Historical well-temeraments, 612, and 412

It seems that Werckmeister III is not the only well-temperament to be nailed by 612. Here are some others, using data taken from Manual's list of scales:

! young.scl
!
Thomas Young well temperament (1807), also Luigi Malerbi nr.2 (1794)
12
!
256/243
196.09000
32/27
392.18000
4/3
1024/729
698.04500
128/81
894.13500
16/9
1090.22500
2/1

The 612-et version of this is again perfection itself:

[0, 46, 100, 150, 200, 254, 300, 356, 404, 456, 508, 556]

Note that all the steps are even, so 306 also works.

! young2.scl
!
Thomas Young well temperament no.2, ca. 1800
12
!
94.13500
196.09000
298.04500
392.18000
500.00000
592.18000
698.04500
796.09000
894.13500
1000.00000
1092.18000
2/1

Again, the 612-et version is insanely accurate:

[0, 48, 100, 152, 200, 255, 302, 356, 406, 456, 510, 557]

Here is one by Marpurg:

! marpurg2.scl
!
Marpurg 2. Neue Methode (1790)
12
!
109.775 cents
9/8
313.685 cents
81/64
4/3
607.820 cents
3/2
811.730 cents
27/16
1015.640 cents
1105.865 cents
2/1

Once again, 306 would work also:

[0, 56, 104, 160, 208, 254, 310, 358, 414, 462, 518, 564]

Finally, here is an example where 612 does not work, but 412 works
excellently:

! marpurg.scl
!
Marpurg, Versuch ueber die musikalische Temperatur (1776), p. 153
12
!
101.955 cents
200.978 cents
300.000 cents
401.955 cents
500.978 cents
600.000 cents
3/2
800.978 cents
900.000 cents
1001.955 cents
1100.978 cents
2/1

In terms of the 412-et:

[0, 35, 69, 103, 138, 172, 206, 241, 275, 309, 344, 378]

(The 1200-et isn't bad here either.)

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> Finally, here is an example where 612 does not work, but 412 works
> excellently:
>
> ! marpurg.scl
> !
> Marpurg, Versuch ueber die musikalische Temperatur (1776), p. 153

However, 1224 works very, very well, so we still have a nice version of this using 612 as a basic measure:

[0,52,102.5,153,205,255.5,306,358,408.5,459,511,561.5]

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@juno.com>
To: <tuning-math@yahoogroups.com>
Sent: Friday, October 11, 2002 5:31 PM
Subject: [tuning-math] Historical well-temeraments, 612, and 412

> It seems that Werckmeister III is not the only well-temperament
> to be nailed by 612. Here are some others, using data taken from
> Manual's list of scales:
> <snip>

wow, Gene, thanks for these!!!
they'll eventually all become Tuning Dictionary webpages.

my guess is that the reason 612 works so well has something
to do with the fact that these temperaments temper out the
Pythagorean comma. wanna look into that more?

-monz

--- In tuning-math@y..., "monz" <monz@a...> wrote:

> my guess is that the reason 612 works so well has something
> to do with the fact that these temperaments temper out the
> Pythagorean comma. wanna look into that more?

My assumption is that the fact that the Pythagorean comma and 3 are both well represeted by 612 has something to do with it, but that's not the whole story or 665 would dominate.

--- In tuning-math@y..., "monz" <monz@a...> wrote:

> wow, Gene, thanks for these!!!
> they'll eventually all become Tuning Dictionary webpages.

Great. Here are a couple more historical temperaments which can be nicely expressed in terms of shismas:

! kirnberger1.scl
!
Kirnberger's temperament 1 (1766)
12
!
256/243
9/8
32/27
5/4
4/3
45/32
3/2
128/81
895.11200
16/9
15/8
2/1

[0, 46, 104, 150, 197, 254, 301, 358, 404, 456.5, 508, 555]

! kirnberger2.scl
!
Kirnberger 2: 1/2 synt. comma. "Die Kunst des reinen Satzes" (1774)
12
!
135/128
9/8
32/27
5/4
4/3
45/32
3/2
405/256
895.11186
16/9
15/8

[0, 47, 104, 150, 197, 254, 301, 358, 405, 456.5, 508, 555]

Just for kicks, here is the Ellis Duodene:

[0, 57, 104, 161, 197, 254, 301, 358, 415, 451, 519, 555]

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "monz" <monz@a...> wrote:
>
> > my guess is that the reason 612 works so well has something
> > to do with the fact that these temperaments temper out the
> > Pythagorean comma. wanna look into that more?
>
> My assumption is that the fact that the Pythagorean comma and 3 are
>both well represeted by 612 has something to do with it, but that's
>not the whole story or 665 would dominate.

guys:

it's because these tunings distrubute the pythagorean comma in
various ways, typically chopping it into thirds, quarters, sixths, or
twelfths.

clearly the solution itself will have to be a multiple of 12 (since
12-equal forms the "baseline" where the pythagorean comma is tempered
out), and because of the above, it also has to express the
pythagorean comma as a multiple of 12.

in 612, the pythagorean comma is 12, so 612 is the simplest solution.

where the pythagorean comms is chopped into *eighths*, we need to go
to 1224.