V. Pi

How the number shapes the music

π − 3 is 0.14159…, and its continued fraction starts 7, 15, 1, 292. Read as repeat counts:

• 7: a phrase is 7 bars. (That is the 7 in 22/7, the fraction for pi many people learn in school.)
• 15: fifteen phrases, then one single bar: 15 × 7 + 1 = 106 bars. (333/106 is the next fraction for pi.)
• 1: one of those 106-bar blocks, then one more 7-bar phrase: 106 + 7 = 113 bars. (355/113, which is right to six decimal places.)

So the movement is fifteen 7-bar phrases (bars 1–105), one held bar (bar 106), and a last 7-bar phrase, marked B (bars 107–113).

Hear and see this number’s form on the interactive page: compose a piece from π − 3, or watch its levels build up from the beats.

The next term, 292, says what would happen if the piece kept going: this whole 113-bar cycle would play 292 times before the first short cycle. That is Movement VI.

Before you play

• Key and meter. C major in 4/4, written as C, with no key changes. A good place to start the suite if you read well in C.
• The right hand moves in quarters, dotted-quarter-and-eighth figures and half notes, from F above middle C up to the C two octaves above middle C (bar 41). Most phrases open with the same four notes, C–E–C–A; some open with the same shape a third higher, G–B–G–E (bars 29, 36, 71, 78 and 107).
• The left hand takes turns: broken chords in eighth notes, in groups of four, for two phrases (bars 1–14, 43–56 and 85–98), then held chords with tied notes for four phrases (bars 15–42 and 57–84), and again from bar 99 to the end.
• The accents are pi’s own rhythm: one accent every 7 eighth notes. Each bar has 8, so the accent moves one eighth note earlier every bar: the 8th eighth note in bar 1, the 7th in bar 2, the 6th in bar 3. Now and then the gap is 8 instead of 7, about once every 16 accents, because 1/(π − 3) is 7.06, a little more than 7.
• Tricky spots. Bars 29–42 are f, with four-note chords in the left hand. Bar 106 is the held bar: a whole-note chord in the right hand over a held left hand. Count all four beats before the B phrase. The tied chords in the held sections (for example bars 15–16 and 31–33) must not be played again. The last bar has a fermata.

Practice plan

1. Right hand alone. Learn the first phrase (bars 1–7) well; many later phrases begin the same way. ♩ = 60
2. Left hand alone. The broken chords of bars 1–7, then the held chords of bars 15–21. Play along with the accent player and watch the accent walk backward through the bar. ♩ = 60
3. Chunk by phrase. Seven bars at a time. Count phrases on your fingers as you go: the held bar comes after the 15th. ♩ = 66–72
4. Hands together, one phrase, then two. Practice the switch from broken chords to held chords (bars 14–15, 56–57, 98–99). ♩ = 72–84
5. Dynamics and pedal. p, mp at bar 8, p, mf at 22, f at 29, mf at 71, mp at 85, p at 92. Change the pedal with each chord; in the held sections, once a bar is usually right. ♩ = 88–96

Practice with the metronome in 4/4 at ♩ = 96 when you’re ready. At that tempo one bar takes 2.5 seconds, and the movement about 4¾ minutes.

Play along

The player taps pi’s left-hand accents. Watch the filled square slide one step to the left in each new bar, the same as the > marks in the score.

Family notes

For younger players: play the right hand of the first phrase. It is seven bars, the length of one of pi’s phrases.

Count to 7: count eighth notes out loud, “1, 2, 3, 4, 5, 6, 7,” and clap on every 7. Then play the accent player and check: the claps land on the filled squares.

Talk about it:

• Work out 22 ÷ 7 and 355 ÷ 113. Which is closer to 3.14159…?
• Why does the accent move one step earlier every bar? (Hint: 8 − 7 = 1.)
• Zu Chongzhi, a Chinese mathematician, found 355/113 about AD 480. How do you think he did it without a computer?

Keep learning

• Note values (dotted notes, ties)
• The grand staff
• Intervals and chords
• C position
• How to practice