--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
Dear Margo:
>
> Thank you for posting your temperament with one wide fifth based on
> complex integer ratios, a technique that fascinates me, and which I
> have used in rather different stylistic settings.
>
My intrest comes form Werckmeister's:
http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
http://en.wikipedia.org/wiki/Werckmeister_temperament#Werckmeister_IV_.28VI.29:_\
the_Septenarius_tunings
http://launch.groups.yahoo.com/group/tuning/message/69750
in his
"Musicalische Temperatur"
http://diapason.xentonic.org/ttl/ttl01.html
there on p72.


! septenarius.scl
! Werckmeister's #6 in string-lenghts on the monochord
!
C196 C#186 D176 D#165 E156 F147 F#139 G131 G#124 A117 B110 H104...
!
12
!
98/93 ! := 196/186 = C/C#
196/176 ! = C/D
196/165 ! = C/D#
49/39 ! := 196/156 = C/E = (5/4)*(196/195)
4/3 ! := 196/147 = C/F
196/139 ! = C/F#
196/131 ! = C/G = (3/2)*(392/393)
49/31 ! :=196/124 = C/G#
196/117 ! = C/A
98/55 ! =196/110 = C/B
49/26 ! =196/104 = C/H
2/1
!

later
http://www.tuningforktherapy.com/about.html
"In 1834, J.H. Scheibler presented a set of 54 tuning forks covering
ranges from 220 Hz to 440Hz."
http://www.apogeelearning.com/acutone/historytuningfork_1_2.html
"J. H. Scheibler in Germany in 1834 presented a set of 54 tuning forks
covering the range from 220 Hz to 440 Hz, at intervals of 4 Hz."
or
http://www.cosmeo.com/viewArticle.cfm?guidAssetId=A44930C3-53BA-4646-AD2C-B6AA4F\
1A62ED&&nodeid=
"The German physicist Johann Heinrich Scheibler (1777-1838) made the
first accurate determination of pitch corresponding to frequency and
proposed the standard A equals 440 in 1834."

for that purpose of defineing his 440Hz standard
S. appearently referred to above Werckmeister's
original Monochord-lenghts C196...C'98 (Upper-case letttes) on p.73
but now instead in reverse pitch-order
interpreted as absolute frequncies
(here denotated in lower-case letters):

g1 = 49 C'98 C196
d3 = F147 := 49*3
a2 = 55 B110 220 440 (<441 = 147*3) hence Scheibler's choice of 440cps
e3 = D#165 := 55*3
b0 = 31 62 G#124 248 496 (!>! 497 = 165*3) attend the wide 5th !
f#2 = 93 186C#
c#3 = F#139 278 (<279 = 93*3)
g#1 = 52 H104 208 416 (<417 = 139*3)
eb1 = 39 78 E156 := 52*3
bb2 = A117 := 39*3
f1 = 44 88 D176 352 (!>! 351 = 117*3) another wirde 5th !
c3 = G131 (<132 = 44*3)
g1 = 49 C'98 C196 392 (<393 = 131*3)

with the invariant 5ths against change from lenghts to frequency:
1. 392:393 for G-C & c-g
2. 278:279 for F#-C# & f#-c#

completely:
C392:393~G131:132~D352:351~AEH416:417~F#278:279~C#G#496:495~D#B440:441~F-C

c392:393~gd440:441~ae496:495~bf#278:279~c#416:417~g#eb-bb352:351~f131:132c

!reverseSeptenarius.scl
!
W's monochord-lengths backwards in reverse direction as frequencies
!
139/131 ! c#3/c3
147/131 ! d3 /c3
156/131 ! eb3/c3
165/131 ! e3 /c3 =(5/4)*(132/131)
176/131 ! f3/ c3 =(4/3)*(132/131) some scholars prefer 175 instead 176
186/131 ! f#3/c3
196/131 ! g3 /c3 = (3/2)*(392/393)
208/131 ! g#3/c3
220/131 ! a3 /c3 in order to meet Scheibler's absolute preference
234/131 ! bb3/c3
246/131 ! b3 /c3
2/1
!


get rid of the 2 wide-5ths ib that
in order to obtain a real 'well-temerature' "Wohl-Temperatur"
by the modified chain of 5ths:

C 392:393 G D 440:441 A E 494:495 B 740:741 F# C# 1664:1665 G#
Ab Eb Bb 350:351 F 524:525 C

!rev_Sept_Well_mod.scl
!
without the wide 5ths in Werckmeister's original ratios
12
!
555/524 ! c#5/c5
147/131 ! d3 /c3
156/131 ! eb3/c3
165/131 ! e3 /c3 =(5/4)*(132/131)
175/131 ! f3/ c3 =(4/3)*(525/524) some scholars prefer 175 instead 176
185/131 ! f#3/c3
196/131 ! g3 /c3 = (3/2)*(392/393)
208/131 ! g#3/c3
220/131 ! a3 /c3 Scheibler's fork a4=440cps
234/131 ! bb3/c3
247/131 ! b3 /c3
2/1
!

but that sounds
-at least in my ears-
all to much alike 12-EDO.

> There is one small revision that might appeal to me, and I would be
> very curious as to your opinion on what I might propose.
>
> Looking at the sizes of fifths, I noticed that at D-A there is a fifth
> considerably narrower than any other, at only about 694.1 cents, or
> almost 8 cents smaller than pure, with a ratio of 884/592. This is a
> bit more than 1/3 Pythagorean comma of tempering, and I wondered if
> the step A (Scala step 9) might be raised slightly to make this fifth
> a bit closer to pure without seriously compromising any other
> interval.
>
> In this proposed variation on your tuning, A is raised to 885/529 with
> respect to C, or 885/592 with respect to D, forming a fifth D-A which
> is narrow by almost exactly 1/4 Pythagorean comma -- indeed, the
> accuracy of 885/592 in approximating this common degree of temperament
> is amazing! The main complication I would see, if you like the result,
> is that your pitch standard would then become a4 = 442.5.

fully agreed!
that's an good idea,
as already mentioned in:
http://launch.groups.yahoo.com/group/tuning/message/76113
"!well_Violin2Piano.scl
...
885/523 ! A = 442.5Hz*2 absolute a4
"
>
hence i do accept yours improvement as welcome:
> ! ProposedVariationOnSparschuh442wideFrench5th.scl
> !
> Proposed revision: step 9 (A) at 885/529, 890.9 cents -- Margo Schulter
...
> 885/529
...
meanwhile i do consider
264.5C4 as to harsh for the middle-C4,
a better choice -in my ears, at least on my piano- would be:
264.3


Perhaps you also like an further "septenarian" refinement too:
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) through all 8 Bs on the piano-keys
F#; 93F#2 := B0*3
C#; 279C#4 := F#2*3
G#; 209G#3 418G#4 836G#5 (< 887 = C#4*3)
Eb; 627Eb5 := G#3*3
Bb; 235Bb3 470Bb4 940Bb5 1880Bb6 (< 1881 = Eb5*3)
F; (C4:3=88.1 C5:3=176.2 C6:3=352.4 <) 352.5F4 705F5 := Bb3*3
C; (G1:3=33 ... G4:3=264 <) 264.3C4
G; 99G2 = 33*3

Chromatically on the keys in frequencies as absolute-pitchs:

264.3_C4
279___C#4
296___D4
313.5_Eb4
330.75E4
352.5_F4
372___F#4
396___G4
418___G#4
442.5_A4
470___Bb4
528.6_C5

!sparschuh885A5.scl
!
c880:881g296:297d295:296a3968:3969bf#c#836:837g#eb1880:1881bbf3524:3525c
!
12
!
2790/2463
2960/2463
3135/2463
6615/4926 ! (5:4)(882:881)
3525/2463 ! (4:3)(3525:3524)
3720/2463
3960/2463 ! (3:2)(880:881)
4180/2463
4425/2463
4700/2463
4960/2463
2/1
!

What do you think about that now?

Yours Sincerely
A.S.