Joerg Fiedler on Prelleur JI violin chart

Yesterday I read a curious article by J. Fiedler in the Basler
Jahrbuch fuer Historische Musixpraxis in which he took the diagram of
violin fingerings from the Prelleur textbook (early 18th century) and
calculated what intervals the indicated finger positions should
correspond to.

This is subject to many caveats - for example, the diagram may have
been prepared by a mathematician, not the musician who wrote the text;
gut strings are not uniform; finger positions do not correspond
directly to pitches because the placing of the finger increases the
string tension; the paper of the book may have warped over time.

Taking these into account he finds the following cent values for the
notes in the first octave of the G string. (The same intervals exist
above the D, A and E strings, appropriately labelled.) The estimated
error is between 3c (for the lowest notes) and 6c (for the highest).

[G 0]
G# 67
Ab 114
A 204
A# 267
Bb 316
B 383
Ab 435
C 500
C# 582
Db 617
D 705
D# 764
Eb 815
E 887
Fb 933
E# 976
F 1024
F# 1094
Gb 1136

What to make of it? Clearly not any regular system...

~~~T~~~

Seems to be a mixture of JI with meantone.

----- Original Message -----
From: "Tom Dent" <stringph@gmail.com>
To: <tuning@yahoogroups.com>
Sent: 04 Nisan 2007 �ar�amba 16:20
Subject: [tuning] Joerg Fiedler on Prelleur JI violin chart

>
> Yesterday I read a curious article by J. Fiedler in the Basler
> Jahrbuch fuer Historische Musixpraxis in which he took the diagram of
> violin fingerings from the Prelleur textbook (early 18th century) and
> calculated what intervals the indicated finger positions should
> correspond to.
>
> This is subject to many caveats - for example, the diagram may have
> been prepared by a mathematician, not the musician who wrote the text;
> gut strings are not uniform; finger positions do not correspond
> directly to pitches because the placing of the finger increases the
> string tension; the paper of the book may have warped over time.
>
> Taking these into account he finds the following cent values for the
> notes in the first octave of the G string. (The same intervals exist
> above the D, A and E strings, appropriately labelled.) The estimated
> error is between 3c (for the lowest notes) and 6c (for the highest).
>
> [G 0]
> G# 67
> Ab 114
> A 204
> A# 267
> Bb 316
> B 383
> Ab 435
> C 500
> C# 582
> Db 617
> D 705
> D# 764
> Eb 815
> E 887
> Fb 933
> E# 976
> F 1024
> F# 1094
> Gb 1136
>
> What to make of it? Clearly not any regular system...
>
> ~~~T~~~
>
>
>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> Yesterday I read a curious article by J. Fiedler in the Basler
> Jahrbuch fuer Historische Musixpraxis in which he took the diagram
of
> violin fingerings from the Prelleur textbook (early 18th century)
and
> calculated what intervals the indicated finger positions should
> correspond to.
>
> This is subject to many caveats - for example, the diagram may have
> been prepared by a mathematician, not the musician who wrote the
text;
> gut strings are not uniform; finger positions do not correspond
> directly to pitches because the placing of the finger increases the
> string tension; the paper of the book may have warped over time.
>
> Taking these into account he finds the following cent values for the
> notes in the first octave of the G string. (The same intervals exist
> above the D, A and E strings, appropriately labelled.) The estimated
> error is between 3c (for the lowest notes) and 6c (for the highest).
>
> [G 0]
> G# 67
> Ab 114
> A 204
> A# 267
> Bb 316
> B 383
> Ab 435
> C 500
> C# 582
> Db 617
> D 705
> D# 764
> Eb 815
> E 887
> Fb 933
> E# 976
> F 1024
> F# 1094
> Gb 1136
>
> What to make of it? Clearly not any regular system...

I presume the second Ab is really Cb. And is Fb really
supposed to be lower than E#? Fix those two and add
a B# and it would make more sense.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> >
> > The estimated
> > error is between 3c (for the lowest notes) and 6c (for the highest).
> >
> > [G 0]
> > G# 67
> > Ab 114
> > A 204
> > A# 267
> > Bb 316
> > B 383
> > [C]b 435
> > C 500
> > C# 582
> > Db 617
> > D 705
> > D# 764
> > Eb 815
> > E 887
> > Fb 933
> > E# 976
> > F 1024
> > F# 1094
> > Gb 1136
> >
> > What to make of it? Clearly not any regular system...
>
> I presume the second Ab is really Cb.
YES

> And is Fb really
> supposed to be lower than E#?

Yes, as it is in any meantone with a fifth less flat than 1/3-comma.
The main point is that Fb is sharper than E and E# is flatter than F.

> add
> a B# and it would make more sense.
>

Why would a range between Fb and B# make more sense than one between
Fb and E#?

Anyway, the first point to notice is that the fifths
C#-G#
A-E
F-C
Gb-Db
are very flat - about a comma - well out of range (given measurement
errors) for any historical regular tuning; and
F#-C#
is pretty flat too. On the other hand there are a lot of pure (with
errors) thirds.
So one start off with the following JI scheme:

G# D# A# E#
E B F ?C#?
C G D A
Ab Eb Bb F
Fb Cb Gb

where each level is a Pythagorean sequence and each column a sequence
of pure thirds. Though C#, in particular, and quite a few distant
sharps and flats don't really fit the scheme within the expected error.

But that's not the whole story I see. Consider the following values :
G-E# 976
G-A# 267
G-D# 764
G-C# 582
G-Db 617
G-Gb 1136
G-Cb 435
G-Fb 933
almost every one of them hits a septimal interval right on the nose!!
It almost looks as if the more complicated 5-limit JI intervals have
been substituted by septimal ones.
Very strange in the context...

~~~T~~~

Tom Dent schrieb:

And this:

> G# D# A# E#
> E B F ?C#?
> C G D A
> Ab Eb Bb F
> Fb Cb Gb

makes it look like a chart for violas, where at least the open strings are pure.

klaus

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

> But that's not the whole story I see. Consider the following
values :
> G-E# 976
> G-A# 267
> G-D# 764
> G-C# 582
> G-Db 617
> G-Gb 1136
> G-Cb 435
> G-Fb 933
> almost every one of them hits a septimal interval right on the
nose!!
> It almost looks as if the more complicated 5-limit JI intervals have
> been substituted by septimal ones.
> Very strange in the context...

Seriously strange, but true. However, meantone, if this
weird thing is taken to be an irregular version of it,
does come close to septimal intervals.

G-E# 976 7/4=968.8
G-A# 267 7/6=266.9
G-D# 764 14/9=764.9
G-C# 582 7/5=582.5
G-Db 617 10/7=617.5
G-Gb 1136 27/14=1137.0
G-Cb 435 9/7=435.1
G-Fb 933 12/7=933.1

To this we may add

G-G 0 1=0.0
G-A 204 9/8=203.9
G-Bb 316 6/5=315.6
G-B 383 5/4=386.3
G-C 500 4/3=498.0
G-D 705 3/2=702.0
G-Eb 815 8/5=813.7
G-E 887 5/3=884.4
G-F 1024 9/5=1017.6

At this point we have a JI version of 17 out
of the 20 notes. We might add

G-G# 67 25/24=70.7
G-Ab 114 16/15=111.7
G-F# 1094 15/8=1088.3

We now have the following 7-limit JI scale:

[25/24, 16/15, 9/8, 7/6, 6/5, 5/4, 9/7, 4/3, 7/5, 10/7, 3/2, 14/9,
8/5, 5/3, 12/7, 7/4, 9/5, 15/8, 27/14, 2]

This has twelve pure fifths, and three septimal meantone
style fifths of 112/75, flat by 224/225. It also has
a very flat 125/84 fifth, flat by 125/126, and three
more "fifths" flat by a comma. Tempering out both
225/224 and 126/125 leads to meantone, and converts
the scale into Meantone[20] in whatever tuning. Tempering
out 225/224 only would also make sense, this would
make the 125/126 and 80/81 fifths equal, so that there
would be four very flat fifths, but not so bad as
a comma flat. The other fifths might be, eg, 700 cents.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> We now have the following 7-limit JI scale:
>
> [25/24, 16/15, 9/8, 7/6, 6/5, 5/4, 9/7, 4/3, 7/5, 10/7, 3/2, 14/9,
> 8/5, 5/3, 12/7, 7/4, 9/5, 15/8, 27/14, 2]
>
> This has twelve pure fifths, and three septimal meantone
> style fifths of 112/75, flat by 224/225. It also has
> a very flat 125/84 fifth, flat by 125/126, and three
> more "fifths" flat by a comma.

A comma basis for <31 49 72 87| is {81/80,126/125,1029/1024}.
Tenney reducing the above (or a chain of 19 fifths, for that
matter) leads to the same scale, except for the first two
intervals, which now are 21/20 and 15/14. Now we have thirteen
pure fifths, two fifths flat by 224/225 and four fifths flat
by a comma.