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Gauss-Comma 9801/9800 = 121*81/25/8

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

2/21/2008 8:07:03 AM

Gene wrote on the:
http://www.xs4all.nl/~huygensf/doc/intervals.html
" 9801/9800 kalisma, Gauss' comma "

= (99/98)*(100/99)

that's the quotient of

"99/98 small undecimal comma" over
"100/99 Ptolemy's comma"

He remarked about that in:
http://www.robertinventor.com/tuning-math/s__11/msg_10675-10699.html
"
Lehmer unfortunately does not give a cite, but he claims Gauss
mentioned 9801/9800, so I think I'll propose gaussisma for that, and
lehmerisma for 3025/3024.
"
Appearently 9801/9800 occurs in:
http://www.arithmetique.net/Problemes_divers.htm
"
Gauss a proposé une simplification de l'expression suivante:

10^59*(1025/1024)^5*(1048576/1048575)^8*(6560/6561)^3*(15624/15625)^8*(9801/9800)^4.

Son résultat est une puissance de 2. Saurez-vous la retrouver ?
Solution : 2^196
"

overtaken as "Problème 68" of that book:
http://www.arithmetique.net/Florilege_de%20_100_problemes_(1).htm

Appearenty Gauss knew already the:
www.plainsound.de/research/notation.pdf
"
...alteration of the 41-limit Schisma.
(32/41)·(81/64)·(81/80) = (6561/6560) = 3^8/41/5 ≈ ± 0,3 Cents.
"

which was probably already used before in
Werckmeister's 41-limit epimoric decompostion of
the Schisma into the subfactors:

32805/32768 = 5*3^8/2^15 = (6561/6560)*(1025/1024)

/tuning-math/message/13675

correct in that at begin and end of the cycle:

C 2187=3^7 instead the wrongly 'C 2173'

with the tempered 5th C-G : 6560/6561 flatter than just 3/2
and 3rd G-B : 1026/1025 sharper than pure 5/4.

!epimoricWerckmeister3.scl
!
PCdistribution: C 6560/6561 G 204/205 D 152/153 A E B 512/513 F#...F C
12
!
256/243 ! c# limma
272/243 ! d
32/27 ! d#
304/243 ! e (5/4)*(1216/1215) with "1216/1215 Eratosthenes' comma"
4/3 ! f
1024/729 ! f#
3280/2187 ! g (3/2)*(6560/6561) with "41-limit schisma"
128/81 ! g#
1216/729 ! a
16/9 ! a#
152/81 ! b = g*(5/4)*(1026/1025)
2/1
!
makes an PC down 2^19/2^12 expressed in terms of 41,19,17-limit
for Werckmeisters most famous "8 just 5ths" in the distribution:

C 41*5*16/3^8 G 17*4/41/5 D 19*8/17/9 A E B 2^8/19/27 F#C#G#D#BbFC

Does anyone in that group here know a better reference
to the above quotated original Gauss source?
than that unspecific one:

http://books.google.com/books?id=fDMAAAAAQAAJ&pg=PA502&lpg=PA502&dq=gauss+9801+9800&source=web&ots=mm6cX-ggtZ&sig=bUB9RPJDtqx9LFV5ho6aBm9VbS0#PPA470,M1

A.S.