III. The Fifth

How the number shapes the music

A perfect fifth (C up to G) is the frequency ratio 3 : 2. How much of an octave is that? The answer is log₂(3/2) = 0.58496…, and its continued fraction is 1, 1, 2, 2, 3, 1, 5, …. Stopping it after each term gives the best fractions for the fifth: 1/2, 3/5, 7/12, 24/41, 31/53. The 7/12 is our piano: a fifth is 7 of the 12 half steps in an octave. The next good fraction, 31/53, is even closer, which is why musicians have long been curious about a 53-note octave.

Unlike Golden and Silver, the terms here change, so each level of the music follows a different rule. The phrases are 5 and 2 bars. Two 5s and a 2 make a 12-bar section. Three 12s and a 5 make 41 bars, and one more 12 makes 53. So the sections run 12, 12, 12, 5, 12: the denominators of the fractions above, 5, 12, 41 and 53, are built right into the form.

Hear and see this number’s form on the interactive page: compose a piece from log₂(3/2), or watch its levels build up from the beats.

The score marks the second and third 12-bar sections A, the 5-bar one B, and the last 12 bars, the extra section that turns 41 into 53, rare block.

Before you play

• Key and meter. F major (one flat, B♭) in 4/4, written as C. The B section (bars 37–41) changes to B♭ major, with two flats, and bar 42 changes back.
• The right hand plays half notes and quarters, with a few dotted-quarter-and-eighth figures. In the first 36 bars every 5-bar phrase begins the same way: a half-note C, then C and E. The B section and the rare block start a fourth higher, on F, and climb above the staff to high C and past it (bars 40 and 48).
• The left hand plays broken chords in eighth notes for most of bars 1–17 and then plays chords on the beat with tied eighths. From bar 37 to bar 51 the chords are full, three and four notes at a time.
• The accents come every 2 or 3 eighth notes: in bar 1 on the 3rd, 5th and 8th eighth notes. By the accent rule, this pattern matches Silver’s for the first five bars, because 1 − log₂(3/2) = 0.415 is very close to √2 − 1 = 0.414; the two part ways at bar 6.
• Tricky spots. Bars 37–51: thick left-hand chords at pp in the rare block, which takes control to keep soft. C♯ appears in bars 46 and 51. Allegretto is a little quick, so learn the chord bars slowly first. The last bar has a fermata.

Practice plan

1. Right hand alone, phrase by phrase. Notice how often a phrase starts with the same three notes; that makes memorizing easier. ♩ = 60
2. Left hand alone. Broken chords first (bars 1–5), then the chord bars. For bars 37–51, block each chord silently, then play it. ♩ = 56–60
3. Chunk by section: 5 + 5 + 2 bars each. Learn the B section and the rare block (bars 37–53) as their own pieces. ♩ = 66
4. Hands together, one phrase at a time, then whole sections. ♩ = 66–76
5. Dynamics and pedal. p, mp, mf, f at bar 13, mf, then pp at bar 42 and p to end. Change the pedal with each new chord. ♩ = 80–88

Practice with the metronome in 4/4; its speed trainer can raise the tempo for you a few beats at a time.

Play along

The player taps the left-hand accents of the fifth’s number at Allegretto, 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 fits under one hand position for most of the way.

Hear a fifth: on a piano, play C and the G above it together. Then count the keys from C up to G, white and black: there are 7 half steps, and 12 in the whole octave. That is the 7/12 from the continued fraction.

Talk about it:

• Why might a piano with 53 keys in every octave be hard to build and to play?
• The form is 12, 12, 12, 5, 12. Where do you hear the music change most?
• Golden and Silver repeat one rule forever. Why does this number’s form use a different rule at each level?

Keep learning

• Scales and the circle of fifths
• Intervals and chords (the perfect fifth)
• Key signatures (F major and B♭ major)
• Sharps and flats on the keyboard
• The circle of fifths on the Rhythm of Numbers page