Full thread (5 messages)
From: Tom Dent (2009-01-30)
Subject: mod meantone 7 11 13 19
A fairly good approximate version of the scale:
round the cycle
(0) (-4) 698 (-5) 195 (-5) 892 (-5) 389 (-4) 1087 (-5) 584 (0) 86 (+3)
791 (+3) 296 (+3) 1001 (0) 503 (-5)
lined up:
86
195
296
389
503
584
698
791
892
1001
1087
1200
has 1- or 2-cent approximations to 7/5, 13/11, 14/11, 19/15, 19/16 -
two of each.
C-F# is mapped from 7/5 and 13/11 . 19/16 giving a comma 1235/1232 to
be divided into three somehow.
F-Bb is reached via F-Ab-C-F#-Bb, thus (approximations to) 13/11 .
19/15 . 7/5 . 14/11, which is 4/3 times 12103/12100 - schismatic
enough to map to 1/1.
Now I reckon some Bach organ works could be interesting...
~~~T~~~
From: Andreas Sparschuh (2009-02-01)
Subject: Re: mod meantone 7 11 13 19
--- In [email protected], "Tom Dent" <stringph@...> wrote:
> A fairly good approximate version of the scale:
> round the cycle
> (0) (-4) 698 (-5) 195 (-5) 892 (-5) 389 (-4) 1087 (-5) 584 (0) 86 (+3)
> 791 (+3) 296 (+3) 1001 (0) 503 (-5)
>
> lined up:
> 86
> 195
> 296
> 389
> 503
> 584
> 698
> 791
> 892
> 1001
> 1087
> 1200
>
That's when using:
http://www.sengpielaudio.com/calculator-centsratio.htm
http://superspace.epfl.ch/approximator/
more-or-less about:
21/20 (~ 84.467...)
19/17 (~192.557...)
19/16 (~297.513...)
5/4 (~386.313...)
115/86 (~503.070...)
217/145 (~697.970...)
30/19 (~790.755...)
43/72 (~892.392...)
41/23 (~1000.788...)
15/8 (~1088.268...)
when allowing an imprecision ~3 Cents
as deviation from accuracy.
> has 1- or 2-cent approximations to 7/5, 13/11, 14/11, 19/15, 19/16 -
> two of each.
>
> C-F# is mapped from 7/5 and 13/11 . 19/16 giving a comma 1235/1232 to
> be divided into three somehow.
>
> F-Bb is reached via F-Ab-C-F#-Bb, thus (approximations to) 13/11 .
> 19/15 . 7/5 . 14/11, which is 4/3 times 12103/12100 - schismatic
> enough to map to 1/1....
From: Tom Dent (2009-02-02)
Subject: Re: mod meantone 7 11 13 19
--- In [email protected], "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In [email protected], "Tom Dent" <stringph@> wrote:
>
> > A fairly good approximate version of the scale:
> > round the cycle
> > (0) (-4) 698 (-5) 195 (-5) 892 (-5) 389 (-4) 1087 (-5) 584 (0) 86 (+3)
> > 791 (+3) 296 (+3) 1001 (0) 503 (-5)
> >
> > lined up:
> > 86
> > 195
> > 296
> > 389
> > 503
> > 584
> > 698
> > 791
> > 892
> > 1001
> > 1087
> > 1200
> >
> That's when using:
> http://www.sengpielaudio.com/calculator-centsratio.htm
> http://superspace.epfl.ch/approximator/
> more-or-less about:
>
> 21/20 (~ 84.467...)
> 19/17 (~192.557...)
> 19/16 (~297.513...)
> 5/4 (~386.313...)
> 115/86 (~503.070...)
> 217/145 (~697.970...)
> 30/19 (~790.755...)
> 43/72 (~892.392...)
> 41/23 (~1000.788...)
> 15/8 (~1088.268...)
>
> when allowing an imprecision ~3 Cents
> as deviation from accuracy.
>
- You missed 7/5 (~582.51) = 21/20 * 4/3 !
~~~T~~~
From: Andreas Sparschuh (2009-02-08)
Subject: Re: mod meantone 7 11 13 19
--- In [email protected], "Tom Dent" <stringph@...> wrote:
> (0) (-4) 698 (-5) 195 (-5) 892 (-5) 389 (-4) 1087 (-5) 584 (0)
> 86 (+3) 791 (+3) 296 (+3) 1001 (0) 503 (-5)
>
> lined up:
> 86
> 195
> 296
> 389
> 503
> 584
> 698
> 791
> 892
> 1001
> 1087
> 1200
>
> Now I reckon some Bach organ works could be interesting...
Dear Tom,
For Bach i tune modern organs in absolute "septenarian" pitches as:
G2: (7*7=49 98 <) 99 198 396 := 440HZ*(9/10) an minor-tone below 440Hz
~ ~ -5.839...
D4: (49*3=147 294 < 295 <) 296 (< 297 := 99*3)
~ ~ -5.857...
A5: (49*9=441 882<)885=2*442.5 Metronome @150 beats/min vs. 440-fork
~ ~ -5.879...
E6: 1323 := 49*27 2646 [> 2645 := 5*C5]
~ ~ -3.056...
B6: [G2*5=495 990 1980 <] 1981 3962 = 7*566 (< 567*7 = 3969 := 49*81)
~ ~ -0.874...
F#6: 1485 := G2*15 with F#-G as an just 16:15 ~111.731... semitone
~ ~ +0.388...
C#5: 557 1114 2228 4456 (> 4455)
~ ~ +1.035...
G#3: 209 418 836 1672 (> 1671)
~ ~ +2.759...
D#3: 157 314 628 (< 627)
~ ~ +1.837...
A#5: (471 942 <) 943
~ ~ +0.612...
F6: [A5/5 =: 177 354 708 1416 >] 1415 2830 (< 2829)
~ ~ -5.310...
C5: (33 66 132 264 528 <) 529 1058 2116 4232 (< 4245)
~ ~ -3.276...
G2: 99
lined up in arising order:
c" 529 tenor_C5
#" 557
d" 592
#" 628
e" 661.5
f" 707.5
#" 742.5
g" 792
#" 836
a" 885Hz
#" 943
b" 990.5
!Sparschuh2009organ885Hz.scl
!
for neoBaroque pipe-organs with fusing 3rds C-E, G-B & F-A
12
!
557/529 ! ! # ~ 89.292...
592/529 ! ! D ~194.795...
628/529 ! ! # ~296.996...
1323/1058 ! E ~386.968...
1415/1058 ! F ~503.355...
1485/1058 ! # ~586.948...
792/529 ! ! G ~698.679...
836/529 ! ! # ~792.282...
885/529 ! ! A ~890.891...
943/529 ! ! #~1000.788...
1981/1058 ! B~1094.068...
2/1
!
!
That tuning enables for the major 3rds C-E, G-B & F-A
E7/(5*C5) = 2646/2645 ~+0.654...c "sharp" above 5/4
B6/(5*G4) = 1981/1980 ~+0.874...c "sharp" above 5/4
(A8/5)/F6 = 1416/1415 ~+1.223...c "sharp" above 5/4
the pleasant sounding "pulling-effect"
as still preserved in some old meantone organs
http://www.arpschnitger.nl/snoordb.html
http://www.arpschnitger.nl/sgro02.html
"The âpulling effectâ in the tuning can be heard very clearly in the
long chords and is produced by the mutual influence of the pipes which
are arranged in thirds on the windchest."
...as already historically described and desired by
http://en.wikipedia.org/wiki/Arnolt_Schlick
and specified by Werckmeister as less than about ~ 0.1 %
for the 5th partial 5/4.
1001/1000 = 1.001 ~+-1.730...cents
hence the above tuning remains staying well within that range
at least for the three most basic 3rds: C-E, G-B & F-A.
Attend that
http://arxiv.org/abs/physics/0506094
http://www.sea-acustica.es/WEB_ICA_07/fchrs/papers/nla-04-002.pdf
consider barely the fundamental 1/1
without discussing the higher partials of the overtone series as
http://www.aes.org/e-lib/browse.cfm?elib=7007
bye
A.S.
From: Andreas Sparschuh (2009-02-08)
Subject: well-tempered version of: Re: mod meantone 7 11 13 19
--- In [email protected], "Tom Dent" <stringph@> wrote:
Tom's meantone was:
> > lined up:
> > 86
> > 195
> > 296
> > 389
> > 503
> > 584
> > 698
> > 791
> > 892
> > 1001
> > 1087
> > 1200
> >
For my well-tempered version
-without the wide 5ths-
attend a few changes that replaces some of the pitches:
> lined up in arising order:
>
> c" 529 tenor_C5
_ #" 558 (> 557)
> d" 592
_ #" 627.75 (< 628)
> e" 661.5
_ f" 707.5 (< 706)
_ #" 744 (> 742.5)
> g" 792
_ #" 837 (> 836)
> a" 885Hz
_ #" 941.5 (< 943)
_ b" 992 (> 990.5)
!Sparschuh2009well885Hz.scl
!
modern pianos with an fusing 3rd: C-E ~+0.654...c "sharp" above 5/4
12
!
558/529 ! ! # ~ 92.397...
592/529 ! ! D ~194.795...
2511/2116 ! # ~296.307...
1323/1058 ! E ~386.968...
706/529 ! ! F ~499.681...
744/529 ! ! # ~590.442...
792/529 ! ! G ~698.679...
837/529 ! ! # ~794.352...
885/529 ! ! A ~890.891...
1883/1058 ! # ~998.032...
992/529 ! ! B~1088.487...
2/1
!
!
That well-tuning has for the
major 3rds C-E, G-B & F-A the sharpnesses
E7/(5*C5) = 2646/2645 ~+0.654...c "sharp" above 5/4
B4/(5*B0) = 496/495 ~+3.494...c "sharp" above 5/4
(A6/5)/F4 = 354/353 ~+4.897...c "sharp" above 5/4
but none wide 5ths as the organ counterpart.
bye
A.S.