The science, mathematics, and philosophy of rhythm

Zebra finches use a "critic" in the brain to differentiate between the rhythm of songs of other birds and, through this, learn songs.
Like the ebb and flow of the ocean, 

A rhythm emerges from the pen,

I capture it, imagine it,

before it disappears. 

Appropriate rhythm in writing means making sense of the relation between words and phrases. Stress, repetition, fluctuation, rhyme, meter, pattern, juxtaposition, and harmony all come together in the way aesthetic and intellectual properties of rhythm. For a philosopher examining the roots of semantics and language or neuroscientist uncovering our true nature, rhythm eludes us. I've written a bit on the subject with respect to symmetry. Let's delve into rhythm's secrets philosophically, mathematically, and scientifically.

Through much of my scientific writing, I pay close attention to how lengthy my sentences are. Too many long sentences at once can lose the reader in monotonous, cumbersome passages. I especially fall into this trap frequently with description and exposition. As I prize so many details and methods of understanding an argument through my writing, it can be difficult for me to determine how much is too much. On the other hand, short, astute sentences can cause the reader to feel the writing is abrasive, clumsy, or over-simplified. Even as we alternate between different types of writing, such as the brevity of Twitter to the nuance of academic prose, it's easy to succumb to habits and forget about the appropriate rhythm with which to write. For these reasons, rhythm is both something we plan in advance while we begin writing, but constantly re-evaluate through reflection and speculation.

In the realm of aesthetics, philosophers have debated the role of rhythm since the Classical era. In Book III of Plato's The Republic, Socrates clarified that rhythm and meter are what separate poetry from pure prose. Pre-Medieval philosopher St. Augustine developed a theory of aesthetics based on ideas of rhythm in De Musica. In congruence with the theologian's religious beliefs, God is the origin of rhythm. We discover these mathematical truths, pre-determined by God, of rhythm, much the same way Plato believed humans collectively remembered them.

Emerson's poem "Merlin" showed the use of rhyme and meter to create rhythm, particularly between lines that moved back and forth between his own sensations and the way to craft meters of poetry from them. Emerson specifically, in this section, showed how the rhyme fit so naturally that it seems like part of human prose. This similarity to Socrates' idea of rhyme and meter separating prose from poetry  allows Emerson to create a deeper meaning of nature through distilled rhythm.

Thy trivial harp will never please

Or fill my craving ear;

Its chords should ring as blows the breeze,

Free, peremptory, clear.

No jingling serenader’s art,

Nor tinkle of piano strings,

Can make the wild blood start

In its mystic springs.

We teach ourselves rules and tips on creating great writing to take into account the effect of the rhythm on the piece. To imagine and care for these aspects of the reader gives the writing an intrinsic property of the writing itself that is only observed at a scale larger than individual words. Rhythm comes from how words interact with each other, yet remains limited by our conventions of writing. It emerges when you take a step back from your computer screen and look at the whole picture. Rhythm itself can be seen as the manifestation of the observable quantity of periodicity or frequency. The conventions that limit it would be our perceptions of reading and understanding writing united through the passage of time. As such, it reveals deeper features of our subjective perceptions, such as the stress, intonation, and tempo of speech itself. We can understand linguistic patterns such as Rhythm elusively hides as a feature composite of different parts of writing interacting with one another (such as between the first half and the second half of a sentence). Yet we speak of it as something deeper than the combined intrinsic content of words themselves. In this sense, it's similar to an emergent phenomena, much like evolution selecting certain genetic traits over others in a population. 


What makes these six clave patterns fundamental are that they reveal maximally even rhythms and maximum sum of pairwise distances between the points as vertices on a tetrahedron.
In a scientific context, we rely on our empirical observations of rhythm to determine higher truths of rhythm.  Canadian computer scientist Godfried Toussaint concludes that, due to similarities between the shape and geometry and musical rhythm. By appealing to measures of evenness between six clave patterns (Shiko, Son, Soukous, Rumba, Bossa, Gahru), Toussaint argues that the similarities between these patterns and findings in mathematics show that the clave is significant mathematically. That a mathematical algorithm such as Euclidean distance could generate music raises questions such as what sort of mathematical or empirical technique governs what "good" rhythm sounds like.


Placing the six intervals in histogram form reveal patterns among themselves that may dictate the nature of rhythm as a whole.
In computer science, we can find fundamental features among strings of numbers such that the patterns of these features give rise to Euclidean rhythms (dictated by the Euclidean distance, or the straight line distance between the points when arranged on vertices of a circle) of music. In these cases, the numbers dictate the span between the beginning of successive notes and, thus, represent the simplest way to represent rhythm. Such a fundamental mathematical discovery requires much more empirical evidence before showings its truth, however, in all of music. Toussaint's claim that Euclidean rhythms that are reverse Euclidean strings appear to have a much wider appeal. French mathematician Jean-Paul Allouche showed these Euclidean strings of numbers share similarities with to combinatoric (or the combinations gievn by) sequences of words. Other mathematicians and computer scientists have developed algorithms to construct Euclidean rhythms from Euclidean strings. Toussaint argues that the Euclidean algorithm (that finds the greatest common divisor of two numbers) can generate rhythm timelines by using the two numbers as an input to the Euclidean algorithm. The two numbers would dictate the beginning of each note in the rhythm and the span between notes. 

In the field of cognitive neuroscience, we can study the ways humans and other organisms produce and evaluate rhythms. Using zebra finches as model organisms, computational neuroscientists Kenji Doya and Terrence J. Sejnowski discovered a "critic" within the brain of a zebra finch differentiates between songs. Then, upon similarity between the songs between birds, NMDA receptors (a glutamate receptor in nerve cells) are activated that allow the bird to learn the "correct" song. Scientists Philipp Norton and Constance Scharff also found patterns between the nerve and muscle cells that corresponded with elements of notes and duration of the notes themselves. This corresponds to Toussaint's study of the beginning of each note and the span between them, both fundamental components of rhythm. This research holds value for finding similar discoveries in the human basis of rhythm as well.

Much the same way a poet translates nature into word, a scientist would find quantifiable metrics of music that can raise questions for musicology, geometry, and, with sufficient empirical evidence, neuroscience. Now take a deep breath and let the waves beat upon the seashore. Detect the rhythm and move along like before. 

Sources

Doya, Kenji & Terrence J. Sejnowski (1999). The New Cognitive Neurosciences. II. MIT Press. pp. 469–482.

G. T. Toussaint, "The Euclidean algorithm generates traditional musical rhythms"Proceedings of BRIDGES: Mathematical Connections in Art, Music, and Science, Banff, Alberta, Canada, July 31 to August 3, 2005, pp. 47–56.

Norton, Philipp & Constance Scharff (2016). “Bird Song Metronomics”: Isochronous Organization of Zebra Finch Song Rhythm. Frontiers in Neuroscience.