1. Middle A: In the case of C major (one of the easiest modes in music), “do-re-mi-fa-sol-la-ti” corresponds to C, D, E, F, G, A, B respectively in representation. Chosen as a standard note for tuning musical instrument, the “middle A” corresponds to “la” in C major. Although it should have a frequency of 440 Hz by the ISO 16 standard [5], the tune is sometimes set at 442 Hz in some wind bands to cater the wind instruments. 2. Editor’s note: The number 1.0136 is given by 312 / 219, i.e. taking the fifths 12 times while reducing the octaves seven times. while lowering the key may convey sorrow or tranquility. In addition, by lowering the key of a song, a singer whose voice range is too low to cover the high pitch can now sing the song. After understanding the concept of key change, you would discover that the equal temperament adapts to key change perfectly because the ratio of the frequencies between any adjacent notes is a constant [1, 2]. Pianists, for example, only need to move up every note for one key on a piano keyboard tuned with equal temperament to complete the key change, and the finite number of keys on the keyboard is sufficient to cover all notes required for any key changes. On the other hand, the seven notes in Pythagorean temperament don’t suffice. Instead of having a constant ratio, adjacent notes in Pythagorean temperament have a ratio of either 9:8 or 256:243 [2]. We have to continue Pythagoras’ calculation to create more and more notes so that key changes can be performed perfectly from any note. By extending his calculation beyond the first octave, we wish the value will return to the starting point 440 Hz at some point, so that we can get a finite number of notes. Nevertheless, this has been proved impossible, due to the fact that (3/2)n is never a power of two, so we will need an infinite number of black keys for a musical instrument to perform key changes, which is simply not practical [1]. Although Pythagoras was able to get close to the desired frequency 440 Hz, there was still a small discrepancy known as the "Pythagorean comma" [2]. This slightly higher frequency ratio of 1.0136:1 posed challenges for musicians and mathematicians until the invention of equal temperament (Footnote 2). Historical Controversies Over the Invention of Equal Temperament One interesting coincidence is that the equal temperament was invented by the Chinese mathematician, physicist and music theorist Zhu Zaiyu in 1584, and given a mathematical definition by the Flemish mathematician Simon Stevin around the period between 1585–1608 [3]. There are still controversies on who should receive the credit and whether the development was independent [3, 4], but we may never know the truth. Nevertheless, one thing you can take away is that anything we take for granted today may have been the outcome of the struggle of our predecessors for thousands of years, and there may actually be a scientific reason behind it. From Pythagoras' exploration to the invention of equal temperament, these mathematicians have shaped the music we enjoy. So next time you sing "do-re-mi-fa-sol-la-ti", remember the mathematical journey that led to these familiar notes. 引言 我們從幼稚園時期就學會唱「do-re-mi-fa-sol-la-ti」， 而在樂理中，這個概念被稱為音律或音階。我們還學過所有 聲音本質上都是衝擊耳膜的振動，這些振動的頻率決定了 音高。那麼，你有否思考過樂理家是如何從數軸上無窮多的 選擇中，為「do-re-mi-fa-sol-la-ti」裡的每個音符選定音 高呢？在這篇文章裡，我們將介紹「畢氏音律」─ 一種普遍 認定為由古希臘數學家畢達哥拉斯提出的早期音樂音階 [1, 2]。我們還會探討樂理家和數學家後來如何發展出「十二平 均律」，這種音階自19 世紀以來一直都是西方音樂最廣泛 使用的音階 [1]。 畢氏音律 首先讓我們了解「八度」這個概念：在數學上，如果兩個 音符的頻率比為2:1，它們就是相差一個八度，例如標準「中 音A」的頻率為440 Hz（註一），亦即每秒振動440次，那 麼比它高一個八度的A的頻率則為880 Hz。當同時演奏這 兩個音時，重疊的聲音聽起來會和諧得使人腦認為它們是 「相同」的音符，只是後者的音高較高而已，這種相似性被 稱為「等價八度」[1, 2]。 因此，就創造一個音階而言，我們只需要考慮一個八度，
RkJQdWJsaXNoZXIy NDk5Njg=