I. Golden

How the number shapes the music

φ − 1 is 0.618…, and its continued fraction is 1, 1, 1, 1, … with no end. A 1 means “one of these, then one of those,” so each block of music is simply the two blocks before it, joined together. That is the Fibonacci rule: 3 + 5 = 8, 5 + 8 = 13, and so on up to 89.

In this movement the phrases are 5 bars and 3 bars long. A 5 and a 3 make an 8-bar section; a single 5-bar phrase makes the short 5-bar section. Then the sections line up in the same Fibonacci order: 8, 5, 8, 8, 5, 8, 5, 8, 8, 5, 8, 8, 5. Eight 8s and five 5s: 64 + 25 = 89 bars.

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

The score marks each 5-bar section B. The 8-bar sections are marked A or rare block; both are 8 bars of a 5 and a 3. Because every term of this number is 1, no section ever repeats more than twice in a row. Look at the strip: two long sections, then a short one, but never three long ones together.

Why the same 8 bars get two names. In the other movements the short block really is rare: in Pi it comes once after 292 long ones. The golden ratio's terms are all 1, so every level is one long block followed by one short block, and the sizes slide up the Fibonacci numbers. Phrases are 5 and 3 bars. Sections are 8 (5 + 3) and 5. One level up, the blocks are 13 (8 + 5) and 8. So an 8-bar unit is the long block, A, at one level and the short block at the level above, and the score names it by whichever level's boundary falls there. That is the self-similarity of the golden ratio: each level is the one below it, scaled up. Here the "rare block" is not rare at all. It comes about once for every 1.6 long blocks.

Before you play

• Key and meter. G major (one sharp, F♯) in 3/4. Three of the B sections, at bars 9, 43 and 64, change to a key signature with no sharps (C major), and the next section changes back. Watch for the natural sign at the start of those bars.
• The right hand sings the tune: dotted quarter, eighth, quarter (long, short, step) is the rhythm you meet most. It starts on D above middle C and climbs as high as the C two octaves above middle C around bar 32.
• The left hand has two patterns. In bars 1–5 and many of the B sections it plays broken chords in six eighth notes: a low note, then up and down through the chord. Elsewhere it plays chords on the beat, often with a tied eighth note that holds across the beat.
• The accents. The little > marks under the left hand carry the number’s rhythm. They come every 2 or 3 eighth notes, so they don’t always land on a beat: in bar 1 they fall on the 3rd and 6th eighth notes, in bar 2 on the 2nd and 5th. Lean into them a little; don’t punch them.
• Tricky spots. Bars 30–42 have full three- and four-note chords in the left hand, with f at bar 35: practice the chord shapes alone first. Sharps outside the key (D♯ and C♯) turn up around bars 18–20, 27–28, 52 and 81–83. The last bar (89) has a fermata: hold it as long as feels right.

Practice plan

1. Right hand alone, one phrase at a time. Start with bars 1–5. Count “1 2 3” out loud. Metronome ♩ = 56
2. Left hand alone. Play the broken chords of bars 1–5, then the chord bars 6–8, and feel the accents land in their places. Use the player below to hear where they go. ♩ = 56
3. Chunk by section. Take one section a day: an 8-bar one is a 5-bar phrase and a 3-bar phrase. Mark the A, B and rare-block starts in pencil on your copy. ♩ = 60–66
4. Hands together, slowly, one phrase at a time. Join two phrases, then a whole section. ♩ = 66–72
5. Dynamics and pedal. Add the markings (p at the start, mp, f at bar 35, mf at bar 56, pp at bar 69). Pedal is up to you; the score says to change it with the harmony, usually once a bar. ♩ = 76–84

Use the site’s metronome in 3/4, and raise the tempo a few beats at a time only when a passage is clean.

Play along

This player taps the left-hand accent pattern, built from the number, at the marked tempo or slower. The filled squares match the > marks in the score. Play the left hand with it, or just listen and count.

Family notes

For younger players: try the right hand of the first phrase alone. It is five bars, all in the right hand, and it uses only six different notes. Press Hear it, then play it at your own speed.

Tap the rhythm: one person taps every eighth note softly on a table (six taps a bar) while another claps on the filled squares in the player. Then swap.

Talk about it:

• Count the bars in each section of the strip. Can you find the Fibonacci numbers 3, 5, 8 and 89?
• Why do you think the short section never comes twice in a row?
• Sunflower seeds and pinecone scales also come in Fibonacci numbers. Can you find a pinecone and count its spirals?

Keep learning

• Beats, measures and time signatures (3/4 time)
• Key signatures (G major and the change to C major)
• Note values (dotted notes)
• Dynamics, tempo and articulation (accents, Andante cantabile)
• The pedals