11 ts Each Note? 符的音高？ However, this exceeded the desired octave range (i.e. greater than 880 Hz), so he divided it by two to get the note equivalent to it at 495 Hz. Pythagoras repeated this process of multiplying by 3/2, and dividing by two if the resulting frequency exceeds 880Hz, until he obtained a musical scale consisting of seven nonequivalent notes which is enough to play simple melodies [1]. He rearranged those frequencies in order, creating a musical temperament very similar to the one we use today (Table 1). Equal Temperament However, the seven notes in Pythagorean temperament are just enough for playing simple melodies. Before we examine the problem of Pythagorean temperament, let’s look at the modern system called "equal temperament". This temperament Table 1 The frequencies of notes and their ratios with respect to the note A in Pythagorean temperament. The values are rounded off to the nearest integer. divides an octave into 12 equal musical intervals. Keep in mind that our brain perceives the distance of musical interval by ratio instead of difference. Therefore, the frequencies of each note in a scale should have an exponential relation, with a ratio r between each pair of adjacent notes satisfying r12 = 2, i.e. r = 21/12. By multiplying the starting frequency by the ratio r = 21/12 for 12 times, we obtain the frequencies of all the notes within an octave (Table 2). Key Change So, why is equal temperament preferred over Pythagorean temperament? You may have heard of a musical jargon called “key change” before. Actually, the mathematical implication of key change is to multiply the frequency of each note of a melody by a constant number. After performing this trick, human brains will still perceive the two melodies as the same since the musical interval (i.e. the frequency ratios) between any two adjacent notes are retained [1]. For example, a melody that plays 440Hz, 660Hz, and 733.3Hz in order is considered equivalent to a melody that plays 550Hz, 825Hz and 916.6Hz. Key change in music usually helps musicians express their feelings: Changing to a higher key in the midway of a piece of music can express excitement or encouragement, Table 2 The frequencies of notes and their ratios with respect to the note A in equal temperament. The values are rounded off to the nearest integer. The frequencies in shaded cells are played by the black keys of a piano. Ratio 1 9/8 81/64 4/3 3/2 27/16 243/128 2 Frequencies(Hz) 440 495 557 587 660 743 835 880 Ratio 1 21/12 22/12 23/12 24/12 25/12 26/12 Frequencies(Hz) 440 466 494 523 554 587 622 Ratio 27/12 28/12 29/12 210/12 211/12 2 Frequencies(Hz) 659 698 740 784 831 880
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