Musical arithmetic

When I was a small boy in the 1920s, any conversation about music interested me, and I took every opportunity to ask questions. ‘What do you mean by a violin is tuned in fifths?’ My friend's mother adjusted the pegs and played the open strings. ‘Those are fifths,’ she said. ‘Why are they called fifths?’ ‘I don't know. That's what they are always called.’ My cousin had had piano lessons. He played middle C on the piano and counted along the white keys ‘One, two, three, four, five,’ to the G. ‘That's why it's a fifth.’ He continued from the G to the C above, counting again. ‘And that's a fourth.’ ‘So you just count along the white ones?’ ‘Yes,’ he said, and played a series of fifths: CG, DA, EB, FC, GD, AE, paused, and played BF#. ‘But why did you play the black one?’ ‘I'm not sure,’ he said, and then with a flash of inspiration, ‘If you count all the keys, black and white [he demonstrated], a fifth is always eight of them.’ ‘And a fourth?’ He counted again. ‘That's always six of them.’ Then he added ‘And a fifth plus a fourth makes an octave – that means eight, you know.’ I was good at arithmetic. ‘So there's fourteen notes altogether in an octave.’ ‘I don't think so, [counting] only thirteen. Anyhow, an octave's twelve semitones.’ And if the White Rabbit had hopped out of the piano at that moment it would not have surprised me.

None of it made any sense. Afterwards I went to the piano on my own and counted along five white notes for a fifth. I was happy with CG, DA and so on but this time I counted and played BF instead of BF#. It didn't sound right. BF# did. So a fifth was something I heard from the right pairs of piano notes. It was the something I'd heard from the pairs of violin strings, though it was a completely different kind of sound to the piano; and a violinist could adjust the strings until she heard that they were producing this something.