RE: Re: summary of Woolhouse's _Essay on Musical Interva ls.. ._

I wrote,

>> I'd have to say that here (which is where it matters),
>> Woolhouse is thinking of 5 as the _odd_ limit, since in an
>> odd limit all intervals are considered equivalently to their
>> inversions.

Joe Monzo wrote,

>I'll say that I have to agree. This is clearly a 5-odd-limit.
>Hmmm... then again, it *could* be a 5-prime-limit too,
>couldn't it?...

I don't think so. In what sense do you think it could?

>Well... the whole point of Woolhouse's book is to present
>and explain tunings that he feels are better than 12-tET
>(or 12-EDO), and considering the fact that he goes thru all
>the trouble of determining 7/26-comma meantone, and finding
>the more practical 50-EDO close approximation to it, and then
>finally accepts 19-EDO as more practical still, it seems
>awfully strange that he uses such strong language when he
>says 'This scale [12-tET] is, *WITHOUT DOUBT* [emphasis mine],
>the best one for such instruments...'.

>Sounds to me like, just for that one moment, he was succumbing
>to the status-quo simply because of the improbability that anyone
>would go thru the trouble to build an instrument to his
>specifications. Maybe it's just the way I'm reading it.

To me, he's being perfectly consistent, because at the time of his writing,
virtually any piece in the repertoire would have called for more than 12
written pitches, so on an instrument with only 12 notes, any of the "better"
meantone tunings would not work -- only a closed 12-pitch system, such as
12-tET, would.

>>> It should be obvious that the 5-limit lattice can be extended
>>> infinitely in both dimensions by adding on additional cells
>>> just like this one (or parts of it), thus proving Woolhouse's
>>> statement.

>> Not sure what it proves.

>It proves what he said in the part I quoted just before that:

> [Woolhouse 1835, p 45]
>
>> the *third-minor*, *third-major* and *fourth*, are the only
>> concords necessary to be considered, because the others are
>> merely the inversions of these, and we know that any error
>> which may increase or diminish a concord, will have precisely
>> the same effect in decreasing or increasing its inversion

>You can see plainly on the lattice that all these intervals
>and their inversions can be added on infinitely in any direction
>without involving any other intervals, which was his point.

I think you're misunderstanding something.

>A side note: those who have looked at my webpage of this will
>see that I've changed 'tET' to 'EDO' in every case. I did
>that because Woolhouse specifically states that the 'octave'
>is always to be a 1:2 ratio.

Seems to me like 'tET' would be better then, since 'tET's always assume a
1:2 octave, at least around here (where we discuss such things as
27.35-tET).

>> Smith's ideal tuning was almost exactly 5/18-comma meantone,
>> which is close to 50-tET but on the other side of it relative
>> to 7/26-comma meantone.

>How do you know that?! Do you have access to Smith's book?!
>Tell us more!!

Well, at first I thought it was exactly 5/18-comma meantone, based on
Mandelbaum. For a derivation of 5/18-comma meantone as an "optimal" tuning
(by different criteria from Woolhouse's, of course), see my posting from
Brett Barbaro's address in TD 134.10 (5/7/99). But then I read Jorgenson,
who gave Smith's equation (supposedly derived from beat rates), and the
solution to that equation is ever-so-slightly different from 5/18-comma
meantone. I actually don't know who's right. Someone should check Barbour as
well.

>Paul, let me know if you mind my
>adding your postings to the webpage - I think they're
>valuable commentary

Go ahead!

>I'd love to be able to figure out an accurate lattice diagram
>of the actual Woolhouse/Erlich 7/26-comma meantone, to accompany
>the simple one of its 5-limit JI implications. Actually, my
>lattice formula can probably already do it...

I have no idea what you mean.

I wrote,

>TD 134.10 (5/7/99).

That date should be 4/7/99.

I wrote,

>> Smith's ideal tuning was almost exactly 5/18-comma meantone,
>> which is close to 50-tET but on the other side of it relative
>> to 7/26-comma meantone.

Joe Monzo wrote,

>How do you know that?! Do you have access to Smith's book?!

In TD 163.10 (5/5/99), I (again using Brett Barbaro's address) wrote,

>Tonight I finally got a hold of copies of Mandelbaum's dissertation and
Jorgenson's tome. The first few pages of both >contain information on the
proposed tuning system of Robert Smith. According to Mandelbaum, Smith's
ideal fifth in his
>1759 _Harmonics_ is flat by, you guessed it, 5/18 of a comma. Mandelbaum's
description makes it seem as if Smith >might have used the same derivation
as me. According to Jorgenson, however, Smith's ideal fifth derived from
equal-beating
>considerations and satisfies the equation 3x^3+4x=16. That implies a fifth
0.0027 cents smaller than that of 5/18-comma >meantone. Though this is
really splitting paramecial hairs, anyone know who's right about Smith?