Name
Date
DIRECTIONS: An octave is a ratio of 2; a fifth is 3 to 2. Fill the middle column: each row is the row above × 3/2. Use a calculator.
| How many 3:2 steps | How much higher than the start | Check |
|---|
| 1 | | 1.5000 |
| 2 | | 2.2500 |
| 3 | | 3.3750 |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | | |
| 8 | | |
| 9 | | |
| 10 | | |
| 11 | | |
| 12 | | |
Now work out 2 × 2 × 2 × 2 × 2 × 2 × 2 — seven octaves — and write it here: ______________
Twelve steps of 3:2 come to 129.746. Seven octaves come to 128.
They should be the same note. Twelve fifths land 1.36% above seven octaves: about 23 hundredths of a semitone, enough to hear.
It cannot be fixed. Powers of 3/2 are made of threes; powers of 2 are not. They never meet.
- Say why this is not a mistake in your arithmetic, a fault in one piano, or something better measuring would fix.
- A tuner tunes twelve perfect fifths, one after another, all the way round, and arrives back at the note they started from. What will be wrong with it?
- You cannot have both. Describe the two things somebody could choose to do about it, before sheet 20 tells you what was chosen.