--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> Comments below!
>
> > a'440Hz a220 A110
> > e'330 e165
> > b'495
> > 1485/1484 f#"742 f#'371
>
> Instead of 496 and 372. This avoids having a wide e-b fifth and is
> better for G major and D major.

right, in order to get rid of the oversharp wolfs in W's original
stringlength numbers, alike in his famous #3 the 'quaternarius'
has only 4 flattend and 8 pure 5ths:
A E B>F# C# G# Eb Bb F C>G>D>A

>
> > 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 is about 1/3 comma...
>
...flattend than a pure 5th: 3/2.
So far about that strongest detuned 5th C>G.
Consider accordingly the belonging 3rd C>E in the major-chord C>E>G:
with compareable sharpness:
e165 e'330 e"660 e'"1330/1315=5*c"263
shortening by common factor 5 yiels
a tempering of C>E about 264/265,
so that the C-major chord consists in

C(1/1)-E*(265/264)-G*(262/263)
or
4:(5*(265/264)):(6*262/263)
instead barely 4:5:6 without the idea of tempering
3rds and 5ths in the same range of magintude.

Many 'experts' do recommend to sharpen the 3rds about the amount in
amplitude alike the corresponding 5ths become flattened, so that the
beatings of 3rds and 5ths beat almost the same in reverse directions.
Skilled organ-builders use that effect in order to demonstrate
that their robust organs survive even such impressive resonances.
It appears that already
http://en.wikipedia.org/wiki/Arnolt_Schlick
knew that old well-known tuning-trick/method in his instructions.

> Better to have 315/350 f'350 f175 and 525/524 c''524 c262 131 ... ?
Good idea, if you intend to stay nearer to W's original version,
when somehow aiming to approximate "ET" however.

>
> Then C-G is pure and
makes that only sense according the above demands
when C-E is also pure chosen?

>
> >
Correction of: !septenarius440Hz.scl

> > 351/263 ! F# [should be 271] for Gene: 371 is the correct pitch.

> or:
>
> !septenarius440Hzmk2.scl
> !
> TD's septenarius @ middle c'=262Hz or a'=440Hz
> !
> 12
> !
> 278/262 ! C# short 138/131
> 294/262 ! D short 147/131
> 312/262 ! Eb short 156/131
> 330/262 ! E short 165/131
> 350/262 ! F short 175/131
> 371/262 ! F#
> 393/262 ! G
> 416/262 ! G# 208/131
> 440/262 ! A 220/131
> 468/262 ! Bb 234/131
> 495/262 ! B
> 2/1

The pure middle c' in reference to a'=440Hz becomes in the just case
440Hz*3/5 = 264Hz. Hence i do prefer the nearer 263Hz instead
yours lower 262Hz, which appears just a little bit to flat lowered
in my personal taste, especially when having just intonation
in mind or ear instead the virtual "et".
Most professional and skilled tuners do to keep
the frequent keys with few accidentials somehow purer than
the less used keys with many accidentials, so that the
strange keys got more pythagorean 3rds, by purer
or even just pure 5ths inbetween them.

>
> (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 ...)
those irrational numbers are far to complicated for solving the
problem. Who needs the advanced precision of 4 decimal digits
in the octave from c' to c"?

But if you want to approach "et" whatsoever, then 262 would be the
better choice for converging "et" as the above approximation suggest.

>
> Can one use the nice ratio 63/50 = 1.26 to build a 'septenarian' >near-
> equal tuning?
I.m.o: the nearer one draws to approach "et",
the most frequently used 3rds get to much worse detuned

Luckily nobody can tune irrational intervals in practice,
so that the worsest case: "et" remains barely a theoretically fiction,
excluded from real implementation on a real sounding instrument.
Simply try out how well can you reproduce by yours ears:
on the one hand:
a pure 5th ratio 3:2=1.5 ~702cents
and on the other hand:
sqrt(2) = 600 Cent "et"-tritous, that's geometrically interpreted:
http://mathworld.wolfram.com/PythagorassConstant.html

Experimental result:
There is no psychoacusitcally evidence for departening
the ratio of a 5th 3:2 for the benefit of sqrt(2) ET-tritone.
Quoting Herrmann Helmholtz: "The ear prefers simple ratios."


> eg Eb-G = 150:189 ...
> via Eb150 Bb225 (675) F337 (1011) C505 (504) G378=189

But by that procedure one does also loose to much of the 'Baroque'
key-characteristics.
http://www.societymusictheory.org/mto/issues/mto.95.1.4/mto.95.1.4.code.html
http://de.wikipedia.org/wiki/Tonartencharakter
>
> 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#.
>
hmm, is it still apt to call that 'baroque'-style?
As far as i understood late "Baroque" post-meantone instructions:
There i found a tendency to keep the 5ths inbetween the accidentials
F#-C#-G#-Eb-Bb
purer more pure (within the upper black keys on the piano)
than in the ordinary F-C-G-D-A-E-B: that got generally more tempering.

In extreme form i do start from my prototype model.
The procedure consists in a chain of 11 almost pure 5ths,
that contains the JI pitches, but also schismic Pythagorean-enharmonics:

Here the chain
F-A-C-E-G-B-D
are all 3 pure major 4:5:6 chords
in exact beatless just proportions:

A 440. 220 110 55
E 165
B 495.
F# 1485
4455 C# 4454 2227
6681 G# 6680 3340 1670 835
2505 Eb 2504 1252 626 313.
Bb 939
2817 F 2816 1408 704 352. 176 88 44 22 11
C 33
G 99
D 297. / 296 148 74 37
111 A 110 55

A E B F#~C#~G#~Eb Bb~F C G D~~~~~~~~~~~~~~~~~~~~~~~~~~~~A

The schisma 32805/32768=5*3^8/2^15=
(4455/4454)(6681/6680)(2505/2504)(2817/2816)
is tempered out by the subdivsion in to that
product of 4 superparticular factors.
Respectively the
SC=81/80=(297/296)(111/110) into 2 parts at one @ D>A 40:27

Rearranging same pitches in ascending order yields:

C' 264Hz middle-C
C# 278.375
D' 297
Eb 313
E' 330
F' 352
F# 371.25
G' 396
G# 417.5
A' 440Hz reference pitch
Bb 469.5
B' 495
C" 528

so far about the 11 other frquencies that i percieve instantly
also in mind immediatley when hearing a 440Hz tuning-fork by ear,
or simpy when imagening that pitch-levels enwraped when reading
musical scores in any fitting tuning.

schismatic_just440Hz.scl
!
sparschuh's-schisma-subdivision(4455/4454)(6681/6680)(2505/2504)(2817/2816)
!
2227/2112 ! C#
9/8 ! D
313/264 ! Eb
5/4 ! E
4/3 ! F
45/32 ! F#
3/2 ! G
835/528 ! G#
5/3 ! A=440Hz
313/176 ! Bb
15/8 ! B
2/1

But, how about that almost similar alternative one at the moment on my
piano?

A 440. 220 110
330 E 329.
B 987
2961 F# 2960 1480 740 370. 185
C# 555
1665 G# 1664 832 416. 208 104 52 26 13
Eb 39
Bb 117
F 351.
1053 C 1052 526 263.
789 G 788 394. 197
591 D 590 285./284 147
441 A 440. (or 3*285=885 A 880 440.)

with strongest tempering @ D>A: 885/880=(285/284)(441/440)=177/176
but still less than SC^(1/2) ~161/160 or ~162/161,
hence rather tolerable than
the ancient Erlangen-monochord or Kirnberger#1,
that charge a full SC on D>A alike the above 'schismatic_just.scl'.

So far my reccomendation for those who prefer to stay nearer at JI
than to the i.m.o. over-detuned "ET", that i do meanwhile consider as
outdated intuneable fiction.

sparschuh_gothic_style440Hz.scl
!
12
!
555/526 ! C# 277.5 Hz
285/263 ! D
312/263 ! Eb
329/263 ! E
351/263 ! F
370/263 ! F#
394/263 ! G
416/263 ! G#
440/263 ! A reference-pitch 440Hz
468/263 ! Bb
987/526 ! B 493.5 Hz
2/1

on the keys

+-----------
| C 263 middle-C
+--|277.5=C#
| D 285
+--|312=Eb
| E 329
+-----------
| F 351
+--|370=F#
| G 394
+--|416=G#
| A 440
+--|468=Bb
| B 493.5
+-----------
| C'526
&ct.

If you dont't like any of that, it's up to you to create yours own
personal version, according yours private preferences.

A.S.