## Re: A Rational Well Temperament

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• ... That s made by construction in 5ths: C = 90 45 G = 3C = 135 D: (3G=405) 404 202 101 A: (3D=303) 302 151 E: (3A=453) 452 226 113 B: (3e=339) 338 169
Message 1 of 63 , Apr 9, 2008
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--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:

> > > > I think it is
> > > > 90:95:101:107:113:120:127:135:143:151:160:169:180

That's made by construction in 5ths:
C = 90 45
G = 3C = 135
D: (3G=405) > 404 202 101
A: (3D=303)> 302 151
E: (3A=453) > 452 226 113
B: (3e=339) > 338 169
F# (3B=507) < 508 254 127 attend the broade layperson's dog-5th
C# (3F#=381) > 380 190 95
G# (3C#=285) < 286 143 another even worser wide dog-5th
Eb (3G#=429) > 428 214 107
Bb (3Eb=321) > 320 160 80 40
F = 3Bb = 120 60 30 15
C = 3F = 45 cycle closed

>... there is the wide fifth 143/95 which is tempered by 286/285,
that sound none well, but is still only good,
if you really intend there an 'open' 5th alike in meantonics.
> > >
> > > Noting that 285=3*95, I think the smallest base number that
allows all
> > > fifths to be tempered less than 321/320 will turn out to be over
107.
> >
> > 125:132:140:148:157:167:176:187:198:210:222:235:250
>
in 5ths cycle:
C 125
G (375) > 374 187
D (561) > 560 280 140 70
A 210 105
E (315) > 314 157
B (471) > 470 235
F# (1175) < 1184 592 148 74 37 even worser wide than the above ex.
C# 222 111
G# (333) < 334 167 another problematic wide 5th
D# (501) > 500 250 125

in order to fix such ugly broade-5th bugs, just consult my:
http://www.strukturbildung.de/Andreas.Sparschuh/Mainz_1999.jpg
without such dog-5ths defects,
as found in some 'esotheric' reinterpretations,
that do not appear in my 1999 original 'discovery' version.

> Wait! There's even better one
> 101:107:113:120:127:135:143:151:160:169:180:191:202
> No fifth differs more than 1/4 pythagorean comma from just.

That's expanded:
C: 101
G: (303) > 302 151
D: (453) > 452 226 113
A: (339) > 338 169
E: (507) < 508 254 127 at least barely only one broade 5th
B: (381) > 382 191
F# (573) > 572 286 143
C# (429) > 428 214 107
G# (312) > 320 160 80 40 20 10 5
Eb 15
Bb 45
F: 135
C: (405) > 404 202 101
>
> And with this
>
> 131:139:147:156:165:175:185:196:208:220:234:247:262
> No fifth differs more than 321/320 from just.
>
C 131
G (393) > 392 196 98 49
D 147
A (441) > 440 220 110 55
E 165
B (495) > 494 247
F# (741) > 740 370 185 {Proposal in order to get rid of the dog}
C# (555) < 556 278 139 { here you'd better let 555 unchanged }
G# (417) > 416 ... 13 {...832 1664 < (1665 = 3*555) }
Eb 39
Bb 117
F: (351) > 350 175 instead W's choice 176=11*2^4
C: (525) > 524 262 131

{considering that little change converts your's originally
'open'-tuning into an
"well"-temperament in the sense of W. & Bach, without the dog}

Attend:
http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
http://en.wikipedia.org/wiki/Werckmeister_temperament#Werckmeister_IV_.28VI.29:_the_Septenarius_tunings

A.S.
• ... Hi George, ... in deed, even that *.scl-file contained some unfixed bug. ... Many thanks again for that repair. ... when considering more evaluated digits,
Message 63 of 63 , May 8, 2008
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--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

Hi George,
> > epimoric decomposition
> >
> > F 1058:1059 C 528:529 G 296/297 D 295/296 A 294/295 E 3968:3969 B
> > B F# C# G# Eb 7532:7533 Bb 5648:5649 F
> >
> >Cents approximation,
>
> >F~ -1.635 ~C~ -3.275 ~G~ -5.839 ~D~ -5.859 ~A~ -5.879 ~E~ -0.436 ~B
> > B F# C# G# Eb~ -0.2298 ~Bb~ -0.306 ~F
> >
>
> I believe there are still a few mistakes.
in deed, even that *.scl-file contained some unfixed bug.

> From the sizes of the
> fifths you give, I think that perhaps you meant this:
>
Now -as far as i can see- those ratios appearto be correct:
> 558/529 ! C#
> 592/529 ! D
> 2511/2116 ! Eb = 627.75/529
> 1323/1058 ! E = 661.5/529
> 706/529 ! F
> 744/529 ! F#
> 792/529 ! G
> 837/529 ! G#
> 875/523 ! A = 442.5Hz*2 absolute a4
> 1883/1058 ! Bb = 941.5/529
> 992/529 ! B
>
Many thanks again for that repair.

> This will result in:
when considering more evaluated digits,
as calculated by "Google"s arithmetics, which yields:

> F~ -1.636 ~C
(1 200 * ln(1 058 / 1 059)) / ln(2) = ~-1.63555425...

> ~C~ -3.276 ~G
(1 200 * ln(528 / 529)) / ln(2) = ~-3.27575131...

>~G~ -5.839 ~D
(1 200 * ln(296 / 297)) / ln(2) = ~-5.83890621...
arises that deviation here due to your's rounding procedere?

~D~ -5.784 ~A
1 200 * ln(295 / 296)) / ln(2) = ~-5.85866566...
arises that deviation here due to your's rounding procedere?

~A~ -5.953 ~E
(1 200 * ln(294 / 295)) / ln(2) = ~-5.8785593...

same question as for D~A?
Or what else could be the reason for the tiny
discrepancy amounting about tiny 1/10 Cents
inbetween ours calculations of the relative deviations
in the tempered 5ths flatnesses?

~E~ -0.436 ~B
(1 200 * ln(3 968 / 3 969)) / ln(2) = ~-0.436243936...

> B F# C# G# all just pure 5ths

Eb~ -0.2298 ~Bb
(1 200 * ln(7 532 / 7 533)) / ln(2) = ~-0.229835254...

Bb~ -0.306 ~F
(1 200 * ln(5 648 / 5 649)) / ln(2) = ~-0.306494477...

at least we both do agree now in all others 5ths except D~A~E.

What do you think about that well-temperement,
with an almost JI the C-major chord:

C:E:G = 4 : 5*(2646/2645) : 6*(529/528)

?

Yours Sincerely
Andreas
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