Pop on through to the other side

Speaking of breakthroughs, I had my own little one last
night - between
waking and sleeping, incidentally, Paul. You do math
when you're awake
and music in your sleep; for me it's the other way
around. :-) Someone
certainly has noticed this one before but it was new to
me, and is bound to
be new to someone here who can make good use of it.

Any superparticular ratio n:d goes into (n+1):(d-1) just
under 3 times (for d>2).
What this means to tuning theory is that a chain of 3 of
any superparticular can
be used to approximate a certain larger interval.
Impose an interval of
equivalence and you're getting somewhere. I first
became aware of it through
the 7 member ring of Golden horagram #2 of Wilson's
Scale Tree, which has
a generator of approx. 10:9, a stack of three of which
approximates 11:8 quite
satisfactorily.

Here's a list within reasonable bounds. 4:3 has to be
stretched awfully far to
approximate 5:2; and three 5:4's = 6:3 = 2:1, closing
the scale too early to be
useful. So we start with 6:5. The columns:

A is the superparticular, n:d.
B is A in cents.
C is (n+1):(d-1), the larger ratio approximated by a
chain of 3 of A (in reduced
terms).
D is C in cents.
E is D/B; notice how it approximates more closely as n
(and d) gets higher.
F is D-(3*B), the difference in cents between the larger
ratio being approximated
and the chain of 3 superp's.
G is a generator that offers a compromise by stretching
B up, and D down, by
the same amount.

A B C D E F G
6:5 315.64 7:4 968.83 3.069389039 21.90 321.12

7:6 266.87 8:5 813.69 3.048988365 13.07 270.14

8:7 231.17 3:2 701.96 3.036477791 8.43 233.28

9:8 203.91 10:7 617.49 3.028236978 5.76 205.35

10:9 182.40 11:8 551.32 3.022514925 4.11 183.43

11:10 165.00 4:3 498.04 3.018377187 3.03 165.76

12:11 150.64 13:10 454.21 3.015286892 2.30 151.21

13:12 138.57 14:11 417.51 3.012917277 1.79 139.02

14:13 128.30 5:4 386.31 3.011060009 1.42 128.65

15:14 119.44 16:13 359.47 3.009577081 1.14 119.73

16:15 111.73 17:14 336.13 3.008374084 0.94 111.97

17:16 104.96 6:5 315.64 3.007384646 0.78 105.15

18:17 98.95 19:16 297.51 3.006560983 0.65 99.12

19:18 93.60 20:17 281.36 3.005867986 0.55 93.74

20:19 88.80 7:6 266.87 3.005279378 0.47 88.92

21:20 84.47 22:19 253.80 3.004775177 0.40 84.57

22:21 80.54 23:20 241.96 3.004339970 0.35 80.62

23:22 76.96 8:7 231.17 3.003961708 0.30 77.03

It could be carried further. Of interest is that two
ratios from column A appear
also in C. Three 17:16s approx 6:5, three of which
approx 7:4. Three 23:22s
approx 8:7, three of which approx 3:2.

Looks like it also works as n:(n-2) approximates
(n+1):((n-2)-1) in a chain of
two. 5:3 has to be stretched too far; 7:5 approx 2:1,
so we start with two
stretched 9:7s approx 10:6 = 5:3.

A B C D E F G
9:7 435.08 5:3 884.36 2.032615585 14.19 439.81
11:9 347.41 3:2 701.96 2.020549673 7.14 349.79
13:11 289.21 7:5 582.51 2.014151508 4.09 290.57
15:13 247.74 4:3 498.04 2.010345049 2.56 248.60
17:15 216.69 9:7 435.08 2.007894835 1.71 217.26
19:17 192.56 5:4 386.31 2.006224115 1.20 192.96
21:19 173.27 11:9 347.41 2.005033583 0.87 173.56
23:21 157.49 6:5 315.64 2.004155139 0.65 157.71
25:23 144.35 13:11 289.21 2.003488381 0.50 144.52
27:25 133.24 7:6 266.87 2.002970302 0.40 133.37
29:27 123.71 15:13 247.74 2.002559730 0.32 123.82
31:29 115.46 8:7 231.17 2.002228828 0.26 115.54
33:31 108.24 17:15 216.69 2.001958226 0.21 108.31
35:33 101.87 9:8 203.91 2.001734105 0.18 101.93
37:35 96.20 19:17 192.56 2.001546392 0.15 96.25
39:37 91.14 10:9 182.40 2.001387605 0.13 91.18

Some interesting patterns in there. Again, some ratios
from A appear in C.
And notice that every other ratio in C reduces to a
superparticular, several
of which provide interaction with the first list.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html

--

"David J. Finnamore" wrote:

> Any superparticular ratio n:d goes into (n+1):(d-1) just
> under 3 times (for d>2).

That's just *over* 3 times, which is obvious when looking at the chart.
I was thinking "3*(n/d) < (n+1):(d-1)."

Aack. I've got to get my email preferences fixed so the lines don't wrap
funny. Sorry about that.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

Hi David

> Any superparticular ratio n:d goes into (n+1):(d-1) just
> under 3 times (for d>2).
> What this means to tuning theory is that a chain of 3 of
> any superparticular can
> be used to approximate a certain larger interval.
> Impose an interval of
> equivalence and you're getting somewhere. I first
> became aware of it through
> the 7 member ring of Golden horagram #2 of Wilson's
> Scale Tree, which has
> a generator of approx. 10:9, a stack of three of which
> approximates 11:8 quite
> satisfactorily.

This may be an easy way to see it, and maybe suggest other
ideas?

(n+1)/n squared is (n^2+2n+1)/n^2

(n+2)/(n-1) divided by (n+1)/n
is (n^2+2n)/(n^2-1)

Denominator and quotient of first ratio are both one more
than those for the second ratio, so they will be very close
and get closer as n increases.

> Looks like it also works as n:(n-2) approximates
> (n+1):((n-2)-1) in a chain of
> two. 5:3 has to be stretched too far; 7:5 approx 2:1,
> so we start with two
> stretched 9:7s approx 10:6 = 5:3.

Could write that as (substituting k = n-1)
(k+1):(k-1) approximates (k+2) : (k-2) in a chain of two:

(k+1):(k-1) squared is (k^2+2k+1)/(k^2-2k+1)
and then that is close to (k^2+2k)/(k^2-2k)
= (k+2)/(k-2)

Hope this may help if there are any more of these relationships
to be discovered.

Robert

I wrote:

> > Any superparticular ratio n:d goes into (n+1):(d-1) just
> > under 3 times (for d>2).

Robert Walker wrote:

> This may be an easy way to see it, and maybe suggest other
> ideas?
>
> (n+1)/n squared is (n^2+2n+1)/n^2
>
> (n+2)/(n-1) divided by (n+1)/n
> is (n^2+2n)/(n^2-1)
>
> Denominator and quotient of first ratio are both one more
> than those for the second ratio, so they will be very close
> and get closer as n increases.

Oh, cool.

> > Looks like it also works as n:(n-2) approximates
> > (n+1):((n-2)-1) in a chain of
> > two. 5:3 has to be stretched too far; 7:5 approx 2:1,
> > so we start with two
> > stretched 9:7s approx 10:6 = 5:3.
>
> Could write that as (substituting k = n-1)
> (k+1):(k-1) approximates (k+2) : (k-2) in a chain of two:
>
> (k+1):(k-1) squared is (k^2+2k+1)/(k^2-2k+1)
> and then that is close to (k^2+2k)/(k^2-2k)
> = (k+2)/(k-2)

Nice. Always helpful to understand why something works, not only *that* it works. I don't know whether there are any more useful relationships like this or not. The fact that we've moved from chains of 3 to chains of 2 suggests that the next step would yield "chains" of 1 - itself.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

Hey that's really neat!

But be careful about mixing up linear and logarithmic points of view
in your descriptions. e.g. cubed vs. 3 times.

At the next microtonal party I go to: "Gimme a superparticular, any
one. .. Ok I'll bet you that 3 of those ..." etc. :-)

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> Hey that's really neat!
>
> But be careful about mixing up linear and logarithmic points of
view
> in your descriptions. e.g. cubed vs. 3 times.

Ooo. Right. Thanks.

> At the next microtonal party I go to: "Gimme a superparticular, any
> one. .. Ok I'll bet you that 3 of those ..." etc. :-)

Amaze your friends! Impress the ladies!

David J. Finnamore