DIRECTIONS: Read what was decided and what it costs, then answer.
Spread the error evenly. Only the octave stays perfect.
Split the octave into twelve equal steps. Twelve of them must multiply to exactly 2, so each step is the twelfth root of two: 1.059463.
Every other interval is now slightly wrong, by the same amount in every key.
Interval
The pure ratio
What you actually get
Out by
The octave
2
2
nothing
The fifth (seven steps)
1.5000
1.4983
-2.0 cents
The major third (four steps)
1.2500
1.2599
+13.7 cents
A cent is a hundredth of a step. 2 is too small to notice; 14 is not.
The fifth is out by only about 2.0 cents and the third by about 13.7. Say which of those matters more in practice, and why the answer is not simply "the bigger number".
Somebody Has to Lose — What It Bought
Why there are twelve notes · Page 2 of 2
What it bought: every key sounds the same. In the old tunings some keys were lovely and some unusable, and changing key meant retuning.
Say what was gained by this compromise, and name somebody who lost something by it.
A singer or a violinist is not stuck with fixed notes — they can put a note exactly where they want it. Say what you would expect a good one to do when playing on their own, and what changes when they play with a piano.