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.

Book III, Chapter VIII

As seen in the previous pedagogical on the double octave, we have a rather troublesome problem: as soon as we tried to extend our system beyond one octave (as may be useful in choral singing, for example), our intervals began to get out of tune with what came before! In fact, this problem isn't merely one of academia, but must have arisen in the development of human culture, when humans began to accompany the singing voice with a non-living instrument. While pitches which the voice sings are flexible and not fixed to one particular tuning, a musician, such as a lute player, must fix the tuning of his strings. So what could be the cause of the obstacle with which we are dealing?

Aha! Here's the problem! Rather than a nice, neat musical building block (or unit half step) of which all of the intervals are composed, we instead derived three different sized half steps (semitone, diesis, limma), incommensurable with one another, which, when compiled into a single system began to give us commas left and right! If we're going to have to have a practicable musical system, we're going to have to resolve this problem, and find a unit interval which will eliminate the discrepancy.

For example, the interval of the major third (4/5 of the string), is made up of four of these different half steps. If instead of this troublesome variety, I could find a half-step of sorts which would fit exactly four times into 4/5, then the problem we encountered when crossing from one octave to the next (“F” to “a” in the previous pedagogical) would be eliminated, because each interval would be perfectly commensurable with every other. Let's see if any of the three that we already have will do. Begin with 15/16 (the semitone). If we take 15/16 four times, we would get 50625/65536, or about .77, an interval larger than the .8 (4/5) we were looking for. Let's take our smallest half step, the diesis. 24/25 taken 4 times give us .85, an interval much smaller than what we need! Well, maybe our mid-sized half step, the limma (128/135) will do it. But even that yields .81, missing the mark by 80/81.

Lucky for us, Vincenzo Galilei around 1589 discovered the perfect half step, which evens out the this nasty little discrepancy (click here for more on Galilei). He took neither the semitone, nor the limma, nor anything else generated from geometry, but settled on the ratio that does the job: 17/18. If we take this interval four times, we get .797, something which is only 3/1000 off! That's well within “observational error”, so to speak.

So how would this work with our string? Galilei simply took his original string, divided it into 18 parts, and plucking a length which is 17/18ths of the string produces a'perfect' half-step. Then, take this new length, divide it into 18 parts, and take 17 of that, and you have the next half step, which proportion is exactly equal to the one before, giving us an equal interval. Perfect! The best news is that if you do this 12 times, you arrive at a string length for your octave which differs imperceptibly from ½! For instance, if my string were 100,000 units long, my 12th division of 17/18th would produce a length of 50,363, which is imperceptibly different from 50,000. Listen to the two examples in the animation: one is the “Kepler” tuning, and one is the “Galilei” tuning. See if you can tell which is which.

Before proceeding, the readers should indeed perform the divisions for themselves on a monochord or stringed instrument, and after they're done, compare each tone with the tone which was generated from Kepler's perfect geometric divisions.

Only with careful listening can a difference be detected, the greatest in fact being the D, or 2/3 division, and certainly without the other tones for comparison, no difference could be noticed at all.

The diagram (from pg. 197 of the English edition) above shows the string lengths for each of the notes of both hard and soft melody, beginning with a whole string of 100,000 units. The column on the left is Galilei's system of equal tempering, and on the right is what Kepler derived from the constructible figures.

So now the question is: if this new, equal half step makes every musical interval perfectly commensurable with every other, ironing out the discrepancy we saw in the previous pedagogical, and it doesn't seem too far off of what we genereated before, why doesn't Kepler praise Galilei for solving our problem? Well, what was Kepler seeking in his study of harmony? If you think back to the beginning of Book III, we called a division of our string a harmonic mean because it was a particular singularity on the string where each part resulting from the division was consonant both with the whole string, and with each other. In other words, though we had three components, the harmonic division formed a kind of unity, a proportion which was completely consonant with itself. For example, taking your monochord, place the bridge at the 2/3 division (or the note D, if the whole string were G). If you play first 2/3 of the string, and then 1/3 you get two tones which form consonances with the whole string, G, and are also in a perfect octave relationship with each other (Click Here:
).

What has Galilei produced in his system, which our ears seem to accept? Taking the “Galilei Divisions” of your string, place the bridge of the monochord at Galilei's “D”. Now put him to the test, and see what relationship exists between the two parts of the division (Click Here:
)
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What has now become of the principle of harmony? Galilei has produced a system which fooled our ears when a very simple melody, a scale, was played, but he was quickly exposed in our experiment as having destroyed the principle which was the cause of the harmonic divisions in the first place. In fact, this points to the real issue of epistomology which underlies Kepler's entire life's work: while Kepler saw what he was hearing as something which could give him insight into how the Creator composed the universe, others (like Galilei) took the sounds of the notes at their mere face value. Now that Galilei's fraud has been exposed, listen to the example to hear what happens as we continue this equal half step up to two octaves (Click Here:
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You can hear what limitations exist in Galilei's system, as he attempted to smooth over the cracks which arise from a lawful process of trying to construct a musical system. As we will see in Book V, these various “cracks”, or commas become a necessary part of the Creator's celestial composition. It is only by discarding the thinking of Galilei, as J.S. Bach does, that man is able to imitate the compositional method of his Creator.