Mathematical induction is a method on proving a statement is true
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The second harmonic is the one you get from the string vibrating in halves. Each half of the string vibrates at twice the frequency of the whole string. The 2:1 relationship between the pitches of the first and second harmonics is called an octave. (I know that the word suggests the number eight, not the number two. Don’t worry about it.) The pitch that’s an octave above middle C has a frequency of 2 Hz, and it is also called C. Both of these notes have the same letter name because in Western convention, notes an octave apart from each other are considered to be “the same note“. The important concept here is that you can move up an octave from any pitch by doubling its frequency. You can also move down an octave from any pitch by halving its frequency.