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Search: id:A143800
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| A143800 |
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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. |
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+0
1
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| 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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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In music, these are known as harmonics.
Observe that log2(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.
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LINKS
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Wikipedia, Harmonic series(music)
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FORMULA
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a(n) = round(log2(n)*12)
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EXAMPLE
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For n = 3, a(3) = round(log2(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.
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PROGRAM
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(Other) //language: C/C++ #include <stdio.h> #include <math.h> //for log #define round(x) (x<0?ceil((x)-0.5):floor((x)+0.5)) int main(){ const NUMBER_OF_TERMS = 100; //change for more or fewer terms const TONES_PER_OCTAVE = 12; //change for different number of increments making an octave long harmonic; for(harmonic=1; harmonic<=NUMBER_OF_TERMS; ++harmonic){ printf("%ld ", (long) round(log(harmonic)/log(2)*TONES_PER_OCTAVE)); } return 0; }
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CROSSREFS
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Equals to round(log2(A000027)*12), since A000027 represents the natural numbers.
Sequence in context: A100541 A056688 A107911 this_sequence A003335 A030609 A053752
Adjacent sequences: A143797 A143798 A143799 this_sequence A143801 A143802 A143803
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KEYWORD
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easy,nonn
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AUTHOR
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Cyril Zhang (cyril.zhang1(AT)gmail.com), Sep 01 2008
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