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Going from D-sharp to A would result in a ratio of 7/5 again. However, if only rational numbers are used, then going from A to D-sharp would result in a ratio of 7/5. The resultant ratio would also be equal to 2. I can go from A to A directly, with the ratio of frequencies being 2, but I can also go from A to D-sharp, with six half-steps, and then from D-sharp to A, with six more half-steps. For example, from A to A (an octave higher), there are twelve half-steps. With an equal tuning system, the ratios between two intervals with the same number of half-steps will be the same. The octave is divided into twelve half-steps, with the frequencies of each half-step and their rational approximations shown below: Musical instruments are tuned to successive twelfth-roots of 2 so that the ratio between two similar intervals is the same, thus ensuring overall harmony when different instruments play together. Thus, one final change is required for our theory: Two frequencies are harmonious if the ratio of their frequencies is sufficiently close to a rational number, with a small denominator in its reduced form. 554:440 is approximately equal to 5:4, and 659:440 is also approximately equal to 3:2, within an error of 1%. Upon closer examination, the frequencies involve closely approximate rational numbers with small denominators. Yet, it is one of the most harmonious musical pieces. 87 No.7 initially only uses the pitches A, C-sharp, and E, with frequencies of 440 Hz, 554.37 Hz, and 659.26 Hz respectively. Yet, many pairs of frequencies still sound harmonious. None of the ratios between two different notes are rational numbers. The most common tuning system today is known as equal-temperament, with the ratios of frequencies being successive twelfth-roots of 2: Yet, musical instruments are rarely tuned such that the ratio of frequencies of two notes is a rational number with a small denominator. Perhaps 2:1 and 5:4 are harmonious as their denominators are small, while 14:13 and 28:23 are cacophonous as their denominators are large. From this observation, we can come up with a new theory: Two frequencies are harmonious if the ratio of their frequencies is a rational number, with a small denominator in its reduced form. It seems that our theory has been defeated, as 14:13 and 28:23 are both rational numbers, but produce cacophonous sounds. To test our theory, let us listen to some more frequencies: Conversely, if the ratio of the two frequencies is an irrational number, they will be cacophonous, such as √3:1 or π:1. It seems that as long as the ratio of the two frequencies is a rational number, then they are harmonious. Set the first tab to 1 and the second tab to Frequency 2, and click “Play” on both tabs) (To listen to these frequencies, open two tabs of. But what makes sounds different from others? Why is listening to an orchestral performance pleasing and harmonious, while your neighbour singing in the shower is irritating and cacophonous? Let’s examine the frequencies of harmonious and cacophonous sounds.
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So far, we have discussed what sounds are, and how we can hear them. These vibrating air molecules collide against your eardrum, and you hear them as a musical note
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If the sound wave stretches air 500 times and squeezes it 500 times per second, its frequency is 500 Hz. This is the frequency of the sound wave, measured in Hertz (Hz). As the sound wave moves, air is alternately stretched out and squeezed, many times per second. These regions are called rarefactions and compressions respectively. Sound waves propagate throughout the air, stretching out the air at some places and squeezing the air together at other places. These molecules vibrate at certain frequencies, which gives rise to the pitch of the musical note. When you hear a beautiful work of music, what makes it pleasing? How do the different frequencies of music combine to create the overall harmony? Musical notes, like all other sounds, are vibrations of molecules in the air.