B) to be tempered by 1/6 of a Pythagorean comma. The rest of the fifths around the "back side" of the circle were to be tuned pure. Young’s is essentially a transposition of this system. The tempered fifths are C—G—D—A—E—B—F#, with the rest of the fifths tuned pure. Using the modern "cent" units (1/100 of an ET semitone, or 1/1200 of an octave), the two temperaments may be expressed as follows:

Fig 1. Vallotti and Young "Circulating" Temperaments in Cents

Vallotti

C C# D Eb E F F# G G# A B B C

0 94 196 298 392 502 592 698 796 894 1000 1090 1200

Procedure: Temper F–C–G–D–A–E–B by 1/6 of a Pythagorean comma, then tune the rest of the 5ths pure to F and B.

Listen to this example

Young

C C# D Eb E F F# G G# A Bb B C

0 90 196 294 392 498 588 698 792 894 996 1090 1200

Procedure: Temper C—G—D—A—E—B—F# by 1/6 of a Pythagorean comma, then tune the rest of the 5ths pure to C and F#.

Listen to this example

Cents charts are especially useful for comparison with ET. ET semitones are found by definition at the even 100s all the way up the scale, so numbers slightly below the 100s represent lower pitches and numbers above, higher pitches. As mentioned above, the two temperaments are similarly constructed but, as can be seen, they are not identical. This is my first problem–granted, a minor one–with the citing of Vallotti/Young as a temperament in use today. The two temperaments are not the same. The "naturals" of the keyboard are all the same except for the F, but the "accidentals" are different. In my experience, many people are not aware of the differences between these temperaments and often use the term "Vallotti/Young" even when they are actually tuning Vallotti. This is semantic distinction but, still, there’s no reason not to get it right. Furthermore, "Vallotti–Young," as it appears in some places to refer to Young as a transposition of Vallotti, seems an unnecessary obfuscation.

So why are these systems so popular anyway? There are three main reasons:

1. Ease of setting. Any system that includes pure fifths is easier to tune than a system with only tempered fifths.

2. Circulation. The amount of tempering is satisfying but there is enough flexibility that they qualify as circulating temperaments. Thus, modern musicians can program whatever pieces they want in whatever order and, using Vallotti or Young, can be confident that each piece will sound acceptable.

3. Key differentiation. This was more important to Baroque musicians than modern ones. Besides the horrid thirds and the difficulty of setting it, the other reason Baroque musicians avoided ET was because it didn’t differentiate among the keys. With Vallotti and Young, every key has a different amount of tempering and thus a slightly different "flavor." The concept of key characteristics, which arose during the Baroque era, originated because the keys really are different under irregular systems like these, a feature that was valued by musicians of the time.

Vallotti Young

Click on a diagram to view a full size image

These are all valid and valuable attributes. These temperaments really are useful and can contribute to good–sounding performances. So why would I say–perhaps in an attention–grabbing overstatement–that I hate Vallotti? What’s not to like?

T he first thing is the "fudge" factor. It may seem dogmatic to point this out, but a

temperament that was almost published in 1779 and another published in 1800 are not the best historical choices for Baroque music. Granted, Vallotti claimed to have worked out his system by 1728 and, presumably, anyone who happened to travel to Padua and ask him about it after that might have used it, but it wasn’t widely known like the published systems of Werckmeister and Neidhardt, if at all. Even Young himself seems oblivious to Vallotti's work: "As far as I know, most of these observations are new" (p. 146). If Vallotti’s temperament weren’t so universally used today, perhaps I wouldn’t care, but it has become so completely accepted as a good, historical solution, that people have stopped noticing that the historical evidence for its use in Baroque music is not that strong. An analogy is the situation in Baroque trumpet performance today, where three and even four tone holes for facilitating certain harmonics have become universally used on slender or non–existent historical evidence, but they work so well that hardly anyone one wants to "rock the boat." It’s time to question the historical position of Vallotti for Baroque music and, if it’s weak, then to explore some of the alternative circulating temperaments that were known earlier in the 18th century, like those of Johann Georg Neidhardt.

My second complaint about Vallotti–and Young, by extension–comes from my position as an educator. It may seem unfair to criticize a temperament on the basis of complexity when the musical results achieved by professional musicians are often excellent. But so long as young musicians continue to be trained in ET, this will be a problem and we might as well face it: the notes in Vallotti are too hard to find. Hah! What a ridiculous assertion, you say. You tune it and then you play. Well, that may be true for keyboard players, who decide on a temperament that suits the music, spend a little time setting the temperament on the keyboard, then forget about it. It is in ensemble situations that I have a problem with Vallotti. From key to key, accidental to accidental, there are six different sizes of semitone in Vallotti and Young. Professionals who get used to performing with the same temperament all the time come to grips with this and eventually learn where to find the notes, but for student performers, it’s a frustrating situation, frequently aggravated by someone telling them how important it is to be performing in a "Baroque" temperament. It’s not just Vallotti and Young that share this problem, of course; any irregular temperament will have similar difficulties. But again, it’s Vallotti as the universal solution, chosen by harpsichordists who use it all the time for their solo playing, know how to tune it (and may not want to learn a new temperament), and unwittingly force the unsuspecting members of the ensemble–from sonata soloist to Baroque orchestra–to play it their way. It’s a clear case of the blind leading the blind. Keyboard players don’t see the problems of non–keyboard instruments performing in Vallotti, and the other instrumentalists and vocalists don’t know what the problem is–they just know it’s hard to be in tune.

Vallotti Young

Click on a diagram to hear the semitones

Lastly, as heretical as it may sound, I would like to suggest that Vallotti and Young are not the best choices for Baroque music for musical reasons. Although recognized as an asset, "circulation" can be a liability as well. The liability of ET, as we have seen, is M3rds that are uniformly bad. In Vallotti and Young, there are three M3rds that are quite good, two that are mediocre, two that are as bad as ET, and five supposedly usable M3rds that are actually worse than ET! In the home key of C for Vallotti and G for Young, there are few problems, but venture very far from those centers and the consonances, though usable, are not very satisfying.

Vallotti Young

Click on a diagram to hear the Major Thirds

Is the flexibility of Vallotti and Young too much of a compromise for Baroque music? Is there some other historical temperament that, especially given a little more circumspection in the choice of keys for a given concert program, would produce a better result and be easier for an ensemble?

I believe the answer is 1/6 comma meantone, a regular meantone system where all the

fifths but one (the wolf) are narrowed by 1/6 of a sytonic comma. The resultant temperament has several keys that are equally consonant, several M3rds of good quality, predictable semitones of two different sizes–one for diatonic and one for chromatic semitones–as well as other musical assets. Threaded throughout the discussion in Bruce Haynes’s superb 1991 Early Music article (pp. 357–81), "Beyond temperament: non–keyboard intonation in the 17th and 18th centuries," is the frequent re–appearance of 1/6 comma meantone. Already widely known today as Silbermann’s temperament (after the organ builder and friend of J. S. Bach), it corresponds to ideals for temperaments given by Telemann, Tosi, Quantz, and Geminiani, among others, and is described by the French theorist Sauveur (1707) as the temperament of "ordinary musicians". The historical evidence as presented by Haynes speaks for itself, and is a stunning endorsement. What better temperament to use as a standard for Baroque music than the one employed by everyday musicians at the time? So, from an historical standpoint for music from the earlier 18th century, 1/6 comma beats Vallotti and Young. Its cents chart (with rounded approximations) is given here:

Fig 1. Vallotti and Young "Circulating" Temperaments in Cents

Fig 2. 1/6 Comma Meantone Temperament in Cents

C C# D Eb E F F# G G# / Ab A Bb B C

0 88.6 196.7 304.9 393.5 501.6 590.2 698.4787 / 806.5

895.1 1003.3 1091.9 1200

Procedure: Temper all 5ths by 1/6 of a syntonic comma, leaving the wolf at Eb—G# or at Ab—C#.

Listen to this example with Ab | Listen to this example with G#

Furthermore, it is clear from the many references to major and minor (diatonic and chromatic) semitones and particularly the distinction between sharp and flat versions of a note in woodwind fingering charts well into the 18th century–not to mention Peter Prelleur’s violin fingerboard illustration of 1731–that some form of regular meantone temperament is being assumed as a standard. Vallotti and Young have major and minor semitones in some places, it’s true, but the whole point of a circulating temperament is that the notes can remain fixed and still be usable. Distinctions between G# and Ab are unnecessary because the compromise position of the note in Vallotti and Young is meant to serve for both harmonic contexts. In other words, if Baroque musicians were thinking in terms of circulating temperaments as a standard, then they wouldn’t need to distinguish between enharmonic notes in fingering charts, and they wouldn’t need to build keyboards with split keys, for which there is still some scattered evidence in the 18th century. In addition, the difference between the major and minor semitone in 1/6 comma meantone is very close to a syntonic comma, which lends an extra degree of melodic elegance in a system based on divisions of that interval. The fact that diatonic semitones are always the same size and chromatic semitones are always the same size, however, creates a predictability as to where the notes will be found that Vallotti and Young simply cannot provide. Furthermore, this regularity enables fretted instruments to play fairly easily with the temperament–setting major and minor semitones on the fingerboard–whereas trying to set frets to match an irregular temperament can be hopelessly frustrating.

1/6 comma semitone with Ab 1/6 comma semitone with G#

Click on a diagram to hear the examples

The large number of good M3rds, meanwhile, and the keys that use them, mean that the possibilities for transposition are superior in 1/6 comma over Vallotti and Young. Pitch standards like Chorton (A=ca.460) and Kammerton (A=ca.415) in Germany must have required frequent transposition by whole tone up or down. In 1/6 comma, this can often be done with no change in the relative position of the notes. But in an irregular system, the tempering is always different for a transposed key from its written version. "Hey, don’t forget, we’re playing in G at A=415 but the organ is in Young at A=460 with the player transposing to F, so that G is going to be lower (or the B and D higher) than you expect, heh, heh." Such conversations were unnecessary with 1/6 comma, but must have taken place, even among musicans familiar with the temperament, when irregular temperaments were in use.

1/6 comma 5th diagram with Ab 1/6 comma 5th diagram with G#

Click on a diagram to view a full size image

1/6 comma M3rd with Ab 1/6 comma M3rd with G#

Click on a diagram to hear the examples

The good M3rds of 1/6 comma are, of course, balanced by four that are so wide as to be unusable (although only 5 cents wider than the worst and supposedly usable M3rds of Vallotti and Young). The classic case is the need for both G# and Ab. Most instruments (and singers) aren’t going to have a problem with that since they can make diatonic semitones, etc. wherever they are needed–hence the distinctions in the fingering charts noted above and references by some theorists to a "55 division" where the octave is divided into 55 commas (basically a fully extended 1/6 comma meantone)–but continuo keyboards with only twelve notes to the octave are stuck with whichever note was chosen when the temperament was set. There are some practical solutions to this problem in performance, some of which may be possible in various circumstances, and all of which can help to extend the utility of 1/6 comma:

1. Choose pieces that need only one note or the other.

2. Retune the note as needed for a piece or group of pieces.

3. Avoid the "wrong" notes and leave them to more flexible instruments.

4. Tune the notes differently in various octaves and avoid the "wrong" versions in the realization.

5. Tune the notes differently on each manual, using one manual for flat–leaning keys and another for sharp–leaning keys.

6. Use a keyboard with split keys.

Finally, there is one last musical reason why I think 1/6 comma is so satisfying. Complex harmonies so typical of Baroque music–including 7th chords and diminished triads–simply sound more satisfying in 1/6 comma. I could never understand why this should be so until I realized that the tritone in 1/6 comma meantone has about as much claim to harmonic purity as could be made. At 590.2 cents (again, a rounded approximation), the 1/6 comma tritone is virtually a combination of the pure (5:4) M3rd at 386.3 cents, and the pure (9:8) major whole tone at 203.9 cents. This is a rationalization, of course, since the interval itself (45:32) does not occur at an audible place in the harmonic series, and yet it may explain why the interval and its reciprocal

diminished 5th (64:45) are so completely satisfying, and thus why the tension and release of Baroque harmonic progressions sound especially fine in 1/6 comma.

Vallotti Young 1/6 comma with Ab 1/6 comma with G#

Click on a diagram to hear the examples

N othing can diminish the flexibility and utility of Vallotti and Young as keyboard

temperaments. They may, in fact, suit your needs precisely. But what I have tried to do here is to show that there may be better choices, especially for Baroque ensemble music, and that the unexamined use of a convenient solution is not necessarily the best musical path. It’s not Vallotti per se that I "hate"; it’s Vallotti’s ubiquitous use in situations where it is clearly neither the best historical nor the best musical choice.

A NOTE ON THE SOUND EXAMPLES:

The sound examples were produced on a Korg M1 keyboard, down a half step from modern pitch and with temperament approximations set to the nearest whole cent. Neither the whole cent approximations nor the artificial harpsichord sound are ideal for hearing the subtleties of temperaments. I did this merely to ensure that the temperament of the instrument didn’t drift while the examples were being recorded, thus rendering the examples completely useless. I highly recommend tuning these temperaments on your own harpsichord, using either a high quality electronic tuner or, alternatively, the beat–counting method used by many professionals.

Click below to hear performances in each temperament

Sarabande in G Minor Vallotti Young 1/6 comma with Ab

Equal Temperament

Prelude No. 1 in C Major Vallotti Young 1/6 comma with Ab

Equal Temperament

Select Bibliography 

Barbieri, Patrizio. "Violin intonation: a historical survey." Early Music 19 (1991), 69–88.

Barbour, J. Murray. Tuning and Temperament: a Historical Survey. East Lansing: Michigan State College Press, 1951.

Blackwood, Easley. The Structure of Recognizable Diatonic Tunings. Princeton: Princeton University Press, 1985.

Haynes, Bruce. "Beyond temperament: non–keyboard intonation in the 17th and 18th centuries." Early Music 19 (1991), 357–81.

Jones, William. Physiological disquisitions; or, Discourses on the natural philosophy of the elements. London: J. Rivington, 1781.

Jorgensen, Owen. Tuning the historical temperaments by ear: a manual of eighty–nine methods for tuning fifty–one scales on the harpsichord, piano, and other keyboard instruments. Marquette: Northern Michigan University Press, c1977.

____. Tuning: containing the perfection of eighteenth–century temperament, the lost art of nineteenth–century temperament, and the science of equal temperament, complete with instructions for aural and electronic tuning. East Lansing: Michigan State University Press, 1991.

Leedy, Douglas. "A personal view of meantone temperament." The Courant 5 (1983), 3–19.

Lindley, Mark. "A suggested improvement for the Fisk organ at Stanford." Performance Practice Review 1 (1988), 107–32.

____. "Interval" and "Temperament." New Grove Dictionary . New York and London, 1980.

____. "Simmung und Temperatur." In Geschichte des Musiktheorie 6: Hören, Messen und Rechnen in der Frühen Neuzeit. Darmstadt: Wissenschaftliche Buchgesellschaft, 1987.

Lloyd, Ll. S., and Hugh Boyle. Intervals, Scales, and Temperaments. New York: St. Martin’s Press, 1963.

Vallotti, P. Francescantonio. Trattato della Moderna Musica. Padova: Basilica del Santo, 1950.

Young, Thomas. "Of the temperament of musical intervals." Philosophical Transactions of the Royal Society of London for the year 1800. London, The Royal Society, 1800.