Ratwolf magic triads

The ratwolf fifth is the eleventh root "f" of 128/(20/13),
f=(416/5)^(1/11), of size 695.84 cents. Then (5/13)f^3 is an
approximate 9/7, the ratwolf 9/7 of 433.3 cents, only 1.785 cents
flatter than a just 9/7, and the ratwolf 6/5 is of course 4/f^3.
Taking the product gives us (4/f^3) (5/13 f^3) = 20/13; and the
ratwolf magic major triad is the 1--9/7--20/13 chord composed of a 9/7
follwed by a 6/5, while the ratwolf magic minor chord is 1--6/5--20/13.

These magic chords are based on (9/7)(6/5)~20/13, and
(9/7)(6/5)/(20/13) is 351/350, so this is a characteristic chord of
any system using 351/350 as a comma. Ets for this include 19, 22, 31,
46, 50, 53, 58, 72, 80, 81, 84, 99, 111, 130 and 152, from which we
may cook up a multitude of linear temperaments. This includes the
13-limit meanpop (a version of meantone) temperament one gets by
combining any two of 31, 50, or 81, and in fact simply using the
12-note 50-et meantone scale gives a good meanpop. The 20/13 for the
50-et is exactly 744 cents, compared to 745.786 cents for 20/13.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> The ratwolf fifth is the eleventh root "f" of 128/(20/13),
> f=(416/5)^(1/11), of size 695.84 cents. Then (5/13)f^3 is an
> approximate 9/7, the ratwolf 9/7 of 433.3 cents, only 1.785 cents
> flatter than a just 9/7, and the ratwolf 6/5 is of course 4/f^3.
> Taking the product gives us (4/f^3) (5/13 f^3) = 20/13; and the
> ratwolf magic major triad is the 1--9/7--20/13 chord composed of a
9/7
> follwed by a 6/5, while the ratwolf magic minor chord is 1--6/5--
20/13.
>
> These magic chords are based on (9/7)(6/5)~20/13, and
> (9/7)(6/5)/(20/13) is 351/350, so this is a characteristic chord of
> any system using 351/350 as a comma. Ets for this include 19, 22,
31,
> 46, 50, 53, 58, 72, 80, 81, 84, 99, 111, 130 and 152, from which we
> may cook up a multitude of linear temperaments. This includes the
> 13-limit meanpop (a version of meantone) temperament one gets by
> combining any two of 31, 50, or 81, and in fact simply using the
> 12-note 50-et meantone scale gives a good meanpop. The 20/13 for the
> 50-et is exactly 744 cents, compared to 745.786 cents for 20/13.

Can someone please explain the preference for 'cents'? What is wrong
with simple coefficients? Is there some exotic exponential function
involved? And even if there is why not just quote 2^x where x is the
exponent?

--- In tuning@yahoogroups.com, "Peter Wakefield Sault" <sault@c...>
wrote:

> Can someone please explain the preference for 'cents'? What is
wrong
> with simple coefficients? Is there some exotic exponential function
> involved? And even if there is why not just quote 2^x where x is
the
> exponent?

Its a log function, which is useful. What you describe is log base 2,
which works fine, except for the fact that most people use cents. I
was dividing the octave into 612 parts before I came here.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Peter Wakefield Sault" <sault@c...>
> wrote:
>
> > Can someone please explain the preference for 'cents'? What is
> wrong
> > with simple coefficients? Is there some exotic exponential
function
> > involved? And even if there is why not just quote 2^x where x is
> the
> > exponent?
>
> Its a log function, which is useful. What you describe is log base
2,
> which works fine, except for the fact that most people use cents. I
> was dividing the octave into 612 parts before I came here.

Well I know you're going to think me an awkward sod, which indeed I
am so you wouldn't be wrong. I am determined to use what seems to me
to be the most rational descriptives. I use pitch numbers, vibration
numbers and coefficients (which may be simple or logarithmic reals)
to describe what goes on in my little world. I don't particularly
want to know about subdominants and other BDSM terminology. Nor will
I surrender to synthesizer manufacturers and their 'cents'. I'll not
buy any of their stuff again until they make something which is plain
and simple vibration number addressable. They could do worse than
dump MIDI, too...

- Peter

"Music is a very abstruse subject requiring the use of many Greek
words in its description." - Vitruvius (a Roman architect).