On Teaching

Gavin Black is director of the Princeton Early Keyboard Center in Princeton, New Jersey. He can be reached by e-mail at [email protected].

Intervals, tuning, and temperament, part 2
Last month I wrote about some of the fundamentals underlying the art of keyboard temperament: aspects of the nature of musical sound and of intervals, the overtone series, and the so-called circle of fifths. This month I want to discuss keyboard temperament itself, using last month’s column as a foundation. I will talk about why temperament is necessary, what the major approaches to temperament have been over the centuries, some of what the different systems of temperament set out to accomplish, and about how different temperaments relate to different historical eras. Next month I will also discuss the practicalities of tuning and a few miscellaneous matters related to tuning and temperament.
As I said last month, my main point is to help students become comfortable with tuning and temperament and to develop a real if basic understanding of them, regardless of whether they are planning to do any tuning themselves. Before describing some of the essential details of several tuning systems, I want to review how we discuss tuning and how our thinking about tuning is organized, so that the descriptions of different temperaments will be easy to grasp.
1) For purposes of talking about tuning, octaves are considered exactly equivalent. (This of course is no surprise, but it is worth mentioning.) The practical point of this is that if I say, for example, that “by tuning up by a fifth, six times in a row, I get from C to F#” I do not need to say that I also have to drop the resulting F# down by three octaves to get the simple tritone (rather than the augmented twenty-fifth); that is assumed. To put it another way, simple intervals, say the perfect fifth, and the corresponding compound intervals, say the twelfth or the nineteenth, are treated as being identical to one another.
2) Intervals fall into pairs that are inversions of one another: fifth/fourth; major third/minor sixth; minor third/major sixth; whole tone/minor seventh; semitone/major seventh. For purposes of tuning, the members of these pairs are interchangeable, if we keep direction in mind. For example, tuning up by a fifth is equivalent to tuning down by a fourth. If you are starting at C and want to tune G, it is possible either to tune the G above as a fifth or the G below as a fourth. It is always important to keep track of which of these you are doing or have just done, but they are essentially the same.
3) When, in tuning a keyboard instrument, we tune around the circle of fifths, we do not normally do this:

but rather something like this:

going up by fifths and down by fourths—sometimes up by fourths and down by fifths—in such a way as to tune the middle of the keyboard first, thus creating chords and scales that can be tested.
4) In tuning keyboard instruments we purposely make some intervals impure: that is, not perfectly (theoretically) in tune. When an interval is not pure it is either narrow or wide. An interval is wide when the ratio between the higher note and the lower note is greater than that ratio would be for the pure interval; it is narrow when the ratio is smaller. For example, the ratio between the notes of a pure perfect fifth is 3:2, that is, the frequency of the higher note is 1½ times the frequency of the lower note. In a narrow fifth, that ratio is smaller (perhaps 2.97:2), in a wide fifth it is larger (perhaps 3.05:2). Here’s the important point—one that students do not always realize until they have had it pointed out: making an interval wide does not necessarily mean making some note sharp, and making an interval narrow does not necessarily mean making some note flat. If you are changing the higher note in an interval, then raising that note will indeed make the interval wider and lowering it will make the interval narrower. However, if you are changing the lower note, then raising the note will make the interval narrower and lowering it will make the interval wider.
5) Tuning by fifths (or the equivalent fourths) is the theoretically complete way to conceive of a tuning or temperament system. This is because only fifths and fourths can actually generate all of the notes. That is, if you start from any note and tune around the circle of fifths in either direction, you will only return to your starting note after having passed through all of the other notes. If you start on any given note and go up or down by any other interval, you will get back to your starting note without having passed through all of the other notes.1 For example, if you start on c and tune up by major thirds you will return to c having only tuned e and g#/a♭. There is no way, starting on c and tuning by thirds, to tune the notes c#, d, d#, f, f#, g, a, b♭, or b. Tuning is sometimes done by thirds, but only as an adjunct to tuning by fifths and fourths. Any tuning system can be fully described by how it tunes all of the fifths.
6) As I mentioned last month, tuning two or more in a row of any interval spins off at least one other interval. For example, tuning two fifths in a row spins off a whole tone. (Starting at c and tuning c–g and then g–d spins off the interval c–d). Tuning four fifths in a row spins off a major third. (Starting at c and tuning c–g, g–d, d–a, a–e spins off the interval c–e). The tuning of the primary intervals—pure, wide, or narrow—utterly determines the tuning of the resulting (spun-off) interval. For example, tuning four pure perfect fifths in a row spins off a major third that is wider than the theoretically correct 5:4 ratio: very wide, as a matter of human listening experience. Tuning three pure fourths in a row (c–f, f–b♭, b♭–e♭, for example) spins off a minor third that is narrower than the theoretically correct 6:5.
So, what is temperament and why does it exist? Temperament is the making of choices about which intervals on the keyboard to tune pure and which to tune wide or narrow, and about how wide or narrow to make those latter intervals. Temperament exists, in the first instance, because of the essential problem of keyboard tuning that I mentioned last month: if you start at any given note and tune around the circle of fifths until you arrive back at the starting note, that starting note will be out of tune—sharp, as it happens—if you have tuned all of the fifths pure. The corollary of this is that in order to tune a keyboard instrument in such a way that the unisons and octave are in tune, it is absolutely necessary to tune one or more fifths narrow. This is a practical necessity, not an esthetic choice. However, decisions about how to address this necessity always involve esthetic choices.
There are practical solutions to this practical problem, and the simplest of them constitutes the most basic temperament. If you start at a note and tune eleven fifths, but do not attempt to tune the twelfth fifth (which would be the out-of-tune version of the starting note), then you have created a working keyboard tuning in which one fifth—the interval between the last note that you explicitly tuned and the starting note—is extremely out of tune. If you start with c and tune g, d, a, etc., until you have tuned f, then the interval between f and c (remember that you started with c and have not changed it) will be a very narrow fifth or very wide fourth. The problem with this very practical tuning is an esthetic, rather than a practical, problem: this fifth is so narrow that listeners will not accept it as a valid interval. Then, in turn, there is a practical solution to this esthetic problem: composers simply have to be willing to write music that avoids the use of that interval. This tuning, sometimes called Pythagorean, was certainly used in what we might call the very old days—late middle ages and early Renaissance. As an esthetic matter, it is marked by very wide thirds (called Pythagorean thirds) that are spun off by all of the pure fifths. These thirds, rather than the presence of one unusable fifth, probably are why this tuning fell out of favor early in the keyboard era.
The second-easiest way to address the central practical necessity of keyboard tuning is, probably, to divide the unavoidable out-of-tuneness of the fifths between two fifths, rather than piling it all onto one of them. For example, if in the example immediately above you tune the last interval, namely b♭–f, somewhat narrow rather than pure, then the resulting final interval of f–c will not be as narrow as it came out above. Perhaps it will be acceptable to listeners, perhaps not. Historical experience has suggested that it is right on the line.
In theory, what I just called the “unavoidable out-of-tuneness” (which is what theorists of tuning call the “Diatonic Comma” or “Pythagorean Comma”) can be divided between or among any number of fifths, from one to all twelve, with the remaining fifths being pure. The fewer fifths are made narrow—that is, “tempered”—the narrower each of them has to be; the more fifths are left pure (which is the same thing), the easier the tuning is, since tuning pure fifths is the single easiest component of the art of tuning by ear.2 The more fifths are tempered, the less far from pure each of them has to be; the fewer fifths are left pure, the more difficult the temperament is to carry out by ear.
Temperaments of this sort, that is, ones in which two or more fifths are made narrow and the remaining fifths are tuned pure, and all intervals and chords are usable, make up the category known as “well-tempered tuning.” There exist, in theory, an infinite number of different well-tempered tunings. There are 4083 different possible ways to configure the choice of which fifths to temper, but there are an infinite number of subtly different ways to distribute the amount of out-of-tuneness over any chosen fifths. From the late seventeenth century through the mid to late nineteenth century, the most common tunings were those in which somewhere between four and ten or eleven fifths were tempered, and the rest were left pure. In general, in the earlier part of those years temperaments tended to favor more pure fifths, and later they tended to favor more tempered fifths. The temperament in which all twelve fifths are tempered and the ratio to which they are all tempered is the same (2.9966:2) is known as equal temperament. It became increasingly common in the mid to late nineteenth century, and essentially universal for a while in the twentieth century. It was well known as a theoretical concept long before then, but little used, at least in part because it is extremely difficult to tune by ear.
In well-tempered tunings and in fact any tunings, the choices about which fifths to temper affect the nature of the intervals other than fifths. The most important such interval is the major third. The importance of the placement of tempered fifths has always come largely from the effect of that placement on the thirds. Historically, in the period during which well-tempered tuning was the norm, the fifths around C tended to be tempered so as to make the C–E major third close to pure, in any case almost always the purest major third within the particular tuning. This seems to reflect both a sense that pure major thirds are esthetically desirable or pleasing and a sense that the key of C should be the most pleasing key, or the most restful key, on the keyboard. In general, well-tempered tunings create a keyboard on which different intervals, chords, and harmonies belonging to the same overall class are not in fact exactly the same as one another. There might be, for example, major triads in which the third and the fifth are both pure, alongside major triads in which the fifth is pure but the third a little bit wide, or the fifth pure but the third very wide, or the fifth a little bit narrow and the third a little bit wide. It is quite likely that one of the points of well-tempered tuning was to cause any modulation or roaming from one harmonic place to another on the keyboard to effect an actual change in color—that is, in the real ratios of the harmonies—not just a change in the name of the chord or in its perceived distance from the original tonic.
In equal temperament, all intervals of a given class are in fact identical to one another, and each instance of a chord of a given type—major triad, minor triad, and so on—is identical to every other instance of that chord except for absolute pitch. Next month I will discuss ways in which the esthetic of each of these kinds of temperament fit in with other aspects of the musical culture of their times.
The other system of tuning that was prevalent for a significant part of the history of keyboard music—from at least the mid sixteenth century through the seventeenth century and, in some places well into the eighteenth—is known nowadays as meantone tuning. (This term was not used at the time, and is now applied to a large number of different tunings with similar characteristics.) In a meantone tuning, there are usually several major thirds that are unusably wide and one or more fifths that are also unusable. In fact, the presence of intervals that must be avoided by composers is greater than in Pythagorean tuning. However, this is in aid of being able to create a large number of pure or nearly pure major thirds. This was, perhaps, as a reaction to the earlier Pythagorean tuning with its extremely wide thirds, considered esthetically desirable during this period. The mathematics behind the tuning of thirds tells us that, if two adjacent thirds are both pure, say c–e and e–g#, then the remaining third that is nestled within that octave (see above), in this case a♭–c, will be so wide that no ears will accept it as a valid interval. Therefore only two out of every three major thirds can be pure—that is, eight out of the twelve—and, if they are tuned pure, the remaining major thirds will become unusable. This, of course, in turn means that composers must be willing to avoid those intervals in writing music. It is striking that composers were willing to do so with remarkable consistency for something like two hundred years.
The distribution of usable and unusable thirds in meantone is flexible. For example, while it is possible to tune c–e and e–g# both pure, as mentioned above, it is also possible to tune c–e and a♭–c pure, leaving e–g# to be unusable. In the late Renaissance and early Baroque keyboard repertoire, there are, therefore, pieces that use g# and piece that use a♭, but very few pieces that use both. There are pieces that use d# and pieces that use e♭, but very few pieces that use both. There are many pieces that use b♭ and a few that use a#, but almost none that use both. There are very few keyboard pieces from before the very late seventeenth century that do not observe these restrictions. This is powerful evidence that whatever was accomplished esthetically by observing them must have been considered very important indeed.

Gavin Black is the director of the Princeton Early Keyboard Center www.pekc.org. He can be reached by e-mail at [email protected].

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.

Gavin Black is director of the Princeton Early Keyboard Center in Princeton, New Jersey. He welcomes feedback by e-mail at <[email protected]>. Expanded versions of these columns with references and links can be found at <http://www.pekc.org&gt;.

Registration and teaching—Part II
In last month’s column, I emphasized the usefulness of starting off the teaching of registration with a clear explanation of the meaning of the foot-designation of organ stops and with a set of demonstrations of that meaning. This is a necessary foundation for understanding everything about combining stops and about choosing organ sounds for music. Once a student clearly understands the meaning of all the numbers on the stop knobs, it is time for that student to begin exploring the art of combining stops. This starts with developing an awareness of what the stop pitch levels imply about the structure of stop combinations, and continues with the development of an ear for the aesthetic nature of different sounds, and then with the acquisition of knowledge about registration practices in different schools of organ composition or in the work of specific composers.
The concept that it is OK to combine stops that are not at the same pitch level as one another, and that the resulting sound will be (or at least can be) a coherent musical sound at a coherent pitch level, is not self-evident. In fact, it is counterintuitive to most people who have not already become well versed in organ registration. It seems, if anything, self-evident that this kind of mixing will result in obvious parallel octaves and fifths, and also in a generalized jumble of pitches, which would at a minimum make clarity impossible (because notes that you play in the tenor register, for example, would produce pitches proper to the treble register, etc.). Since the blending of stops at different pitch levels in fact can work the way it does because of the overtone series, it is useful to explain something about overtones to students. It is certainly not necessary to go into all of the scientific details—the physics of the creation of overtones, the reasons for inharmonicity of overtones in certain situations, or even what the notes of the overtone series are, above the first few. However, it is a good idea to review the basics:
1) Almost all musical sounds produced acoustically have many frequencies blended together. (It usually takes a computer to produce a sound at exactly one frequency.)
2) These frequencies are (usually) a) a given frequency and b) other frequencies that are multiples of that first frequency. (Of course we use the lowest frequency to identify the note, as in “A 440.”)
3) These multiples produce sounds that are related to the lowest frequency by common musical intervals: octave, octave-and-a-fifth, two octaves, two-octaves-and-a-third, etc.
On most organs it is possible to find individual notes on some stops in which some specific overtones can be heard as separate pitches. These can be used to demonstrate the existence of overtones and the pitch levels of some of them. Gedeckts, flutes, and quintadenas are often the most fruitful for this, and notes in the octave and a half or so below middle c are the most promising, because they are the easiest to hear. Usually it is possible to find a pipe or two in which the twelfth is clear (quiet, perhaps, but clear), others in which the seventeenth is, and others in which the octaves are. To someone who has never tried to listen to overtones before, these sounds are usually hard to hear at first, but then suddenly “come in.” The teacher can help with this, first by making sure to zero in on the pipes with the clearest individual overtones, and then by briefly playing, singing, or whistling the actual note corresponding to the overtone that you wish to help the student to hear. This will attune the student’s expectation to that pitch, and it will probably only be necessary for the first few notes.
(A further exercise in listening to overtones is this: play a simple melody on one stop. Try to hear and follow the counter-melody created by the clearest and most noticeable overtones. For example, consider the notes of the fugue subject of Bach’s E-flat major fugue:
b-flat – g – c – b-flat –
e-flat – e-flat – d – e-flat
Depending on what the overtones of each pipe happen to be doing, a counter melody could arise that went like this:
d – d – g – d – g – g – f# – g
or that went like this:
f – b – e – f – b-flat –
b-flat – f# – b-flat
or any number of other possibilities. It will be different for each different stop on which you play the melody. The “extra” melody will be quiet, and usually it will range from one to three octaves above the “official” melody. It is quite possible that these inherent counter-melodies are one source of the human invention of counterpoint. This is all a bit of a detour from learning techniques of registration as such, but it is a useful exercise both for learning to listen carefully to sound and for remembering that sounds themselves are complex and interesting, often doing more than we might at first expect.)
Once a student understands the basic concept of overtones (and believes in them!), it is easy for him or her to understand the blending of stops of different pitches: a 4? stop blends with and reinforces the first upper partial of an 8? stop, a 22?3? stop the second upper partial, a 2? stop the third, etc. One advantage of going through all of this quite systematically is that it answers the question of how in the world it can make sense to combine stops that don’t even produce the same letter-name notes as one another. This is certainly the thing that seems the least intuitive and the most questionable about registration to many of those who are not yet experienced with the organ.
(This can be true especially if someone stops to think about all of the pitches that are present in a thick texture. For example, a G-major 7th chord played on a registration that includes a 22?3? stop includes the pitches g, a, b, c, d, f, f#. If you throw in a tierce you add a d#. That this would be acceptable makes a lot more sense if you know that all of those “extra” pitches are present anyway as overtones.)
So the most basic description of the structure of the art of combining organ stops, and the most useful as a starting point, goes something like this: that, as long as you have one or more 8? stops present in your combination of stops, anything and everything higher than 8? pitch has the potential to blend with the 8? sound. In so doing, it will change the nature of the sound by changing the overall balance of the overtones, and by changing the volume, but it will not upset the pitch identification of the notes that you play.
A simple exercise to demonstrate this would be as follows:
1) choose a keyboard that has more than one 8? stop and several higher-pitched stops.
2) draw the louder (loudest) 8? stop.
3) play a simple passage—a chord progression or a bit of a hymn is good—adding and taking away various 4? and higher stops at random.
4) after a while, remove the 8? stop. The student will hear the music suddenly jump up in pitch.
5) repeat all of this with a softer 8? stop.
Anyone performing or listening to this exercise will certainly notice that not all of the combinations work equally well. Some of the sounds that could blend in theory will not seem to blend very well in practice, perhaps because a 4? or higher stop is too loud or too bright (or for that matter out of tune) or because a given 8? stop is too thin or weak or has something about its intrinsic overtone development that conflicts with rather than supports the addition of higher-pitched stops. These considerations are extremely important. They are also subjective and in the end belong to the realm of artistic judgment or discretion. A student listening to or trying out this exercise should be encouraged to notice aesthetic aspects of each sound. However, the main point for the moment is that the dropping of the 8? pitch makes a sound that is utterly different in kind from the adding or dropping of any higher-pitched stops.
Of course, it might occur to a student, or a teacher might want to mention for completeness if nothing else, that it is perfectly possible to use sounds that omit 8? stops, for some special reason or in some special way. The simplest of these is the use of a 4? or higher sound to play the music at an octave or more higher than the written pitch. Also fairly common is the use of a non-8? registration accompanied by the moving of the hands to a different position on the keyboard to bring the pitch in line with original expectations. These are useful things to bear in mind as a performing organist, but they are special cases that can best be thought about at a slightly later stage in learning, and that should certainly not distract a student from developing the most thorough possible understanding of “normal” stop combination and registration. The same can be said about the use of 16? sound in multi-voiced or chordal manual playing. This, in theory, just transposes the music down an octave, but often doesn’t—for some psychoacoustic or just plain acoustic reasons—quite sound like that.
So far we have developed a rather scientific approach—perhaps too scientific for some people’s taste—to the teaching or learning of registration. We have asked students to think very clearly about the pitch designations of stops, about overtones, about what overtones imply about the use of different pitches of stops, and about how to make sure that a sound is grounded in unison pitch. We have not yet talked about either how to choose registrations that “sound good” (or “beautiful” or “appropriate” or anything else) or about how to respect composers’ wishes or any other way to tailor sounds to pieces. We have also barely mentioned stop names, or even names of important categories of stops, diapasons, flutes, reeds, and so on. Nor have we mentioned any rules or even ideas about how or whether to combine stops of different types, or for that matter of the same type.
Organs have lots and lots of sounds. For example, by my calculations, allowing only for sounds that include 8? pitch and leaving cornets and “céleste” stops out of any ensemble, but taking into account couplers, the Grand Orgue of the Mander organ at St. Ignatius Loyola in New York—a well-known, recent, large but not gargantuan organ—commands 121,889,158,594,564 different sounds. A hypothetical medium-sized organ in which three manuals have 25 to 30 stops would have about 200 million to about a billion 8?-based sounds available in the manual divisions. If the pedal division of such an organ had eight stops, then, assuming normal couplers, the pedals would have a quarter of a trillion different sounds available.
Harpsichords, on the other hand, have rather few sounds. Most large harpsichords have seven to ten different available sonorities all together. Many very fine and versatile harpsichords have only three. In planning registration for a piece on the harpsichord, it is always possible to use what I consider to be the soundest and most artistically thorough approach: simply trying the piece out on every possible sound, listening carefully and with attention, and deciding which sound you like best.
This approach is almost always impossible on the organ. It is always impossible on any organ but the very smallest. However, it seems to me that it is still—albeit only in an underlying theoretical way—the best approach, and the right concept to have in the back of one’s mind when working out registrations. That this is true can, I think, be almost proven logically. If you are using a given registration, whether it comes from an editor, or from your teacher, or from something that you jotted down in your copy years before, or from any other source, but there is in fact a different registration that you would like better if only you heard it, then you should in theory be using that other registration. Therefore, ideally, one would always hear every registration before making a final choice.
(I am not right now dealing with the extra-musical quasi-ethical considerations of authenticity that arise when a specific registration comes from the composer. I will address that at least briefly next month.)
The purpose of taking a student very systematically through what I described above as a scientific approach to the technique of registration—the feet, the overtones, the combinations at different pitch levels—is to allow the student then to feel free to try anything and everything (again, knowing that there won’t really be time for everything!) without fear of doing something that really, in some concrete way, doesn’t and can’t work. This will enable the student to be relatively independent of outside guidelines, and increase the chance that the student will contribute something new and interesting to the world of the organ. It will also almost certainly provide the student with a great measure of out-and-out fun, and keep the job of practicing as interesting as it can be.
Next month I will talk about ways to practice listening to the more subjective, sound-quality-oriented aspects of the blending and combining of stops. I will also talk about helping students to begin to relate sound to other aspects of their concept of a piece of music, and to both structural and historical considerations.?

The musical character of an historic tuning can be difficult to grasp— and the mathematics involved can be daunting. Modern descriptions of tunings use the mathematical concept of the “cent” because it is independent of a reference frequency.

Cents simply represent the convenient division of the octave into twelve equal intervals of 100 cents each. The use of cents, however, has absolutely no relationship to the natural harmonic series, i.e., cents have no relationship to the consonance or dissonance of the intervals we hear. To make this point clear, the equally tempered third is 400 cents (the pure third is 386 cents) and the equally tempered fifth is 700 cents (the pure fifth is 702 cents), and our ears tell us that the third is very impure. Cents tell us nothing about the purity of the interval. In the middle octave of the compass, the 700-cent fifth sounds like a warm celeste at about one beat per second, whereas the 400-cent third sounds a harsh ten beats per second. The purity and consonance of an interval improves with fewer beats, and the dissonance of an interval increases with more beats. The relationship of a tuning system to the natural harmonic series is represented by its beat rates. It tells you how the tuning will sound.

Pythagoras noted 2,500 years ago that if you tuned G pure to C, D pure to G, A pure to D, and continued this series of pure fifths to arrive again at C, the initial note C and the final note C would be different. These dissonant tones would be in the ratio of 81/80—this is known as the “Pythagorean comma.” In modern equal temperament we divide this error and dissonance equally across all twelve notes in the octave, and no intervals other than the octave are pure without beats.

Classes of tunings

The consonance of harmonic purity is alluring. Early compositions took advantage of tunings that featured both consonant purity and dissonant tension. These are the basic classes of tunings in a nutshell: Pythagorean tuning is the oldest and is based on the purity of fifths; meantone was developed in the Renaissance and is based on the purity of thirds; equal temperament became ubiquitous in the mid-nineteenth century, allows the use of all keys, and is based on an equal impurity in all keys without any pure fifths or thirds.

Meantone was a prevalent tuning for a very long period in the history of the pipe organ. J. S. Bach favored tunings that allowed free usage of all 24 major and minor keys; Bach was known to be at odds with the organbuilder Gottfried Silbermann, who used meantone tuning. Although equal temperament was gaining favor in the late eighteenth century, meantone was known to be in use in English churches well into the nineteenth century. What was its appeal?

There are eight pure major thirds in 1/4-comma meantone. The interval of the fifth in meantone is only slightly less pure than the fifths in equal temperament, which has no pure intervals. The appeal of meantone was a wonderful sense of harmonic purity and a deep, rich sonority. The natural harmonics, when played together, create sub-tones representing the fundamental of the harmonics. The interval of the pure fifth C–G produces a sub-tone one octave below the C. The interval of the pure third C–E produces a sub-tone two octaves below the C. This is a primary source of bass tone deriving from nothing more than the pure thirds of meantone tuning. The famous three-manual and pedal organ by the elder Clicquot at Houdan has a satisfying depth of tone, but it has no 16′ stops, not even a single 16′ stop in the Pedal. The 16′ tonal gravity is entirely the result of the tuning.

The later trend towards equal temperament produced sounds that lacked the gravity of meantone, and organbuilders responded in two ways. First, organ specifications often featured manual 16′ stops. Second, the new Romantic voicing style and higher wind pressures provided real fundamental power. The end of the eighteenth century saw a profusion of transitional “well temperaments,” which tried to bridge the gap between meantone and equal temperament. All of these attempted to preserve some harmonic purity while affording some degree of the freedom of equal temperament, but the results were largely unsatisfactory on both counts. It is worth taking a closer look at some of the early tunings, uncompromised by later efforts to dilute their character.

Comparing triads

We can visualize the sonority of major and minor triads as shown in illustration 1. The upper triangle of notes, B–F# and B–D, represents the B-minor triad. The B-major triad is shown in the lower triangle. Beat rates are shown between the notes, e.g., the minor third B–D dissonantly beats 26.7 times per second when playing B in the middle octave with the D above. A pure or nearly pure interval is represented by a green line connecting the notes. Intervals with more beats are represented by lines of different colors as seen in the table to the right, where very dissonant intervals are red, and violet intervals represent extreme dissonance, also known as the “wolf.” These colors allow us to “see” the relative consonance or dissonance of these intervals—numbers are more difficult to interpret at a glance. Black arrows point in the directions in which we will find intervals of fifths, major thirds, and minor thirds. Minor triads are shaded gray.

Comparing tunings

We can expand this model to include all 24 major and minor triads. And with this expanded model we can quickly compare different tunings based on pure fifths, pure thirds, and equal temperament, all of which are shown in illustration 2. (Beat rates are referenced to the 2′ middle octave, a = 440Hz.)

The first example in illustration 2, Kirnberger I (not to be confused with Kirnberger II or III) is a late Baroque tuning that features nine pure fifths, three pure major thirds, and two pure minor thirds. This is a variant of Pythagorean tuning and has the tonal color required for very early music. It also plays much of the later literature with radiant harmonic purity.

The second example shown in illustration 2 is the 1/4-comma meantone devised by Pietro Aaron in 1523. It is a wonderful representative of the class of tunings that emphasize the purity of major thirds. Also note the extreme dissonance in the “wolf” intervals in violet, the price paid for the purity in the thirds. A glance at this example will show why older organs tuned in strict meantone had no bass octave keys for C#, D#, F#, or G#. Many variants exist that rearrange the dissonant and consonant intervals, and it is important to match compositions created in meantone with their proper meantone variations. (The important reference for this is Claudio Di Veroli’s Unequal Temperaments, Theory, History and Practice, 3rd Edition.)

As the demand arose to have more freedom in the use of more remote keys in the eighteenth century, a virtual flood of attempts arose to trade off the purity of the meantone third for less dissonance in the more remote keys. These are known as the “well temperaments.” As noted earlier, these attempts mostly disappoint; harmonic purity was watered down to the point where the sense of consonance disappeared when any real sense of freedom emerged in the more remote keys. The logical consequence was, of course, the rise of equal temperament, which is ubiquitous today.

The third example shown is equal temperament. This tuning has a wealth of nearly pure fifths, but no interval has real purity, without beats. The major thirds are quite impure and very dissonant. Minor thirds are worse. We have simply grown to tolerate this dissonance through familiarity with it. The pure, or nearly pure, triad is rarely a part of modern keyboard experience. We pay a very dear price in the sonority of our music with the freedom we gain to access the tonality of any key.

We can make early compositions sound as exciting to us as they did to their composers if we play them in their appropriate tunings. The musical impact of a tuning is determined by its consonances and dissonances, and these sounds are described by beat rates, not “cents.” This model hopefully provides a more intuitive way to understand the variety of tuning styles for the pipe organ.

References

Di Veroli, Claudio. Unequal Temperaments, Theory, History and Practice, 3rd Edition. Bray, Ireland: Bray Baroque, 2013. Available as an eBook on Lulu.com.

Jorgensen, Owen. Tuning the Historical Temperaments by Ear. Marquette,  Michigan: Northern Michigan University Press, 1977.

McNeil, Michael. The Sound of Pipe Organs. Mead, Colorado: CC&A LLC, 2012.

Back in the 19th century, two great scientific figures wrote books on the science of sound. The earlier was Hermann Helmholtz with his On the Sensations of Tone, 1863; the later was Lord Rayleigh in his The Theory of Sound, 1877. Both books are large and heavy tomes. These scientists were both highly skilled and thorough in their investigations. Indeed, these works are for most of us, formidable. Much theory, much mathematics, and much ponderous discussion. Nevertheless, they both set forth a theory of sound that has stood the test of time. As with any such intricate disquisition, the public has taken from the work of these two men what they want and, in a sense, "simplified" the principles. Since those books were written, the science of acoustics has made several substantial and significant advances.

Over the last several years, I have encountered two presentations of simple acoustic theory under somewhat formal circumstances: one was a lecture given at the convention of American Institute of Organbuilders held in Virginia, October 1997. The other is in a fine book of the popular press, Music, the Brain and Ecstacy, by Robert Jourdain (Avon). This book is well written and in its later pages examines deeply and interestingly the psychology of the musical experience. But in the earlier pages it gives an explanation of sound, like the speaker at the convention of organbuilders, that reads completely from the old theories.

In brief, the older and still popular theory says that all sound can be analyzed into component harmonics. All the harmonics in this composite sound are integral multiples of the bass frequency. Thus, given an A=440 hertz (cycles per second), its harmonics are 2x440=880, 3x440=1320, 4x440=1760, and so forth. Furthermore, these harmonics, like a pharmacist's compound, are added at various dynamic levels: the second harmonic is, say, 42% of the fundamental, the third harmonic is, say, 86% of the fundamental,and so forth. In other words, if you were to compound a given group of harmonics in their proper proportions to the fundamental, you will, perforce, re-create the original sound.

 Here is the basis on which Hammond constructed his now famous organ. Little sliders above the keyboards--there are nine of them--can be drawn out to one of 8 different positions to represent the various intensities of each harmonic and--voila--there you have the oboe, the violin, or whatever other sound your mind can dream up. Anyone who has played around with a Hammond knows that no such miracle takes place. Rather, the instrument produces some sterile sounds that can beguile briefly but ultimately do not imitate anything but a Hammond organ. The Hammond has been used extensively in the field of popular music. It is portable and not too expensive. And Hammonds do last a long time! Their basic sound-producing mechanism is a whirling shaft--spinning at a constant speed to keep the frequencies constant--with many notched wheels (one for each frequency); electromagnetic pickup heads pointed at each wheel feed the signal from the wheels through the sliders into an amplifier and eventually the loudspeakers. Another fact about the Hammond is that the harmonics are borrowed from the tempered scale of the bass frequencies. Therefore, the pitches of the nonunison "harmonics" are not in tune with the fundamental--not true harmonics--and therefore they cannot form composite sounds because they will beat with the fundamental.

The idea of compounding harmonics to produce composite tones is not new. The major chord is nothing more than the first, third, and fifth harmonics of the basic key note. Intricate harmonies such as in the Bach chorales or the barbershop quartet are merely other combinations of harmonic pitches to make a composite sound. In fact, the major and minor scales we use for our music are composed of the harmonics of a fundamental note. (The C-major scale, by the way, is based on the harmonics of F!) All the notes of the C-major scale can be played or sung simultaneously without producing any beats provided the notes are accurately tuned as harmonics of the frequency of the generating tone, F. The common theory of sound believes that by properly proportioning harmonics, one can make a human being think he hears the strident blare of the trumpet or the soothing coo of the flute. But later research and better understanding of acoustic phenomena have altered these earlier theories.

One of my colleagues, Ralph Showers, at the University of Pennsylvania, way back in 1943, was preparing a master's thesis on sound. I talked with him about synthesizing tones by the old theory. He laughed at me. I exclaimed, "You mean you can't put the proper proportions of harmonics together and recreate sounds of violins, oboes, and others?" He told me that in analyzing the sound spectra of the various instruments, he could find no clearly singular spectrum for any one instrument. "They went all over the place! The spectrum of the violin looks like the spectrum of the oboe; the clarinet looks like a trumpet," he said. I was crushed; I thought it was all so simple and neat. Further conversation with Ralph only made things worse; the whole idea of synthesizing tones by adding proportions of harmonics was nice in theory, but not in practice.

The harmonic synthesis principle, even if true, leaves a whole class of musical sounds out of the "equation." What about the struck instruments, the bell, the cymbal, the drum, the glockenspiel? These are all enharmonic instruments that do not obey the "harmonic" law at all.

Then I took a graduate course in acoustics and learned a fuller approach to the theory of sound. Its premise is that all elements that produce sound vibrate because they have been temporarily deformed. The restoring forces of the deformed bodies are due to one or the other of two forces: tension, or stiffness. The violin string is deformed by contact with the moving resin-laden bow and pushed from its normal position until the restoring force of tension in the string exceeds the grip of the resin on the string, at which point the string flips back. Continued bowing thereby sets the string into vibration. The triangle is struck with a little rod and is thus deformed, as with the violin string, but it returns to normal by virtue of the stiffness in the structure of the triangle. Unlike the violin, whose tone can be maintained as long as the bow keeps moving, the tone of the triangle gradually falls off as the energy of the initial blow is gradually dissipated into the air and the metal of the triangle. All instruments of music are governed by one or the other of the two restoring forces: tension or stiffness.

In order to study these phenomena, the physicist, in his formal mathematical way, writes down a differential equation that says: the vibration of a body is proportional to the sum of two restoring forces, tension and stiffness, as a function of time. Fortunately, the physicist finds that the musical instruments fall into either one or the other class: tension or stiffness, but rarely both (see pianoforte later). The equation is then simplified by dropping either the term for tension or the term for stiffness. If tension is the dominant restoring force, the equation reduces to the harmonic series of integral multiples of the bass frequency: one times, two times, three times, the fundamental frequency. The various frequencies are sometimes called upper partials, a good general term. But they can also be called, quite specifically, harmonics.

The class of harmonic instruments is large. The typical tension instruments are all the bowed string instruments, also the guitars, zithers, clavichords, harpsichords and other plucked instruments (the pianoforte is not included here; it's in a class by itself!). Also, it turns out that the wind instruments function also in the harmonic system because a column of air acts like a string under tension. Thus, all the wind instruments are also harmonic devices: the oboe, the flute, the trumpet, the human voice, the french horn, and the organ pipe, for example.

When stiffness is the dominant restoring force, the equation leads to another set of equations known as Bessel Functions. (Friedrich Bessel was a Prussian astronomer, 1784-1846, who developed a series of equations that have been named for him and which are much like the trigonometric equations.) Without going further into the abstruse, the Bessel equations describe the higher frequencies that form the upper partials of the sounds of stiffness-dominated instruments. The frequencies of these upper partials are not integral multiples of the fundamental; the Bessel multipliers are called transcendental numbers. These multipliers are long decimal numbers that have an infinite number of decimal places. They are not harmonic! They should properly be called overtones. These sounds do not blend quietly into the mélange of the harmonic orchestral sounds of the tension-dominated instrumental sounds.

The class of stiffness-dominated instruments is not as large as the tension-dominated. They include the triangle, the chime, the bell, the glockenspiel, the xylophone, the celesta, all the drums, the cymbals, and other struck-type instruments. The triangle with its single ting! sounds out clearly over the entire symphony orchestra. What is the loudest instrument in the football band? Not the trumpets. No, it's the glockenspiel, that brightly plated, lyre-shaped instrument that is played with a little mallet. Its ting sings out over everything else. It is nonharmonic and therefore does not get lost in the rest of the sounds.

So, we can summarize: All instruments are divided into two main classes--harmonic (tension-dominated) and nonharmonic (stiffness-dominated).

Now a special note on the pianoforte. Pianos, as they are commonly called, used to be in almost every American home. Acoustically, the piano is a total hybrid. Did you ever try to replace a piano string? Quite unlikely. But I assure you it is far from easy. All the strings in a piano are stiff and difficult to handle. They are unwieldy in their stiffness. And they must be wrapped around a tuning pin that is about 1/4≤ in diameter! Unlike the thin, delicate strings of the harpsichord, the piano string is definitely a thing of stiffness. And have you heard of the tremendous tension that exists in a piano when it is all strung up and tuned? There are many tons of tension between the tuning block in the front and the terminals at the rear of the cast-iron frame that holds all the strings. With these two facts, yes, the piano carefully sits astride the stiffness-tension systems. It is not quite either; it is both. Thus the piano--like the triangle--can, in a concerto, quite dominate the orchestral scene. At the same time, the pianoforte is not a member of the symphony orchestra. Often one sees pianos as part of high-school orchestras, but these pianos are used to fill in the parts of missing instruments.

Let's change gears rather abruptly. There have been two significant experiments performed in the last 30 years that have made us reconsider the old gospel significantly. One of these experiments might seem to have nothing to do with music. It has to do with language. (But isn't music a language?) A branch of the U.S. military service was curious about the limitations of frequency range on radio communication through earphones. What would the absence of certain frequencies in the communicating system do to the accuracy with which the serviceman would hear the messages? How accurately do the pilots hear the messages through the radios and earphones that guide them?

A test was carried out at MIT in which they had three rooms. Listeners were placed in rooms A and C. An articulator--a person who spoke English clearly--was placed in room B. Room B was connected to room A through an amplifier that passed only frequencies of 400 cycles per second and lower. (The unit, cycles per second, has been replaced with the single word hertz.) Room C was connected to room B by a similar amplifier that passed only frequencies of 400 hertz and higher. The decision on 400 hertz was to separate vowels from consonants. Realize that the vowels, though spoken, are pronounced as if they were sung. Vowels are sustainable; they can be made into continuous sound. These are the tones one makes when singing; they are, for a man's voice, below 400 hertz. The high A of the tenor is 440 hertz. But the fricative sounds--the s's and sh's, the explosives, p's and t's--are all in the upper frequencies, frequencies above 400 hertz.

 This test setup neatly divided the English language into vowels in room A and consonants in room C. The articulator in room B then read a list of words while the listeners in the other two rooms wrote down what they heard. Results showed that room A, with its vowels, heard about 20% of the words correctly; room C, with its consonants, heard 80% of the words correctly. In other words, 80% of the intelligibility of language lies in the consonants, not the vowels. Choruses and singers in general find it difficult to make the words they are singing intelligible to the listener because they sing mainly on vowels. However, there are a few singers, notably Rosemary Clooney and Bing Crosby, who do (did) an incredible job of pronouncing their consonants so distinctly as they sing that we can all understand what they are singing about.

Another experiment--this in music directly--was performed with only two rooms. Room A contained a number of experienced musicians; room B contained a number of familiar sound sources such as violins, trumpets, oboes, and singers. Room A was connected to room B by a fine, high-fidelity sound system. In this experiment, the various sound sources in room B played notes. In room A, the auditors wrote down what they thought was the source of the various sounds they heard. Their answers were nearly all correct.

Then they did a second phase of the experiment. In each case, now, the microphone was turned off until the source had already started its sound and the microphone was turned off before the sound had stopped. In other words, the auditors in room A heard only the steady-state sound without any initial or terminal transients. The "consonants" were excluded and only the "vowels" were sent to room A. The results were nearly all in error. Experienced musicians could not tell the "vowel-sound" of the oboe from that of the violin! In other words, in music as in language, it is the consonants of the "language" that make for intelligibility, not the vowels. As we grow older, our ears lose their high frequency sensitivity; we drop off the consonants so that we fail to understand what other people are saying.

The significance of these two experiments cannot be overstated. Notice now that the entire sound of those "loudest" instruments--triangle, glockenspiel--lead to an idea that sounds silly when posed, yet is not frivolous. Why does a composer use ties to sustain notes across a bar? One might answer unwittingly; "In order to make the note longer." But wait! May it not be that he wishes to avoid a new transient at that moment? Syncopation is another example; by not accenting the beat and accenting the offbeat note, one achieves syncopation. The force of the syncopation is enhanced by the transient sounding powerfully on the offbeat!

The articulation--the breaking up of the music into its parts and pieces by the presence of transients--is the way to understand the details of a work. Mozart used the trumpet--much to the trumpeter's distress because it made Mozart's trumpets parts so simple‚ largely as a source of juicy transients. The ta-ta-ta-tah of the trumpet is really a percussion effect. The trumpet has long been the alarm, the tocsin, that brings us all to attention. The wonderful cackle of the oboe is the source of its joy and rhythm. The breathy concomitant sounds with the flute are its own joy.

Once I was demonstrating a new organ to a group of curious people. While I was bubbling about the wonderful tones of the new instrument, I picked one stop and put my finger on a single note and held it. I continued to rhapsodize about the beauty of the tone as I held the note ever longer. Suddenly, one of the listeners burst out, "Play it again!" He wanted another transient to savor again the "tone" of the pipe, not listen just to the steady-state tone.

Jazz, the great American music, is known for its rhythm and especially its syncopation. What are the instruments most useful to jazz? The trumpet, the saxophone--which has some wonderful transients--the trombone, the clarinet. Notice, the violins are missing. Why? Because the violins and their cousins have weak transients. Listen to a violinist trying to play jazz and you will hear a musician with an instrument that has weak transients doing his best. He will employ every device known to music to overcome his lack. He will play late (agogic accent), he will play off pitch, he will play with vibrato. These are the devices also used by vocalists, who also have weak transients. How often have you heard a jazz singer arrive at the last note singing well flat of the note and then, as he/she sustains the note, gradually begins to add vibrato and slide up to the pitch? A string quartet is, at best, inept at jazz.

Bach's favorite instrument was the clavichord; he also liked the harpsichord. The sounds of both these instruments consist largely of transients. In fact, they are noted for their lack of sustained sound. The distinctive ting of both these instruments makes the counterpoint brilliantly clear. When Silbermann showed Bach his latest pianoforte, Bach is said to have made some remarks that apparently caused a falling-out between these two men. Perhaps Bach's comments had to do with the less pronounced transients of the new pianoforte compared with those of the clavichord and the harpsichord. The counterpoint was not as easily heard as on the earlier instrument. So too, the "chiff" of the unnicked organ pipe provides a transient that also helps to make the counterpoint clear.

One must conclude from these experiments and from personal experience tha the definition of musical tone and its place within the fabric of music is more dependent on the opening and closing transients of the sounds than on the old theories of composite tone that consists of harmonics or overtones in varying proportions. Yes, there are harmonics; yes, there are overtones; yes, they are there in varying quantities. But between the actual music and the laboratory analysis, there is a long distance. We live on transients for most of our interest in music. Many of us may think that the pear-shaped tones are the sine qua non of music. Certainly, the mellifluous sustained tones of a great choir of strings in an orchestra, or the sustained sound of a large chorus of human voices is a stirring sound, but both are sounds that need to be ventilated frequently with consonants to make them totally meaningful.

We must modify the old credo of sound to bring in these new understandings. Transients form the intelligence in music. Music must have transients to be interesting. It is the consonants, not the vowels, that make for articulate, intelligible, and moving music. Otherwise, we have ululation!

What are the questions?

An old adage says that the more experience you have in a field, the more you realize how little you know. This thought lurks at the back of my mind, ready to spring forth without notice. You hear a teenager say, “that’s the best movie ever made,” and you wonder how someone so young can be so sure. Then, pain of pains, you are reminded of similar cocksure statements you made when you were young. I knew so much when I was 18, 20, 22 years old that it was hard to imagine there would be more to know. Thank goodness for the inexorable professors who really did know more than I, and for the mentors who encouraged me in what I did know and never failed to point out those that were still mysteries to me. Whispered aside: A colorful and I think underused word in the English language is moil. The American Heritage Dictionary (Houghton Mifflin Company, 2000) gives “intr.v. 1. To toil, slave. 2. To churn about continuously. n. 1. Toil, drudgery. 2. Confusion, turmoil. (Note that moil is part of turmoil—what do you suppose tur means?) With that definition in your minds: I’ve been toiling and moiling (churning, drudgery, confusion) in the organ business since my first lessons as a young teenager and my first experiences in a workshop. There are completed projects and past performances of which I am very proud, and at least as many (God help us there are not more) that I’d like to forget. But that brings us to another most valuable adage: we learn from our mistakes. So as much as we’d like to forget them, we owe it to ourselves to keep their memory fresh lest they be classified as wasted pain. As I work in my shop I hear little voices saying, “if you do that . . . ” When I fail to listen to those voices I cut my finger or break the piece I’m working on. My friends might chuckle and say, “of course he’s hearing little voices—we’ve known that for years.” But the fact is, I think those little voices are the younger me seeing the scar on my hand caused twenty years ago by exactly the same obtuse motion. Those little voices are not signs of going over the edge, but are pearls of wisdom—that elusive and unquantifiable commodity that comes only from experience. And aren’t some of our best learned lessons those that rise from the smoldering coals of our mistakes? The master watches the motions of the apprentice and reaches for the Band-Aids® minutes before they are needed. The parent wishes to be able to spare the child inevitable pain, realizes that advice will not be heard, and has the Kleenex® on the kitchen table an hour before the school bus arrives.
I started by noting that the more you know, the less you know. A cubist view of that statement says that experience in a field reveals more questions than answers. If you really understand the questions, then you are getting somewhere. Often as I write I suppose I’m giving answers, or at least relating my experiences and observations as actualities. This time, I thought I’d give some questions, try to put them in context, and invite you to cogitate and moil over them. As always, I invite your comments: .

1. Which is better, tracker or electric action?

I grew up in the heart of the famed Revival, immersed in both new and antique pipe organs, believing tracker action to be the root of all that is good. As a young adult I had wonderful opportunities to work on massive electro-pneumatic instruments and was exposed to brilliant players doing magical things with them. I was startled when I realized that I was preferring the flexibility of fancy registration gizmos and the orchestral possibilities of these wonderful organs. Now I know I’m interested in good organs. As long as an instrument is well-conceived and well-built, it doesn’t make a whit of difference what kind of action it has. What do you think?

2. Why do some historical styles of organs have developed pedalboards and pedal divisions while others don’t?

The organs of 17th- and 18th-century France have simple and awkward pedalboards in comparison to those of northern Europe, and the music written for them reflects that. François Couperin le Grand (1668–1733) and J. S. Bach (1685–1750) were contemporaries—a quick glance shows the difference—most of Couperin’s music is notated on two staves. I’ve written before about the reproduced engraving that hangs over my desk (from l’Art du Facteur d’Orgues, Dom Bedos de Celles, 1766). It depicts a large 18th-century French organ shown in cross-section, with an organist playing. He is wearing a powdered wig (good thing it was tracker action, think of that powder clogging up the keyboard contacts), a heavy formal coat with long tails and buttoned cuffs, an equally heavy vest under the coat, and a sword whose tip was right next to his feet on that primitive pedalboard. A sword? No wonder they didn’t use the pedals. One fast flourish and your feet would be bleeding. Imagine the teacher saying, “Go ahead, take a stab at it.” And, to protect himself from injury he was wearing heavy boots. No Capezios here.

3. How do historical styles evolve?

It’s relatively easy to identify and study the differences between, for example, 18th-century French and German organs, but what caused the development of those differences? Was it the wine? Was it the spätzel?

4. Where did the different pitches of organ stops come from?

There is a simple answer—8' is the fundamental tone, 4' is first pitch of the overtone series, 22?3' is the second, and so on through 2', 13?5', 11?3', 11?7'. 102?3' is two octaves below 22?3' so 102?3' is the second overtone of 32' pitch—that series continues with 8', 62?5', 51?3'etc. The overtone series was perhaps first heard clearly in the tone of a big bell. The experienced listener can hear fifths and thirds clearly in the tone of such organ stops as an Oboe, Clarinet, Krummhorn, or Trumpet—in fact, those stops get their color from those strong overtones. That’s why you can hear the pitch of a Tierce so much more clearly against a reed than against warm and fuzzy Gedackt. (When I’m tuning those stops I have the habit of humming and singing parallel intervals and arpeggios inspired by the overtones —another example of the little voices in my head.) But the real question is how the perception of those overtones in the sound of an organ pipe led the early builders to experiment with creating individual stops that doubled overtones.

5. Is chiff a good thing?

During the aforementioned Revival many organbuilders experimented with “chiff, ” that characteristic chiffy consonant that starts the speech of an organ pipe. Every musical tone has some sort of attack that precedes the vowel of the note, and an organ pipe can be voiced to have lots of chiff or virtually no audible chiff. It’s a matter of personal preference, but if some people like it can it be all bad?

6. How does a modern church justify the cost of purchasing and maintaining a pipe organ?

Hardly an organ committee comes and goes without grappling with this one. A committee member asks, “with all the hunger and suffering in our community, why shouldn’t we use the money for a food pantry?” Our church buildings with their fancy windows, silver chalices, statuary, paintings, and pipe organs are expressions of our faith. Our culture is loaded with examples of historical expressions of faith through art—think of the liturgical music of Mozart and Bach, the sculptures of Michelangelo, the buildings designed by Bernini and Henry Vaughan. Are we better able to fund a soup kitchen from a building that makes obvious to our neighbors the strength of the bonds that tie us together as a community of faith?

7. How does a chestnut become a chestnut?

Given the production cycle of this publication, I am writing in mid-December, these few hours sequestered, escaping the tyranny of commercialized versions of our favorite Christmas carols.
Otherwise, I’m racing around the countryside tuning organs (plenty of opportunity to be humming arpeggios next to Krummhorns). Several of the churches I visit are presenting “Messiah Sings.” Handel’s masterpiece is a fantastic artwork. It’s easy to understand how it would filter down through generations as a perennial international favorite. But it’s very difficult music. The choir members in these churches have no idea how difficult it is. I’m sure they wouldn’t dream of tackling Handel’s Israel in Egypt, another masterwork that’s equally majestic and equally difficult to perform. Why is that?
Many parish organists will agree with my assertion that you could successfully plan and play a thousand weddings, fully pleasing all the families involved, with a repertory of ten pieces. We could all name the same list: Wagner, Mendelssohn, Schubert’s Ave, Jesu Joy, Clarke, Purcell, Stookey (“there is love . . . ”). You play through ten unfamiliar pieces for a bride and groom with no response, and they light up with the first six notes of Jesu Joy (boom-da-da dee-da-da . . . ). It doesn’t matter if you’re in Boston, Seattle, San Antonio, Milwaukee, or London. Why is that?
How many of us look forward to playing those wonderful sassy French noël variations—the ones with the non-existent pedal parts? I see volumes of Daquin and Balbastre on organ consoles all across New England. How many congregants recognize them as seasonal music? We erudite organists associate them with Christmas as readily as reindeer and O, Holy Night. Why is that?

8. Why did it take so long to develop equal temperament?

(Please do not interpret this as an indication of personal preference!)
Equal temperament is the most common system of tuning keyboard instruments and was not commonly used until at least the late nineteenth century. Pythagoras (6th century, BC) is credited with the development of the concept of tempering, of dividing the circle of fifths into the octave, a feat that is technically impossible. If you start on a single note and tune pure fifths around the circle of fifths, when you complete circle returning to C from F, you have nothing like a fifth. So over the centuries, various musicians, mathematicians, and theorists toiled and moiled developing systems that would divide that discrepancy over more and more of the intervals, allowing more of the twelve possible keys to be useful—or usable. The advent of Pythagorean tuning was natural, but I wonder why he or one of his contemporaries didn’t solve the problem by dividing the difference over all the intervals from the very beginning. That would have changed the development of music dramatically.
Some of these questions have real answers. Some of these questions have different answers, depending on whom you ask. I’ve given comments to introduce each of the questions that may lead a reader to deduce that I have an opinion. And those of you that know me personally may be able to read what you know to be my opinions, whether I know them or not. Why is that?
The questions frame the debate. If there’s a debate over a specific question, does it follow that there is no right or wrong answer?
Here’s an exercise that illustrates the elusiveness of correct answers. Take a well-known church building: St. Thomas Church, Fifth Avenue, New York. Consider two well-known and successful organbuilders, respected for the toil and moil of their respective careers: Ernest Skinner and Taylor & Boody. Imagine what each would consider the ideal organ for the space. Now tell me, who’s right?