Something easy

1. Take the function 1/x from x=1 to x=2
2. Take trapezoidal approximations of integrals centered at x=5/4
and x=7/4 (use the derivative at those points for the slope of the
trapezoids, but it doesn't really matter what slope you use, it just
looks nicer). So there are two trapezoids, one between 1 and 3/2 and
one between 3/2 and 2

3. Calculate this:

First trapezoid: 1/2 * 4/5 = 2/5
Second trapezoid: 1/2 * 4/7 = 2/7

Total: 24/35. Let's use this to approxmate ln(2). Which is nice
because I need to divide out by ln(2) to convert to log(base2)

(2/5)/(24/35)= 7/12
(2/7)/(24/35) = 5/12

Which is logbase(2) of a perfect fifth and a perfect fourth in 12t-
ET. It's really just the old Riemann sums of course. So, just like 2^
(7/12) approximates a perfect fifth, trapezoids approximate the
integral. Of course, perfection would be 2^(log2(3/2))=3/2. I don't
know, it's not much but I need to start somewhere.

Interesting that 35/24 appears on Gene's 7-limit lattice. And that
it approximates 1/ln(2) which I guess is log(2)e

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> 1. Take the function 1/x from x=1 to x=2
> 2. Take trapezoidal approximations of integrals centered at x=5/4
> and x=7/4 (use the derivative at those points for the slope of the
> trapezoids, but it doesn't really matter what slope you use, it
just
> looks nicer). So there are two trapezoids, one between 1 and 3/2
and
> one between 3/2 and 2
>
> 3. Calculate this:
>
> First trapezoid: 1/2 * 4/5 = 2/5
> Second trapezoid: 1/2 * 4/7 = 2/7
>
> Total: 24/35. Let's use this to approxmate ln(2). Which is nice
> because I need to divide out by ln(2) to convert to log(base2)
>
> (2/5)/(24/35)= 7/12
> (2/7)/(24/35) = 5/12
>
> Which is logbase(2) of a perfect fifth and a perfect fourth in 12t-
> ET. It's really just the old Riemann sums of course. So, just like
2^
> (7/12) approximates a perfect fifth, trapezoids approximate the
> integral. Of course, perfection would be 2^(log2(3/2))=3/2. I don't
> know, it's not much but I need to start somewhere.
>
> Interesting that 35/24 appears on Gene's 7-limit lattice. And that
> it approximates 1/ln(2) which I guess is log(2)e
>

Just for fun, here are the convergents to ln 2 using this method:
(1 trapezoid, 2 trapezoid, 3 trapezoid, 4 trapezoid)

2/3 .666666
24/35 .685714
478/693 .689755
13344/19305 .69122
Actual .69314718

These produce just divisions of the octave, obtaining 1-ET, 12-ET,
239-ET and 6672-ET

First: 2^1 for the octave C-C' Just (2-limit)
Second: 2^7/12 and 2^5/12 for C-G-C'Just (3-limit)
Third: 2^99/239, 2^77/239 and 2^63/239 for C-F-A-C' Just (5-limit)
Fourth: 2^2145/6672, 2^1755/6672, 2^1485/6672 and 2^1287/6672 for
C-E-G-Bb-C' Just (7-limit)

(I know, just easy peezy calculus)

Hi Paul,

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:

> (I know, just easy peezy calculus)

That really gave me a chuckle in this "lofty" forum,
because i have a 4-year-old piano student who, when she first
tries to play a new piece which turns out to be not difficult
for her, likes to tell me that it's "easy peezy".

:-)

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Paul,
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
>
> > (I know, just easy peezy calculus)
>
>
> That really gave me a chuckle in this "lofty" forum,
> because i have a 4-year-old piano student who, when she first
> tries to play a new piece which turns out to be not difficult
> for her, likes to tell me that it's "easy peezy".
>
> :-)
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software

I teach piano too, had five students at one point, but nobody under
12. (Well not since high school anyway) The full phrase is "easy
peezy lemon squeezy" which I actually got from a PBS special on
Stephen Hawkings!
>