## Corrected version of preferred.... Re: absoute-pitch.. on the violin & piano

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• ... 11-limit septenarian comma versus the SC ... 3^4/5/16 = (11*3^2/7^2/2)*(7^2*3^2/11/5/2^3) i do call the ratio: 441/440
Message 1 of 63 , May 2, 2008
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--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> from Werckmeister's
11-limit septenarian comma versus the SC
> >
> > 81/80 = (99/98)*(441/440)
3^4/5/16 = (11*3^2/7^2/2)*(7^2*3^2/11/5/2^3)

i do call the ratio:
441/440
http://www.petersontuners.com/index.cfm?category=85&sub=89
as "Werckmeister's 11-limit septenarian schisma"
Scheibler later in the early 19.th century used that interval
for defineing our todays still actual 440cps standard:
http://www.1911encyclopedia.org/Musical_Pitch
> >
Werckmeister's 11-limit-septenarian-comma becomes when
> > divided into 3 epimoric subparts:
> >
> > 99/98 = (297/296)*(296/295)*(295/294)
> >
That 3-fold decompostion of W's-comma can be used for
> > tempering the 5ths G-D-A-E flattend by the corresponding amounts:
> >
> > G 296/297 D 295/296 A 294/295 E
> >
> > yielding on the violin empty stings the absolute pitches:
>
G2=99Hz lowest violin__G3=198__string
G3=148 __G4=296__(<297=3*G2)
A3=221.25 __A4=442.5__ (<444=3*G3)
highest violin__E5=661.5__string (<663.75=3*A3)

> > g3: 198 cps = 99*2 := 220*(9/10) a minor-tone below a3=440Hz/2
> > d4: 296
> > a4: 442.5 := 885/2
> > e5: 661.5 := 1323/2
> >
or
When generalized to a dozen 5ths-cirlce on my own old piano:
> >
> > C_5: 523Hz (>522 264 132 66 33) 'tenor-C'
> > G_2: 99 (((> 98 49=7*7 taken from Werckmeister's "septenarius")))
> > D_4: (297>) 296 (>295 (>294 147=49*3)
> > A_5: 885 (>882 441=49*9)
> > E_6: 1323 = 49*27
> > B_0: (49*81 = 3969>) 3968 ... 496...31 use all 7 'B's on the keys
> > F#_2: 93
> > C#_4: 279 a semitone above 'middle-C'
> > G#_5: 837
> > Eb_7: 2511
> > Bb_6: (7533>) 7532 3716 1883
> > F_4: (5649>) 5648 2824 1412 706 353
> > C_5: (1059>) 1058 529 = 23^2 cycle returned back to 'tenor-C'
> >

Sorry, the old previous meassge contains here some typo-errors:
I had confused that due to a mistake in all to much hurry.
Simply forget about the wrong numbers:

again the now corrected version:

!well_Violin2Piano.scl
!by A.Sparschuh
temper from violin empty strings G 296/297 D 295/296 A 294/295 E
12
! middle_C 264.5Hz = 529cps/2
!
558/529 ! C#
598/529 ! D
2511/2116 ! Eb = 627.75/529
1323/1058 ! E = 661.5/529
706/529 ! F
724/529 ! F#
792/529 ! G
837/529 ! G#
885/523 ! A = 442.5Hz*2 absolute a4
1883/1058 ! Bb = 941.5/529
992/529 ! B
2/1
!
!
the relative deviation of the
5ths corresponds to the following 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

into the 8 superparticular subfactorization of the PC=3^12/2^19.

> Either I don't understand this,
or if you prefer the same distribution of the PC=~23.46cents
in logarithmically values as 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

correspodning to the above 8 epimoric ratios.

> or something is very wrong with the
> numbers.
In deed -i have to agree-
my first data were somewhat out of control.

Many thanks for making me aware of my blunder,
that had urgently demanded some bug-fixing.

> The fifths D-A and A-E are tempered >23 cents,
not anymore , but now
all that both 5ths are less tempered than PC^1/4 =~ 5.865 Cents
compareable to G-A in Werckmeister's#3.

> and D-F# is
> tempered by >55 cents.
that 3rd: D-F# is barely 186/185 ~9.33Cents wide
but attend the 3rd C-E with barely 2646/2645 ~0.654Cents wider
than 5/4, hence almost nearly to pure JI.
>
> Or is this a joke (since you said, "have a lot of fun")? :-)
That was never intened as hoax, even in its faulty version.
so,
that patched revision is really meant seriously adjusted
for properly usage.

Yours 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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