Music · Piano · Original music
The Continued-Fraction Suite
Six short pieces for piano by Teddy Rook, one number each. The numbers decide how long the phrases are, when the sections change, and where the left hand leans. Watch the video, download the score, and learn to play them.
The Continued-Fraction Suite is original music by Teddy Rook: six pieces for piano, each one built from a single number. It began on the Rhythm of Numbers page, where pi is played as a beat, and grew into music you can sit down and play.
How a number becomes music
Any number can be written as a continued fraction: a list of whole numbers that says how to build it, step by step. Pi minus 3 starts 7, 15, 1, 292. The golden ratio is 1, 1, 1, 1, forever.
Each piece reads those whole numbers as repeat counts. A 7 means “seven of these, then one of those.” Bars group into phrases, phrases into sections, and sections into the whole piece, and the list of numbers says how many of each. So every movement has a usual section (marked A in the score), a shorter one that turns up now and then (B), and sometimes a rare block that the number saves for a surprise.
Inside each bar, the same number taps out the rhythm of the left-hand accents. In the score they are the little > marks under the bass notes. The melody walks the number’s decimal digits. You don’t need any of the math to enjoy the music, but it is fun to hear the pattern once you know it is there.
Watch and listen
Teddy Rook’s video of the Continued-Fraction Suite. It loads from YouTube only when you press play.
The score
The full score of all six movements, engraved by Teddy Rook. Print it, read along with the video, or play from it at the piano. Free to download.
⬇ Download the score (PDF, 16 pages, 1.7 MB)Open it in a new tab
The six movements
Each card opens a piano lesson for that movement: what is on the page, how its number shapes the music, a practice plan, a play-along, and ideas for families.
- Movement I: Golden
φ − 1 = [0; 1, 1, 1, 1, …]
A gentle, singing piece in 3/4 and G major built on the golden ratio: 89 bars of 8-bar and 5-bar sections in Fibonacci order.
The piano lesson → - Movement II: Silver
√2 − 1 = [0; 2, 2, 2, …]
A slow, singing Adagio in D major built on √2 − 1, whose continued fraction is all 2s: 12-bar sections with a 5-bar B section after every second one.
The piano lesson → - Movement III: The Fifth
log₂(3/2) = [0; 1, 1, 2, 2, 3, 1, 5, …]
A lively Allegretto in F major built on the musical fifth, log₂(3/2): 53 bars, because 53 is the next stop after 12 on the ladder of fifths.
The piano lesson → - Movement IV: Euler
e − 2 = [0; 1, 2, 1, 1, 4, 1, 1, 6, …]
A waltz in A minor built on e − 2, whose continued fraction grows 2, 4, 6, 8: 7-bar sections, a short 4-bar B, and a form that keeps stretching.
The piano lesson → - Movement V: Pi
π − 3 = [0; 7, 15, 1, 292, …]
A Moderato in C major built on π − 3: fifteen 7-bar phrases, one held bar and one more phrase, 113 bars in all, the 113 of the famous fraction 355/113.
The piano lesson → - Movement VI: Pi, bar 32,996
π − 3 = [0; 7, 15, 1, 292, …]
Fifty bars from deep inside pi’s full form: the end of the 292nd 113-bar cycle and the start of the first short cycle, the rare block that comes about 23 hours in.
The piano lesson →
Where the music came from
The Rhythm of Numbers is the workshop behind the suite: play pi as a beat, see its levels, and compose your own piece from any number. Its levels section shows how a continued fraction turns into bars, phrases and sections, and Compose a piece builds one for you.
New to reading music? Start with Piano Basics and Reading Music, then come back. Keep time with the metronome.