- --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> from Werckmeister's

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)

> > 81/80 = (99/98)*(441/440)

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:

That 3-fold decompostion of W's-comma can be used for

> >

> > 99/98 = (297/296)*(296/295)*(295/294)

> >

> > tempering the 5ths G-D-A-E flattend by the corresponding amounts:

G2=99Hz lowest violin__G3=198__string

> >

> > G 296/297 D 295/296 A 294/295 E

> >

> > yielding on the violin empty stings the absolute pitches:

>

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

or

> > d4: 296

> > a4: 442.5 := 885/2

> > e5: 661.5 := 1323/2

> >

When generalized to a dozen 5ths-cirlce on my own old piano:> >

Sorry, the old previous meassge contains here some typo-errors:

> > 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'

> >

please read always 529 instead there formerly faulty 523.

I had confused that due to a mistake in all to much hurry.

Simply forget about the wrong numbers:

and study instead of that

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,

about the amounts:

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

In deed -i have to agree-

> numbers.

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

that 3rd: D-F# is barely 186/185 ~9.33Cents wide

> tempered by >55 cents.

but attend the 3rd C-E with barely 2646/2645 ~0.654Cents wider

than 5/4, hence almost nearly to pure JI.>

That was never intened as hoax, even in its faulty version.

> Or is this a joke (since you said, "have a lot of fun")? :-)

so,

now after that proof reading/checking/correction

that patched revision is really meant seriously adjusted

for properly usage.

Yours Sincerely

A.S. --- 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,

> > about the amounts:

>

> >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