Woolhouse and 730-EDO

From: "monz"

> the first mention i know of for arc measurement is
> from W. S. B. Woolhouse:
>
> http://tonalsoft.com/monzo/woolhouse/essay.htm
>
> look under the section "730-EDO as a basic unit":

I knew the idea of looking at the octave (or any other period) as a circle divisible into arcs is too obviously to have originated with some guy in early 21st century Texas. I remember reading about someone who did that long ago, and Harrison's pi-based meantone also comes to mind.

Now about 730-tone. The fifth is 427 degrees, the third is 235. It's weak in 7- and 11-limit, however: 7/4 is 589 and 11/8 is 335, but both intervals are a third of a degree sharp. 665-tone (fifth = 389, third = 214) has a decent approximation of 7/4 (537 degrees), though 11/8 is horrible, almost exactly halfway between 305 and 306.

The comma in 730-tone is 13 degrees, while in 665-tone it's 12 degrees, which of course is divisible by 3 and 4. I'm wondering why Woolhouse liked 7/26-comma meantone so much (it of course lies between third- and quarter-comma).

<vanity>Now here's a meantone I like: set the generator to 697.085365 cents (lower the fifth by one-sixth of 896/729), so the augmented fourth is 7/5. 105-TET is great for this.</vanity>

~Danny~

> <vanity>Now here's a meantone I like: set the generator to 697.085365 > cents
> (lower the fifth by one-sixth of 896/729)

I meant 3584/3645, which is the difference between 729/512 and 7/5. (Where in the hell did I come up with 896/729?)

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:

> Now about 730-tone. The fifth is 427 degrees, the third is 235. It's
weak in
> 7- and 11-limit, however: 7/4 is 589 and 11/8 is 335, but both
intervals are
> a third of a degree sharp. 665-tone (fifth = 389, third = 214) has a
decent
> approximation of 7/4 (537 degrees), though 11/8 is horrible, almost
exactly
> halfway between 305 and 306.

All of which is part of why I used to use 612 instead.

> The comma in 730-tone is 13 degrees, while in 665-tone it's 12 degrees,
> which of course is divisible by 3 and 4. I'm wondering why Woolhouse
liked
> 7/26-comma meantone so much (it of course lies between third- and
> quarter-comma).

It's what you get by applying an equally-weighted least squares
analysis in the 5-limit.

> <vanity>Now here's a meantone I like: set the generator to
697.085365 cents
> (lower the fifth by one-sixth of 896/729), so the augmented fourth
is 7/5.
> 105-TET is great for this.</vanity>

I was calling that the Aaron Johnson fifth a few days back. Do you
have a prior claim? It's good for playing the Tristan chord in.

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
> > <vanity>Now here's a meantone I like: set the generator to 697.085365
> > cents
> > (lower the fifth by one-sixth of 896/729)
>
> I meant 3584/3645, which is the difference between 729/512 and 7/5.
(Where
> in the hell did I come up with 896/729?)

I dunno, but it seems easier simply to call it (56/5)^(1/6).

From: "Gene Ward Smith":

>> <vanity>Now here's a meantone I like: set the generator to
> 697.085365 cents >> (lower the fifth by one-sixth of 896/729), so the augmented fourth
> is 7/5. >> 105-TET is great for this.</vanity>
> > I was calling that the Aaron Johnson fifth a few days back. Do you
> have a prior claim? It's good for playing the Tristan chord in.

Nope, I just came up with the idea a few hours ago. ;)