--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> Or, if anyone has any recommendations for recordings of classical
> works done in 53-tet or other temperaments, or even perhaps JI...

Dear Mike & all others,

for Newton's "horogramm" in 53:
http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindley7.gif

in the
http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
style, i do reccomend the epimoric
stepwise cycle of 5ths modulo octaves in Bosanquet's notation
in terms of the corresponding absolute pitches:

0; c-_-4 = 1 ... c-_4=256Hz unison as general reference to the unit
1; g-_-2 = 3 ! 5th
2; d-_-1 = 9 ! major-tone
3; a-_0 = 27 ! Pythagorean 6th
4; e-_2 = 81 ! ditone
5; b-_3 = 243 ! octave:limma
6; gB_6 = 729 ! tritone
7; dB_7 = 2,187 ! apotome 4,374 [< 4,375 = 5*a\_5] the 'ragisma'
8; aB_3 = 205 410 820 1,640 3,280 6,560 (<6,561 = 3^8)
9; eB_5 = 615
10; bB_5 = 1,845
11; f\_7 = 2,767 5,534 (<5,535)
12; c\_7 = 2,075 4,150 8,300 (<8,301)
13; g\_4 = 389 778 1,556 3,112 6.224 (<6,225)
14; d\_6 = 1,167
15; a\_5 = 875=5*f._3 ; 1,750 3,500 (<3,501) JI 3rds: F. -> A\
16; e\_7 = (41 82 164 328 656 1,312 2,624<) 2625 = 5*c._3
17; b\_8 = (123 ... 7.872<) 7875 = 5*g._6
18; gb_10 = (369 ... 23,616<) 23,625 = 5*d._8 last JI 3rd among 4
19; db_6 = 1107
20; ab_4 = 415 830 1,660 3,320 (<3321) neoBaroque tuning-forks
21; eb_6 = 1,245
22; bb_6 = 1,867 3,734 (<3,735)
23; f._3 = 175 350 700 1,400 2,800 5,600(<5601) instead W's "176"
24; c._5 = 525 tenor_C5 ; middle_C4 = 262.5 Hz
25; g._6 = 1,575
26; d._8 = 4,725
27; a._9 = 14,175
28; e._11 = 42,525
29; b._12 = 127,575
30; f#_12 = (1,495 ... 95,680<) 95,681 ... 382,724 (<382,725)
31; c#_8 = 4,485
32; g#_9 = 13,455
33; d#_11 = 40,365
34; a#_12 = 121,095
35; f/_12 = 90,821 181,642 363,284 (<363,285)
36; c/_13 = 136,231 272,462 (<272,463)
37; g/_12 = 102,173 204,346 408,692 (<408,693)
38; d/_13 = 153,259 306,518 (<306,519)
39; a/_4 = 449 ... 459,776 (<459,777)
40; e/_6 = 1,347
41; b/_7 = 4,041
42; f&_9 = 12,123 := f#/ with '&'='#'*'/' about 6 commata sharper
43; c&_11 = 36,369
44; g&_12 = 109,107
45; d&_11 = 40,915 ... 327,320 (<327,321)
46; a&_9 = 15,343 ... 122,744 (<122,745)
47; f+_11 = 46,029 := f//_11 with '+':= '//' double comma elevation
48; c+_12 = 69,043 138,086 (<138,087)
49; g+_10 = 25,891 ... 207,128 (<207,129)
50; d+_12 = 77,673
51; a+_12 = 116,509 233,018 (<233,019)
52; ( e+ = f- )_13 = 177,763 349,526 (<349,527) enharmoic change
53=0'; b+_3 = c-_4 = 256Hz=2^8 ... 2^19=524,288 (<524,299) back home

that cycle matches almost
http://en.wikipedia.org/wiki/53_equal_temperament
it also subdivides the
"Mercator's Comma. Mercator's Comma is of such small value to begin
with (~3.615 cents)"
into the above 23 epimoric subfactors instead of 53 equal ones.
Attend within that the schisma 32805:32768 inbetween:

...Gb 2624:2625 Db 3320:3321 Ab Eb 3734:3735 Bb 5600:5601 F...

gaining JI heptatonics for C-major in

1. major and minor 3rds:
[F.] 5:4 [A\] 6:5 [C.] 5:4 [E\] 6:5 [G.] 5:4 [B\] 6:5 [D.]

2. as scale of whole&semi-tones:
[C.] 9:8 [D.] 10:9 [E\]16:15[F.] 9:8 [G.] 10:9 [A\] 9:8 [B\]16:15[c.]

or in commatic ascending order, as in Newton's 1664 drawing:


!septenarian53well.scl
Sparschuh's 53 generalization of Werckmeister's septenarius
53
2075/2048 ! 1; c\_7 : 2^11
525/512 ! 2; c._5 : 2^9 ~tenor_C in ET in reference to a4=440Hz
136231/131072! 3; c\_13 : 2^17
69043/65536 ! 4; c+_12 : 2^16
2187/2048 ! 5; dB_7 : 2^11 apotome
1107/1024 ! 6; db_6 : 2^10
4485/4096 ! 7; c#_8 : 2^12
36369/32768 ! 8; c&_11 : 2^15
9/8 ! 9; d-_3 : 2^3 Pythaogorean major-tone
1167/1024 !10; d\_6 : 2^10
4725/4096 !11; d._8 : 2^12
153259/131072!12; d\_6 : 2^10
77673/65536 !13; d+_12 : 2^16
615/512 !14; eB_5 : 2^9 (5:4)*(123:128)
1245/1024 !15; eb_6 : 2^10 (5:4)*(249:256)
40365/32768 !16; d#_11 : 2^15 (5:4)*(8073:8192)
40915/32768 !17; d&_11 : 2^15 (5:4)*(8183:8192)
81/64 !18; e-_2 : 2^6 Pythagorean ditone
2625/2048 !19; e\_7 : 2^11 = C.*(5:4) JI 3rds in [C.]->[E\]
42525/32768 !20; e._11 : 2^15
1347/1024 !21; e/_6 : 2^10
174763/131072!22;(e+=f-)_13:2^17 = (4:3)*(524,289:524,288) enharm.ch.
2767/2048 !23; f\_7 : 2^11
175/128 !24; f._3 : 2^7 instead of Werckmeister's "176" choice
90821/65536 !25; f/_12 : 2^16
46029/32768 !26; f+_11 : 2^15
729/512 !27; gB_6 : 2^9 tritone
23625/16384 !28; gb_10 : 2^14 = D.*(5:4) JI 3rds in [D]->[F#\]
95681/65536 !29; f#_12 : 2^16
12123/8192 !30; f&_12 : 2^16
3/2 !31; g-_-2 : 2 the initial 5th step at begin
389/256 !32; g\_4 : 2^8
1575/1024 !33; g-_5 : 2^10
102173/65536 !34; g/_12 : 2^16
25891/16384 !35; g+_10 : 2^14
205/128 !36; aB_3 : 2^7
415/256 !37; ab_4 : 2^8 neoBaroque tuning-fork in 415Hz
13455/8192 !38; g#_9 : 2^13
109107/65536 !39; g&_12 : 2^16
27/16 !40; a-_0 : 2^4 Pythagorean 6th
875/512 !41; a\_5 : 2^9 = F.*(5:4) JI 3rds [F.]->[A\]
14175/8192 !42; a._9 : 2^13
449/256 !43; a/_4 : 2^8
116509/65536 !44; a+-12 : 2^16
1845/1024 !45; bB_6 : 2^10
1867/1024 !46; bb-6 : 2^10
121095/65536 !47; a#_12 : 2^16
15343/8192 !48; c&_9 : 2^13
243/128 !49; b-_3 : 2^7 Pythagorean 7th or octave:limma
7875/4096 !50; b\-8 : 2^12
127575/65536 !51; b._12 : 2^16
4041/2048 !52; b/_7 : 2^11
2/1 ! 53==0 ; ( b+ = 2*c+ )_3 Helmholtz's enharmonics: B// = C\\
!
!
that yiedls -compared against 53EDO- an intersting
53-comma key-charcteristics
http://www.wmich.edu/mus-theo/courses/keys.html
because all the 3rds range in fine graduation inbetween

8192:6561 ~384Cents (schimatic 3rd) <<<???<<< and 5:4 ~386Cents

Attend for instance the 3rd [A\] -> [DB]
that becomes about an
http://en.wikipedia.org/wiki/Ragisma
flattend.
That small interval was also historically also used for coin-making:
http://de.wikipedia.org/wiki/Karlspfund
"Bei historischen Längenmaßen liegt der Variationskoeffizient im
allgemeinen unter 1/500, was eine Genauigkeit von ± 0,2 % bedeutet. So
gelten bei den Längenmaßen z.B. 1/2400 oder 1/4374, also die 7-glatten
Ratios 2401 : 2400 und 4375 : 4374, sowie ihre Reziprokwerte nicht als
eigentliche Ratios, sondern nur als Kommata."

Yours Sincerely
A.S.