Boethius

There’s more to Boethius (480–524 AD) than I knew! Sure, he wrote On the Consolation of Philosophy while imprisoned, later to be executed. And sure, it contains worthy stoic chestnuts like

Nothing is miserable unless you think it is so; and on the other hand, nothing brings happiness unless you are content with it.

But how did he get there? In fact his story is quite dramatic. He was born in Rome a few years after the collapse of the Western Roman Empire. After mastering Latin and Greek in his youth, he rose to prominence as a statesman during the Ostrogothic Kingdom, becoming a senator by age 25, and later a personal advisor to the king, Theodoric the Great.

He tried to translate all the Greek classics into Latin. Though his project was unfinished, it helped the works of Aristotle survive in the West. This is probably the most important thing he did.

Alas, he became very unpopular among members of the Ostrogothic court after he denounced their corruption. He was imprisoned by Theodoric in 523, and tortured and executed a year later.

But here’s what I hadn’t known: earlier he wrote about math and music, including the math of tuning systems!

Perhaps most importantly, he invented the system of using letters for notes. He started with the lowest note he cared about and called it A. Then came B, C, D, E, F, G, H, I, K… yes, this was before the letter J was invented! Much later, this system got changed to the one we’re familiar with. But if you look at a standard 88-key grand piano you’ll see the lowest note is still A.

He also discovered some sophisticated concepts in tuning theory. I recently discussed the Pythagorean comma, a glitch that shows up in the Pythagorean tuning system:

I’ve also discussed the syntonic comma, a glitch that shows up in just intonation:

These glitches are close but not equal. Some tuning systems try to exploit this fact, using one of these commas where you should really use the other. But this gives rise to a kind of meta-glitch: a glitch between glitches! It’s almost undetectable, since the ratio of the Pythagorean and syntonic commas is

But it’s there nonetheless.

It seems that Boethius was the one who first thought about this meta-glitch. Apparently he discussed it in the third book of his De Institutione Musica, and even gave it a name: the schisma. It’s so cool to imagine an advisor to a Gothic king doing this fancy math.

(Wikipedia claims that Boethius also discovered another musical fraction, the diaschisma, but that this was named much later by the German physicist and mathematician Helmholtz. Part of the fun of music theory is that it brings together famous figures from very different eras.)

I also just learned that most of On the Consolation of Philosophy was set to music in the Middle Ages! The melodies were considered lost because their notation relied on now-forgotten oral traditions. But Sam Barrett at Cambridge has tried to reconstruct them—and the ensemble Sequentia, who put out an amazing complete works of Hildegard von Bingen, performed his versions in 2016. You can learn more about this here:

Later, in 2018, they released an album of this music called Boethius: Songs of Consolation, which you can listen to here.

Three days ago I posed a puzzle on Mastodon: can you figure out what’s going on in this picture from his De Arithmetica?

This version of the picture was modernized by Martin Kullman. You can see the whole book by Boethius here.

The only really good answer to my puzzle came from David Egolf who wrote:

I don’t have a Mastodon account, but I wanted to respond to what you posted about there. You mentioned a somewhat mysterious figure from a book by Boethius.

I was able to read the version of the book here:

• Gottfried Friedlein, Anicii Manlii Torquati Severini Boetii De institutione arithmetica libri duo, De institutione musica libri quinque, 1828.

To do this, I took screenshots of the text, converted them to “copy-pastable” text using https://www.imagetotext.info/ and then translated the text to English using chatGPT.

I believe the square is illustrating some simple properties of certain arithmetic and geometric sequences. Namely, if you take two terms a(n) and a(m) of an arithmetic sequence, then their sum a(n)+a(m) is equal to a(n-k) + a(m+k). Similarly, if you take two terms g(n) and g(m) of a geometric sequence, then their product g(n)g(m) is equal to g(n+k)g(n-k).

Each column of this square is an arithmetic sequence, and each row of the square is a geometric sequence. By the way, the reason Boethius is talking about this is because he is interested in studying “even-ness” and “odd-ness”. Here, he is interested in numbers that are “somewhat even” but not “minimally” or “maximally” even. These are the numbers that have at least two factors of 2, and have at least one prime factor besides 2. The first column is the “least even” numbers satisfying these criteria – the odd multiples of 4 (skipping 4×1). The entries in the second column from the left have three factors of 2, and so its entries are the odd multiples of 8 (skipping 8×1). The “even-ness” continues to increase as one moves to the right.

The “arcs” on the outside of the square illustrate the property of arithmetic and geometric sequences I mentioned above. I first give some examples relating to geometric sequences. For example, 12 · 96 = 1152 = 24 · 48. We also have that 24 · 96 = 48 · 48 = 2304. Similarly, moving to the second row (which corresponds to the “inner arcs” on the top of the drawing), we have 20 · 160 = 3200 = 40 · 80. We also have 40 · 160 = 6400 = 80 · 80.

On the left of the square, we have some examples of the above
mentioned property of arithmetic sequences. On the leftmost arcs, we have some examples illustrating properties of the arithmetic sequence in the leftmost column. For example, 20 + 36 = 56 = 28 + 28 and 12 + 36 = 20 + 28 = 48. Moving to the “right-most arcs” on the left side, these now correspond to properties of the arithmetic sequence in the second column from the left. For example, 112 = 40 + 72 = 56 + 56 and 96 = 40 + 56 = 24 + 72.

After having had all the fun of working from the Latin, I now see that there is a nice presentation of the key ideas in English here:

• Dorothy V. Schrader, DE ARITHMETICA, Book I, of Boethius, The Mathematics Teacher 61 (1968), 615–628.