Yahoo Tuning Groups Ultimate Backup tuning Rational well-temperament and Stanhope well-temperament

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@...> wrote:
>
> ...and obtained a well-temperament which is in essence the same as > one Scala has down asdue to the third earl of Stanhope.
> It would be interesting to know
> more about Stanhope and his temperings.
>
> The transposition I give is not the prettiest, but the good range is
> around 1/1, and as we can see by comparing, is a lot like Stanhope.
>
> ! stanhope.scl
> !
> Well temperament of Charles, third earl of Stanhope (1806)
> 12
> !
> 256/243
> 196.09000
> 32/27
> 8192/6561
> 4/3
> 1024/729
> 3/2
> 128/81
> 890.22500
> 16/9
> 4096/2187
> 2/1
>

Hi Gene,
Here comes an link to Stanhope's (1806) original Paper in facsimile:

http://books.google.de/books?id=kYAweZtA0SAC&pg=PA291&lpg=PA291&dq=stanhope+tuning&source=bl&ots=yAYgneF9k-&sig=UFBZbbLuZGxt3bhYeT5rYu3_2rc&hl=de&ei=juPNS_KaIo6VOOOu5aEP&sa=X&oi=book_result&ct=result&resnum=2&ved=0CA4Q6AEwAQ#v=onepage&q=stanhope%20tuning&f=false

Consider there the coarse monochord string-lengths,
as shown on p.311(Google-scan) = p.21(Original-document)
"
C 120 1st-bass_C
# 113
D 107
# 101
E 96
F 90
# 85
G 80
# 75
A 71
# 67
B 64
C 60 middle_C
"

That yields in terms of an modern "scala"-file:

!StanhopeMonochord.scl
Stanhope's (1806) monochord string lenghts compiled by A.Sparschuh
12
!
120/113 ! C#
120/107 ! D
120/101 ! Eb
120/96 ! E
120/90 ! F
120/85 ! F#
120/80 ! G
120/75 ! G#
120/71 ! A
120/67 ! Bb
120/64 ! B
2/1
!
![eof]

Probaly, I assume, that seqence was obtained by an 'Werckmeister-Collatz' procedure, alike the "Septenarius"?:

Hence here my own proposal of an potentially
reconstruction backwards in 4ths:

C : 15 30 60 :=middle_C
F : 45 90
Bb : 67 134 (< 135 := F*3)
Eb : (G#/3 =: 25 50 100 <) 101 202 (> 201 := Bb*3)
G# : 75
C# : 113 226 (> 225 := G#*3)
F# : 85 170 340 (> 339 := C#*3)
B : 1 2 4 8 16 32 64 128 256 (> 255 := F#*3)
E : 3 6 12 24 48 96
A : (9 18 36 72 >) 71 !(sic) that's even more than an Pyth.-Comma
D : 107 204 (< 213 := A*3)
G : 5 10 20 40 80 160 320 (< 321 := D*3)
C : 15 30 60 ...

Try to read also John Farey's (1809) cirtique in:
http://www.informaworld.com/smpp/content~content=a911256127&db=all
That includes an calcution of the beating-rates.

Here comes my own 'Septenarian' refinement of Stanhope's idea in 5ths:

5Eb=4.9 ... eb'313.6
4Bb=14.7 ... bb'470.4
FF=44.1 ... f'352.8
c=132.3 c'264.6 middle_C
(Werckmeister's choice 7*7=49 < 49.4 < 49.6 99.2 198.4 396.8<) g'396.9
(49*3 = 147 < 147.6 <) d148.2 d'296.4
(49*9 = 441 <) a'=442.8 Hz
e'=330.75 e"661.5 e'''1323 = 49*27 = c'*5
(31 ... 496 <) b'=496.125 b"992.25 = e'*3 = g*5
F#=93 f#196 f#'392
C#=69.7 c#139.4 c#278.8 (<279 := F#*3)
g#=209 g#'418.2 := c#*3
5D#=4.9 ... d#'313.6 d#"627.2 (< 627.3 := g#*3)

when lined up in ascending order:

c' 264.6 middle_C
#' 278.8
d' 296.4
#' 313.6
e' 330.75
f' 352.8
#' 372
g' 396.9
# 418.2
a' 442.8 Hz
#' 470.4
b' 496.125
c" 529.2 tenor_C

a la scala

! Sp7Stanhope.scl
Sparschuh's (2010) septenarian variant of Stanhopes (1806) idea
12
!
1394/1323 ! C# (256/243) * (6273/6272 ~+0.276...Cents sharper )
494/441 ! D (10/9) * (247/245 ~+14.075...Cents sharper )
32/27 ! Eb 2^5/3^3 Pythagorean minor-3rd
5/4 ! E
4/3 ! F
620/441 ! F# (45/32) * (3968/3969 ~-0.436...Cents flattend )
3/2 ! G
697//441 ! G# (128/81) * (6273/6272 ~+0.276...Cents sharper )
82/49 ! A (5/3) * (246/245 ~+7.052...Cents sharper )
16/9 ! Bb
15/8 ! B
2/1
!
! [eof]

Attend the characteristic deviations from JI only @: C#, D, F# and G#.

So far today about an over 200-years old historically tuning,
that was popular @ Beethoven's time and that is still worth
of playing B.'s master-works in that temperament,
that is located remarkable near JI in many aspects.

bye
A.S.

Rational well-temperament and Stanhope well-temperament

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗a_sparschuh <a_sparschuh@...>