A143800 In acoustics, using 12-tone equal temperament, the rounded number of semitones in the interval perceived when a vibrating string is divided into n congruent segments.

0, 12, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 54, 55, 56, 56, 57, 58, 58, 59, 59, 60, 61, 61, 62, 62, 63, 63, 63, 64, 64, 65, 65, 66, 66, 66, 67, 67, 67, 68, 68, 68, 69, 69, 69, 70, 70, 70, 71, 71, 71, 71, 72, 72, 72, 73, 73, 73, 73, 74

Offset: 1

Author

Cyril Zhang, Sep 01 2008

Keywords

• easy
• nonn

Comments

In music, these are known as harmonics.

Observe that log_2(n) produces irrational numbers for all n that are not powers of 2, and that dividing a string in half produces an octave interval.

Therefore the only harmonics that are perfectly in tune (equal to an interval in 12-TET) are the octaves, which correspond to all harmonics n that are powers of 2.

Examples

			For n = 3, a(3) = round(log_2(3)*12) = round(19.0195500086539...) = 19 Therefore dividing a string in three equal parts will result in a tone approximately 19 semitones higher, or an octave and a perfect fifth.
		

Links

• Wikipedia, Harmonic series (music)

Programs

Maple

a:= n-> round(12*log[2](n)):
seq(a(n), n=1..70);  # Alois P. Heinz, Nov 07 2019

Formula

a(n) = round(log_2(n)*12).