Introduction Known as the temperament or scale in music theory, we learn to sing “do-re-mi-fa-sol-la-ti” as early as kindergarten. We also learn that all sounds are essentially produced by the vibrations that hit our eardrum, whose frequency decides the pitch. Then, have you ever thought about how music theorists chose a pitch for each note in “do-re-mifa-sol-la-ti” from an infinite number of options on a number line? In this article, we will introduce you to “Pythagorean temperament,” an early musical scale often attributed to an ancient Greek mathematician, Pythagoras [1, 2]. We will also delve into how music theorists and mathematicians later developed “equal temperament,” which has become the most widely used musical scale in western music since the 19th century [1]. How Did Music Theoris Decide the Pitch of 樂理家如何決定每個音 By Jane Yang 楊靜悠 Pythagorean Temperament First of all, let's understand the concept of an "octave." Mathematically, two sounds are considered an octave apart if their frequencies have a ratio of 2:1. For example, the standardized “middle A” has a frequency of 440 Hz (Footnote 1), or vibrations per second, while the next A which is an octave higher has a frequency of 880 Hz. When they are played at the same time, they sound so consonant that the human brain perceives them as the “same” note but the latter in a higher pitch. This phenomenon is called “octave equivalence [1, 2].” Therefore, to create a musical scale, we only need to consider an octave, or one cycle of “do-remi-fa-sol-la-ti”. We can then multiply or divide the frequencies of the notes in an octave by any power of two to obtain a higher or lower octave because of octave equivalence. Pythagoras also discovered two notes that are a "fifth" apart, meaning their frequencies have a ratio of 3:2, also sound pleasant when played together. Hence, he decided that the task was to create as many ratios of 3:2 and 2:1 as possible to provide convenience for composers. Obviously, Pythagoras should have no access to the accurate frequency of each note, so the tuning was probably completed by hearing the pitch and comparing its relative distance to the base note. However, for a better understanding, let’s unveil the ancient method based on our modern understanding. To decide a frequency to each of the notes in an octave, Pythagoras started with the note A at 440 Hz and multiplied its frequency by 3/2 to obtain the note at 660 Hz. By multiplying 3/2 again, he got 990 Hz.
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