When Testing Questioned Philosophy

In the first post in this series, I ended by focusing a bit on Galileo who started to make the idea of testing what it eventually would be recognized as today. That’s the same thing as saying Galileo effectively produced one of the first attempts to make science as it is known today. Let’s continue this path of investigation.

Folks with some historical background may wonder why I start with Galileo here. Consider this, from David Wootton’s The Invention of Science:

Modern science was invented between 1572, when Tycho Brahe saw a nova, or new star, and 1704, when Newton published his Opticks, which demonstrated that white light is made up of light of all the colours of the rainbow, that you can split it into its component colours with a prism, and that colour inheres in light, not in objects.

This is definitely true and I think Galileo — born in 1564, died in 1642 — fits well within that “between” range. Wootton continues:

There were systems of knowledge we call ‘sciences’ before 1572, but the only one which functioned remotely like a modern science, in that it had sophisticated theories based on a substantial body of evidence and could make reliable predictions, was astronomy, and it was astronomy that was transformed in the years after 1572 into the first true science.

Indeed also true. In these posts, I’m framing things around Galileo and he was actually not all that concerned about astronomy at all. Galileo was much more concerned about motion and forces, although his interests and investigations ultimately aligned with those who were concerned more with astronomy. One more bit from Wootton:

What made astronomy in the years after 1572 a science? It had a research programme, a community of experts, and it was prepared to question every long-established certainty (that there can be no change in the heavens, that all movement in the heavens is circular, that the heavens consist of crystalline spheres) in the light of new evidence.

Galileo was also very willing to question established certainties and orthodoxies but also to provide a basis for why it made sense to do so. It’s interesting to consider how this came about because I believe it showcases the evolution of a test-focused thinker.

The Philosophy of Music

In 1558, a guy named Gioseffo Zarlino published a book called Le istitutioni harmoniche (“The Principles of Harmony”). Zarlino was very into the whole idea of Pythagorean musical theory. This theory can probably be stated most simplistically by saying that harmonies were inflexibly controlled by mathematical proportions.

Essentially the original Pythagorean system of harmony had three basic musical intervals. Each of these was defined by the numerical ratio between anything that could produce those notes, such as lengths of string. So there was the octave, a 2:1 ratio. There was the perfect fifth, a 3:2 ratio. There was the perfect fourth, a 4:3 ratio.

Music theory during this period relied on theorems drawn from geometry, hence the reliance on mathematical proportions.

Thus a combination of a fifth and a fourth produces an octave: 3:2 × 4:3 = 12:6 = 2:1. The difference between a fourth and fifth is 3:2 divided by 4:3, which is 9:8. That’s called a whole tone interval.

Zarlino embraced the Pythagorean tradition of what’s called diatonic tuning.

This can get really complicated fast but making it simple: Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. What results is the seven tones of the diatonic scale: C, G, F, D, A, E and B. These comprise what’s called a major scale in Pythagorean tuning. The above image shows one way of forming the diatonic scale using pure fifths.

What does any of this have to do with Galileo or his forerunner attempts at testing?

The relevance here is that Galileo’s father, Vincenzo, studied with Zarlino. So Vincenzo initially was tutored in the dogma of musical theory based on the Pythagorean ideas of a rigid mathematical system. This was, in essence, a philosophy — really, an orthodoxy — of music theory.

But then Vincenzo came upon a philologist named Girolamo Mei. Mei introduced Vincenzo to the work of a guy named Aristoxenus, who was a Greek music theorist back in the fourth century BCE. Aristoxenus insisted math had little to do with music. Instead, according to Aristoxenus, you should rely on your own senses to decide what music was most aesthetically pleasing.

That view, obviously, conflicted with the Pythagorean. As these views of music theory conflicted over the course of time, the cornerstone for those who accepted the mathematical route was a debate over finding the best mathematical ratios of the lengths of strings producing what are called “consonances.” These are sounds, like the octave, that are said to be most pleasing to the ear.

So was it math or more of a situational thing?

Well, Mei, very much a believer in the views of Aristoxenus, pointed out to Vincenzo that the “equal temperament” approach adopted by practicing musicians — those people alive and working at the time; actual practitioners — to tune their instruments wasn’t consistent with the Pythagorean doctrine favored by theorists, which specified precise ratios for the intervals.

Consider again our image but with a slight addition:

Comparing the diatonic scale with an equal-tempered scale, you could see that the difference between the tone positions increases by what’s called two cents at each fifth. A cent is a unit of measure for the ratio between two frequencies.

So if the entire chroma — twelve consecutive fifths — was tuned using only fifths, the difference would become 1/8 step which is approximately 24 C. Even to an untrained ear, this would not really be a pleasing sound. In other words, the mathematical system seemed to have a discrepancy.

This discrepancy is called the Pythagorean comma, and it was one of the primary reasons why other tuning systems, like temperaments, were studied in the medieval ages and why some doubted the orthodoxy.

Understand the Discrepancy

If there are discrepancies, you often have to figure out what they actually are. This is how you know something is, in fact, a discrepancy.

So let’s say you start from a frequency of 100 Hz. Then after seven octave jumps (multiplication by 2) you reach 12800 Hz. Likewise if you start from the same place, after twelve fifth jumps (multiplication by 3/2) you reach 12974.634 Hz. The numbers 12800 and 12974.634 are very close and the difference between them is very small. But this illustrates the Pythagorean comma. The question might be: but what is it actually illustrating?

This illustrates the idea of trying to tune an instrument by ear, using the alleged mathematical purity of intervals as a guide. While the above talks about seven octaves and twelve fifths, any notes you reach can also be brought down by one or more octaves as well.

To illustrate this, you can bring all the notes down into the same octave. For the sake of simplicity, let’s use that whole tone I mentioned earlier, which you can get by ascending two perfect fifths, then descending an octave — for example, from C to D). In terms of ratios, this is equivalent to multiplying a frequency by:

(3/2) × (3/2) × (1/2) = (9/8)

The cents I mentioned earlier are actually logarithmic units and thus the same formula can be expressed as an addition:

702 cents + 702 cents – 1200 cents = ~204 cents

Let’s look at the line of fifths. This is more so you can see how the notes are named if music theory isn’t your thing.

Notice how each successive fifth must be named with the letter that’s five letters “higher.” Note for music purists: this is the case even if a given fifth has to be modified via a sharp or flat.

Now you can start to create a scale from whole tones. Since each whole tone is two fifths away, you would be using the note name two places to the right along the line of fifths. Which always will use the next letter in the alphabetic sequence. We don’t need to calculate any actual frequencies here. Instead we’ll just calculate the frequency ratio and number of cents difference between each note and the starting note.

  • C = (9/8)0 = +0 cents
  • D = (9/8)1 = +204 cents
  • E = (9/8)2 = +408 cents
  • F# = (9/8)3 = +612 cents
  • G# = (9/8)4 = +816 cents
  • A# = (9/8)5 = +1020 cents
  • B# = (9/8)6 = +1224 cents

This is very approximate in some ways but you can see that the “one-octave” scale actually overshot an octave by one Pythagorean comma. There’s a distinct difference between the sound of a B# and a C and if you try to substitute one for another you will end up with something that sounds clearly out of tune.

The Testing of Music

Obviously I wasn’t there when Mei and Vincenzo were working with each other but it’s likely that discrepancies like this were pretty much front and center of Mei’s thinking.

Importantly, however, Mei didn’t just ask Vincenzo to believe. Instead, Vincenzo was encouraged to test all this for himself. The prevailing assumption at the time was that, just as the ratio of lengths of two identical strings with the same tension and mass per unit length, tuned an octave apart, would be 2:1, the ratio of the tensions of two identical strings of equal length, tuned an octave apart, would be 2:1 as well.

Hey, that’s testable!

Testing Led to Observations

In one test, Vincenzo used two different lutes, one tuned to the requirements of equal temperament, and the other tuned per the dictates of Zarlino and thus the Pythagorean orthodoxy. Vincenzo also tested this assumption with a simple experiment involving hanging weights from strings and what he found was that, in fact, the ratio of tensions was 4:1. This provided convincing evidence that, indeed, consonant sounds were not determined solely by abstract mathematical ratios.

Likewise, there were other discrepancies in the established system. For example, sticking with the whole tones I’ve been using, an interval of two whole tones is 9:8 × 9:8 = 81:64, which is almost but not quite the interval that we now call the major third, 5:4 = 80:64. That’s clearly a small difference but, as with the example above, one that trained musicians and those with sensitive hearing could most definitely distinguish.

Observations Led to Assessments

Thus Vincenzo Galilei came to the conclusion that what musicians did in practice was not what Zarlino said they did, or should do. Vincenzo argued that musicians, by ear, adopted a scale in which each of the notes of an octave was separated from the next by the same interval — which meant that none of the intervals were perfect according to Pythagorean accounting.

In 1581, Vincenzo published Dialogo della musica antica et della moderna (“Dialogue on Ancient and Modern Music”), which explicitly countered the ideas of Zarlino and the Pythagorean orthodoxy. Zarlino, like the Pythagoreans whose message he believed in, essentially were saying it was the philosophy that mattered; it was the actual mathematics that came out of that philosophy. For Zarlino and like-minder thinkers, music was not the reality of lived experience.

Historically, it seems at least likely that Vincenzo continued and refined these experiments around 1588, which is the exact time when Galileo was living at home and tutoring local students in mathematics. Historians of science consider it likely that Galileo may have helped his father with the experiments. Thus Vincenzo influenced his son to pursue pragmatic experimentation in his science as a means of testing hypotheses.

Questioning the Philosophy

Let’s back up a bit here.

Aristoxenus, incidentally, was roundly criticized by Socrates, someone whom we mainly know of only through Plato. The view of Socrates, and thus likely Plato, was that performers who tuned their instruments by practice rather than according to the principles of philosophy were people who “prefer their ears to their intelligence.” It literally was a case that you should distrust what your ears tell you is pleasing or “correct” — or that has the right quality — and instead conform to the philosophy.

The Pythagoreans get a little bit of bad press in all this but, interestingly, Plato criticized Pythagoras for being too eager to understand the empirical practices of working musicians rather than focusing strictly on the elementary numerical ratios. So far as Plato was concerned, those ratios were the only way to truly understand harmony as the Creator intended it. In short, it was Plato — and not Pythagoras — who really pushed the point of reducing musical harmonies to simple numbers.

Galileo, around the age of twenty, and thus in 1584 or 1585 and after the publication of his father’s book, is thought to have participated with his father in experiments on vibrating strings that demonstrated the inadequacy of Zarlino’s teachings on harmony. As stated earlier, this may have also been occurring in 1588 as well when Galileo was teaching mathematics.

Early in his life, therefore, Galileo became aware of an incontrovertible case in which the desire of Plato and others to make reality conform to idealized mathematics really just didn’t work. It could serve as a crude approximation to get people going. But once said people were going, they tended to abandon the philosophy and do something that more comported with actual observation.

Galileo learned that if experiments, as well as the real-life habits of musicians, were at odds with whatever some philosophy said, then you had demonstrable and empirical evidence that the philosophy was wrong. (Or, again, at best approximate.)

Galileo had to move beyond Plato and Platonic style thinking but, more importantly, he had to move beyond just trusting a non-empirical philosophy and beyond blindly following an orthodoxy.

Thus were two key traits of an experimentalist, and thus a tester, set.


This article was written by Jeff Nyman

Anything I put here is an approximation of the truth. You're getting a particular view of myself ... and it's the view I'm choosing to present to you. If you've never met me before in person, please realize I'm not the same in person as I am in writing. That's because I can only put part of myself down into words. If you have met me before in person then I'd ask you to consider that the view you've formed that way and the view you come to by reading what I say here may, in fact, both be true. I'd advise that you not automatically discard either viewpoint when they conflict or accept either as truth when they agree.

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