# How Mathematics Has Helped Develop Music

Issac Carstensen. 16 December 2016.

Many people question how mathematics is practically used on a day to day basis, however, what those people don’t know is that mathematics has helped to construct the fundamentals of music. Music is thought of as an expressive subject and isn’t typically associated with mathematics, but without mathematics, music wouldn’t sound as it does today.

It may surprise many people to know that the 1st ever form of tuning musical instruments was created by Pythagoras, the famous Greek mathematician. He found that different strings would naturally vibrate when another string was played, which depended on various factors such as their length. There were 2 ratios that defined whether a string would vibrate or not, 2:1 and 3:2. So for example, if a string has the length of 1m, and was hit by something, a string with the length of 2m would also vibrate. These 2 ratios caused him to create a tuning system with these ratios:

A scale consists of 7 different notes out of a possible 12. The 8th note it equivalent to the 1st note, just at a higher pitch, due is due to the ratio of 2:1. The problem with this system was that the 2 different ratios could be equal to each other when applied. In this case, say that the 1st note had a value of 100Hz, if you multiplied that value by 1.5 twelve times (or multiply it by), then the new value is 12974.6337891. This new note is theoretically the same note as the previous one, just at a higher pitch, but when this value is divided by 2 multiple times, the value that is closest to the original 100Hz is 101.364326477Hz. This meant that this system was unsuitable as the music would gradually get more out of tune the more the ratio of 3:2 was used. There have been many different tuning systems that have tried to improve on the Pythagoras tuning system over the past 2 millennia, but they all led to the current tuning system, known as Equal Temperament. The concept of this tuning system was that all notes should have an even interval between each other, meaning that nothing sounds wrong or out of tune. Since there are 12 different notes, this caused the ratio between each note to be 1:25/12.

Since 25/12 is an irrational number, it’s very difficult to compare it against the original Pythagoras system or any other tuning system. This caused musicians to create a different form of measuring musical intervals, which is known as cents. Cents are calculated using logarithms, and are based around equal temperament, so the interval between 1st and 2nd note in equal temperament is 100 Cents, the 1st and 3rd is 200 cents etc… The formula for cents is cents = 1200log2r, where r is the ratio.

So overall, music that exists nowadays is due to mathematics, and that’s only one example. There are multiple different things that maths has affected and developed that many people wouldn’t even realise.

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