# Six Notes all Nice Ratios

There's a claim often made, that Maths and Music go together, which is great if you like both - but if, like some of us here, you know you're keen on Maths but haven't so far succeeded in mastering any instrument including a singing voice don't feel bad about it.

It seems very possible that music (and mathematics) has its own way of talking and it's easy to feel on the outside of that conversation. The encouraging thing is that mathematics can help us understand music, and maybe the other way round too.

This is the first of three problems based around this idea (the other two are Pythagoras' Comma and Equal Temperament ):

The Greeks, Pythagoreans especially, were interested in the notes made by plucking a collection of strings of different lengths (I'm guessing that the strings all had the same tension - maybe by hanging equal weights, beyond the section being tested).

The Pythagoreans noticed that nice simple ratios of string length made nice sounds together. For example a length with a half length sounded good, and lengths in a ratio of three to two sounded good also. It didn't seem to matter what the lengths were, so long as there was a nice simple ratio between them.Now making music on two strings is a bit limited, so what they tried to find was a collection of lengths that would all sound good together. For convenience, no length would be more than double the shortest length. They settled on a six point scale (a set of agreeable notes). We'll call the longest length Note 1, and make that length our unit. The shortest length (half a unit) we'll call Note 6.

The length which makes a ratio of two to three with the length for Note 1 turned out to be the fourth note in their scale.

What might be good fractions for the Notes 2, 3, and 5 ?

Hint 1 : Remember, we need nice simple ratios between any two strings in your scale.

For example 5/9 of the unit would probably sound good with Note 1 and 6/7 would also sound OK with Note 1, but 5/9 and 6/7 would probably not sound good together because their length ratio (35:54) is not simple.

Hint 2 : Once you have the ratios for all your strings - you should be able to show that taking any pair of strings, you only once have a ratio with numbers bigger than 10.

Hint 3: You have ratios based on halves and thirds - what else could you try?

We had a good collection of offers:

David from St.Albans, Adam from Ystalyfera, Adrianos, Julian from Wilson's School, and Andrei from Bucharest all sent in their chosen sets of fractions with their reasons.

Here's one set in a table : it shows, for example, that 3/4 is 15/16 of 4/5 (check that for yourself)

And here's the spreadsheet file I used to check the fraction sets which people sent in .(Excel file )

If you check some other possible solutions then you will find that Andrei has found the best set.

The topic of music can make a good connection between science and mathematics

The nature of sound and the working of the ear are rich areas of applied mathematics.

The ratio emphasis follows from harmonics or overtones and rests on ideas like lowest common multiple.

One teacher has offered the following comment :This Stage 4 work makes practice in multiplying and dividing fractions purposeful. And in school invites collaboration between the music and mathematics departments.