--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> This suggests something around 53-ET as a first approximation. 53-ET
> has some useful properties,....

how about
http://launch.groups.yahoo.com/group/tuning/message/81808
as second approxiamtion with other useful properites?

Represented in
extended Bosanquet-Helmholtz notation
by the additional accidentials:

'/' one comma sharper up
'//' two commata sharper up '+'
and respectively
'\' one comma flattend down
'\\' two commata flattend down '-'

Begin with:
0: C- 1 C\\ absolute fundamental root 1/1 == 3^0/2^0 @ start
1: G- 3 G\\ 3^1
2: D- 9 D\\ 3^2
3: A- 27 A\\ 3^3
4: E- 81 E\\ 3^4
5: B- 243 B\\ 3^5
6: GB 729 Gb\ 3^6 last just Pythagorean 3-limit-smooth ratio 729/512
7: DB(547 1094 2188 4376>) 4375 (>4374 2187 = 3^7) poppy-seed ragisma
8: AB 1641 := 547*3 Ab\
9: EB 1231 2646 4924 (> 4923 := 1641*3) Eb\
10: BB 1847 3694 (> 3693 := 1231*3 ) Bb\
11: F\ 2771 5542 (> 5541 := 1847*3 )
12: C\ 4157 8314 (> 8313)
13: G\ 1559 3118 6236 12472 (> 12471)
14: D\ 1169 ... 4678 (> 4677)
15: A\ 877 ... 3508 (> 3507) absolute normal-pitch A\_4=438.5 Hz
16: E\ 329 ... 2632 (> 2631)
17: B\ 987
18: Gb 2961
19: Db 2221 4442 8884 (> 8883 := 2961*3)
20: Ab 833 ... 6664 (> 6663 := 2221*3)
21: Eb 625 1250 2500 (> 2499 = 833*3) = 5^4 = 5*5*5*5 first 5-smooth
22: Bb 1875 = 5^4*3
23: F_ 703 1406 2812 5624 (<5625 = 5^4*3^2)
24: C_ 527 middele_C4 263.5 Hz
25: G_ 1581
26: D_ 2371 4742 (< 4743)
27: A_ 889 ... 7112 (< 7113) ~@ Herbert Karajan's pitch a'=444.5Hz
28: E_ 2667
29: B_ 125 = 5^3 = 5*5*5 first schisma ready versus 21:
30: F# 375
31: C# 1125
32: G# 1687 3374 (< 3375 = 5^3*3^3 )
33: D# 1265 ... 5065 (< 5061)
34: A# 1897 3794 (< 3795)
35: F/ 2845 5690 (< 5691)
36: C/ 4267 8534 (< 8535)
37: G/ 25 50 100 200 400 800 1600 3200 6400 12800(<12801) 5^2=5*5
38: D/ 75 second schisma ready on 37:
39: A/ 225
40: E/ 675
41: B/ 253 ... 2024 (< 2025 = 5^2*3^4)
42: F& 759 F#/
43: C& 569 1138 2276 (< 2277) C#/
44: G& 1707 G#/
45: D& 5 ... 5120 (< 5121) D#/ third schisma ready;simplest 5-smoothie
46: A& 15 A#/
47: F+ 45 F//
48: C+ 135 C//
49: G+ 405 G//
50: D+ 1215 D+
51: A+ 911 1822 3644 (<3645 = 5*3^6) A// attack the last schisma
52: E+ 683 1366 2732 (<2733) E//
53: B+ 1 ... 2048 (< 2049) B//=C\\ returned back to root C-_-4 = 1/1

or lined up in ascending order
as scala-file ratios ! vs. note-names in absolute-pitches

! poppy_seed53tone.scl
!
Sparschuh's 7-lim.dyadic 53-tone ragismatic 4375:4374 poppy-seed-comma
!
! 1/1 ___ ! @ 00: C- 512Hz C\\ tenor-C\\5, 9-octave above the root 1/1
4157/4096 ! A 01: C\ 519.625
527/512 !_! B 02: C_ 527
4267/4096 ! C 03: C/ 533.375
135/128 !_! D 04: C+ 540 C//
4375/4096 ! E 05: DB 546.875 Db\ := 4375/8 = 7*5^3/2^3
2221/2048 ! F 06: Db 555.25
1125/1024 ! G 07: D# 562.5
569/512 !_! H 08: C& 569 C#\
9/8 !_____! I 09: D- 576 D\\
1169/1024 ! J 10: D\ 584.5
2371/2048 ! K 11: D_ 592.75
75/64 !___! L 12: D/ 600
1215/1024 ! M 13: D+ 607.5 D//
1231/1024 ! N 14: EB 615.5 Eb\
625/512 !_! O 15: Eb 625 = 5^4
1265/1024 ! P 16: D# 630.5
5/4 !_____! Q 17: D& 640 D#/
81/64 !___! R 18: E- 648 E\\
329/256 !_! S 19: E\ 658
2667/2048 ! T 20: E_ 666.75
675/512 !_! U 21: E/ 675
683/512 !_! V 22: E+ 683 E// = F\\ = F- temper out Mercator's comma
2771/2048 ! W 23: F\ 692.75
703/512 !_! X 24: F_ 703 F
2485/2048 ! Y 25: F/ 711.25
45/32 !___! Z 26: F+ 720 F//
729/512 !_! a 27: GB 729 Gb\
2961/2048 ! b 28: Gb 740.25
375/256 !_! c 29: F# 750
759/512 !_! d 30: F& 759 F#/
3/2 !_____! e 31: G- 768 G\\ quinte
1559/1024 ! f 32: G\ 779.5
1581/1024 ! g 33: G_ 790.5
25/16 !___! h 34: G/ 800 Mersenne, Sauveur & Werckmeister's CammerThon
405/256 !_! i 35: G+ 810 G//
1641/1024 ! j 36: AB 820.5 Ab\
833/512 !_! k 37: Ab 833
1687/1024 ! l 38: G# 843.5
1707/1024 ! m 39: G& 853.5 G#/
27/32 !___! n 40: A- 864 A\\
877/512 !_! o 41: A\ 877 = 2*438.5Hz @ 90 MetronomeBeats/min vs.440Hz
889/512 !_! p 42: A_ 889
225/128 !_! q 43: A/ 900
911/512 !_! r 44: A+ 911 A//
1847/1024 ! s 45: BB 923.5 Bb\
1875/1024 ! t 46: Bb 937.5
1897/1024 ! u 47: A# 948.5
15/8 !____! v 48: A& 960 A#/
243/128 !_! w 49: B- 972 B\\
987/512 !_! x 50: B\ 987
125/64 !__! y 51: B_ 1000 = 10^3 = 1kHz psycho-acustical normal-pitch
253/128 !_! z 52: B/ 1016
2/1 !_____! @'53: B+ 1024 = 2^11 B//_5 = C\\_6 = C-_6 the sopran-C\\
!
!
So far my 'theoretically-notation',
but practically i do recommend
out of that theory the plain absolute fixed pitch-frequencies alone

@ 00: C- 512Hz root
A 01: C\ 519.625
B 02: C_ 527
C 03: C/ 533.375
D 04: C+ 540
E 05: DB 546.875
F 06: Db 555.25
G 07: D# 562.5
H 08: C& 569
I 09: D- 576
J 10: D\ 584.5
K 11: D_ 592.75
L 12: D/ 600
M 13: D+ 607.5
N 14: EB 615.5
O 15: Eb 625
P 16: D# 630.5
Q 17: D& 640 third
R 18: E- 648
S 19: E\ 658
T 20: E_ 666.75
U 21: E/ 675
V 22: E+ 683
W 23: F\ 692.75
X 24: F_ 703
Y 25: F/ 711.25
Z 26: F+ 720
a 27: GB 729
b 28: Gb 740.25
c 29: F# 750
d 30: F& 759
e 31: G- 768 quinte
f 32: G\ 779.5
g 33: G_ 790.5
h 34: G/ 800
i 35: G+ 810
j 36: AB 820.5
k 37: Ab 833
l 38: G# 843.5
m 39: G& 853.5
n 40: A- 864
o 41: A\ 877 = 2*438.5Hz @ 90 MetronomeBeats/min vs.440Hz
p 42: A_ 889
q 43: A/ 900
r 44: A+ 911
s 45: BB 923.5
t 46: Bb 937.5
u 47: A# 948.5
v 48: A& 960
w 49: B- 972
x 50: B\ 987
y 51: B_ 1000
z 52: B/ 1016
@'53: B+ 1024 or sopran_C-

Labeled here more restrictive
only in one- or two-letter note-name nomenclauture,
in order to keep my rational-53 basic concept more simple.

bye
A.S.