History of 50 equal

Monz lists

1558 Zarlino (nearly identical to 2/7-comma meantone, the first
meantone to be described with mathematical exactitude)

1710 Konrad Henfling

1759 Robert Smith (as an approximation to his ideal 5/18-comma
meantone system)

1835 Wesley Woolhouse (almost identical to his 7/26-comma 'optimal
meantone')

1940s Tillman Schafer

Googling does not net me anything useful about Konrad Henfling, and
what turns up about Tillman Schafer seems to be 19-et and 31-et, not
(19+31)-et. I've got Robert Smith's book in front of me, and he
definately and explicitly mentions 50-et, though he is not proposing
that it be used in practice. It is an approximation to 5/18-comma
meantone, which he obtains by a process of weighted averaging over
three octaves worth of consonances, and is almost exactly ten times
closer to 50-et than Zarlino's 2/7-comma. Smith also averges over four
octaves and obtains 11/40-comma meantone, which is not very different,
being well approximated by 50-et also.

I'd be interested to know more about Konrad Henfling, but it seems
quite possible that Robert Smith is the first to discuss 50-et and to
notice, as Woolhouse does later, that it is a reasonable choice as an
optimum or near optimum.

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

> I've got Robert Smith's book in front of me, and he
> definately and explicitly mentions 50-et, though he is not proposing
> that it be used in practice. It is an approximation to 5/18-comma
> meantone, which he obtains by a process of weighted averaging over
> three octaves worth of consonances, and is almost exactly ten times
> closer to 50-et than Zarlino's 2/7-comma.

If I am decyphering Smith properly, which I am not sure of, the point
of this averaging process was to find an approximation to the fifth
with brat -9/4, which is one that I put on my list of "magic
meantones" a while back:

/tuning/topicId_42599.html#42599

5/18 comma meantone has major thirds flat by 1/9 comma, whereas the
brat -9/4 meantone has major thirds flat by 0.110983 comma. Smith at
one point (page 172) mentions that what he's trying to approximate by
1/9 comma is to have the major third flat by 0.11024 comma, but that
is something he didn't find by solving a quartic equation (which is
how it really should be done, if I am interpreting this correctly) but
by screwing around with averagings.

Anyway, I propose that a Smith fifth S is really supposed to be the
positive real root of 3S^4 - 4S - 9 = 0, which gives something very
close to both 50-equal and 5/18 comma.

hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > I've got Robert Smith's book in front of me, and he
> > definately and explicitly mentions 50-et, though he
> > is not proposing that it be used in practice. It is
> > an approximation to 5/18-comma meantone, which he obtains
> > by a process of weighted averaging over three octaves
> > worth of consonances, and is almost exactly ten times
> > closer to 50-et than Zarlino's 2/7-comma.
>
> If I am decyphering Smith properly, which I am not sure of,
> the point of this averaging process was to find an
> approximation to the fifth with brat -9/4, which is one
> that I put on my list of "magic meantones" a while back:
>
> /tuning/topicId_42599.html#42599
>
> 5/18 comma meantone has major thirds flat by 1/9 comma,
> whereas the brat -9/4 meantone has major thirds flat by
> 0.110983 comma. Smith at one point (page 172) mentions
> that what he's trying to approximate by 1/9 comma is to
> have the major third flat by 0.11024 comma, but that
> is something he didn't find by solving a quartic equation
> (which is how it really should be done, if I am interpreting
> this correctly) but by screwing around with averagings.
>
> Anyway, I propose that a Smith fifth S is really supposed
> to be the positive real root of 3S^4 - 4S - 9 = 0, which
> gives something very close to both 50-equal and 5/18 comma.

Paul Erlich has pointed out Robert Smith's work to me,
and in fact he only just a few months ago sent me a photocopy
of an article on Smith's tuning ... i believe it is from
Jorgensen's book. i read it with much interest, but just
haven't had time to write anything about it yet.

-monz

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

> Anyway, I propose that a Smith fifth S is really supposed to be the
> positive real root of 3S^4 - 4S - 9 = 0, which gives something very
> close to both 50-equal and 5/18 comma.

From another remark he makes, it seems as if the positive real root of
3F^3 + 4F - 16 = 0 may be what is intended. These are very close; F is
only 0.003431 cents flatter than S. Both may be regarded as
well-described by 5/18-comma and well-apporximated by 50-equal. F is
29.99910 steps of 50, S is 29.99924 steps, and 5/18 comma is 29.99921
steps.

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

> From another remark he makes, it seems as if the positive real root of
> 3F^3 + 4F - 16 = 0 may be what is intended. These are very close; F is
> only 0.003431 cents flatter than S. Both may be regarded as
> well-described by 5/18-comma and well-apporximated by 50-equal. F is
> 29.99910 steps of 50, S is 29.99924 steps, and 5/18 comma is 29.99921
> steps.

Jorgensen gives the Smith fifth as 3F^3 + 4F - 16 = 0, I find.

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

> Jorgensen gives the Smith fifth as 3F^3 + 4F - 16 = 0, I find.

Another tidbit from Jorgensen is that he deciphers Harrison in the
same way as Lucy does, with a fifth of 600+300/pi cents; his cited
source being the same, Smith's _Harmonics_.

hi Gene,

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

> Another tidbit from Jorgensen is that he deciphers
> Harrison in the same way as Lucy does, with a fifth of
> 600+300/pi cents; his cited source being the same,
> Smith's _Harmonics_.

i mention on my webpage that LucyTuning resembles
88-edo and 3/10-comma meantone:

http://tonalsoft.com/enc/lucy.htm

-monz