--- 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 "Nat�rlich" (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
1024/729 ! 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?-,
please let me know about that.

A.S.