leaderboard1 -

Exploring the Sound of Keyboard Tunings

April 1, 2016
Default

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 16stops. 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.

Related Content

March 18, 2024
The celebration “These people will be your friends for life,” Karel Paukert pronounced to his organ class at Northwestern University in the mid-1970s…
March 18, 2024
That ingenious business Great Britain’s King George III (1738–1820), whose oppressive rule over the American colonies led to the American…
March 18, 2024
Robert Eugene Leftwich Robert Eugene Leftwich died January 13, 2024. He was born July 2, 1940, in Texas and grew up in Longmont, Colorado. He…