VI. Pi, bar 32,996

How the number shapes the music

Movement V used the first three terms of pi’s continued fraction, 7, 15 and 1, to build a 113-bar cycle. The fourth term is 292. Read as a repeat count, it says: “play the 113-bar cycle 292 times, then one short cycle of 106 bars.”

A short cycle is the same as a long one without its closing 7-bar phrase: fifteen 7-bar phrases and a held bar, 106 bars instead of 113. The long cycles fill 292 × 113 = 32,996 bars. At ♩ = 96, each 4/4 bar takes 2.5 seconds, so that is about 22.9 hours of music before the first short cycle. This is pi’s rare block: it comes once in 33,102 bars (292 × 113 + 106), and 33,102 is the bottom of the next great fraction for pi, 103,993/33,102. The 292 is unusually large, and that is exactly why 355/113 is such a good fraction: pi’s pattern holds for a very long time before it slips.

This movement plays bars 32,968 to 33,017, 50 bars around that turning point:

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.

• Bars 32,968–32,996: the end of the 292nd cycle. Three 7-bar phrases, the held bar (32,989), and the B phrase (32,990–32,996), here in F major.
• The mark “bar 32,996”: it sits on the bar line where those 32,996 bars end. The score numbers bars from 1, so the first bar of the short cycle is printed 32,997. (On the Rhythm of Numbers page the same moment is “beat 32,996,” counting the first beat as beat 0.)
• Bars 32,997–33,017: the first three phrases of the short cycle. The score ends here, with a fermata. The short cycle itself would end at bar 33,102, after its held bar, with no B phrase.

On paper a short cycle starts just like a long one; it is only shorter at its end. Teddy Rook marks the moment anyway: the B phrase before it turns to F major, and the short cycle opens quietly, p and then pp, with held chords.

Before you play

• Key and meter. C major in 4/4. The B phrase (bars 32,990–32,996) has a key signature of one flat, F major; bar 32,997 cancels it with a natural sign.
• Read the bar numbers. They are five digits long. It helps to think of them by their last two digits: “68” to “17.”
• The first two phrases (bars 32,968–32,981) have the same notes as the first two phrases of Movement V (only the dynamics differ). If you learned bars 1–14 of Pi, you already know them.
• The left hand plays broken chords in eighth notes up to the held bar, then held chords with ties for the rest of the movement. The accents are pi’s every-7-eighth-notes rhythm, the same as in Movement V.
• Tricky spots. The change into F major (B♭ in bars 32,990–32,996) and back (B♮ from bar 32,997). G♯ in bars 33,003 and 33,010. The pp chords from bar 33,004 take a light, even touch. The held bar 32,989 gets its full four beats.

Practice plan

1. Start from Movement V. Bars 32,968–32,981 match Pi’s bars 1–14; review them there. ♩ = 72
2. Hands separately, the new music: bars 32,982–32,996 (the forte phrase, the held bar and the B phrase in F major). ♩ = 60
3. Hands separately, the short cycle: bars 32,997–33,017, held chords in the left hand. ♩ = 60
4. Hands together, one 7-bar phrase at a time, then across the turning point: from the held bar 32,989 to bar 33,003. ♩ = 72–84
5. Dynamics and pedal. p, mf at 32,975, f at 32,982, then p at 32,997, pp at 33,004 and p at 33,011. Let the short cycle sound like a hush. ♩ = 88–96

Keep time with the metronome in 4/4.

Play along

The player taps pi’s left-hand accents at the marked tempo, numbered from bar 32,968. As in Movement V, the accent slides one eighth note earlier each bar.

Family notes

For younger players: the first phrase here is the same as the first phrase of Pi. Learn it on the Pi lesson with the right hand alone, then play it here and imagine you have been playing for 23 hours.

Feel the size of 292: clap a steady beat and count to 7 again and again, with a loud clap on each 7. That is one phrase. It would take 15 of those, plus one, plus one more phrase, to make one cycle, and 292 cycles to reach this movement.

Talk about it:

• If you started playing at breakfast and never stopped, about when would you reach bar 32,996?
• Why does a big number in the continued fraction mean a long wait for the surprise?
• The golden ratio’s terms are all 1. Would its music ever have a 23-hour wait?

Keep learning

• Key signatures (the change to F major)
• Sharps, flats and naturals (the natural sign that cancels it)
• Dynamics, tempo and articulation (fermata, pp)
• The Composer on the Rhythm of Numbers page, with its “Jump to the rarest block” button