Mum 
*sings a note* I’m guessing either a C or a D. 

*goes to piano* Oh, it’s a C sharp. 
Dad 
Well, it’s the average of your guesses. 
Me 
Only if it’s a linear relationship. 
Dad 
Of course it is. The C above a middle C has twice the frequency. 
Me 
Then it wouldn’t be linear. It would increase by powers of two. 
Dad 
… 
Me 
*consults Wikipedia* Yep, powers of two. Twelve semitones, so to get the frequency of the next semitone, multiply by the twelfth root of two. Middle C is roughly 262 hertz; Tenor C is 523 or so and Soprano C is 1047. 
Dad 
*does the maths* And the fifth, the seventh semitone, is roughly halfway. Maybe that’s why it’s the perfect cadence. 
Me 
*gets calculator* *nearly blasphemes* 2(^{12}√2)^{7} ≡ 2(2) ∕ (^{12}√2)^{5}!!!! Modelling 2 as n hertz. And the best thing is that the answer is 2.996614154…which rounds to 3, which is halfway between 2 and 4. Aaaaghhh *the romance of logic overwhelms* 
Dad 
So our brains aren’t adding a fixed value between each semitone. They’re actually making a complex geometric calculation. 
Me 
What do you think, Mum? 
Mum 
I stopped listening when you mentioned averages. 
Me 
Yet you’re the only one of us who can sing in tune. 
Dad and I 
*return to calculations* *major excitement* 
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