Fractions you can clap
Tonight's idea
Music is written in notes, and the name of each note says how long it lasts. A whole note is the unit. A half note lasts half as long, a quarter note a quarter as long, an eighth note an eighth. Each is half the one before.
A dot after a note adds half of its length again. A dotted quarter note lasts \(\tfrac14 + \tfrac18 = \tfrac38\). A dotted half note lasts \(\tfrac12 + \tfrac14 = \tfrac34\).
Music is cut into bars of equal length. The time signature at the start says how long a bar is, and it looks like a fraction. In 4/4 each bar holds four quarter notes, a total of 1. In 3/4 it holds three quarter notes, a total of \(\tfrac34\). In 6/8 it holds six eighth notes. These fractions measure written length, in whole notes. How many seconds a note takes depends on the tempo, the speed of the music.
Whatever notes you choose, each bar must add up exactly to its total. Too little leaves a hole. Too much will not fit.
Watch one: does the bar add up?
In 4/4, try: dotted quarter, eighth, quarter, quarter. Write everything in eighths.
\[\tfrac38 + \tfrac18 + \tfrac28 + \tfrac28 = \tfrac88 = 1.\]A full bar. It is the bottom row of the picture. To clap it, count the eight eighths aloud as "1 and 2 and 3 and 4 and". Clap on 1, on the "and" after 2, on 3 and on 4.
Now try: half, dotted quarter, quarter. That is \(\tfrac48 + \tfrac38 + \tfrac28 = \tfrac98\). One eighth too long, so it is not a bar of 4/4.
Your turn
- (a) How long is a dotted eighth note, as a fraction of a whole note? (b) How many eighth notes last as long as a dotted half note? (c) How many sixteenth notes last as long as a dotted quarter note? (A sixteenth is half an eighth.)
- Which of these are full bars of 4/4? For any that are not, say how much is missing or left over. (a) half, quarter, eighth, eighth; (b) dotted half, eighth; (c) quarter, dotted quarter, eighth, quarter.
- One note is missing from each bar. Name it. (a) A bar of 3/4: dotted quarter, ____, quarter. (b) A bar of 6/8: quarter, eighth, ____. (c) A bar of 4/4: eighth, dotted half, ____.
- (a) Show that a bar of 3/4 and a bar of 6/8 hold the same fraction of a whole note. (b) Fill each with six eighth notes. In 3/4 they are counted in three pairs. In 6/8 they are counted in two threes. Which single note is as long as one pair? As one three? (c) A tune is 24 bars of 6/8. If an eighth note takes the same time in both, how many bars of 4/4 last just as long?
Stretch: two dots
A second dot adds half of what the first dot added. How long is a double-dotted half note? Which single note fills the rest of a 4/4 bar? If you could keep adding dots for ever, what length would the note creep towards?
Answers
- (a) \(\tfrac18 + \tfrac{1}{16} = \tfrac{3}{16}\). (b) \(\tfrac34 = \tfrac68\), so 6. (c) \(\tfrac38 = \tfrac{6}{16}\), so 6.
- (a) \(\tfrac48 + \tfrac28 + \tfrac18 + \tfrac18 = \tfrac88\). Full. (b) \(\tfrac68 + \tfrac18 = \tfrac78\). An eighth is missing. (c) \(\tfrac28 + \tfrac38 + \tfrac18 + \tfrac28 = \tfrac88\). Full.
- (a) \(\tfrac68 - \tfrac38 - \tfrac28 = \tfrac18\): an eighth note. (b) \(\tfrac68 - \tfrac28 - \tfrac18 = \tfrac38\): a dotted quarter note. (c) \(\tfrac88 - \tfrac18 - \tfrac68 = \tfrac18\): an eighth note.
- (a) \(\tfrac68 = \tfrac34\): the same fraction of a whole note. (b) A pair of eighths is \(\tfrac28 = \tfrac14\), a quarter note. Three eighths is \(\tfrac38\), a dotted quarter note. So 3/4 has three beats in a bar and 6/8 has two, which is why they feel different. (c) \(24 \times \tfrac34 = 18\) whole notes, and a bar of 4/4 is one whole note: 18 bars. At a different tempo the answer in seconds would change, but the written length would not.
Stretch. \(\tfrac12 + \tfrac14 + \tfrac18 = \tfrac78\). An eighth note fills the bar. Each new dot adds half of the gap that is left before a whole note, so the length creeps towards 1, a whole note, which is twice the plain half note. It never quite gets there.