You Are Reading the First 6 FREE Chapters (470 pages)

The Pythagorean Comma: Octave Interval and 2:1 Ratio

Start with Middle C at 261.6 Hz and keep doubling the frequency to preserve the 2:1 simple frequency ratio that defines an octave interval. Table 13 shows what you get.

TABLE 13 Consecutive Octave Intervals, Going Up Seven Octaves

Note Frequency (Hz)
  • Middle C
  • C, one octave up
  • C, two octaves up
  • C, three octaves up
  • C, four octaves up
  • C, five octaves up
  • C, six octaves up
  • C, seven octaves up from Middle C
  • 261.6
  • 523.2
  • 1,046.4
  • 2,092.8
  • 4,185.6
  • 8,371.2
  • 16,742.4
  • 33,484.8

Have a look at the last frequency in Table 12 and compare it with the last frequency in Table 13.

They’re both supposed to be the note C, seven octaves above Middle C, right? So the two frequencies are supposed to be exactly the same, aren’t they?

But they ain’t.

The ratio between them, 33,941.6 Hz : 33,484.8 Hz, boils down to a ratio of 1.0136:1, instead of 1:1.

Dang, for the third time.

That ratio of 1.0136:1 is called the Pythagorean comma. (In music, a tiny interval is called a comma.)

The Pythagorean comma caused all sorts of havoc with instrument tuning for more than 2,000 years after Monty Pythagor died of laughter, without telling anybody how to fudge the Pythagorean comma and stay in tune.

(Chapter 5 discusses some clever jiggery-pokery, called equal temperament, that gets around the Pythagorean comma and cures all problems with scales forever. Well, sort of.)

< Previous   Next >