47Fractions: Comparing and Adding

Problem 47.1

Weekend problem — note values are fractions that must add up exactly, rhythm counting hides a famous sequence, and parity rules the Balkan beat

Music is written in fractions. A whole note lasts $1$ measure (in the common “four-four” time); a half note lasts $\frac12$; then come the quarter note $\frac14$, the eighth note $\frac18$ and the sixteenth note $\frac{1}{16}$. One inviolable law: the durations inside a measure must add up to exactly the measure’s total — $1$ in four-four time. Every question below is this chapter’s arithmetic (Theorem 47.4, Method 47.5) set to music; on the way you will meet a counting sequence discovered by Indian scholars a thousand years ago.

Part I — Notes that must add up.

1. Check that each of these measures is legal in four-four time: (a) half $+$ quarter $+$ quarter; (b) quarter $+$ eighth $+$ eighth $+$ half.
2. A copyist wrote: half $+$ quarter $+$ eighth. What fraction of the measure is written, and which single note completes it?
3. How many sixteenth notes fill one whole measure? How many sixteenths make one eighth note?
4. A dot after a note adds half the note’s value: a dotted half is $\frac12 + \frac14$, a dotted quarter is $\frac14 + \frac18$. Compute both values.
5. A waltz is written in three-four time: each measure must total $\frac34$. Check that a single dotted half fills a waltz measure, and write down two other legal waltz measures using only quarters and eighths.

Part II — The copyist’s workshop.

6. A four-four measure contains: dotted quarter $+$ eighth $+$ quarter. What is missing?
7. Which lasts longer, a dotted eighth or a quarter note? Compare via sixteenths.
8. A tie glues durations together into one long sound: a half note tied to an eighth lasts how long? How many sixteenths is that?
9. The notes of a measure add up to $\frac{11}{16}$; the remainder is silence (a rest). How long is the rest?
10. A drummer fills one quarter note ($\frac{4}{16}$ of the measure) using only eighths ($\frac{2}{16}$) and sixteenths ($\frac{1}{16}$), order mattering: for instance eighth–sixteenth–sixteenth, or sixteenth–eighth–sixteenth. Check that both examples are legal, then list all the possible rhythms. How many are there?

Part III — A thousand-year-old sequence, and an odd measure.

11. Let us count the drummer’s rhythms for longer notes. Every rhythm ends either with a sixteenth or with an eighth; explain why this gives the rule: (rhythms filling $n$ sixteenths) $=$ (rhythms filling $n - 1$) $+$ (rhythms filling $n - 2$). Starting from $1$ rhythm for one sixteenth and $2$ for two, build the table up to $n = 8$: how many rhythms fill a half note?
12. The numbers you found — $1, 2, 3, 5, 8, 13, 21, 34$ — were published by the Indian scholar Hemachandra around the year 1150, counting exactly such rhythms, two generations before Fibonacci wrote them down in Italy. State their defining pattern in one sentence, and check it on the last three entries of your table.
13. “Swing” style replaces two equal eighths by a long–short pair: $\frac16$ then $\frac{1}{12}$. Verify that the swap is legal (same total duration), using the common denominator $12$.
14. A triplet squeezes three equal notes into one quarter note. How long is each note (check your answer by multiplying it by $3$, Proposition 47.8)? Same question for three equal notes filling a half note.
15. Balkan dance music uses seven-eight time: each measure totals $\frac78$. Verify that dotted quarter $+$ quarter $+$ quarter fills it. Then explain why no combination of half notes and quarter notes alone, however many, can ever fill a seven-eight measure. (Count in eighths and think about even and odd.)