In The Republic, Plato emphasized the importance of studying mathematics, music, and astronomy. These subjects, he argued, begin with the study of physical things and draw the soul upward toward higher, immaterial things. However, he asserted, they were only worth studying if connections between them could be understood. This claim can be seen as prophetic in light of Divine Revelation and the advancements of modern science and engineering.

The biggest step forward in astronomy since Plato's time came with Isaac Newton. Unlike previous astronomers, Newton did not merely identify the geometric shapes that the stars, or planets, traced out; he explained why they moved. According to Newton, the speeds and directions of objects can only be changed by forces. The force of gravity pulls planets towards the sun. However, finding which path the planet takes presents a problem. To find where a planet will be at any time, you need to know how fast it moves. To know how fast it moves, you need to know how strong and in which direction the force pulls. To know the strength and direction of the force, you need to know where the planet would be at any time (the thing you were originally trying to find). Newton invented the branches of mathematics known as calculus and differential equations to solve these problems.

This same physics that can be applied to astronomy can also be applied to music, for example, in movement of a vibrating string. The number of times per second that the string moves back and forth, which is known as the frequency and recorded in units of hertz (Hz), measures how high pitched the sound will be. But determining what frequency a given string would have presents a problem similar to the problem in astronomy. To know the position of the string at any time, you need to know the forces at any time, and to know the forces, you need to know the position of the string. This problem can also be solved with Newton's differential equations.

This is best done by looking at the harmonics, the various waves that can be seen when one end of a taut string is shaken by a machine, while the other end is held fixed. In most cases it is hard to see the string move. However, if the machine has a certain precise frequency, known as the fundamental frequency, the differential equations prove that a very large vibration is present with the center of the string moving up and down the farthest (the first harmonic). This phenomenon explains why singers can hit notes that shatter glass and why the gargantuan Tacoma Narrows Bridge was blown down by winds less than 45 mph.

Even more stunningly, if the frequency of the machine is exactly twice the fundamental frequency, the differential equations reveal that the string will move in a wave, called the second harmonic, as shown in the figure below. Even though nothing is holding the center, it remains motionless, while both sides vibrate up and down dramatically. If the frequency of the machine is exactly three times the fundamental frequency, you will find two motionless nodes as shown below (the third harmonic). Frequencies that are exactly four times, five times, etc., all yield similar waves, but if the machine is even slightly off, no motion will be visible.

Of course, typical vocal cords or guitar strings do not have machines attached to them. However, a mathematician named Joseph Fourier showed that when a string is plucked, it will produce exactly the same sound as if all the harmonics were played at the same time. The string will vibrate back and forth in a combination of waves at different frequencies, which get transferred by the air to organs in our ears called cochleae. These organs are stiff enough that they do not usually vibrate in response to waves, but if a part of the cochlea has a fundamental frequency that is the same as the frequency of the wave that entered, it will undergo a large vibration, sending a signal to the brain. The brain responds to these signals by producing the sensation of a high- or low-pitched noise depending on what part of the cochlea the signal came from.

If a string with a fundamental frequency of 100 Hz is plucked, the wave heard will be in a combination of the first harmonic at 100 Hz, the second harmonic at 200 Hz, the third at 300 Hz, and so on. The parts of the cochlea corresponding to those frequencies will vibrate. A shorter string might have frequencies of 150 Hz, 300 Hz, 450 Hz, etc., each of which would also cause vibrations in the cochlea. If these two strings are both plucked, either simultaneously or in short succession, some of the frequencies in the first series will also be in the second: 300 Hz, 600 Hz, 900 Hz, etc. On the other hand, if a string had frequencies 110 Hz, 220 Hz, etc., it would not overlap much with either of the two previous ones, and the parts of the cochlea that vibrated in response would be almost entirely different.

It is quite a mechanical challenge for our brains to process information. When seemingly incomprehensible data becomes understandable, our brains respond with delight. This is what happens when there is overlap between frequencies of waves. Of course, the sound is not processed consciously, and the delight takes the form of a sense of beauty. When two notes do not have many overlapping frequencies, the brain must process more information and, in response, it produces a feeling of dissonance.

Remarkably, if you take any frequency, and multiply it by 2^(1/12) to get a new frequency, multiply that number by 2^(1/12) and so on, you will get a collection of frequencies that have a lot of overlap. This collection is what we call musical notes. These notes, which one might see depicted for instance in a hymnal at Mass, make up all music. The beauty in these hymns or in any other music is written into the biology of the human ear and the structure of numbers themselves.

When Plato wrote about connecting mathematics, astronomy, and music, no one knew that scientists and engineers would one day develop a mathematical theory that could explain the motions of celestial bodies and musical notes. However, Plato could reason to the conclusion that there is a unity in all higher things. In light of Divine Revelation, we can see that the unity that connects astronomy, music, and all other studies of the material world flows from the unity of the Creator's purpose. This order in the physical universe was further elevated at the Annunciation, when the second Person of the almighty Trinity submitted to the same physical laws that govern stars and strings.