Medieval Music

In 2016, after 20 years of research, Cambridge University medieval music specialist Sam Barrett used the rediscovered leaf of an 11th-century manuscript to reconstruct music as it would have been heard a thousand years ago.

Melodies in those days were not recorded as precise pitches but relied on the memory of musicians and on aural traditions that died out in the 12th century. “We know the contours of the melodies and many details about how they were sung, but not the precise pitches that made up the tunes,” Barrett said. The missing leaf, appropriated by a Germanic scholar in 1840, contained vital neumes, or musical symbols, that allowed him and his colleagues to finish their reconstruction of Boethius’ “Songs of Consolation” as it was performed in the Middle Ages.

“There have been times while I’ve been working on this that I have thought I’m in the 11th century, when the music has been so close it was almost touchable,” Barrett said. “And it’s those moments that make the last 20 years of work so worthwhile.”

Secret Message

Jonathan Swift’s Journal to Stella, a collection of 65 letters written to his friend Esther Johnson, contains some puzzling passages, such as this one:

“He gave me al bsadnuk lboinlpl dfaonr ufainfbtoy dpionufnad, which I sent him again by Mr. Lewis.”

How should the obscured phrase be read?

Click for Answer

Image: Flickr

In painting backdrops for the dioramas at the Denver Museum of Nature and Science in the 1970s, artist Kent Pendleton hid eight elves. “It was just kind of my own little private joke,” he said in 2018. “The first one was so small that hardly anyone could see it, but it sort of escalated over time, I guess. Some of the museum volunteers picked up on it and it developed a life of its own.”

The museum’s field guide currently lists nine hidden finds, but there are more — the exact number is not known.


“The only way to keep ahead of the procession is to experiment. If you don’t, the other fellow will. When there’s no experimenting there’s no progress. Stop experimenting and you go backward. If anything goes wrong, experiment until you get to the very bottom of the trouble.” — Thomas Edison

Finger Numerals

Writing in the north of England in the early 8th century, the Venerable Bede described a Roman system of finger counting:

1 = the little finger bent at the middle joint
2 = the ring and little fingers bent at the middle joints
3 = the middle, ring, and little fingers bent at the middle joints
4 = the middle and ring fingers bent at the middle joints
5 = the middle finger only bent at the middle joint
6 = the ring finger bent at the middle joint
7 = the little finger closed on the palm
8 = the ring and little fingers closed on the palm
9 = the middle, ring, and little fingers closed on the palm
10 = the tip of the index finger touching the middle joint of the thumb
11 to 19 = the actions denoting each numeral from 1 to 9 plus that of 10
20 = the thumb tucked between the index and middle fingers, so that the thumbnail touches the middle joint of the index finger
21 to 29 = the actions denoting each numeral from 1 to 9 plus that of 20
30 = the tips of the thumb and index finger touching and forming a circle or ring
40 = the thumb and index finger standing erect and close together
50 = the thumb bent at both joints and held against the palm
60 = the index finger closed over the thumb
70 = the first joint of the index finger resting over the first joint of the thumb, which is held nearly straight
80 = the tip of the index finger resting on the first joint of the thumb
90 = the thumb bent over the first joint of the index finger

The signs for 100, 200, 300, and so on are the same as 10, 20, 30, but made by the right hand; and the signs for 1,000, 2,000, 3,000 and so on are the same as 1, 2, 3 but made by the right hand. “To add two numbers, one simply signed the first, then made the mental arithmetical calculation and reproduced the gesture corresponding with the correct sum,” writes Angus Trumble in The Finger: A Handbook (2010). “The process was cumulative; to add a further number to the sum of the first two, you proceeded to represent the gesture corresponding with the new total, and so on. Likewise, the task of subtraction merely threw the whole system into reverse. It was perfectly clear to anyone observing you carry out these separate procedures whether the job in hand was one of addition or subtraction.”

Trumble says that at the end of the 19th century Wallachian peasants were discovered to have preserved a few methods of digital multiplication and division that had been preserved throughout the Roman empire. Here’s one.

The Georgia Guidestones
Image: Wikimedia Commons

In Elbert County, Georgia, stands a granite monument bearing 10 guidelines inscribed in eight languages:

  1. Maintain humanity under 500,000,000 in perpetual balance with nature.
  2. Guide reproduction wisely — improving fitness and diversity.
  3. Unite humanity with a living new language.
  4. Rule passion — faith — tradition — and all things with tempered reason.
  5. Protect people and nations with fair laws and just courts.
  6. Let all nations rule internally resolving external disputes in a world court.
  7. Avoid petty laws and useless officials.
  8. Balance personal rights with social duties.
  9. Prize truth — beauty — love — seeking harmony with the infinite.
  10. Be not a cancer on the earth — Leave room for nature — Leave room for nature.

It was commissioned in 1979 by a man using the pseudonym Robert C. Christian, who seemed to have considerable resources. A ledger nearby explains that the structure operates as an astronomical calendar and that a time capsule lies 6 feet beneath.

It’s thought that the inscription lists the principles that will be needed to rebuild a devastated civilization. A tablet reads, “Let these be guidestones to an Age of Reason.”

“Settling a Bill”

Four sharpers having treated themselves to a sumptuous dinner at the Hotel Montreuil, were at a loss how to settle for it, and hit on the following plan: They called for the waiter and asked for the bill. One thrust his hand into his pocket as if to draw his purse; the second prevented him, declaring he would pay; the third did the same. The fourth forbade the waiter from taking any money from either of them, but all three persisted. At last one said: ‘The best way to decide is to blindfold the waiter, and whoever he first catches shall settle the bill.’ This proposal was accepted, and while the waiter was groping his way around the room they slipped out of the house one after another.

Western Literary Messenger, May 1854


Choose one of these cards and fix it clearly in your mind. Then open the answer box.

Click for Answer

Our System

“If x is the population of the United States and y is the degree of imbecility of the average American, then democracy is the theory that x × y is less than y.” — H.L. Mencken, A Mencken Chrestomathy, 1949

The Precarious Picture

Suppose you want to hang a picture by a string that’s attached at two points on the back of the frame. How can you arrange the string on two nails such that the picture will fall if either nail is removed?

One solution is above. I don’t know who first asked the question; I first saw it in Mathematical Mind-Benders, by Peter Winkler, who got it from Giulio Genovese, a mathematical graduate student at Dartmouth, who’d seen it in more than one source in Europe.

But it opens up a surprisingly rich discussion — see the paper below for some entertainingly complex variants.

(Erik D. Demaine et al., “Picture-Hanging Puzzles,” Theory of Computing Systems 54:4 [2014], 531-550.)