--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> --- In tuning@yahoogroups.com, Afmmjr@ wrote:

>> Neidhardt
in using algebraically irrational numbers
>>took up theoretically from exactly where W. had left off.
while W stayed alwas reamaining within the rational fraction concept,
which was citizied by N as 'inferior' versus his own
abstract calculations using Simon Stevins ET invention,
that W refused in reference to the Pythagorean concept of
http://en.wikipedia.org/wiki/Commensurability_%28mathematics%29
of all possible intervalls to the unit 1.
N. abonded that traditional demand, still indispensable for W.


henc I do agree, that:
> > Werckmeister III is different systems.
against N's mathematically more advanced
way of computation in his 'sectio-canonis'.

> Each key is intervallically
> different from another.
yielding just
http://de.wikipedia.org/wiki/Tonartencharakter
that W considered still as essential.
N. firstly advocated ET without that feature,
needed time depart from ET
for accepting the traditional view also for himself too.


> Some keys embody Pythagorean tuning.
That's right observed,
W. intended fully aware just that sound in the rare keys
for a change in the 'variationibus' when modulating.
>
> Hum, I don't hear F# major and C# major having much intervallic
> difference myself. Rather too much Pythagoras.
Appearently W. designed that nice effect consciously wanted
especially for fast melodic transitions,
alike violinist change to pythagorean when playing runs rapidly.

>
>> W's later turn towards ET
Simply wrong, because:
He never refers to that new mathematically concept, not even
squareroots do appear nowhere in any of his writings,
not to mention logarithms.
Alike later Kirnberger W. never cared about that
modern "mathematical-stuff", as JSB
inbeween them also too refused that.
>>and/or diluted
>> meantones may have acknowledged this.
W. considered that outdated practice simply as: 'wrong'.
JSB refused Silbermann's 'barbaric 3rds!'
Who needs even today an much over-broadly based
wolf 5th of ~704Cents?
inept placed inbetween Bb and F unduly, absolute needless, even
faulty labeled as an alleged theoretically "diminished 6th"?
in 55ET, 'a tuning that had never existed' in the Baroque era.
>
>> Werckmeister VI, a basically near ET tuning found with
>> measurements like one/seventh of a Pythagorean comma.
Long ago disproved early 20th century scholary nonsene!
>
TD:
> Oh, please! Werckmeister VI is based on (complicated) j u s t
> intervals, it is nothing to do with divisions of a comma, the text >of
> Musicalische Temperatur explicitly says as much. Unfortunately many
> 20th century 'historians' starting with Dupont & Barbour have > >utterly
> misread the text and imagine (simply because of the number 7 popping
> up) that it has something to do with 1/7 comma.
Cave!
That both time-honored authors should be touched only with croucher
tools due to careless deformation of historical source texts: All in
all: Obsolete out-dated faulty below todays scientific standards.

> Well, that isn't even a good approximation.
but even worser:
Theirs deceptive description misleads astray
about W's concept behind his plain rational number arithmetics.

> Actually the 'septenarius' is better approximated by fifth-comma
> steps
There's no need for improving him:
Why using barely approximations instead
staying in his own original values
given concrete in absolute monochord-stringlengths:

196 C 1
186 C# 98/93
176 D 49/44
165 Eb 196/165
156 E 49/39
147 F 4/3
139 F# 196/139
131 G 196/131
124 G# 49/31
117 A 196/117
110 Bb 98/55
104 B 49/26
98 C' 2

obtained from tempered 5hs-circle

196 C (_393_)392;196 start
131 G 393/3:= 131(132,66,33)
176 D (_351_)352;176;88,44,22,11:= 33/3
_117_A :=351/3
156 E 78,_39_:= 117/3
104 B (_417_)416,208;104;52,26,13:= 39/3
139 F# (_279_)278;139:= 417/3
186 C# 93:= 279/3
124 G# (_495_)496,248;124;62,31:= 93/3
110 Bb (_441_)440,220;110;55:= 165/3 not Scheibler's pitch: 440cps
_147_F := 441/3
196 C 98,49:= 147/3 returned to begin

so that seven 5ths are tempered by the pure rational fractions:

C*392/393*G*132/131*D*352/351*A>E>B*416/417*F#*278/279*C#>G#*496/495*Eb>Bb*440/4\
41*F>C


Analog it's also possible to fit the corresponding four 5ths of his
#3(1691) the 'quaternarius' according in the same manner,
My actual interpretation sounds:

C 6560/6561 G 204/205 D 152/153 A>E>B 512/513 F#>C#>G#>Eb>Bb>F>C

expanded in absolute frequencies:

273.375 C ((17))2187:= 3^7
410 G (17*3=51,102,204)205;410;820,1640,3280,6560(6561:= 3^8)
306 D (19,38,76,152)153:= 17*9
456 A 57:= 19*3
342 E 171:= 19*9
256.5 B (1,...,512)513:= 19*27
384 F# 3
288 C# 9
432 G# 27
324 Eb 81
486 Bb 243
364.5 F 729:= 3^6
273.375 C 2187:= 3^7

that's in ascending pitch order

273.375 C 1 middle C
288 C# 256/243
306 D 272/243
324 Eb 32/27
342 E 304/243
364.5 F 4/3
384 F# 1024/729
410 G 3280/2187 coeval Cammer-tone ~410cps
432 G# 128/81
456 A 1216/729 coeval Choir-tone ~456cps
486 Bb 152/81
512.5 B 16/9
546.75 C' 2

Further refinement in
http://launch.groups.yahoo.com/group/bach_tunings/


> Oh well. There's only so long I can go on about the same stuff - the
> sheer multiplicity of historical existence, compared to the scarcity
> of historical sources;
The original sources are still worth to study and recheck again and
again, against meanwhile unsustainable claims and questionable
allegations,
alike the historical W3 would consist in the later (20th-century)
foisted modern PC^(1/4) of ~ 6Cents variant or even worser 12-ET.

> the need (nevertheless) to actually read >those
> sources and see what they do or don't say.
In order to get at least partially rid of historically obsolete
ballast, that meanwhile has got mouldy.
>
http://www.strukturbildung.de/Andreas.Sparschuh/