IV. Euler

How the number shapes the music

e − 2 is 0.71828…, and its continued fraction is 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, …: two 1s, then the next even number, again and again. That is why the score calls this form one that “keeps stretching”: further in, the big terms keep growing.

The phrases are 4 and 3 bars, so a 7-bar section is a 4 and a 3, and the short B section is a single 4-bar phrase. Then the 4 in the continued fraction takes over: four 7-bar sections and one 4-bar B make a 32-bar block. The 1s after it say “one of these, then one of those”: 32 + 7 = 39, and 39 + 32 = 71 bars. The best fractions for e − 2 have exactly these denominators: 5/7, 23/32, 28/39, 51/71.

Hear and see this number’s form on the interactive page: compose a piece from e − 2, or watch its levels build up from the beats.

The score marks the usual 7-bar sections A, the 4-bar ones B, and the two 7-bar sections in the middle (bars 33–46) rare block. The first is the lone 7 that turns 32 into 39; the second begins the last 32-bar block.

Before you play

• Key and meter. A minor in 3/4, a waltz (Tempo di valzer). There is no key signature; the G♯s are written in each time, and there are many of them.
• The right hand moves in quarters and eighth-note pairs, most of the time between E and E above middle C. Every A section starts with a pair of eighths, C and B, then C; the B sections and rare blocks start the same way a fifth higher, on G and F. The B sections and rare blocks go higher, to the top of the staff and above it.
• The left hand plays broken chords in six eighth notes in some bars (1–4, 8–11, 47–50, 54–57) and waltz chords on the beat in the rest, often a held note under a chord. Bars 29–43 have full three-note chords.
• The accents come every 3 or 4 eighth notes, so they drift across the bar: in bar 1 on the 4th eighth note, in bar 2 on the 2nd and 5th, in bar 3 on the 3rd and 6th. Against the waltz’s strong beat 1, they give the music a gentle lilt.
• Tricky spots. The G♯s: bars 4, 6, 10, 13, 18, 20, 25–27 and 70, among others. In bars 29–43, the left-hand chords change on every beat. The last bar has a fermata in both hands.

Practice plan

1. Right hand alone, one 4-bar phrase at a time, counting “1 2 3” with a little lean on 1. ♩ = 60
2. Left hand alone. Broken chords in bars 1–4, then the waltz chords in bars 5–7. Play the player below and find where each accent falls. ♩ = 60
3. Chunk by section: 4 + 3 bars. Practice the middle, bars 29–46, as its own piece. ♩ = 66–72
4. Hands together, phrase by phrase, then two sections at a time. ♩ = 72–84
5. Dynamics and pedal. p, mp, mf, f at bar 19, then mp, p, pp at bar 51 and p at the end. A light pedal once a bar suits a waltz. ♩ = 88–96

Set the metronome to 3/4 with an accent on beat 1, and let the left hand feel the dance.

Play along

The player taps the left-hand accents of e − 2 at the waltz tempo, or slower. The filled squares match the > marks under the left hand in the score.

Family notes

For younger players: play the right hand of the first phrase alone. It uses only three different notes, B, C and E.

Dance it: waltz around the room, stepping on beat 1 of each bar, while someone plays the accent player. Clap on the filled squares.

Talk about it:

• Write the pattern 1, 2, 1, 1, 4, 1, 1, 6, … and keep going. What are the next six terms?
• Pi’s continued fraction has no pattern anyone has found. Why is it surprising that e’s does?
• The form grows from 7 to 32 to 39 to 71 bars. If the piece kept going, how long would the next block be? (The next term is 6: try six 71s and a 39.)

Keep learning

• Time signatures (3/4, the waltz)
• Sharps, flats and naturals (the G♯s)
• Key signatures (A minor, the relative minor of C)
• The bass clef
• Finger numbers