## New septenarius for a'=440 Hz, was Re: Werckmeister's Septinarius temperament

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• ... agreed, hence i do return to W s original 131. ... also right, hence so the resulting ratios get even more simple: The calculations benefit from that by
Message 1 of 26 , Apr 17, 2007
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--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> > 1113/1112 c#"556 c#'278 c#139
> > 417/416 g#208 G#104
> > eb'312 eb156 Eb78
> > bb'468 bb234 Bb117
> > f'351
> > 1053/1052 c"526 c'263/262 131
> > g'393/392 196 98 49=7*7
>
> I don't think this is so good, you have C-G tempered by 262/263
> which
>
> Better to have 315/350 f'350 f175 and 525/524 c''524 c262 131 ... ?
>
also right, hence so the resulting ratios get even more simple:
The calculations benefit from that by less computational0 overhead.

start traditional @ pitch-class GAMMA=G alike in:

http://www.celestialmonochord.org/log/images/celestial_monochord.jpg

GG 49 := 7*7 (GAMMA-ut, the empty string in the picture)
D 147
3D441 > a'440 a220 A110 AA55Hz=the AA-string of a double-bass
e 165
3e495 > b'494 b247
3b741 > f#"740 f#'370 f#185
c# "555
3c#"1665 > g#"'1664 g#"832 g#'416 g#208 G#104 GG#52 GGG#26 GGGG#13
EEb 39
Bb 117
3Bb351 > f'350 f175
3f525 > c"524 c'263 c131
3c393 > g'392 g196 G98 GG49=7^2 returned

!septenarius_GG49Hz.scl
sparschuh's version @ middle-c'=262Hz or a'=440Hz
12
!absolute pitches relativ to c=131 Hz
555/524 ! c# 138.75 Hz
147/131 ! d
156/131 ! eb
165/131 ! e
175/131 ! f
185/131 ! f#
196/131 ! g
208/131 ! g#
220/131 ! a 440Hz/2
234/131 ! bb
247/131 ! b
2/1

That results on my old piano in the first/lowest octave:

AAA 27.5 Hz lowest pitch, on the first white key on the left side
BBBb29.25 next upper black key
BBB 30.875 http://en.wikipedia.org/wiki/Double_bass "at~30.87 hertz"..
CC_ 32.875
CC# 34.6875 := c"555/16
DD_ 36.75
EEb 39 := GGGG#13Hz*3
EE_ 41.25 ..."E1 (on standard four-string basses) at ~41.20 Hz
FF_ 43.75
FF# 46.25
GG_ 49 := 7*7 Werckmeister's/Scheibler's initial septimal choice
GG# 52 = GGGG#13Hz*4
AA_ 55 = 440Hz/8 ; 3 octaves below Scheibler's choice

http://mmd.foxtail.com/Tech/jorgensen.html
#133: "Johann Heinrich Scheibler's metronome method of 1836"

http://www.41hz.com
"41 Hz is the frequency of the low E string on a double bass or an
electric bass." if it has none additional 5th string
for midi(B0)=BBB 30.875 Hz an 2nd above AAA 27.5 Hz,
the lowest A on the piano, without attending or even careing
"string-imharmonicty"
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN0000760000S1000S22000004&idtype=cvips&gifs=yes

Consider the 3rds qualities in violins empty sting G-D-A-E block order:

1: G > B > Eb> G
2: D > F#> Bb> D
3: A > C#> F > A
4: E > G#> C > E

1: G > B > Eb > G.
absolute analysis:
GG 49.
5*49 = 245 < b'247 < 248 124 62 31
5*31 = 155 < eb156 Eb78 EEb39
5*39 = 195 < g196 G98 GG7*7.
relative diesis 128/125 subpartition:
G 123.5/122.5 B 96/95 Eb 196/195 G
G ~14.1 cents B ~18.1 Eb ~ 8.86c G

2: D > F# > Bb > D.
abs:
d147. < 148 74 37
5*37 = f#185 < 186 93
5*31*3=3*155 < 156*3 78*3 39*3 = Bb117
5*39*3=3*195 < 196*3 98*3 49*3 = d147.
rel:
D 148/147 F# (1/9+85)/(84+1/9) Bb 196/195 D
D ~ 11.4c F# ~ 20.5 cents Bb ~ 8.86cents D

3: A > C# > F > A.
abs:
AA55. A110 < 111 = 37*3
5*111= c#"555 = 5*111 < 112*5 56*5 28*5 14*5 7*5
5*35 = f175 < 176 88 44 22 11
5*11 = AA55.
rel:
A 111/110 C# 112/111 F 176/175 A ; with all 3 factors superparticular
A ~ 15.7c C# ~ 15.5c F ~ 9.86c A

4: E > G#> C > E.
abs:
e165. < 166 83
5*83 = 415 < g#'416 g#208 G#104 GG#52 GGG#26 GGGG#13 §§
5*13 = 65 130 < c131 < 132 66 33
5*33 = e165.
rel:
E (6/7+118)/(117+6/7) G# 131/130 C 132/131 E
E ~ 14.6 cents ~ G# ~ 13.3 cents C ~ 13.2c E
§§ GGGG# 13 Hz has negative "midi"-index G#_-1,
which midi-keyboard supports negative key indices?

Summary:
3rds martix in "Cents"
5ths in top>down order
3rds in left>right direction respectively:

1: G 14.1 B_ 18.1 Eb 8.86 G
2: D 11.4 F# 20.5 Bb 8.86 D
3: A 15.7 C# 15.5 F_ 9.86 A
4: E 14.6 G# 13.3 C_ 13.2 E

Conversely "ET" detunes all 3rds about the same amount:
(128/125)^(1/3) = ~13.7Cents or ~127/126,
Attend that: the "septenarius" fits therefore better than ET to
horns and trumpets in Eb,Bb & F, with inherent natural
3rds Eb>G, Bb>D, & F>A that turn out less than 10Cents out of tune
in the septenarius case.

> (cf. 12ET frequencies: 261.6, 277.2, 293.7, 311.1, 329.6, 349.2,
> 370.0, 392.0, 415.3, 440.0, 466.2, 493.9 ...)

In the "ET" case, it is difficult to resolve the
septenarian root-factors above alike:
11,13,31,37 & 83 below the well known 3,5 & 7 limits.
>
> Can one use the nice ratio 63/50 = 1.26 to build a 'septenarian' >near-
> equal tuning?
on the one hand is:
(63/50)(4/5)=126/125
but but on the other side
the diesis 128/125=(128/127)(127/126)(126/125)
contains 3 factors.
hence:
(63/50)^3 = 2.000376... > 2/1
overstretched octave
or as superparticular ratio:
((63/50)^3)/2= 250047/250000=(7/47+5320)/(5319+7/47)
~1/2 per mille
but a better approximation of the octave delivers 127/126
the factor in the middle:
((5 / 4) * (127 / 126))^3 = ~1.99999802............

> eg Eb-G = 150:189 ...
> via Eb150 Bb225 (675) F337 (1011) C505 (504) G378=189
>
> then continue:
> ... (567) D283 (849) A424=212 (636) E635 (634) B951 (2853/2848)
> F#356=178 C#267 (801) G#400 Eb300
>
> seems to work nicely at late Baroque pitch levels - only three pure
> fifths between Eb-Bb, G#-Eb, F#-C#.
>
that is expanded:
A 424 212
3A636 > E635 > 634 317
B951 > 950 475
3*475=1425 > F#1424 712 356 178 89
C# 267
801 > G# 800 400 200 100 50 25 Werckmeister's tief-Cammerthon 400Hz
Eb 75
Bb 225
675 > F 674 337
1011 > C 1050 505 > 504 252 126 63
G 189
567 > D 566 283
849 > A 848 424

recombining that 5ths-circle in ascending ordered pitches yields:
C 252.5 Hz middle_C
C#267
D 283
Eb300
E 317.5 or better 317?
F 337
F#356
G 378
G#400 Beekman's, Descartes's, Mersenne's & Sauveur's standard-pitch
A 424
Bb450
B 475.5 or better 475?
C'505

T.D. remarked already in his numbers inbetween:

>B951 (2853/2848)
> F#356

that there appears an unsatisfactory irregular gap:
of 2853/2848 = 570.6/569.6 = (951/950)(1425/1424)
induced by the choice of
E 635 > 634 317 instead
635 > E 634 317
That results in the above none-integral superparticular ratio bug.

Hence i do suggest to replace
by the tiny changes
1: E 635-->>>634
2: B 951-->>>950
in order to fix the bug.

so that now all 5th-tempering steps become integral superparticular
ratios, without any exception:

A 424 212 106
3A=318 E 317 instead formerly 317.5
3E=951 B 950 475 instead formerly 951 475.5
3B=1425 F# 1424 712 356 178 89
&ct.
the rest of the circle remains unchanged.
Is that ok?

Analysis of the:
3rds sharpness, -how much wider than 5/4-
per diesis subpartition into superparticular factors,
so that the product of 3 tempered 5ths results an octave
in each of the 4 blocks:

1: G > B > Eb > G.
abs:
G378. 189 > 190 95
5*95 = B 475 < 480 240 120 60 30 15
5*15 = Eb75
5*75 = 3*125 < 126*3 = G378.
rel. 2^7/5^3=128/125=
G 190/189 B 160/159 Eb 126/125 G remember (63/50)(4/5)=126/125
?or formerly in the original version:
?G378. 189 > 190 95
?5*95 = 475 950 < B951 < 960 480 240 120 60 30 15
?......
?rel. 2^7/5^3=128/125=
?G (190/189)(951/950) B 320/317=(2/3+106)/(105+2/3) Eb 126/125 G
That appears i.m.o. much more complicated than my suggested change.

2: D > F# > Bb > D.
abs:
D283. < 284 142 71
5* 71 = 355 < F#356 178 89 < 90 45
5* 45 = Bb225 < 226 113
5*113 = 565 < D556 283.
rel. 128/125=
D (284/283)(356/355) F# 90/89 Bb (226/225)(556/565) D

3: A > C# > F > A.
abs:
A424. 212 106 53
5* 53 = 265 < C#267 < 268 134 67
5* 67 = 335 < F 337 < 338 169
5*169 = 845 < A 848 424.
rel: 128/125=
A 133.5/123.5 C# (268/267)(168.5/166.5) F >>>
>>> F (338/337)((2/3+282)/(281+2/3)) A

4: E > G# > C > E.
E 317. < 320 160 80 40 20
5* 20 = G#100 < 101
5*101 = C 505 < 506 253
5*253 = 1265 < E 1268 634 317.
rel: 128/125=
E (2/3+106)/(105+2/3) G# 101/100 C (506/505)/(422.666.../421.666...) E
?or formerly
?E 635? > 640 320 .....
?....&ct. alike above...
?....
?5*253 =1265 < 1270 635?

try to find out similar improvements in order to reduce the ratios
to less complicated proportions

have a lot of fun in whatever tuning you do prefer
A.S.
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