--- In tuning@yahoogroups.com, Margo Schulter <mschulter@> wrote:

Dear Margo,

Werckmeister's so called
11-limit "septenarian" 99:98 Comma amounts (81:80):(441:440)
11-limit "septenarian" 441:440 Schisma amounts (81:80):(99:98)
if understood that ratios as epimoric bisection of the SC=81:80.

the following well-tuning uses that as epimoric bi-scetion:

F 440:441 G = F 881:882 C 880:881 G inbetween F-C-G

and the trisection
inbetween the violin empty strings G-D-A-E consisting in

G 98:99 E = G 276:279 E subdivided into 3 epimoric factors:

G 296:297 D 295:296 A 296:297 E

and respectively for the schisma B 32768:32805 F within

E 3968:3969 B F# C# G# 2510:2511 Eb 3764:3765 Bb 4704:4705 F

into 4 another superparticular ratios
in order to conclude the cycle of 5ths by
the totally amount of an PC=3^12/2^19.


meanwhile my good-old piano got tuned in that refinement :

F 881:882 C 880:881 G 296:297 D 295:296 A 296:297 E

with the interim product inbetween FCGDAE = 80:81 = SC and

E 3968:3969 B F# C# G# 2510:2511 Eb 3764:3765 Bb 4704:4705 F

for yielding the Schisma=2^15/5/3^8=32768:32805.

that's expanded:
> Start at an minor-tone (10:9) below Scheibler:
>
> G; (7*7=E6:27=49 E8:27=98 <) 99G2 198G3 396G4 := 440Hz*(9:10)
> D; (E6:9=147 E8:9=294 < 295=A5:3 <) 296D4 (<297 = G2*3)
> A; (E6:3=441 E8:3=882 <) 885A5 just as Margo proposes
> E; 1323E6
> B; 31B0...3968B7(<3969=E6*3=49) lowest pitch on 5string doublebass
> F#; 93F#2 := B0*3
> C#; 279C#4 := F#2*3
attend the modifications from here on:
G#; 887G#5 := C#4*3
Eb; 1255Eb6 2510Eb7 (< 2511 := G#5*3)
Bb; (F2:3=29.4 .. F6:3=470.4<)470.5Bb4 941Bb5 1882Bb6 (3765:=Eb6*3)
F; (C4:3 = 88.1 <) 88.2F2
> C; (G1:3=33 ... G4:3=264 <) 264.3C4
> G; 99G2 = 33*3
>
Hence the
> Chromatically
improved values
>on the keys in frequencies as absolute-pitchs:
do arise a tiny little bit on the corresponding notes:
>
Correction:
> 264.3_C4
> 279___C#4
> 296___D4
313.75 > instead formerly 313.5_Eb4
> 330.75E4
> 352.5_F4
> 372___F#4
> 396___G4
418.5 > instead formerly 418___G#4
> 442.5_A4
470.5 > instead fromerly 470___Bb4
> 528.6_C5
>
all others pitches remain unchanged.

please forget about my obsolte
> !sparschuh885A5.scl
...
and replace it by the more elaborate:


!ef_JI_MiFa.scl
!
Sparschuh's 80:81 inbetween F-C-G-D-A-E and 32768:32805 in E B...Bb F
!
12
!
2790/2463
2960/2463
6275/2463 ! was > 3135/2463 formerly
6615/4926 ! (5:4)(882:881)
3524/2463 ! (4:3)(882:881) was > 3525/2463 ! (4:3)(3525:3524)
3720/2463
3960/2463 ! (3:2)(880:881)
4185/2463 ! was > 4180/2463 formerly
4425/2463
4705/4700 ! was > 4700/2463 fromerly
4960/2463
2/1
!

attend the exact JI haltone E 16:15 F that is
shifteted 882:881 upwards versus the unison 1:1,
because the 4th F-C and the 3rd C-E
turn out to be the same 882:881 sharper than JI.

Quests:
Who in that group here dares to try that on his/hers own instrument?
Are there any proposals in order to improve even that once more again?

Yours Sincerely
A.S.