Topic: Fwd: Re: I just deduced.....

1 scales

Thread (2 messages)

From: Andreas Sparschuh (2008-09-22)
Subject: Fwd:  Re: I just deduced.....

--- In [email protected], "Andreas Sparschuh" <a_sparschuh@...>
wrote:

--- In [email protected], Brad Lehman <bpl@> wrote:
> 
> I'm wondering where the wolf 5th goes, with its full Pythagorean comma.
> 
> Neidhardt in his 1732 example
> http://harpsichords.pbwiki.com/f/Neidhardt_1732_ascii.html
> put it at C-G.

Hi Brad,
Probably you had meant the schemes in the second section:
http://harpsichords.pbwiki.com/f/Neidhardt_1724_ascii.html

"
3  :  4   F  :  C 
3  :  4   Bb :  F 
3  :  2   D# :  Bb 
3  :  4   G# :  D#
3  :  2   C# :  G#
3  :  4   F# :  C#
3  :  4   B  :  F#
3  :  2   E  :  B 
3  :  4   A  :  E 
3  :  2   D  :  A 
3  :  4   G  :  D 
3  :  2   C  :  G 
"
 
Already Arnold Schlick in 1511 critzised 
such an ancient "11-eyes"-chain of 11 pure 5ths as obsolete. 

N. converts the series into ratios relative to the unison 1:1
"
c_ |2      : 1      
B_ |243    : 128    
Bb |59049  : 32768  
A_ |27     : 16     
G# |6561   : 4096   
G_ |3      : 2      
F# |729    : 512    
F_ |177147 : 131072 
E_ |81     : 64     
D# |19683  : 16384  
D_ |9      : 8      
C# |2187   : 2048 
C_ |1      : 1
"  
attend the reverse direction against "scala"-convention,
because N. present his calculation in terms of string-lenghts.

Later in the text he prefers in stead of his pythagorean-wolf 5th C-G
an division of the PC into 12 different subfactors 
in order to get something more near to Simon-Stevin's 12ET.

"
12. 524288  0.  
11. 524880  1.
10. 525473  2.
_9. 526066  3.
_8. 526661  4.
_7. 527255  5.
_6. 527851  6.
_5. 528447  7.
_4. 529045  8.
_3. 529642  9.
_2. 530241 10.
_1. 530840 11.
_0. 531441 12.
"

He remarks proudly, that his partition of the PC
contains even the schisma at the end of the algorithm:

"Hierbey wird/ nicht ohne Vergnügen/ wahrgenommen/ daß das letzte
Geometrische Zwölfftheil 524880 : 524288 = 32805 : 32768."

tr:
'At this junction here it turns out with pleasure,
that the the last geometrically 12th-part amounts
524880 : 524288 = 32805 : 32768'  

So he got 12 differnt, but almost same impure 5ths 
with only tiny deviations from each others: 

He begins with the schisma:
12. (1 200 * ln(524 880 / 524 288)) / ln(2) = ~ 1.95372079...Cents
11. (1 200 * ln(525 473 / 524 880)) / ln(2) = ~ 1.9548131....Cents
10. (1 200 * ln(526 066 / 525 473)) / ln(2) = ~ 1.95260832...Cents
9.  (1 200 * ln(526 661 / 526 066)) / ln(2) = ~ 1.9569829... Cents
8.  (1 200 * ln(527 255 / 526 661)) / ln(2) = ~ 1.95148975...Cents
7.  (1 200 * ln(527 851 / 527 255)) / ln(2) = ~ 1.95585202...Cents
6.  (1 200 * ln(528 447 / 527 851)) / ln(2) = ~ 1.9536449... Cents
5.  (1 200 * ln(529 045 / 528 447)) / ln(2) = ~ 1.95798752...Cents
4.  (1 200 * ln(529 642 / 529 045)) / ln(2) = ~ 1.9525069... Cents
3.  (1 200 * ln(530 241 / 529 642)) / ln(2) = ~ 1.95683732...Cents
2.  (1 200 * ln(530 840 / 530 241)) / ln(2) = ~ 1.95462798...Cents
1.  (1 200 * ln(531 441 / 530 840)) / ln(2) = ~ 1.95893888...Cents

I.m.h.o:
But nobody can distinct N's apt approx. from 12EDO barely by ear.

A.S.

From: Andreas Sparschuh (2008-09-25)
Subject: Neidhardt's 1724 rational 12-EDO approximation as Scala-file, was: I just deduc.

--- In [email protected], Brad Lehman <bpl@> wrote:

> Neidhardt in his 1732 example
> http://harpsichords.pbwiki.com/f/Neidhardt_1732_ascii.html
> put it at C-G.
> 
> Hi Brad,
> Probably you had meant rather the schemes in the second section:
http://harpsichords.pbwiki.com/f/Neidhardt_1724_ascii.html

N. presents there his distribution of the PC into 12 tempered 5ths 
German original:
"
Es solten aber die Theile des commatis / von Rechts wegen/ 
nicht einander gleich seyn/ 
sondern nach proportion abfallen. /
Derowegen theile ich es auch geometrice in 12 rationes:
"
Translation:
'
The parts of the comma should not turn out /in order to be correct/
be equal each one to anothers, 
but should rather decline in proportion downwards. 
Hence I divide it(the comma) also geometrically into 12 ratios:
'
"
C : 12. 524288  0.  
F : 11. 524880  1.
Bb: 10. 525473  2.
Eb: _9. 526066  3.
G#: _8. 526661  4.
C#: _7. 527255  5.
F#: _6. 527851  6.
B : _5. 528447  7.
E : _4. 529045  8.
A : _3. 529642  9.
E : _2. 530241 10.
D : _1. 530840 11.
G : _0. 531441 12.
"
Hence each of the dozen 5ths deviates unequal from pure 3/2 
by the individual differing amounts of:

F--C : (1200 * ln(32805/32768))/ln(2) =schisma= ~ 1.95372079...Cents
Bb-F : (1200 * ln(525 473 / 524 880)) / ln(2) = ~ 1.9548131....Cents
Eb-G#: (1200 * ln(526 066 / 525 473)) / ln(2) = ~ 1.95260832...Cents
G#-C#: (1200 * ln(526 661 / 526 066)) / ln(2) = ~ 1.9569829... Cents
C#-F#: (1200 * ln(527 255 / 526 661)) / ln(2) = ~ 1.95148975...Cents
F#-B : (1200 * ln(527 851 / 527 255)) / ln(2) = ~ 1.95585202...Cents
B--E : (1200 * ln(528 447 / 527 851)) / ln(2) = ~ 1.9536449... Cents
E--A : (1200 * ln(529 045 / 528 447)) / ln(2) = ~ 1.95798752...Cents
A--D : (1200 * ln(529 642 / 529 045)) / ln(2) = ~ 1.9525069... Cents
D--G : (1200 * ln(530 241 / 529 642)) / ln(2) = ~ 1.95683732...Cents
G--C : (1200 * ln(530 840 / 530 241)) / ln(2) = ~ 1.95462798...Cents
C--G : (1200 * ln(531 441 / 530 840)) / ln(2) = ~ 1.95893888...Cents

Or in the
http://www.xs4all.nl/~huygensf/scala/scl_format.html
representation:

!Neidhard1724rationalETapprox.scl
!
from his "Canone Harmonico" extracted and compiled by A.Sparschuh
!
12
!
99.9946105 ! C#:= 1200 * ln((256/243) * (527 255 / 524 288)) / ln(2)
199.996435 ! D := 1200 * ln(( 9 / 8 ) * (530 241 / 531 441)) / ln(2)
299.996140 ! Eb:= 1200 * ln((32 / 27) * (526 066 / 524 288)) / ln(2)
399.997092 ! E := 1200 * ln((81 / 64) * (529 045 / 531 441)) / ln(2)
499.998720 ! F := 1200 * ln(( 4 / 3 ) * ( 32 805 /  32 768)) / ln(2)
599.995462 ! F#:= 1200 * ln((729/512) * (527 851 / 531 441)) / ln(2)
699.996062 ! G := 1200 * ln(( 3 / 2 ) * (530 840 / 531 441)) / ln(2)
799.998122 ! G#:= 1200 * ln((128/ 81) * (526 661 / 524 288)) / ln(2)
899.994598 ! A := 1200 * ln((27 / 16) * (529 642 / 531 441)) / ln(2)
999.998532 ! Bb:= 1200 * ln((16 / 9 ) * (525 473 / 524 288)) / ln(2)
1099.99411 ! B := 1200 * ln((243/128) * (528 447 / 531 441)) / ln(2)
2/1
!
!

Conclusion:
All that 11 pitches turn out to be located marginally below 12-EDO, 
rangeing in the teeny-weeny little-bity exiguous deviation downwards
of (~-0.001878 ... ~-0.00589)*Cents 
underneath the theoretically multiples of 2^(1/12). 

I.m.h.o:
Alike 12-EDO itself, such close approximations of 12-EDO, 
i really can only reccomend for:
http://en.wikipedia.org/wiki/Atonality
.
But for performing music back from the age of:
http://en.wikipedia.org/wiki/Tonality
i do prefer the apt harmonics of
http://en.wikipedia.org/wiki/Well_temperament
s in order to gain more pronounced
http://de.wikipedia.org/wiki/Tonartencharakter
Which native speaker here can translate that Wiki-article inti engl.?
alike
http://www.wmich.edu/mus-theo/courses/keys.html

bye
A.S.