> However, we typically say a fifth has a 1:1.5 Ratio. This is an approximation due to the overtone series. When a string vibrates (or a horn vibrates, take you

> However, we typically say a fifth has a 1:1.5 Ratio. This is an approximation due to the overtone series. When a string vibrates (or a horn vibrates, take your instrument), it vibrates at a fundamental frequency and several harmonics above it as well. I cannot find it exactly, but I believe it is closer to 1:1.48 to make overtones work. I think you got this backwards. A perfect fifth is precisely 1:1.5 — or, put differently, the third harmonic is 3x times the root/first harmonic's frequency, and, transposed down an octave, that gives us the fifth. Modern instruments use equal temperament tuning, where the frequency ratio between any two consecutive steps is the same, and you go from any note to a note at double the frequency in 12 equal steps — this means each step has to correspond to multiplying the base frequency by the 12th root of 2. In this tuning, the fifth corresponds to 7 steps, which corresponds to a multiplier of 2^(7/12) = 1.498.