50

Here are the TM bases for 50-et up to the 19-limit. It might be noted
that in terms of the 50-equal version of meantone, 20/13 maps to -11,
11/8 to -13, and 16/13 to -15, leading to the 12, 14, and 16 note
wolves of the same size.

5-limit
[81/80, 1207959552/1220703125]

7-limit
[16807/16384, 81/80, 126/125]

11-limit
[81/80, 245/242, 126/125, 385/384]

13-limit
[81/80, 245/242, 105/104, 126/125, 144/143]

17-limit
[81/80, 105/104, 126/125, 144/143, 170/169, 221/220]

19-limit
[81/80, 105/104, 126/125, 133/132, 144/143, 153/152, 170/169]

hi Gene,

have you ever seen this?

http://tonalsoft.com/monzo/woolhouse/essay.htm#temp

Woolhouse and Paul Erlich both (independently, about
160 years apart) discovered that 7/26-comma meantone
is an optimal meantone by one type of measure. Woolhouse
then advocates 50-et as a very good approximation to
that tuning, and then 19-et as a more practical (but
not as close) alternative.

good luck trying to find Woolhouse's book itself if
you want it. the only copy i've ever discovered is
at the U. of Pennsylvania library in Philadelphia.

-monz

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

> Here are the TM bases for 50-et up to the 19-limit. It
> might be noted that in terms of the 50-equal version of
> meantone, 20/13 maps to -11, 11/8 to -13, and 16/13 to -15,
> leading to the 12, 14, and 16 note wolves of the same size.
>
>
> 5-limit
> [81/80, 1207959552/1220703125]
>
> 7-limit
> [16807/16384, 81/80, 126/125]
>
> 11-limit
> [81/80, 245/242, 126/125, 385/384]
>
> 13-limit
> [81/80, 245/242, 105/104, 126/125, 144/143]
>
> 17-limit
> [81/80, 105/104, 126/125, 144/143, 170/169, 221/220]
>
> 19-limit
> [81/80, 105/104, 126/125, 133/132, 144/143, 153/152, 170/169]

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Gene,
>
>
> have you ever seen this?
>
> http://tonalsoft.com/monzo/woolhouse/essay.htm#temp

>
> Woolhouse and Paul Erlich both (independently, about
> 160 years apart) discovered that 7/26-comma meantone
> is an optimal meantone by one type of measure.

It's worth noting that Woolhouse found it. It's not too surprising
Paul did, or I did a little later for that matter. It's what anyone
would find who applied unweighted least squares. Gauss invented this
method as a teenager, proved some important facts about it when it is
used to estimate a noisy signal, and it was still of pretty recent
vintage when Woolhouse used it. I don't know if anyone had used it
before Woolhouse to approximate something which was *not* a noisy
signal, but known and exact quantities; this could be an idea original
to him. Certainly, applying it to music must have been, but you quote
Woolhouse as saying:

[Woolhouse 1835, p 46:]

This system is precisely the same as that which Dr. Smith, in his
Treatise on harmonics [Smith 1759], calls the scale of equal harmony.
It is decidedly the most perfect of any systems in which the tones are
all alike.

Is Smith's tuning 50-equal?

I'd also cut Woolhouse some slack on 5-limit vs 5-odd-limit. The place
where he discusses that you *might* interpret to say that 2 and 3/2
are consonances, but 4, 3 or 6/5 are not, but I would presume his
assumption of octave equivalence would be clear from elsewhere.

I also find this comment interesting:

He then analyzes the resources of a 53-EDO 'enharmonic organ', built
by J. Robson and Son, St. Martin's-lane, but says that the number of
keys is too much to be practicable, and settles again on 19-EDO.

Is this the first time someone built a 53-edo instrument?

Incidentally, convergents to the Woolhouse fifth go 7/12, 11/19,
18/31, 29/50, 76/131, 257/443 ... . It would be interesting to dig up
someone who advocated 131-equal for meantone! 29/50 is slightly (8/27
of a cent) to the south of the smallest poptimal meantone at 47/81,
but 0.19 cents *sharper* than Zarlino's 2/7 comma. It occurs to me
that Zarlino's advocacy of 2/7-comma could be taken as evidence for
Paul's contention that the poptimal range ought to be brought all the
way down to exponent 1; if we did that 50 becomes the smallest
poptimal meantone et. I think I'll at least code it. In any case 50
also has the distinction of being the last to appear in the
convergents to both Zarlino's and Woolhouse's optimal fifths.

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

> but you quote
> Woolhouse as saying:
>
> [Woolhouse 1835, p 46:]
>
> This system is precisely the same as that which Dr. Smith, in
his
> Treatise on harmonics [Smith 1759], calls the scale of equal
harmony.
> It is decidedly the most perfect of any systems in which the tones
are
> all alike.
>
> Is Smith's tuning 50-equal?

It's close, but it's much closer to 5/18-comma meantone than to 50-
equal. Search the tuning list for more info ;) Also see Jorgenson.

hi Gene,

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

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > have you ever seen this?
> >
> > http://tonalsoft.com/monzo/woolhouse/essay.htm#temp
>
> you quote Woolhouse as saying:
>
> [Woolhouse 1835, p 46:]
>
> This system is precisely the same as that which
> Dr. Smith, in his _Treatise on harmonics_ [Smith 1759],
> calls the scale of equal harmony.
> It is decidedly the most perfect of any systems in
> which the tones are all alike.
>
> Is Smith's tuning 50-equal?

yes.

but i haven't read his book.

>
> I'd also cut Woolhouse some slack on 5-limit vs 5-odd-limit. The
place
> where he discusses that you *might* interpret to say that 2 and 3/2
> are consonances, but 4, 3 or 6/5 are not, but I would presume his
> assumption of octave equivalence would be clear from elsewhere.
>
> I also find this comment interesting:
>
> He then analyzes the resources of a 53-EDO 'enharmonic organ', built
> by J. Robson and Son, St. Martin's-lane, but says that the number of
> keys is too much to be practicable, and settles again on 19-EDO.
>
> Is this the first time someone built a 53-edo instrument?

i don't think so, but can't cite anything concrete.

Mercator studied 53edo and recommended it for instruments,
but i don't know what was/wasn't built.

> Incidentally, convergents to the Woolhouse fifth go 7/12,
> 11/19, 18/31, 29/50, 76/131, 257/443 ... . It would be
> interesting to dig up someone who advocated 131-equal for
> meantone!

hmmm ... there's a blank space in my equal-temeperaments
table, just waiting for that! maybe Gene will be the first.

-monz

hi Paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > but you quote
> > Woolhouse as saying:
> >
> > [Woolhouse 1835, p 46:]
> >
> > This system is precisely the same as that which
> > Dr. Smith, in his Treatise on harmonics [Smith 1759], calls
> > the scale of equal harmony.
> > It is decidedly the most perfect of any systems in which
> > the tones are all alike.
> >
> > Is Smith's tuning 50-equal?
>
> It's close, but it's much closer to 5/18-comma meantone
> than to 50-equal. Search the tuning list for more info ;)
> Also see Jorgenson.

hmm ... Woolhouse *did* write exactly what Gene quoted
from my webpage.

it's too much of a pain to search the list archives.
can you give us some details? exactly how did Smith
measure his tuning?

here's a very interesting webpage about Smith's
"Equal Harmony" tuning (described as 50et), and its
application to harpsichords.

http://216.239.41.104/search?q=cache:dAUFEkI--AcJ:www.music.ed.ac.
uk/russell/conference/robertsmithkirckman.html+%22robert+smith%22+%
2250%22+tuning%22&hl=en

or

http://tinyurl.com/6hbwu

-monz

> > But it's much closer to 5/18-comma meantone
> > than to 50-equal. Search the tuning list for more info ;)
> > Also see Jorgenson.
>
>
>
> hmm ... Woolhouse *did* write exactly what Gene quoted
> from my webpage.

There's very little difference between 50-equal and 5/18-comma
meantone, especially if you are doing your computing in the 18th
century. The convergents to 5/18 meantone go 12, 19, 50, 1219 ...,
whereas the convergents to 50-et in terms of fractions of a comma go
1/3, 1/4, 2/7, 3/11, 5/18, 13/47, 18/65... The fifth of 50-equal is a
mere 0.019 cents sharper than 5/18-comma. It would be interesting to
know why Smith chose the tuning he did, because it does seem he could
have intended to close at 50, or to do something equivalent such as
making the ratio between diatonic and chromatic semitones 5/3.