Topic: Historical well-temeraments, 612, and 412

4 scales

Thread (6 messages)

From: Gene Ward Smith (2002-10-12)
Subject: 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.)

From: Gene Ward Smith (2002-10-12)
Subject: Re: Historical well-temeraments, 612, and 412

--- 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]

From: monz (2002-10-12)
Subject: Re: [tuning-math] Historical well-temeraments, 612, and 412

----- Original Message ----- 
From: "Gene Ward Smith" <[email protected]>
To: <[email protected]>
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

From: Gene Ward Smith (2002-10-12)
Subject: Re: Historical well-temeraments, 612, and 412

--- 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.

From: Gene Ward Smith (2002-10-12)
Subject: Re: Historical well-temeraments, 612, and 412

--- 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]

From: wallyesterpaulrus (2002-10-15)
Subject: Re: Historical well-temeraments, 612, and 412

--- 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.