Nonradical fifths

The theoretical intervals derived for tempering are nearly always
algebraic numbers, though there are exceptions--examples being Golden
Meantone and Lucy Tuning, where the fifth is a transcendental number.
An algebraic number is called a radical if it is of the form p^q,
where p and q are rational numbers. Much of the time the algebraic
numbers used are of this form, but sometimes they are not. Some
meantone fifths which are exceptions are:

Holden 4f^4-3f^3-10 697.656 cents, almost exactly 1/5 comma

Keller f^4+2f-8 697.278 cents, almost exactly 5/23 comma

Smith 3f^3+4f-16 695.978 cents, almost exactly 5/18 comma

Wilson f^4-2f-2 695.630 cents, almost exactly 5/17 comma

This might be a good subject for the Encyclopedia of Tuning.

hi Gene,

what's the "f"?

i'm too busy to write new webpages now ... if you could
write up something more detailed about this, i'll make
a page of it, to be uploaded when the new version of
the Encyclopedia is ready.

-monz

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> The theoretical intervals derived for tempering are nearly always
> algebraic numbers, though there are exceptions--examples being
Golden
> Meantone and Lucy Tuning, where the fifth is a transcendental
number.
> An algebraic number is called a radical if it is of the form p^q,
> where p and q are rational numbers. Much of the time the algebraic
> numbers used are of this form, but sometimes they are not. Some
> meantone fifths which are exceptions are:
>
> Holden 4f^4-3f^3-10 697.656 cents, almost exactly 1/5 comma
>
> Keller f^4+2f-8 697.278 cents, almost exactly 5/23 comma
>
> Smith 3f^3+4f-16 695.978 cents, almost exactly 5/18 comma
>
> Wilson f^4-2f-2 695.630 cents, almost exactly 5/17 comma
>
> This might be a good subject for the Encyclopedia of Tuning.