Topic: Rational well-temperament and Stanhope well-temperament

3 scales

Thread (2 messages)

From: Gene Ward Smith (2004-07-28)
Subject: Rational well-temperament and Stanhope well-temperament

I put the 112/75 meantones together as I suggested, and obtained a
well-temperament which is in essence the same as one Scala has down as
due 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.

!  ratwell.scl
7-limit rational well-temperament
12
!
256/243
28/25
32/27
175616/140625
4/3
1024/729
3/2
128/81
3136/1875
16/9
4096/2187
2

! 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

From: a_sparschuh (2010-04-20)
Subject: septenarian Stanhope Re: 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.