Septenarian Re: Bach-Cup tuning decoded

--- In bach_tunings@yahoogroups.com, Miklos Mikael Spanyi <spanyi@...> wrote/asked:
>
> Any hint of a temperament found on the Bach-Cup so far?
>
> > http://www.carus-verlag.com/images-intern/pk/4034110p.jpg
> > http://liposomen.org/jpg/bachpokal.jpg

Investigation result:
A closer look on Bach's personal cup shows,
that there are two hidden spiry twisted 7s
engraved into the antique glass of the goblet.

Hence that allows an "septenarian" interpretation in W's sense:

http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
Quote:
"...Through a clever choice of 196 = 2 × 2 × 7 × 7 for the fundamental number (representing tenor C) he was able to produce fifths and thirds that were not too far from the pure ratios...."

http://en.wikipedia.org/wiki/Werckmeister_temperament#Werckmeister_IV_.28VI.29:_the_Septenarius_tunings
"This tuning is based on a division of the monochord length into 196 = 7-times 7-times 4 parts."

That engraving on the cup corresponds
to an analogous Werckmeisterian superparticular cycle of 5ths:

-6: Ab 49:= 7*7 98 196 .... as overtaken from W.
-5: Eb 147 := 49*3
-4: Bb 221 442 ( > 441 := 49*9)
-3: F 83 166 (331 <) 332 664 ( < 663 := Bb*3)
-2: C (31 62 124 248 496 992 < ) 993 := 331*3
-1: G (23 46 92 < ) 93 := 31*3
+-0: D (23*3 = 69 138 276 < 277 < ) 278 ( < 279 := G*3 )
+1: A (69*3 = 207 414 828 < ) 831 [ > 830 415=83*5 := F*5 ]
+2: E 621 [ > 620.625 = C*5/8]
+3: B 931 [> 930 465 := G*5] 1862 ( < 1863 := E*3 )
+4: F# 349 698 1396 2792 ( < 2793 := F#*3 )
+5: C# 523 1046 ( < 1047 := F#*3 )
+6: G# 49=7*7 98 196 .... 392 784 1568 ( < 1569 := C#*3 )

that yiels when lined up chromatically in ascending order

c" 496.5 tenor_C5
#" 523
d" 556
#" 588
e" 621
f" 664
#" 684
g" 744
#" 784
a" 831
bb 882
b" 931
c"'993 sopran_C6

For conversion into the modern MIDI specifiactions use the
http://www.xs4all.nl/~huygensf/scala/scl_format.html

! Bach_Cup.scl
Septenarian interpretation of J.S.Bach's cup compiled by A.Sparschuh
!
12
!
1046/993 ! C# ~ 91.674983...Cents
1112/993 ! D_ ~195.949398...
1176/993 ! Eb ~292.826925...
1242/993 ! E_ ~387.359461...
1328/993 ! F_ ~503.267429...
1396/993 ! F# ~589.719982...
1488/993 ! G_ ~700.210684...
1568/993 ! G# ~790.871924...
1662/993 ! A_ ~891.665711...
1768/993 ! Bb ~998.703182...
1862/993 ! B_ ~1088.38494...
2/1
!

Attend the small deviations from the exact JI 4:5:6 tri-chord
in the three most common major-chords:

F:A:C = 4: 5*[831/830] : 6*(331/332) = 4 : 5+~2.08Cents : 6-~5.22Cents

C:E:G = 4: 5*[1656/1655]:6*(992/993) = 4 : 5+~1.05Cents : 6-~1.74Cents
and
G:B:D = 4: 5*[931/930] : 6*(278/279) = 4 : 5+~1.86Cents : 6-~6.21Cents

have a lot of fun when playing Bach's works in the tuning of his cup
A.S.

Hi Andreas how are you.
________________________________
>> Any hint of a temperament found on the Bach-Cup so far?
> > http://www.carus-
<http://www.carus-verlag.com/images-intern/pk/4034110p.jpg>
verlag.com/images-intern/pk/4034110p.jpg
> > http://liposomen. <http://liposomen.org/jpg/bachpokal.jpg>
org/jpg/bachpokal.jpg
Investigation result:
A closer look on Bach's personal cup shows,
that there are two hidden spiry twisted 7s
engraved into the antique glass of the goblet.
etc. etc.
_________________________________

Oh no! :-(

We had Kellner after Bach's signet ring.
Then came Lehman after the WTC squiggle.
Now Bach's cup.

It's too much!
I shouldn't have written on temperaments!!
Whatever I did to deserve this, I repent!!!
Won't do it again!!!!
Have mercy!!!!!

:-)

Best wishes,

Claudio