The musical system that has just been built is riddled with impure, inharmonious intervals. Does that mean that it's invalid? Here we will address the task of tempering the musical system, as a prelude for what is to come in the tempered harmonies of Kepler's Solar System.
Listen to this:
Now listen to it again, this time instead of beginning in the scale or key of C, we're going to move every note up a half step, transposing it into the scale built on C#:
The first was harmonious, the second sounded completely out of tune. Now listen to this one, transposed into the key of D:
That was better, but still sounded different than the first, especially this interval:
If all that's happening is that we're beginning on a different note than the original, either higher or lower in the system, but keeping everything else about the music the same, then why would there be such a drastic difference in how it sounds?
Take a closer look at the three scales. The first is the series of intervals you're familiar with, the hard scale as a result of the harmonic, consonant divisions. The second and third contain these impure intervals—actually, the scale on C# has so many of them that music in this key is almost unplayable, as you heard a moment ago. There are many other keys in the system which have this shortcoming, some more, some less, making some of them inaccessible for music.
If you remember, we ran into this problem already, when we first extended Kepler's system to multiple octaves. Here's the example from earlier. Starting the soft scale from the note A, the second of the original octave, the scale proceeds by set of intervals, different than that of the original soft scale, and so contains a number of imperfect intervals. Whereas the intervals which formed the original soft scale, from the knowable, constructible polygons, were harmonic means of the octave, meaning that all of the tones created by the division were consonant with one another, these imperfect divisions are not harmonic means of the octave from A to a.
While many of the scales and modes built on the imperfect intervals are perfectly tolerable for music, and even enrich it, by expanding the palette of colors available to the composer, the question remains open whether these intervals have validity as far as scientific measurement goes, since they are not harmonic divisions of the octave. In other words, when looking for harmonies among the motions of the Solar system, can Kepler consider two motions which form an imperfect interval to be harmonious? This question will force us to examine what we mean by harmony, and will be answered further in Book V.
Hopefully you're coming to see that as soon as we move away from the original hard and soft scales, we can't get away from the imperfect intervals; only the original octave is free of them, and even that's not entirely true—within the first octave, this third between D to F, for example, is an impure soft third, 27/32. In fact, as we'll begin to see more clearly in a moment, the slight variation of the sizes of intervals, and therefore the variability of the tuning of the corresponding notes is not so much an anomaly of the harmonic system, as it is an essential characteristic.
We'll bring out the problem that this poses as far as practical music goes, here, but addressing the real issue at hand more fully will have to wait for Book V.
Let's start by reviewing the example that Kepler gives in his Mysterium Cosmographicum:
In the scale built on G, the note F is five steps higher than B-flat, and five steps lower than the octave C. 5 steps would be a fifth, so to find the appropriate interval for F, I move a 2/3 interval up from 5/6, which would be 5/9. Doing the same down from C, or 3/8, I get 9/16. So, F has a dual role to play—either it's a perfect fifth from B-flat, or a perfect fifth from C, but it can't be both. The necessity for the single note F to be two different pitches, or said otherwise, to change its tuning depending on which other note it's orienting to isn't really a problem for a singer, or violinist, who doesn't have a fixed tuning, and can make small adjustments in pitch from moment to moment—but a pianist, for example, is stuck with whatever tuning his strings are set to. Changing the pitch means opening up the piano and re-tuning that set of strings1. The same issue we just saw was known to the Greeks in this way: If I start from G, and move up a fifth, I arrive at D. Moving up a fifth from there, I arrive at A. A fifth from that is E. Continuing to move up by fifths will eventually get me back to the note g, but 7 octaves higher. The G seven octaves above the original is 1/2 seven times, or 1/128. Taking 12 fifths, which would land me on the same note of the scale, g, is 2/3 twelve times, or 4096/531441, roughly 1/129. Transposing those two G's, one 7 octaves, the other 12 fifths, down, so that it's more comfortable for the hearing, the difference sounds like this. This came to be known as the Pythagorean comma, comma meaning "piece cut off."
Another example, contained within a single octave is this: a hard third from G is B, a hard third from that is D#, and from that is F##, or, g, an octave above where I started. Taking three hard thirds in a row is (4/5)^3, or 64/125. This should be an octave, but a perfect octave would be 64/128's, making this interval too small by the interval 125/128, also called the enharmonic comma.
There are countless other examples of commas which pop up all over the harmonic system—in fact, no pure harmonic interval is commensurable, and matches up with any other: three thirds doesn't equal an octave, 12 fifths, are not 7 octaves. This leaves "cracks" or inconsistencies throughout the system. On top of that, the intervals themselves, taken in terms of steps of the scale, change sizes depending on which note of the scale you start from! The fourth from G is different than the fourth from F. Instead of a coherent system of intervals, it's as if we have a strangely mismatched collection of parts again, putting the harmonic system as a system on shaky ground. But remember, the system was constructed based on the pure harmonic divisions, like 2/3 and 3/4, which are not intervals that were made up, abstractly—the ear (and mind) judged them to be the most harmonic, and Kepler shows that they share the quality of being the six, unique harmonic means of the octave. If maintaining that quality of these intervals means that the system contains cracks, then is it really a shortcoming? But why would intervals with such individual perfection not come together to create an internally consistent system?
We heard what that means practically when we tried to transpose a piece of music from the scale of C, to C#. It took on a totally different, and in this case, unpleasant character.
But even when it was transposed into D those cracks, so to speak, were audible—played in D the piece has a slightly different color, or personality. In this case, the change between C and D could be seen as a potential tool at the composer's disposal, like having several personalities to choose from for a dialogue—but in the case of C#, and many other keys in the system, the "cracks" render these areas off-limits to music.
To resolve the issue, to "fill the cracks", so to speak, so that the system of intervals is consistent, allowing music to be played in all of the keys, while losing as little of the perfection of the harmonic intervals as possible, is not an easy task. Going back to this example (of the enharmonic comma), filling the cracks means that either the octave is made smaller, or the thirds larger, or a little bit of both. The process of mixing in some degree of impurity into the intervals came to be called tempering.
Let's take a closer look at the hard third. You see here that it's broken down into 4 smaller intervals, 4 similar-sounding half steps: 2 semitones, 1 diesis, and 1 limma. This imperfect hard third here from F to a is also made of 4 half steps, but it's composed of 2 semitones and 2 limmas, making the total interval of a third 64/81, just a little bit larger than the original one. The same type of difference exists between all of the impure intervals and the corresponding pure consonances, so that we could reduce the problem to the fact that the half steps are different sizes.
If all of the half-steps were the same, then a third, always spanning four of them, would be the same size no matter where it is in the musical system. Making the half-steps the same size would also make all of the intervals commensurable with one another, such that an octave is 12 half steps, a fourth is 5, a fifth 7, and so on. Then, three tempered thirds would always equal an octave, even though none of them would be a perfect consonance anymore.
A tempering of this type, where all of the half-steps are made the same, is called equal tempering. To determine the size of the required equal half-step, which would get us to the octave after 12 repetitions, and hit as closely as possible to all of the harmonic intervals in between—let's start by trying the three half-steps which already exist as derivatives of the harmonic divisions. You can see right away, that taking 12 semitones is larger than an octave, 12 limmas is smaller than an octave, and 12 dieses is a lot smaller, so none of these half steps will work.
The only specific case that Kepler takes up in Book III is the equal tempering proposed by Vincenzo Galilei. In in his 1581 treatise, Galilei proposed the half-step interval of 17/18—something a little bit smaller than our semitone. The way to construct this on a string, is to divide the whole string into 18 parts, take 17 of them for the first half step. That would sound like this. Then, erasing those divisions, take the length which was 17 as the new whole string; divide it up into 18 parts, and take 17 for the next half step, and so on. By doing this, each interval generated is exactly the same as the last, and after 12 repetitions, the tone you come to is indistinguishable from half the string. Here are the "tempered octave" and the pure octave intervals.
Now, would a piano or other instrument tuned in this way allow us to play in the full set of keys and modes? Listen to the full, two octave scale for an answer.
Aside from not being very practicable, Kepler zeros in on what he sees as the biggest flaw with equal tempering. We can see it here with the fifth. As with the octave, Galilei's tempered fifth is almost indistinguishable from the true fifth. However, with a perfect 2/3 division, all three tones from the division form a unified consonance. With Galilei's tempered fifth, the quality of harmony is lost entirely.
“See another very clever tempering of this sort by Vincenzo Galilei, made not in ignorance of the mathematical size of the notes, but with a particular intention. And I indeed recognize its mechanical function, so that in instruments we can enjoy almost the same freedom of tuning as can the human voice. However for theorizing, and even more for investigating the nature of melody, I consider it ruinous; and the effect of it is that the instrument never truly attains the nobility of the human voice.”
We can scoff at how far off 17/18 is from doing the job, but tempering is not an easy problem. In 1605, 25 years after Galilei, Simon Stevin in the Netherlands used logarithms to calculate a half-step which, after 12 repetitions would come exactly to an octave: the twelfth root of 1/2, or, the number which, times itself 12 times, is 1/2. This is the half step which is used almost universally in piano and other keyboard tuning today. However, even this would not have escaped Kepler's criticism. Yes, all of the intervals are uniform, and the octaves are perfect, but the richness of the intervals is lost to greater or lesser degree b/c the quality of harmony is sacrificed—since all intervals are now imperfect, none has the rich resonance of a perfect, harmonic division.
The other, more significant deficiency of equal tempering was lamented later by Bach's student, Kirnberger, as a protest from art itself. When the tempering is made equal, then the characters of the different keys are lost. Whereas one key may have had particularly harmonious fourths and fifths, another had harmonious thirds and sixths, and other variations which gave slightly different personalities to the different keys, now all keys are exactly the same. A dimension of richness and complexity of the system is lost, which could be compared roughly to going from 3D vision to 2D vision, or losing the color from a painting. Before, the major and minor modes had several variants. Now there is only one type of each; what was a system with a differentiated terrain, of hills and valleys, where each key had its own specific topography, has now been rendered a flat plane. Bach clearly had a sense of these unique characters of the keys, and in their wider implications, as he composed the C prelude, which you've just heard, in C...and not in C#.
As a side note, it should be no surprise that equal tempering made its rise as the prevalent tuning of keyboards, with the rise of atonal music, which the great conductor, Wilhem Furtwaengler described as "walk[ing] through a dense forest. The strangest flowers and plants attract our attention by the side of the path. But we do not know where we are going, nor whence we have come. The listener is seized by a feeling of being lost, of being at the mercy of the forces of primeval existence." In atonal music, the basis of the composition is not ordered thoughts and expressions of the mind behind the notes, but the tones and sounds themselves become the subject. The atonalist therefore desires to eliminate a sense of connection to a home-base key, or a set of keys. In equal tempering, the landscape is flat, there is no topography, no landmarks in the terrain to tell you where you are, relative you where you've started.
So what is the solution, after this long journey? Maintaining all the pure intervals, and the characters of the keys greatly restricts the freedom of transposition and movement. Completely flattening the space with equal tempering sacrifices the harmoniousness of the intervals, and the variety of character. Countless other temperings have been invented since Kepler's time, each having its virtues, and its drawbacks, but none is a perfect solution—and in fact, none can be. Looking among the fractions and calculations of mathematics and harmonic ratios, per se, we end up chasing our tail. No fraction of a fraction perfectly fits the bill of both allowing a freedom of movement throughout the keys, while at the same time maintaining perfect harmonies. But does that mean that the system is flawed? Bach himself, as it is reported, was not attached to one or another set system of tempering as "the perfect" tempering. He started from certain temperings, most likely the ones proposed by Andreas Werckmeister, made his own adjustments to them, and did so, so that the keyboard was best suited to manifest his music. This he considered to be "well" tempered" not as the perfect system of tuning, but a particular concept of having the best substrate to manifest the musical idea—in having a keyboard whose voices could most closely imitate the tuning of a chorus of human voices, each guided by a human mind. The perfection is not in the system of notes, but in the idea, whose shadow is the notes. This places the concept of harmony, once again, on a higher plane than just sounds.
Where we're now left is at the doorstep of mind, which is exactly where Kepler, one-hundred years earlier than Bach, takes us for his solution. As we'll see, the question of tempering, and its solution is not only a musical problem. It's just as much a scientific problem, because it's an issue of the preeminence of mind, idea, over the organization of matter. Kepler doesn't resolve this question within Book III—so, we'll follow him first into Book IV, a direct exploration of mind and the senses, and then, in Book V, into the domain of astrophysics.
1. And what about an organist?!
Book III, Chapter Eight
Johann Phillip Kirnberger, On the Art of Strict Musical Composition