[Math Lair] The Mathematics of the Lost Chord

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Lost chord postcard by Bamforth, ca. 1910. Lost Chord post card, from Bamforth (ca. 1910)

"The Mathematics of the Lost Chord", by Stephen Leacock (1869–1944) can be found in his book My Remarkable Uncle.


Every one is familiar with the melodious yet melancholy song of "The Lost Chord." It tells how, seated one day at the organ, weary, alone and sad, the player let his fingers roam idly over the keys, when suddenly, strangely, he "struck one chord which echoed like the sound of a Great Amen."

But he could never find it again. And ever since then there has gone up from myriad pianos the mournful laments for the Lost Chord. Ever since then, and this happened eighty years ago, wandering fingers search for the Lost Chord. No musician can ever find it.

But the trouble with musicians is that they are too dreamy, too unsystematic. Of course they could never find the Lost Chord by letting their fingers idly roam over the keys. What is needed is method, such as is used in mathematics every day. So where the musician fails let the mathematician try. He'll find it. It's only a matter of time.

The mathematician's method is perfectly simple—a matter of what he calls Permutations and Combinations—in other words, try out all the Combinations till you get the right one.

He proposes to sound all the Combinations that there are, listen to them, and see which is the Great Amen. Of course a lot of the combinations are not chords at all. They would agonize a musician. But the mathematician won't notice any difference. In fact the only one he would recognize is Amen itself, because it's the one when you leave church.

He first calculates how many chords he can strike in a given time. Allowing that for striking the chord, listening to it and letting it die away, he estimates that he can strike one every 15 seconds, or 4 to a minute, 240 to an hour. Working 7 hours a day with Sundays off and a half day off on Saturday, and a short vacation (at a summer school in Mathematics), he reaches the encouraging conclusion that if need be—if he didn't find the Chord sooner—he could sound as many as half a million chords within a single year!

The next question is how many combinations there are to strike. The mournful piano player would have sat strumming away for ever and never have thought that out. But it's not hard to calculate. A piano has 52 white notes and 36 black. The player can make a combination by striking 10 at a time (with all his fingers and thumbs), or any less number down to 2 at a time. Moreover he can, if a trained player, strike any 10, adjacent or distant. Even if he has to strike notes at the extreme left and in the middle and at the extreme right all in the same combination, he does it by rapidly sweeping his left hand towards the right, or his right towards his left. There is a minute fraction between the initial strokes of certain notes, but not enough to prevent them sounding together as a combination.

This makes the calculation simplicity itself. It merely means calculating the total combinations of 88 things, taken 2 at a time, 3 at a time and so on up to 10 at a time.
The combinations, 2 notes at a time 
Are
88×87
1×2
.............................................
3,828
For   3 at a time
88×87×86
1×2×3
......................
109,736
For   4 at a time
88×87×86×85
1×2×3×4
...................
2,331,890
For   5 at a time
88×87×86×85×84
1×2×3×4×5
................
39,175,750
For   6 at a time
88×87×86×85×84×83
1×2×3×4×5×6
.............
541,931,236
For   7 at a time
88×87×86×85×84×83×82
1×2×3×4×5×6×7
..........
6,348,337,336
For   8 at a time
88×87×86×85×84×83×82×81
1×2×3×4×5×6×7×8
.......
64,276,915,527
For   9 at a time
88×87×86×85×84×83×82×81×80
1×2×3×4×5×6×7×8×9
....
571,350,360,240
For 10 at a time
88×87×86×85×84×83×82×81×80×79
1×2×3×4×5×6×7×8×9×10
.
4,513,667,845,896
      For all combinations ....................5,156,227,011,439

This gives us then an honest straightforward basis on which to start the search. The player setting out at his conscientious pace of half a million a year has the consoling feeling that he may find the Great Amen first shot, and at any rate he's certain to find it in 10,000,000 years.

It's a pity that the disconsolate players were so easily discouraged. The song was only written eighty years ago; they've hardly begun. Keep on, boys.


Also by Stephen Leacock are Boarding-House Geometry, The Force of Statistics, and Mathematics for Golfers.

More mathematics humour.

This doesn't have anything to do with Stephen Leacock or mathematics or anything like that, but if you liked the "Lost Chord" postcard at the top of the page, here is another one, this one from 1907:

Lost Chord postcard

If you like postcards, see also postcards and estimation. There is also a picture of a postcard at The Expanding Bridge page.