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## 'Six Notes All Nice Ratios' printed from http://nrich.maths.org/

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.

There are plenty of interesting questions to think about. For
example, why are particular sounds (notes) thought to be good
together?

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 ?