II. Silver

How the number shapes the music

√2 − 1 is 0.414…, and its continued fraction is 2, 2, 2, … with no end. A 2 means “two of these, then one of those.”

The phrases are 5 bars and 2 bars long. Two 5s and a 2 make a 12-bar section: 5 + 5 + 2. The short 5-bar section is a single 5-bar phrase. Then the same rule works one level up: two 12-bar sections and one 5-bar section make 29 bars, and two of those plus one more 12 make the whole piece: 29 + 29 + 12 = 70 bars.

Hear and see this number’s form on the interactive page: compose a piece from √2 − 1, or watch its levels build up from the beats.

The numbers 2, 5, 12, 29 and 70 are the Pell numbers. Each one is twice the one before plus the one before that: 2 × 12 + 5 = 29. They are to √2 what the Fibonacci numbers are to the golden ratio. In the score the 12-bar sections are marked A, the two 5-bar ones B, and the last 12 bars, the extra “one of those” at the top level, rare block.

Before you play

• Key and meter. D major (two sharps) in 4/4, written as C. Both B sections (bars 25–29 and 54–58) change to G major, with one sharp, and the next section changes back.
• The right hand has long, slow notes: dotted quarters, half notes and whole notes. Keep them singing and connected. Most of the tune lies between F♯ above middle C and the A an octave higher; bars 49–50 climb to high D and E with ledger lines.
• The left hand plays broken chords in eighth notes, in groups of four for most of bars 1–17 and 54–68, and repeated chords on the beat with tied eighth notes in between in bars 18–53 and in the short 2-bar phrases.
• The accents come every 2 or 3 eighth notes. In bar 1 they fall on the 3rd, 5th and 8th eighth notes; in bar 2 on the 2nd, 5th and 7th. At this slow tempo you have time to feel each one.
• Tricky spots. Bars 35–51 have full chords of three and four notes in the left hand, many of them tied: practice the chord changes alone, slowly. A♯ appears in bars 63 and 68. The rare block (bar 59) starts pp, very soft, after the louder sections. The last bar has a fermata in both hands.

Practice plan

1. Right hand alone, one 5-bar phrase at a time. Count “1 and 2 and” out loud so the dotted quarters get their full length. ♩ = 52
2. Left hand alone. First the broken chords of bars 1–5, then the chord bars 18–22. Hear the accents with the player below. ♩ = 52
3. Chunk by section: a 12-bar section is 5 + 5 + 2 bars. Learn the 2-bar phrase at the end of each one carefully: it is where the music breathes. ♩ = 56–60
4. Hands together, one phrase at a time, then a whole section. Take bars 35–51 last. ♩ = 60–66
5. Dynamics and pedal. p at the start, mp, mf, f at bar 18, then quieter again down to pp at bar 59. Change the pedal with the harmony, usually once a bar or once a half bar. ♩ = 70–76

Set the metronome to 4/4 and turn on eighth-note subdivisions; slow music is easier to keep steady with a little help.

Play along

The player taps the left-hand accents of √2 − 1 at the marked Adagio, or slower. The filled squares match the > marks in the score.

Family notes

For younger players: play the right hand of the first phrase alone. It moves mostly by step, and the slow tempo gives you time to find each note.

Tap the rhythm: say “1 and 2 and 3 and 4 and” and clap on the filled squares of the player. Do the claps make a pattern? (Look for 2s and 3s between them.)

Talk about it:

• Golden used 1s and this movement uses 2s. How does the strip look different?
• Check the Pell rule yourself: 2 × 5 + 2 = 12, 2 × 12 + 5 = 29. What comes after 70?
• The movement is slow and calm. Does a number with all 2s feel calmer than one with all 1s?

Keep learning

• Key signatures (D major and G major)
• Time signatures (common time, C)
• Intervals and chords (the left-hand chords)
• Dynamics, tempo and articulation (Adagio, pp to f)
• The pedals