Intervals, tuning, and temperament, part 3
In the first two columns on tuning I did not refer at all to names of temperaments—neither the rather familiar terms such as “Werckmeister,” “Kirnberger,” or “Vallotti,” nor less familiar ones such as “Fogliano-Aron,” “Ramos,” or “Bendeler.” It can be interesting or useful for a student to learn something about these historical temperaments; however, there is a reason that I have avoided framing my discussion of temperament with these established tunings. It is much more useful for students to grasp the principles that underlie any keyboard tuning. It is then possible for the student to both understand any specific tuning system—historical or hypothetical—and to invent his or her own, and also to understand some of the practical and artistic implications of different tuning approaches.
Underlying tuning principles
1) It is impossible for all twelve perfect fifths on a normal keyboard instrument to be tuned absolutely pure. This arises out of the mathematics of the fundamental definition of intervals, and it is an objective fact. If you start at any note and tune twelve perfect fifths pure, then the note that you come back to—which is supposed to be the same as the starting note—will be significantly sharp compared to the starting note.
2) Therefore, at least one perfect fifth must be tuned narrow. Anywhere from one to all twelve perfect fifths can be tuned narrow, as long as the overall amount of narrowness is correct.
3) The need to narrow one or more fifths is an objective need, and doing so is the practical side of keyboard temperament. The choice of which fifths to narrow and (bearing in mind that the overall narrowness must add up to the right amount) how much to narrow them is subjective and is the esthetic side of keyboard temperament.
From these principles it is possible to understand, or indeed to re-invent, any of the historical temperaments, each of which is of necessity simply a way of approaching and solving the issues described above.
Major historical tunings
1) Pythagorean tuning. This is the simplest practical approach, in which eleven fifths in a row are tuned absolutely pure, and the remaining fifth is allowed to be extremely narrow: so narrow that human ears will not accept it as a fifth and it has to be avoided in playing.
2) Well-tempered tuning. In this approach, the narrowness of fifths is spread out over enough fifths that the narrowed fifths sound acceptable to our ears. Practical experience suggests that this means over at least three fifths. The fifths that are not narrowed are left pure. All intervals and thus all chords and all keys are usable.
3) Meantone tuning. Here the tuning of fifths is configured in such a way as to generate pure or relatively pure major thirds. When this kind of tuning was in very widespread use (primarily the 16th and 17th centuries), this was a widely and strongly held esthetic preference. In order to generate a large number of pure major thirds, it is necessary to tune a large number of unusable intervals, both thirds and fifths—actually more than in Pythagorean tuning.
4) Equal temperament. In this temperament, each of the twelve perfect fifths is narrowed by exactly the same amount. In this tuning, alone among all possible keyboard tunings, each specific instance of each type of interval—perfect fifth, major third, and so on—is identical to all other instances of that interval.
Tuning intervals
When two close pitches are sounding at the same time we hear, alongside those notes, a beating or undulating sound that is the difference between the two pitches that are sounding. If a note at 440hz and a note at 442hz are played at the same time, we hear a beating at the speed of twice per second. If the two notes were 263hz and 267hz the beating would be at four times per second. This kind of beating sounds more or less like a (quiet) siren or alarm. It is so much a part of the background of what we hear when we listen to music that most people initially have trouble distinguishing it or hearing it explicitly. Normally once someone first hears beats of this kind, it is then easy to be able to hear them and distinguish them.
These beats are a real acoustic phenomenon. They are not psychological, or part of the physiology of hearing: they are present in the air. If you set up a recording in which one stereo channel is playing one pitch and the other is playing a close but different pitch, then if you play those two channels through speakers into the air, they will produce beats that can be heard. However, if you play them through headphones, so that the two notes never interact with one another in the air but each go directly to a separate ear of the listener, then no beats will be created and the listener will hear the two different pitches without beats.
Notes that are being produced by pipes or strings have overtones. When two such notes are played together, the pitches that mingle in the air include the fundamental and the overtones. Any of those component sounds that are very close to one another will produce beats if they are not in fact identical. It is by listening to these beats and comparing them to a template or plan (either no beats or beats of some particular speed) that we carry out the act of tuning.
For example, if we are tuning a note that is a fifth away from an already-tuned note, then the first upper partial of the higher note is meant to be the same pitch as the second upper partial of the lower note. (For a discussion of overtones see this column from July 2009.) If these overtones are in fact identical, then they will not produce any beats; if they are not quite identical they will produce beats. If the goal is to produce a pure perfect fifth, then beats should be absent. If the goal is to produce a narrow perfect fifth, then beats should be present—faster the narrower a fifth we want. In tuning a major third, the same principle applies, except that it is the third upper partial of the higher note and the fourth upper partial of the lower note that coincide.
Listening for beats produced by coinciding overtones is the essential technique for tuning any keyboard instrument by ear. Any tuning can be fully described by a list of beat speeds for each interval to be tuned. For example, in Pythagorean tuning the beat speed for each of the eleven fifths that are tuned explicitly is zero. (The twelfth fifth arises automatically.) Any well-tempered tuning can be described as a combination of fifths that have beat speeds of zero and fifths that have various moderate beat speeds. In equal temperament, all the fifths have beat speeds greater than zero, and they all reflect the same ratio, with higher notes having proportionately higher beat speeds. In most meantone systems, major thirds have no beats or very slow beat speeds, while those fifths that are tuned directly have beat speeds that are similar to those of well-tempered fifths.
These beats have a crucial effect on the esthetic impact of different tuning systems. For example, in Pythagorean tuning, while all of the perfect fifths are pure (beatless), all of the major thirds are very wide and beat quite fast. This gives those thirds, and any triads, a noisy and restless feeling. A triad with pure fifths and pure thirds—a beatless triad—is a very different phenomenon for a listener, even though it looks exactly the same in music notation. Other sorts of triads are different still: those with a pure major third and a narrow fifth, for example, or with all of the component intervals departing slightly from pure.
Temperaments throughout history
General tendencies in the beat structure of different temperaments may explain some things about the history of those temperaments, why they were used at different times, or at least how they correlate with other things that were going on musically at the time when they were current.
Pythagorean tuning
For example, Pythagorean tuning was in common use in the late Middle Ages. This was a time when the perfect fifth was still considered a much more consonant or stable interval than the major or minor third. Thus it made sense to use a tuning in which fifths were pure and thirds were wide enough—buzzy enough—to be almost inherently dissonant.
(But it is interesting to speculate about the direction of causality: did Pythagorean organ tuning suggest the avoidance of thirds as consonant intervals, or did a theory-based avoidance of those intervals suggest that a tuning with very wide thirds was acceptable?)
Meantone tuning
The rise of meantone tuning in the late fifteenth century corresponded with the rise of music in which the major third played an increasingly large role as a consonant interval and as a defining interval of both modal and tonal harmony. A major triad with a Pythagorean third does not quite sound like a resting place or point of arrival, but a major triad with a pure third does. During this same period, the harpsichord and virginal also arose, supplementing the clavichord and the organ. These new instruments had a brighter sound with a more explosive attack than earlier instruments. This kind of sound tends to make wide thirds sound very prominent. This may have been a further impetus to the development of new tuning systems in those years.
Meantone tuning, since it includes many unusable intervals, places serious restrictions on composers and players. Modulation within a piece is limited. In general, a given piece can only use one of the two notes represented by a raised (black) key, and must rigorously avoid the other. Many transpositions create impossible tuning problems. Many keys must, as a practical matter, be avoided altogether in order to avoid tremendous amounts of re-tuning.
Some keyboard instruments built during the meantone era had split sharps for certain notes, that is, two separate keys in, for example, the space between d and e, sharing that space front and back, one of them playing the d#, the other playing the e♭. Composers do not seem to have relied on it more than once in a while to write pieces in which they went beyond the harmonic bounds natural to meantone tuning. These split keys were probably intended to reduce or eliminate the need to re-tune between pieces, rather than to expand the harmonic language of the repertoire.
Meantone was no easier to tune than what came before it, or than other tuning systems that were known theoretically at the time but little used, since by limiting transposition it placed significant harmonic limitations on composers and improvisers, and thus made accompaniment more difficult. Yet it remained in use for a very long time. It seems certain that whatever it was accomplishing esthetically must have seemed very important, even crucial. Many listeners even now feel that the sonority of a harpsichord is most beautiful in meantone.
Well temperaments
In the late seventeenth century, composers and theorists began to suggest new temperaments that overcame the harmonic restrictions of meantone. These were the well-tempered tunings, in which every fifth and every third is usable as an harmonic interval. In order to achieve this flexibility, these tunings do away with most or, in some cases, all of the pure major thirds. This change can be seen as a shift from an instrument-centered esthetic—in which the beauty of the sound of the pure thirds was considered more important than perhaps anything else—to a composer-centered esthetic and philosophy, in which limitations on theoretical compositional possibilities were considered less and less acceptable. There were strong defenders of the older tunings well into the eighteenth century. It is interesting that in one well-known dispute about the merits of meantone as opposed to well-tempered tuning, the advocate of the former was an instrument builder (Gottfried Silbermann) and the advocate of the latter was a composer (J. S. Bach).
The crucial esthetic characteristic of well-tempered tunings is that different keys have different harmonic structures. That is, the placement of relatively pure and relatively impure intervals and triads with respect to the functional harmonies of the key (tonic, dominant, etc.) is different from one key to another. (An interesting experiment about this is possible in modern times. If a piece is recorded on a well-tempered instrument in two rather different keys, say C major and then E major, and the recordings are adjusted by computer so as to be at the same pitch level as one another, then they will still sound different and be easily distinguishable from each other.) Their differences are almost certainly the source of ideas about the different inherent characters of different keys. Lists of the supposed emotional or affective characteristics of different keys arose in the very late seventeenth century, at about the same time that well-tempered tuning took hold.
Equal temperament
In equal temperament, which became common in the mid- to late-nineteenth century, every interval with a given name and every triad or other chord of a particular type is the same as every other interval, triad, or chord of that type. Part of the appeal of this tuning in the nineteenth century was, probably, its theoretical consistency and symmetry. Many people have found the concept of equal temperament intellectually satisfying: it does not have what might be thought of as arbitrary differences between things that, theoretically at least, ought to be the same. Equal temperament took hold in the same era of organ history that included logarithmic pipe scalings—another theoretically satisfying, mathematically inspired idea. During this same time, designers of wind instruments were working to make those instruments sound the same—or as close as humanly possible—up and down the compass. This is another manifestation of a taste for avoiding seemingly arbitrary or random difference.
On an equal-tempered keyboard, the computer experiment described above would result in two indistinguishable performances: it is not possible to tell keys apart except by absolute pitch. The rise and dissemination of equal temperament also coincided with a general worldwide increase in travel. In a world in which equal temperament and a particular pitch standard (say a′=440hz) will be found anywhere and everywhere, a flutist, for example, can travel from Europe to America or Japan or anywhere and expect to be able to play with local musicians.
It is also likely that the general acceptance of equal temperament helped lead to twelve-tone and other atonal music by promoting the idea (and the actual listening experience) that all keys and all twelve semitones were the same.
In equal temperament, no interval is pure, and no interval is more than a little bit out of tune. This is a tuning that, just as a matter of taste or habit, appeals strongly to some people and does not appeal to others. I have known musicians with no training (or for that matter interest) in historical temperaments who could not stand to listen to equal temperament because they found equal-tempered thirds grating; I have known others who can accept the intervals of equal temperament as normal but who cannot tolerate the occasional more out of tune intervals of well-tempered tuning.
At the Princeton Early Keyboard Center website there are links to several resources describing and comparing historical temperaments and discussing further some of what I have written about here.
