The math behind perfect 5th tuning

[Niklas Eliasson]

Robert Scott wrote:

>   For that to happen you would need to have the 3rd partial
>   inharmonicity equal to the 2nd partial inharmonicity (i.e. both at
>   3.26 cents.)  - a neat trick if you could do it.
>

In theory - no problem! If we are talking middle register, just shorten
the lengths with about 20%, keep the same diameter on the strings and
lower the tension so we get the same tones again. Then thats about where
we will end, although the strings will sound like churchbells...
(I couldnt resist expand your calculations a little bit!)

> Getting back to the consequences of perfect 5th tuning, let's look at
> 4ths and 3rds:
>
>   4ths:   r ^ 5 = 1.335916983,   (sharp by 3.26 cents)

Now, thats the problem, since they are sharp and not flat, like the
fifths! And that is the problem we can not get around. In practice, we
get more sharp fourths.

Now, all this rhymes well with experience, as we all know - if we
tune small pianos, the fifths are likely to get more pure. Big grands,
the other way around. C'est la vie.

Greetings

Niklas Eliasson
Linkoping, Sweden