How to explore nearly equal temperament with your piano tuner

Hearing a pipe organ tuned in a sympathetic temperament is
like discovering fine wine after a lifetime of roadhouse coffee. There is
simply no comparison between the delights of pure tuning and the frustration of
cadences that beat unmercifully, no matter what the key or modulation.

When the listener does not hear this woeful tuning,
psychologists call it habituation. In other words, the average person does not
hear the inharmonicity of equal tuning because they know nothing better, and
have come to accept the ragged chords that have echoed in their ears for so
long as normal everyday music. One may take a holiday from equal temperament by
listening to a barbershop quartet for a dose of close harmony. Or take in a
concert on an organ made by an artisan builder who regards tuning as an
integral part of the instrument, reflective of its true baroque heritage. This
journey is worth the expense of rethinking all that we have taken for granted
in years past.

Ironies abound in the world of musical bias and each new
discovery can be delicious. In the late 19th century, we find a reliable bearer
of tempered tuning in that most unassuming of instruments, the reed organ. Free
reeds can hang on to their original tuning at least as well as cone tuned
pipes--in fact, they suffer less from wear and tear. Pump them up, and they
continue to play with the same sweet harmonies that their original tuning gave
them.

There are some aspects of 19th-century tuning that are tantalizing
indeed. Victorian temperaments are nearly equal, which means that in the
tradition of well-tuning, they render harmonious chords in all keys, though not
without individual key color. They are subtle, providing tension and relaxation
behind the scenes, rather than by the blunt contrast of sheep and wolves, as in
baroque temperaments. Their intervals gently progress from calm to agitated,
depending on the complexity and remoteness of each key. Somehow, they walk a
fine line between purity and utility. It seems that their particular strength
is modulation, where the prime keys assert themselves like the sun appearing
through cloud or the calm after a storm.

Of late, Victorian models of tuning have become popular with
both piano technicians and organ builders. The late 19th century was no less
rich in its diversity of temperaments than the 17th and 18th centuries.
Although theorized very early on, equal temperament was a child of the
industrial revolution. Perhaps it was the factory production of musical
instruments that propelled it into nearly universal practice among tuners and
musicians. Studies of ethnomusicology have informed us that the practice of
equal tuning was unique to western civilization and that other cultures
simultaneously developed far more rich and complex modes of intonation.

As we reflect on the revitalization of early music and an
increased regard for performance practice, we take equal tuning less for
granted. The realization that tuning methods have varied tremendously according
to time and place has awakened our ears in such a way that we can now explore
the world of sound and imagination, unfettered by musical prejudice. Take the
challenge: play through the modulations of your favorite 19th-century composer
and see what a "less than equal" temperament does for the music!
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Three practical considerations

If one is going to change an organ from equal to
well-temperament, it should be an operation that is undertaken with
considerable planning. One should consider the nature of an appropriate
temperament and what music will be the primary repertoire. It is important to
look at the objectives of a major change and to evaluate the musical results,
insofar as possible, ahead of time.

The sound of an organ goes a long way to dictate temperament.
Compatibility of organ building style and repertoire are major issues. If equal
temperament is one frustration among many, the organist must decide if a change
to well temperament is going to help change musical values for the better. It
is comforting to know that even a spinet piano can be satisfying when tuned in
a historic temperament. By the same token, there are many organs that will
benefit immensely from the natural harmonicity and increased resonance of a
carefully chosen temperament.

Once the decision is made, one should not use the
"candy store" approach to the selection of a temperament! It is a
good idea to seek out a consultant who has the sounds of various tunings in his
ears. Experience can be most helpful! There are several practical matters that
should be considered when evaluating the pros and cons of various tunings:
balance of thirds, regular or irregular intervals, and shared tuning with equal
temperament.

Balance and width of thirds (in cents)

The reason for tuning in well-temperaments is to achieve key
color. As a composer calls for various keys with a lesser or greater number of
accidentals, the key color is expected to change from pure and restful chords
to vibrating and agitated harmonies. These shifting key colors are relatively
subtle, perhaps even obscure to the layperson, though quite obvious to most
musicians. As one evaluates diverse temperaments, the issues revolve around the
amount of key color desired and the achievement of an even balance that
increases the frequency of beating thirds in accordance with a greater number
of accidentals, both in sharp and flat keys.

Circle of fifths: regular or irregular intervals

This consideration is often overlooked until one makes music
with orchestral and chamber players. Regular intervals ensure the best tuning
of obbligato instruments because the transition from various intervals within
the temperament is predictable and intuitively correct. Some well tunings have
a fine balance of key color, but present such irregular intervals that out of
tune playing by ancillary instruments is unavoidable. It is not a reflection
upon the players--actually, the more experienced and intuitive the players are,
the more likely they are to have difficulty with irregular temperaments. It is
precisely the "anticipatory" nature of "tuning on the fly"
that causes the problem.

Certainly, the best chamber players always tune with the
continuo for each open string or major interval, usually in a circle of fifths.
If that circle of fifths is predictable, things go well. If each successive
fifth is a bit wide or narrow, almost at random, how is an instrumentalist
going to remember the exact tuning? "Regular" temperaments solve this
problem by the use of predictable intervals for the circle of fifths.

Convertible or shared tunings

This is a special consideration where a well temperament
will actually share part of the circle of fifths with equal tuning, usually the
notes A-E-B-F#-C#. These five notes may be tuned exactly the same in both
temperaments! In an equal temperament, the remaining seven notes are tuned in
the same ratio as the first five. However, in a convertible or shared
temperament, the remaining seven notes are altered to the new temperament. The
benefits of a shared tuning are considerable, particularly if the instrument is
to be tuned back and forth between well and equal tuning. This is often the
case with a continuo organ which is featured in various temperaments and often
at various pitches from one concert to another.

Graphs

It is very helpful to see these relationships in a graph, as
well as text and numbers. It has become very common to express numeric
relationships among various temperaments in terms of deviation in cents from
equal temperament. This is not because equal temperament is best or right, but
because each interval is a mathematical division. Thus, a rendering of equal
temperament is not given as a "norm," but as a mathematical point of
reference.

Using an electronic tuning device vs. tuning by ear

It is ironic that tuning in equal temperament became
standard practice about the same time as electronic tuning devices became
commonplace professional tools. At this time, it may be said that most tuning
of musical instruments is done with an electronic reference. That is not to say
that "tuning by ear" is no longer practiced, but aural tuning has a
new perspective, to "test" temperament rather than set it. Before the
reader jumps to any conclusion, it should be emphasized that the "art of
tuning" is still very much intact, and fine piano and organ tuning has not
suffered at all. The very finest tuners still use their ears, and the machines
are just another tool in the box.

Paradoxically, the resurgence of well temperament coincides
with the widespread use of electronic tuning devices and computerized tuning
programs. Virtually every device available offers a synthesis of historic
temperaments that are available at the touch of a button. One might argue that
this enables those without sufficient ear training to "tune" various
instruments--it also enables quick and precise tuning by professional
technicians who have more than enough ear training to do the entire job without
an electronic tuner. It is very advantageous to move from theoretical considerations
to practical application  so easily
and effortlessly. It is a conundrum, but a happy one. Without electronic
assistance, historic tunings would be sufficiently tedious that they might well
be left undone.

Tuning by ear remains indispensable. The name of the game in
tuning is to reduce error--especially cumulative error. "Tests" are
the most important aspect of any tuning. They keep the tuner on the straight
and narrow, and prevent compound or cumulative errors that seriously degrade an
artistic tuning.

Recommended computer programs

Two fine computerized tuning programs are Robert Scott's
TuneLab program, available from Real Time Specialties, 6384 Crane Road,
Ypsilanti, MI 48197 ([email protected]) and Dean Reyburn's CyberTuner,
available from Reyburn Piano Service, 2695 Indian Lakes Road, NE, Cedar
Springs, MI 49319.

These are devices for tuning both historic and equal
temperament. Cost varies from less than $100 to about $900, depending on the
range of software desired. The best feature of these programs is that each
historic temperament file may be edited with a word processor. Other electronic
tuning devices are available, usually with pre-programmed historic
temperaments. The author suggests that they be compared on the basis of
accuracy (up to 1/10 cent) and the ease of programming various temperaments.
After that, there are issues of cost, portability and so forth.

As an aside, one may also consider style of tuning. The
author prefers the use of not one, but two electronic tuning devices--a
portable one to use inside the organ and a fixed unit at the console to monitor
tuning as the job progresses. This keeps the tuner's helper quite busy at both
organ and computer keyboards and reduces cumulative error by a considerable
amount. 

Historic tuning on the Internet

Bicknell, Stephen. A beginner's guide to temperament.

www.users.dircon.co.uk/~oneskull/3.6.04.htm

Bremmer, William. The true meaning of well-tempered tuning.

www.billbremmer.com/WellTemp.html

Foote, Edward. Six degrees of tonality; The well-tempered
piano.

www.uk-piano.org/edfoote/well_te mpered_piano.html

Gann, Kyle. An introduction to historical tunings.

http://home.earthlink.net/~kgann/his tune.html

Greenberg, Bernard S. What does "well-tempered"
mean?

www.bachfaq.org/welltemp.html

Kellner, Herbert Anton. Instructions for tuning a
harpsichord "wohltemperirt." 

ha.kellner.bei.t-online.de/

Palmer, Frederic. Meantone tuning.

home.pacbell.net/jeanannc/mpro/art icles/MeanTone.htm

Rubenstein, Michael. Well vs. equal temperament.

www.ma.utexas.edu/users/miker/tun ing/tuning.html

Taylor, Nigel. Tuning, temperaments and bells; The
ill-tempered piano.

www.kirnberger.fsnet.co.uk/   

If one may indulge in melodrama, one might refer to "The Curse of Equal Temperament" when commenting on the method of tuning that steadfastly refuses to take into account the relationship between an instrument, its music and player. "Equal" is tuning for the sake of tuning, done by successive generations of tuners who practice their craft exactly the way they were taught to do it, no questions asked. And the whole business has been cast in cement by electronic tuning devices--ETDs--in widespread use today!

Looking back over the past two centuries, we can take note of several events that contributed to this situation. They include the invention of tuning forks, the industrial revolution (with its myriad of factories that produced musical instruments) and the emergence of ETDs such as the Conn Strobotuner.

Tuning forks as we know them appeared in the early 1800s. Their fixed pitch enabled a reference to specific frequencies for tuning of musical instruments; tuning practices previously had varied widely by region and nationality. Tuning forks were a valuable resource for the stabilization of tuning everywhere. By the end of the 19th century, they were used for temperament tuning in the great piano houses such as Broadwood and Moore.

The turn of the 19th to the 20th century was surely the golden age of the piano, and in North America the houses of Steinway and Heintzman represented a pinnacle of musicality and at the same time promoted the artisanship of factory craftsmen unparalled in our own times. The revival of the organ as an "authentic" instrument would wait some fifty years, and with it the same emphasis on tuning as an integral part of a musical instrument.

Thinking back on the piano and its artists of the early twentieth century, one can reflect on the incredible tuning of these instruments, made for Rubinstein, Horowitz, Richter, Gilels and so many others. Pianism was almost a cult, and the tuners who worked on these instruments behind the scenes contributed a rare form of art to the piano. They defined its sound, its carrying power and its musicality as surely as the artists who played it so superbly.

With the revival of the tracker organ, tuning once again became an integral aspect of the musicality of these instruments. Temperament is most carefully thought out by artisan organ builders today with or without the help of tuning machines.

Machines? Yes, the same tuning devices that began with tuning strobes evolved into electronic displays of one sort or another, as varied as one might imagine. To some extent, they displaced "aural" tuning, so highly valued within the community of piano tuners and technicians. Unfortunately, some tuning practitioners passed "go" on the Monopoly board and skipped ear training by jumping into machine tuning as a quick means to an end. However, fine tuners the world over incorporated tuning devices into their tool kit as important aids to the musical ear that was already hard at work. It is this kind of practitioner that exemplifies the best in the tuning business.

The "curse" of machine tuning is that it implies that equal tuning is mathematically precise, and that the ear is irrelevant to the outcome of setting a temperament. Semantics are everything, and it is something of an understatement to say that "equal" tuning is not at all equal! An artistic tuning, whether in a baroque temperament for Bach cantatas or a modern tuning for a Rachmaninoff piano concerto, is anything but equal. It is what the music demands. A marvelous example is the use of the Vallotti temperament for performances of Beethoven's "Emperor" Piano Concerto. Yes, it works very well. One can only marvel at the work of the world's best piano tuners on the concert stage. The tuner's ear is alive and well in our finest recordings and live concerts--as it should be in the presentation of our finest pipe organs.

The "blessing" of machine tuning is that it provides the opportunity to record "best" tunings for various instruments and occasions--for tuning devices are not only tone generators of various pitches with an array of mathematical relationships, they are recorders, too. They make possible the quantification of any kind of tuning, from pianos to organs to gamelans. They are, in a sense, the power that destroyed some important aspects of tuning by ear, but they are also the force that brings back aural tuning. This is a happy conundrum that should be exploited for all it is worth.

The tuning device as recorder provides the opportunity to use temperament in an artistic manner to give expression to the best qualities of an instrument (and sometimes, to suppress the worst ones). For example, a concert grand piano in a large hall derives carrying power from vibrations generated within the temperament, as well as the soundboard and case of the instrument. For this reason, mild temperaments with more- and less-pure thirds benefit these pianos if they are speaking in a vibrant hall. On the other hand, a pure temperament can go a long way to smooth out a small piano with short strings that are full of false beats. Try that on your spinet in the choir room. You will be amazed at the improvement in sound!

Some practical considerations for the tuner

For the benefit of the reader who is truly interested in investigating the benefits of 19th-century (or earlier) temperaments with the help of machine tuning, this last of three articles will be devoted to the practical application of tuning techniques. Since it is widely available at low cost, the ETD (Electronic Tuning Device) of choice will be Robert Scott's TuneLab97 software, available at <www.tunelab-world.com&gt;. A basic computer and sound card are also required. With this tuning program, there will be a set of historical temperaments that offer a wide range of options for the tuner. Temperament files are extremely simple. They are notated in cents deviation from an equal distribution in this manner:

Representative Victorian Temperament (Moore)

C   2.5

Cs  0.0

D   1.5

Ds  1.0

E  -1.5

F   2.0

Fs -0.5

G   3.0

Gs  0.5

A   0.0

As  1.5

B  -1.0

Armed with this modification to equal proportional tuning, the tuner can proceed to lay bearings for a temperament. Fear not! I am not going to give the reader blow-by-blow instructions on how to tune. But it is important to note that most tuning failures result from tempering the wrong intervals first! Therefore, with this temperament one can follow the practice of using F, A and Cs tuning forks to divide the circle of fifths into manageable portions, so that one will not choke on a cumulative error. In this case, A and Cs may be set from tuning forks A=440 and Cs=277.18. "A" is used to embark on the white notes in the circle of fifths, and Cs is used for the black notes. As a rule of thumb, the intervals involving black notes are tuned first, pure or nearly so, and the intervals involving white notes are tempered and tuned last. Follow that rule, and you will avoid the trap of "reverse well" tuning.

The tuning fork F=349.23 completes the triad of foundation notes. In well-tempered tuning, "F" will be raised to provide the desired effect of the third F-A. Generally, the F-A and C-E triads will determine the nature of the well-temperament desired, whether mild, moderate or intense, as in the baroque temperaments. This is where the sound and character of the instrument and its music come in.

If one is tuning "equal" temperament, the thirds F-A-Cs¢-F¢-A¢ provide a very useful octave and a third in which to lay the bearings. These thirds will increase their vibrations as they ascend. This is one of the tests used in setting equal temperament. Conversely, in laying well-tempered bearings, the thirds will alternate in vibrancy between white and black keys. F-A will be slower than equal, A-Cs will be the same as equal (13.7 cents wide), Cs-F will be faster than equal, and once again, F¢-A¢ will be slower than equal. So far, the only adjustment has been to sharpen "F" to make a relatively slow third F-A.

Once this has been accomplished, one should tune Cs-Fs-B relatively pure and Cs-Gs-Ds-As-F relatively pure, monitoring the computer screen while one tunes these notes. Then, tune A-E-B and A-D-G-C-F relatively tempered, while monitoring each note on the computer screen. This will provide well tempered bearings, while applying tuning tests to the process. There will be little chance of a cumulative error of any significance.

Since the tuner is applying aural tests as well as reading a computer screen or a dial tuner to monitor progress, this work can be carried out at the organ console or the inside of the organ case, or preferably both. A tuner's assistant can do much more than hold keys. It is very helpful if they monitor an ETD while the tuning is in progress. This prevents errors and speeds up the tuning.

Which temperament to use?

There are literally hundreds of temperaments from which to choose, so it is very useful for each tuner to develop criteria which work for them. Several points are worth consideration.

It is most helpful to adopt a temperament that allows equidistant bearings for the tuning of a circle of fifths. The F-A-Cs method provides this option in both well and equal tempered tunings.

Another consideration is the provision of various degrees of purity within a related group of well tunings. An example of three temperaments that progress from mild to moderate are Moore, Peter Prelleur, and Young (1799). All are based on zero deviation in cents for the notes A and Cs, and increased purity for the triads C-E-G, F-A-C and G-B-D.

One may take into consideration the balance of triads in a symmetrical or non-symmetrical array. A symmetrical array of triads will increase vibrancy in direct proportion to the number of accidentals in each key. Asymmetrical triads will favor certain keys and are more consistent with harpsichord tunings where temperaments are chosen for specific literature.

Blessing or curse: from anathema to good fortune

If one looks upon machine tuning as a curse for its illogical suppression of musical values (modulation being the first victim), the descendants of strobo-tuners must bear a heavy burden of resentment. However, the computer and its dedicated mechanical brethren have rescued those who still tune by ear by providing the means to record their "best" tunings, and experiment with the most musical tunings for each instrument. Credit must be given to a significant group within the Piano Technicians Guild for their unflagging efforts to promote both aural tuning and the use of unequal, "well" and nearly-equal temperaments. A review of the comments of these technicians reveals a dedication to musical performance that stands as an inspiration to organ technicians and tuners as well. Commendation and approbation is also well deserved by artisan organ builders who have often stood alone in a sea of indifference by insisting that temperament and tuning are significantly related to each musical instrument they produce. There are many organ builders who will not resign their instruments to "ordinary tuning and care," but who steadfastly maintain their own instruments so that among other things, the tuning will be preserved. Bravo (!) to these dedicated builders.    

Herbert L. Huestis is a contributing editor of The Diapason.

On October 20, 2003, Dr. Herbert L. Huestis, contributing editor of The Diapason, hosted a "Well-Tuning" seminar for 20 members of the British Columbia Chapter of the Piano Technician's Guild. A collection of keyboard instruments was assembled for the tuning demonstrations: two organs, three harpsichords, and five pianos, including a fortepiano. None of these instruments were tuned in equal temperament.

The tuning demonstration began with a shop organ that serves as a voicing machine, set up with a lightly-winded Italian principal, tuned in 1/4-comma meantone. A number of piano tuners present were hearing meantone tuning for the first time. This initial demonstration was a good way to prepare their ears for the well-tuning temperaments to come.

A second organ was tuned in the Kellner "Bach" tuning, and mention was made of Martin Pasi's recent organ at St. Cecilia's Cathedral in Omaha, where Christian Wegsheider's "20 note wohltemperiert" combines meantone and well-tuning in one instrument featuring octaves with 20 pipes each.

The second part of the presentation was devoted to harpsichord tuning, featuring Vallotti (1750), Young (1799) and Handel (1780) temperaments. The tuning instructions for these three temperaments were explained, using Excel charts as visual aids.

The final segment of this discussion was a presentation of two Victorian temperaments on five pianos, including a fortepiano. Musical examples were played on the Moore and Broadwood temperaments, recorded by Alexander Ellis in 1885. These Victorian tunings were considered "equal" at the time and were collected as a record of the work of tuners in the leading piano manufacturing houses of the day. Several fine pianists were present, and they put the pianos to the test by playing 19th-century music while listening to the increased key color that was available with each temperament. Special accolades went to the playing of Chopin preludes on pianos tuned in the Broadwood temperament. It was interesting that the piano technicians present tended to favor more colorful temperaments, rather than those closer to equal temperament!

Some aspects of machine tuning were discussed, but there was a strong consensus among this group that aural tunings were superior and that electronic devices were most valuable when used to record "best tunings" for specific pianos. At the conclusion of this seminar, it was evident that well-tuning could be a significant factor in bringing out the finest musical qualities of pianos and harpsichords. In western Canada, organ tuning in equal temperament is almost universal, with the exception of the Martin Pasi organ in West Vancouver and a John Brombaugh organ in Victoria. Hopefully, new organs in the area will also share some form of well-tuning as events like this are held to encourage broader thinking in musical terms. However, it was agreed that equal tuning is so entrenched amongst the players that well-tuning of most organs might have to wait for another time and another place.

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.

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.