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• ## 2/7 SC in 'septenarian' "qaternarius"C~G~D~A-E-B~F#.....C, was: Re: continuo...

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• ... Dear Margo, divide alike Zarlino once had done, the SC = 81/80 = (81*7)/(80*7) = 567/560 into 4 arithmetic subparts [2/7 + 2/7 + 2/7] + 1/7 in his manner:
Message 1 of 143 , Nov 25, 2007
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--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Then, again, the example of Zarlino
> and his student Vincenzo Galilei cautions us that a teacher is not
> necessarily responsible for all the views of her/his pupil --
> although Galilei, as it happens, expresses a liking for Zarlino's
> 2/7-comma as the most pleasing keyboard temperament.

Dear Margo,

divide alike Zarlino once had done, the

SC = 81/80 = (81*7)/(80*7) = 567/560

into 4 arithmetic subparts [2/7 + 2/7 + 2/7] + 1/7 in his manner:

567/560 = [ (567/565) * (565/563) * (563/561) ] * (561/560) = 81/80

That yields in Werckmeister's terms of 8 just pure 5ths
an corresponding 4-fold subdivision of the:

PC = 3^12/2^19 = (81/80)*(32768/32805) = SC * schisma

into

C 565/567 G 563/565 D 561/563 A-E-B (560/561)(32768/32805)
F#-C#-G#-D#-A#-Bb-F-C

or 'septenarian' circulating:

begin=7=Ab-Eb-Bb-F-C~G~D~A-E-B~F#-G#=7=end

expanded multiplying consecutive the following lines by factor 3

Ab_-2: 7 cps or Hz
Eb_1: 21 (> 20.777777777...)
Bb_2: 63 (> 62.3333333333...)
F_3: 189 (> 187 = 561/3)
C_5: 567 (> 565 (> 563 (> 561 = 187*3 )))
temper down by: (1 200 * ln(565 / 567)) / ln(2) = ~-6.11744117...Cents
G_6: 1695 (> 1689 (> 1683 = 187*9 ))
temper down by: (1 200 * ln(563 / 565)) / ln(2) = ~-6.13913427...Cents
D_8: 5067 (> 5049 = 187*27)
temper down by: (1 200 * ln(561 / 563)) / ln(2) = ~-6.16098177...Cents
A_10: 15147 = 187*81
E_12: 45441 = 187*243
B_13: 136323 = 187*729...(> B-7: 7/27)
lower (1200*ln((560/561)*(32768/32805)))/ln(2) = ~-5.04245318...Cents
(F#_15: 187*2187>) F#_-5: 7/9
C#_-3: 7/3
G#_-2: 7 cycle closed enharmonic back to start: Ab_-2: 7 cps or Hz

That's chromatic in ascending pitch order as absolute frequencies
when taken modulo 2^n into the middle octave:

c 283.5 = 567/2 "middle_C"
# 298.6666666... = 298+2/3 = 128*7/3
d 316.6875 = 5067/16
# 336 = 21*16
e 355.007812 = 45441/128
f 378 = 189*2
# 398.2222222... = 398+2/9 = 512*7/9
g 423.75 = 1695/4
# 448 = 7*64
a 473.34375 = 473+11/32 = 15147/32 ~Praetorius high Choir-Thone~
# 504 = 63*8
b 532.511719.. = 532+131/256 = 17*11*3^7/2^8
c'567 "tenor_C"

for the corresponding lower Cammerthone version
simply divide each pitch by 9/8 by of an major-tone downwards.

so that:

c_4 becomes 252 Hz = (567*4/9)cps and
a_4 = 1683/4 = 420.75 Hz

in order to replace the my meanwhile outdated 9.9.99
first original "squiggle" 420Hz proposal:
http://www.strukturbildung.de/Andreas.Sparschuh/
by the above new improved version, the now actual:

Rational 2/7-SC-"squiggle" interpretation absolute @ a'=420.75cps

that tempers barely 4 of the dozen 5ths
just in Werckmeister's famous 8-pure 5ths layout,
instead fromerly only 4 pure 5ths once in 1999 at
DA&F#C#G#Eb. Meanwhile, now that turns out in my
in my ears as suspicious to much near
inbetween Kellners modern PC^(1/5) schmeme
or even worser others ahistoric alleged PC^(1/6) claims.

I.m.h.o:
As far as i do see the squiggles now:
There's no reason why JSB should had
depart from W's original layout
in whatsoever interpretation for
C~G~D~AEB~F#...C
you wants to prefer in yours taste.

Never the less:

Try out the rational 2/7-SC variant :

!septenarianFC_G_D_AEB_Fsharp.scl
!
C 565/567 G 563/565 D 561/563 AEB(560/561)(32768/32805)F#C#G#D#A#BbFC
!
12
!
256/243 ! C# ~1.05349794...
563/504 ! D ~1.11706349...
32/27 ! Eb ~1.18518519...
563/448 ! E ~1.25669643...
4/3 ! F ~1.33333333...
565/378 ! G ~1.49470899...
128/81 ! G# 1.58024691...
187/122 ! A ~1.53278689...
16/9 ! Bb ~1.7777777...
1683/896 ! H ~1.87834821...
2/1

as alternative choice when considering JSB's squiggels.

Concluding remark;
Attend that:
Above Zarlino's arithmetic 2/7-SC division should not be
confused with its modern irrational approximation:

(81/80)^(2/7) = ~1.0035556...
(1 200 * ln((81 / 80)^(2 / 7))) / ln(2) = ~6.14465417...Cents
with barely tiny deviation but significant
impact on the representation.

not to mention the even less useful: PC^(2/7)

(1 200 * ln(((3^12) / (2^19))^(2 / 7))) / ln(2) = ~6.70286011...Cents

or for all those,
exactly precisely by ear within 15 minutes?

720TUs/7 = 102+6/7TUs ~102.857143...TUs

Sorry, but:
Personally i don't need for an other logarithmic unit
than the traditional Cents of 1200-EDO.

Anyhow:
have a lot of fun with my new actual
arithmetic 2/7-SC "squiggles"
that fit even matching into Werckmeister's
C~G~D~A&B~F# pattern.

sincerely
A.S.
• ... it is also possible to read Werckmeister s #3 pattern C~G~D~A E B~F#...C in 1/3 SC terms: C 242/243 G 241/242 D 240/241 A E B 32768/32805 F# C# G# D# Bb F
Message 143 of 143 , Mar 28, 2008
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--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
>.... it occurred to me that if the
> "comma" may be either Pythagorean or syntonic, with the schisma
> regarded as not so important, then why not 1/3-syntonic comma
> tempering for the narrow and wide fifths alike?

> ! werckmeisterIV_variant.scl
> !
> Werckmeister IV with 1/3 syntonic comma temperings
> 12
> !
> 85.00995
> 196.74124
> 32/27
> 393.48248
> 4/3
> 45/32
> 694.78624
> 785.01123
> 891.52748
> 1003.25876
> 15/8
> 2/1
>
>
> ! WerckmeisterIV_variant_c.scl
> !
> Werckmeister IV variation, 1/3-SC, all intervals in cents
> 12
> !
> 85.00995
> 196.74124
> 294.13500
> 393.48248
> 498.04500
> 590.22372
> 694.78624
> 785.01123
> 891.52748
> 1003.25876
> 1088.26871
> 2/1
>
> The 1/3-comma variation seems
> to fit this model -- at least if, like Costeley (1570) and Salinas
> (1577), we are ready to accept fifths tempered by this great a
> quantity, as in a regular 1/3-comma meantone or 19-EDO. Zarlino (1571)
> found 1/3-comma temperament "languid," ....

it is also possible to read Werckmeister's #3 pattern

C~G~D~A E B~F#...C

in 1/3 SC terms:

C 242/243 G 241/242 D 240/241 A E B 32768/32805 F# C# G# D# Bb F C

as refinement of his JI tuning presented in his book:
"Musicae mathematicae hodegus curiosus"
FFM 1687: p.71: a'=400cps
extracted from his "Natï¿½rlich" (natural) scale,
there defined in absolute pitch-frequencies:

c" 480 cps
(db 512)
c# 500
d" 540
d# 562.5
eb 576
e" 600
f" 640
f# 675
g" 720
g# 750
ab 768
a" 800 overtaken from Mersenne's reference-tone a'=400Hz
b" 864
h" 900
c"'960

The W3 pattern can be understood as
modification of layout pattern,
in absolute terms,
as cycle of partially tempered 5hts:

Db 1 unison, implicit contained in his absolute "hodegus" tuning
Ab 3
Eb 9
Bb 27
F 81 (>80+2/3 (>80+1/3 (80 40 20 10 5)))
C 243 (>242 (>241 (>240 120 60 30 15)))
G (729 >) 726 (>723 (>720 360 180 90 45))
D 2169 (>2160 1080 540 270 135)
A 405 compare to Chr. Hygens(1629-95) Amsterdam determination:~407 Hz
E 1215
B 3645
F# (10935=32805/3 >) 32768/3 ... 1/3
C# 1 returend back unison again

that's relative in chromatically ascending order as Scala-file:

!Werckmeister3_one3rd_SC_variant.scl
!
Werckmeister's famous C~G~D-A-E-B~F#...C pattern as 1/3 SC + schisma
!C 242/243 G 241/242 D 240/241 A E B 32768/32805 F# C#=Db Ab Eb Bb F C
!
256/243 ! Db=C# enharmonics @ absolute Mersenne's 256cps unison
241/216 ! D
32/27 ! Eb
5/4 ! E
4/3 ! F
121/81 ! G = (11/9)^2 = (3/2)*(243/242)
128/81 ! Ab
5/3 ! A
16/9 ! Bb
15/8 ! B (german H)
2/1

attend:
That one contains more pure intervals than other interpretations.

if you have some better ratios for W3 -even nearer to JI?-,

A.S.
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