## understanding George's: (was, Re:..) Rational Well Temperament

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• ... Hi George, Tom & all others, ... Did you mean by that following corresponding absolute pitches, or which other frequncies had you concrete in mind when
Message 1 of 63 , Apr 23, 2008
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--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
Hi George, Tom & all others,
>
> ! Dent-YN-RWT.scl
> !
> Tom Dent's Young-Neidhardt well-temperament (rationalized by G. Secor)
> 12
> !
> 560/531 ! C#
> 2643/2360 ! D
> 70/59 ! Eb
> 74/59 ! E
> 315/236 ! F
> 1329/944 ! F#
> 883/590 ! G
> 280/177 ! G#
> 890/531 ! A
> 105/59 ! A#
> 887/472 ! B
> 2/1
>
Did you mean by that following corresponding absolute pitches,
or which other frequncies had you concrete in mind when
creating that particular one by the factor 59?

C: 531 := 59*9
C# 560 280 140 70 35 = 7*5
D: 594.675 := 264.3*9/4
Eb 630 315 = 7*5*9
E: 666 333 = 37*9
F: 708 354 177 = 59*3
F# 747.625 = 1329*9/16
G: 794.7 = 88.3*9
G# 840 = 240*3
A: 890 445Hz ? probably yours intend A4=445 cps ?
Bb 945 = 105*9
B: 997.875 = 887*9/8

Hence i do agree with your's observation:
> The brats aren't nearly as simple as with the harmonic temperaments
> we've been looking at lately, but OTOH the fifths are much more
> consistent in size (all tempered <4 cents).
>
Sincerely
A.S.
• ... Hi George, ... in deed, even that *.scl-file contained some unfixed bug. ... Many thanks again for that repair. ... when considering more evaluated digits,
Message 63 of 63 , May 8, 2008
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--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

Hi George,
> > epimoric decomposition
> >
> > F 1058:1059 C 528:529 G 296/297 D 295/296 A 294/295 E 3968:3969 B
> > B F# C# G# Eb 7532:7533 Bb 5648:5649 F
> >
> >Cents approximation,
>
> >F~ -1.635 ~C~ -3.275 ~G~ -5.839 ~D~ -5.859 ~A~ -5.879 ~E~ -0.436 ~B
> > B F# C# G# Eb~ -0.2298 ~Bb~ -0.306 ~F
> >
>
> I believe there are still a few mistakes.
in deed, even that *.scl-file contained some unfixed bug.

> From the sizes of the
> fifths you give, I think that perhaps you meant this:
>
Now -as far as i can see- those ratios appearto be correct:
> 558/529 ! C#
> 592/529 ! D
> 2511/2116 ! Eb = 627.75/529
> 1323/1058 ! E = 661.5/529
> 706/529 ! F
> 744/529 ! F#
> 792/529 ! G
> 837/529 ! G#
> 875/523 ! A = 442.5Hz*2 absolute a4
> 1883/1058 ! Bb = 941.5/529
> 992/529 ! B
>
Many thanks again for that repair.

> This will result in:
when considering more evaluated digits,
as calculated by "Google"s arithmetics, which yields:

> F~ -1.636 ~C
(1 200 * ln(1 058 / 1 059)) / ln(2) = ~-1.63555425...

> ~C~ -3.276 ~G
(1 200 * ln(528 / 529)) / ln(2) = ~-3.27575131...

>~G~ -5.839 ~D
(1 200 * ln(296 / 297)) / ln(2) = ~-5.83890621...
arises that deviation here due to your's rounding procedere?

~D~ -5.784 ~A
1 200 * ln(295 / 296)) / ln(2) = ~-5.85866566...
arises that deviation here due to your's rounding procedere?

~A~ -5.953 ~E
(1 200 * ln(294 / 295)) / ln(2) = ~-5.8785593...

same question as for D~A?
Or what else could be the reason for the tiny
discrepancy amounting about tiny 1/10 Cents
inbetween ours calculations of the relative deviations
in the tempered 5ths flatnesses?

~E~ -0.436 ~B
(1 200 * ln(3 968 / 3 969)) / ln(2) = ~-0.436243936...

> B F# C# G# all just pure 5ths

Eb~ -0.2298 ~Bb
(1 200 * ln(7 532 / 7 533)) / ln(2) = ~-0.229835254...

Bb~ -0.306 ~F
(1 200 * ln(5 648 / 5 649)) / ln(2) = ~-0.306494477...

at least we both do agree now in all others 5ths except D~A~E.

What do you think about that well-temperement,
with an almost JI the C-major chord:

C:E:G = 4 : 5*(2646/2645) : 6*(529/528)

?

Yours Sincerely
Andreas
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