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A scholarly critique of Stephen Leacock’s The Mathematical Problem of the Lost Chord

A scholarly critique of Stephen Leacock’s The Mathematical Problem of the Lost Chord

– a tongue‐in‐cheek essay

“Seated one day at the organ…” begins Arthur Sullivan’s song The Lost Chord. The protagonist tells how he “struck one chord of music like the sound of a great Amen”. Sadly he has so little musical talent that he loses it without trace: “I have sought, but I seek it vainly.” Nor has he much understanding of his instrument, or he would have recognised that the sound comes from the organ pipes, not “from the soul of the organ” nor from its “noisy keys”. He despairs, “It may be that only in Heav’n I shall hear that grand Amen.”

In his Literary Study The Mathematical Problem of the Lost Chord from My Remarkable Uncle and other Sketches (1942), Stephen Leacock proposes a programme for finding the Lost Chord by consecutively sounding 5,156,227,011,439 possibilities at the rate of one every 15 seconds over a considerable number of millions of years. This paper takes issue with his analysis on several counts.

Firstly, Leacock misquotes the opening words of the song as “Seated one day at the piano.” He consequently bases his arithmetic on the 88 notes of the grand piano keyboard, whereas the five manuals and the pedal board of a large organ can boast 337 notes among them. He allows the pianistic technique whereby the performer “can, if a trained player, strike any ten [notes], adjacent or distant … by rapidly sweeping his left hand towards the right, or his right towards the left,” so that the “minute fraction between the initial strokes of certain notes” is “not enough to prevent them sounding together as a combination.” But you can’t do that on an organ, because it has no sustaining pedal.

Secondly, he fails to acknowledge certain chords. He considers only chords that are played with two or more fingers, overlooking the truth that a chord can be a “thing that organists play with one finger” (see the dish towel nearest the bottom of my dirty‐linen basket).

But thirdly, and most fundamentally, he wastefully counts as distinct those combinations of notes that are essentially the same “chord”. Two combinations of notes are essentially the same chord if they are inversions of each other: that is, if they are composed of notes with the same names (all “A”s, for example, being considered equivalent, even though they may be separated by one or more octaves). Further, two combinations of notes are essentially the same chord if one is a mere transposition of the other: that is, they are the same chord in different keys. If we adopt that practical definition of “chord”, then standard chromatic theory tells us that there are not 5,156,227,011,439 chords but a mere 351. Of these,

  1 chord consists of  1 note;
  6 chords consist of  2 notes;
 19 chords consist of  3 notes;
 43 chords consist of  4 notes;
 66 chords consist of  5 notes;
 80 chords consist of  6 notes;
 66 chords consist of  7 notes;
 43 chords consist of  8 notes;
 19 chords consist of  9 notes;
  6 chords consist of 10 notes;
  1 chord consists of 11 notes;
  1 chord consists of 12 notes;

Jimmy Durante lays claim to the dominant minor ninth, one of the 5‑note chords, as being the lost chord (I’m the guy that found the lost chord, 1947). And some cheek he has too, for it is the very chord that Sullivan himself used on the words “sought, but I seek it”.

Lest you may doubt the legitimacy of 11‑note and 12‑note chords on an instrument played by a performer with 8 fingers and 2 thumbs, let me remind you that an organist has in addition two feet, which may standardly play one note each when they are not cracking walnuts (ibid).

However, when I tried each of these 351 chords in turn, including the dominant minor ninth without Schnozzola’s kind permission, none of them “lay on my fever’d spirit with a touch of infinite calm”. On the contrary, my spirit grew all the more fever’d as the ever‐growing chords placed greater and greater demands on the fingers, thumbs, feet, ears and patience of this particular “trained player”.

And then, supremely, just a few seconds after I thought my experiment had ended in dismal failure, I heard a chord that truly captured the aforementioned infinite calm. It is not on the above list! It consists of 0 notes, and I hereby stake my claim that it is the genuine lost chord.

Everything falls into place. The text books are all wrong: there are not 351 chords but 352. The table above becomes perfectly symmetrical: there is one chord with 0 notes just as there is one chord with 12. Just as 0 was the last natural number to be discovered, so now the chord with 0 notes is the last chord to be discovered. And it is perfect, this “one lost chord divine”! Consider. It has every conceivable symmetry. It sounds the same forwards as backwards, the same upside down as upside up. Even the most highly trained ear cannot discern whether it was played yesterday or tomorrow. Like God, there is precisely one of it, even when a Trinitarian does the counting. It sounds the same in every inversion, the same in every key, the same in every dynamic, the same at every tempo, the same on every instrument, the same for every performer, the same for every duration whether that be a 26‑millisecond frame in an MP3 file or the full 4 minutes 33 seconds (Cage, 1952). There is no distortion, there are no distracting echoes. It is perfection itself. Yet the character in Sullivan’s song did not recognise it when he heard it, the loser, because he had stopped listening.

Finally, if certain modernist theologians are correct and this life is all there is, then that chord with 0 notes is indeed what we shall all hear in Heav’n. That even applies to Beethoven, deaf or no, provided that he is no longer afflicted with tinnitus.