Werckmeister 1697 translation

Here is the translation of
/tuning/topicId_77273.html#77273 :

---------------------------
Some are also of the opinion that one must always have a monochord to
hand in tuning, and make thereby a great difficulty. But I answer,
that I do not tune from the monochord, I could also commit a great
foolishness if I now once plucked on the string and wanted to tune a
clavichord or harpsichord [Instrument] by it; with a harpsichord or
clavichord it proceeds well if both strings have one attack, but with
the monochord it would be heavy going since one cannot hear the
shuddering [Tremorem, beating] as with organ pipes when they are being
tuned. But if I dared myself to tune one rank of an organ from the
monochord I would commit the greatest foolishness, since a strong and
gentle tone cannot be handled one with the other, much less a gentle
string with an organ pipe.
But in order for me to have certainty the division in the monochord
shows me how I should proceed; and I can in practice, if I want, let
the fifths all be tempered one with another a very little flat {but
1/4 comma would be absurd}, on the other hand I let all the major
thirds be tempered a little high, the upper compared with the lower
note; now because I know how the land lies, I can soon adjust one or
the other. I may also let the fifths be, as described in the
monochord, some pure and some tempered, which also goes quite well. I
also wanted to give another such small division [Bruch] so that the
fifths are all tempered a little one with another on the monochord;
[I] also wanted to depict the partition but because such divisions are
tiresome to construct with the compass and the ingratitude much too
great, so I have misgivings to write any more about it; primarily
because it requires [a] publisher and expense. But since sensation
[and] reason are the judges, so [one] has from my monochord or
``sensus'' so much information, that he can well find his feet and
manage for himself, and there is the self-evident proof in it, to see
and hear what is certain; any clever fellow gets help already if he
only practises diligently.
Then I have also found some [people] who firstly have not divided up
the monochord accurately, which, given how a prick of the compass
wears out with much turning around, can easily be understood; so also
often the strings are pressed by the bridges so that they do not
remain accurate; since anyway the strings cannot be tuned as purely as
the pipes since one cannot hear the beating [Tremorem] so clearly.
[P] Occasionally one pipe may seem to be pure with another, but it is
not completely pure, because it does not [lie] so to say in the
absolute centre [centri centro] :
---------------------------

... More when I get round to it!
~~~T~~~

On a trip to the library to extend the transcription I noticed I had
missed out a whole line or two of the original passage - it should
read

Damit ich aber Gewißheit habe / so zeiget mir die Abtheilung in
Monochord wie ich verfahren soll / und kann in meiner praxi die
qvinten, wenn ich will / alle mit einander ein gar wenig unter sich
schweben lassen {(aber 1/4 Commatis wehre absurd)} / **hingegen lasse
ich alle Tertias majores den ober gegen dem unter=clavem einwenig in
die höhe schweben** / weil ich nun weiß wo die Zäume hängen / kan ich
einem und dem anderen bald nach helffen.

Trans:
-----------------------------
But in order for me to have certainty the division in the monochord
shows me how I should proceed; and I can in practice, if I want, let
the fifths all be tempered one with another a very little flat {but
1/4 comma would be absurd}, **on the other hand I let all the major
thirds be tempered a little high, the upper compared with the lower
note**; now because I know how the land lies, I can soon adjust one or
the other.
-----------------------------

... just tuning by fifths is not enough, and 'adjust one or the other'
makes a lot more sense referring to the thirds. Between about 1500 and
the 20th century there is one, and only one, useful method for setting
temperaments by ear: tune by fifths and check by major thirds.

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> Here is the translation of
> /tuning/topicId_77273.html#77273 :
>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> translated:
>...I may also let the fifths be, as described in the
> monochord, some pure and some tempered, which also goes quite well. I
> also wanted to give another such small division [Bruch] so that the
> fifths are all tempered a little one with another on the monochord;
> [I] also wanted to depict the partition but because such divisions are
> tiresome to construct with the compass and the ingratitude much too
> great, so I have misgivings to write any more about it; primarily
> because it requires [a] publisher and expense. But since sensation
> [and] reason are the judges, so [one] has from my monochord or
> ``sensus'' so much information, that he can well find his feet and
> manage for himself, and there is the self-evident proof in it, to see
> and hear what is certain; > ... More when I get round to it!
>
Dear Tom,
alike Zarlino & others bevor him,
W. divided the SC=80:81 regulary into:

2 parts: (160/161)(161:162) alike Kirnberger 3 inbetween D-A-E
3 parts: (242:243)(242:241)(241:240) alike Stanhope inbetw: G-D-A-E
http://groenewald-berlin.de/text/text_T041.html
Groenwald confuses in his reprenstation PC versus SC.
see: Lindley's "Stimmung & Temeratur"
or
/tuning/topicId_73833.html#75760
"it is also possible to read Werckmeister's #3 pattern

C~G~D~A E B~F#...C

in 1/3 SC terms:

C 242/243 G 241/242 D 240/241 A E B 32768/32805 F# C# G# D# Bb F C
"
/tuning/topicId_75270.html#75449
"http://diapason.xentonic.org/ttl/ttl01.html
on p.37, Chap. XVII
when considering some arithmetical subdivisions of
81:81 for tempering:
"Wenn ein Comma in zwey Theile getheilt wird /
so stehen in kleinesten Zahlen 162. 161. 160.
In drey Theile sind die kleinsten Termini 243. 242. 241. 240.
So es in vier Theile gemachet; stehen die kleinsten Termini 324. 323
322. 322. 320. Die äussersten sind das comma...."
tr:
'If a comma is divided into 2 parts,
then arise in the smallest numbers 162. 161. 160.
In 3 parts the smallest termini become 243. 242. 241 240.
when made into 4 parts, the smallest Termini get 324. 323. 322. 320.
the outer ones represent the comma...'"

http://harpsichords.pbwiki.com/f/Kirn_1871.html
"Oder wenn man von C nach e 80 : 81 in vier Quinten vertheilen will,
kann es folgender Art geschehen:

C-G 216 : 323 temperirte Quinte = 2/3 - 1/324
216 : 324 reine Quinte
-------------------------------------
G-d 215 1/3 : 322 temperirte Quinte = 2/3 - 1/323
215 1/3 : 323 reine Quinte
-------------------------------------
A-e 214 2/3 : 321 temperirte Quinte = 2/3 - 1/322
214 2/3 : 322 reine Quinte
-------------------------------------
D-A 214 : 320 temperirte Quinte = 2/3 - 1/321
214 : 321 reine Quinte
-------------------------------------

Analogous
in generalizing W's concept of SC subdivisions Neidhardt obtained:
11 an corresponding epimoric subfactors:
(880:881)(881:882)(882:883)*...*(890:891)
so that:
Werckmeister11/Kirnberger11/Neidhardt11's approx. of 12 ET yiels:

F# 32805:32768 C# 890:891 G# 889:890 Eb 888:889 Bb...
...A 882:883 E 881:882 B 880:881 f#

That makes in modern 20th century TUs = PC(1/720) units:

F#
schisma:
(720TU*ln(32768/32805) / ln((3^12) / (2^19)) = ~-59.9607138...TUs
C#
(720TU * ln(890 / 891)) / ln((3^12) / (2^19)) = ~-59.665901...TUs
G#
(720TU * ln(889 / 890)) / ln((3^12) / (2^19)) = ~-59.732979...TUs
Eb
(720TU * ln(888 / 889)) / ln((3^12) / (2^19)) = ~-59.8002081...TUs
Bb
.
.
.

.
.
A
(720TU * ln(882 / 883)) / ln((3^12) / (2^19)) = ~-60.2067818...TUs
E
(720TU * ln(881 / 882)) / ln((3^12) / (2^19)) = ~-60.2750822...TUs
B
(720TU * ln(880 / 881)) / ln((3^12) / (2^19)) = ~-60.3435377...TUs

Espeically for those in that group here that do claim to be able to
tune 12-EDO stepwise in that precision,
or even an alleged "Squiggle"-reinterpretation
in exactly 60TUs steps, without sligthest error in deviation.

I.m.h.o:
It's an modern absurd oversimplification
to impute that Bach had only tempered barely in 60TU steps.
Completely ahsitorically!

Yours Sincerely
A.S.