Abstract
When Pythagoras investigated frequencies of vibrating strings under tension, he found that by dividing the string in half it sounds an octave higher, by dividing it into three he obtained a note called a perfect fifth with its frequency three times that of the original string. By taking J of the string he found it vibrated at a frequency of the perfect fourth with frequency ratio to the fundamental of 4:3. With simple ratios like these he was able to construct a musical scale which subsequently developed into the diatonic scale which has been with us ever since. However, if Pythagoras had listened very carefully he may have wondered whether he should keep the ratio 4 : 3 for the perfect fourth or substitute another with ratio 11 : 8 related to the eleventh harmonic of the vibrating string. The matter is investigated below.
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