Convergent series to log2(3/2)

Here's an example of the sort of convergent series which arises from
the sequence of convergents of a continued fraction, by taking
differences:

Convergents: 0, 1, 1/2, 3/5, 7/12, 24/41, 31/53, 179/306, 389/665 ...

Series: 1-1/2+1/10-1/60+1/492-1/2173+1/16218-1/203490+... = log2(3/2)

The denominators are products of denominators of the convergents:

1-1/(1*2) + 1/(2*5) - 1/(5*12) + 1/(12*41) - ...

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Here's an example of the sort of convergent series which arises from
> the sequence of convergents of a continued fraction, by taking
> differences:
>
> Convergents: 0, 1, 1/2, 3/5, 7/12, 24/41, 31/53, 179/306, 389/665 ...
>
> Series: 1-1/2+1/10-1/60+1/492-1/2173+1/16218-1/203490+... = log2(3/2)
>
> The denominators are products of denominators of the convergents:
>
> 1-1/(1*2) + 1/(2*5) - 1/(5*12) + 1/(12*41) - ...

Oh. Is this what Mark Gould had in mind? Mark?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Here's an example of the sort of convergent series which arises from
> the sequence of convergents of a continued fraction, by taking
> differences:
>
> Convergents: 0, 1, 1/2, 3/5, 7/12, 24/41, 31/53, 179/306, 389/665 ...
>
> Series: 1-1/2+1/10-1/60+1/492-1/2173+1/16218-1/203490+... = log2(3/2)
>
> The denominators are products of denominators of the convergents:
>
> 1-1/(1*2) + 1/(2*5) - 1/(5*12) + 1/(12*41) - ...

The 179/306 convergent gives rise to my favorite system, 612-et.
I define 1/612 octave or 1/51 12-et semitone as a tempered schisma ($).
1C (cent) = precisely 0.51$. Important intervals:

V = 358$ (fifth: 3/2 ~ 701.955001C ~ 357.997050$)
III = 197$ (third: 5/4 ~ 386.313714C ~ 197.019994$)
c = 11$ (comma: 81/80 ~ 21.506290C ~ 10.968208$)
s = 1$ (schisma: 32805/32768 ~ 1.9537208C ~ 0.9963976$)

The last line explains my calling $ the tempered schisma. :)

--- In tuning-math@yahoogroups.com, "Glenn Leider" <GlennLeider@n...>
wrote:

> The last line explains my calling $ the tempered schisma. :)

I used to just call it a schisma. Cents seem to be pretty well
established, but maybe with the "$" notation this could still fly.
Keeping some of the numbers in mind allows you to do easy hand
calculations.