You decide to circumvent the tyranny of the twelve even-tempered semitone musica

Question

You decide to circumvent the tyranny of the twelve even-tempered semitone musical scale of Western civilization by introducing your own 17 tone scale to cover the octave in equal percentage steps. How close can you get to the interval which in Western music is called the "third (1.250)"? (Express your answer as a percentage deviation of the upper note from the just value, keeping the sign and rounding it off to the nearest tenth of a percent.) (Note: The % symbol is not necessary to include here. Ex: if your answer is .1%, write ".1" and not ".001")

Explanation / Answer

12 tone scale: ? Just intonation of a forth interval is 4/3 = 1.333333333 decimal ratio equivalent to 498.044999 cent. Equal temperament interval of a forth is 2^(5/12) = 1.334839854 decimal ratio equivalent to 500.000000 cent. 17 tone scale: Interval of 2^(6/17) = 1.277161684 decimal ratio equivalent to 423.529412 cent. ? Interval of 2^(7/17) = 1.330312058 decimal ratio equivalent to 494.117647 cent. Interval of 2^(8/17) = 1.385674339 decimal ratio equivalent to 564.705882 cent. Help: Conversion of Intervals - ¢ = cent

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