Re: 440 Hz, 41Hz,415Hz ADO, was to monzo : about ado systems : RE: 243/242

Hi Andreas,

On Wed, 18 Jan 2006 "a_sparschuh" wrote:

[snip]
> 1.Many musicians posses absolute pitch for A4=440 Hz frequency,
> 2.but electric- & double-bass players prefer E1=41 Hz as lowest string
> http://www.41hz.com
> 3.and neo-baroquists want G#=415 Hz as their "Cammerthon" a'=415 Hz.
> Here comes an ADO that satifies all that 3 demands in one temperament.
>
> A_1 = 55,110,220,440 Hz start-frequency
> E_1 = 41,82,164/165 lowest bass string 41Hz
> B_3 = 123
> F#4 = 369
> C#6 = 1107
> G#4 = 415,830,1660,3320/3321 neo-baroque Cammerthone a'=415Hz
> Eb4 = 311
> B_3 = 233,466,932/933
> F_5 = 699
> C_3 = 131,262,524,1048,2096/2097
> G_1 = 49,98,196,392/393
> D_3 = 147
> A_4 = 440/441

Impressive!

> PC=531441/528244=3^12/2^19=
> (165/164)(3321/3320)(1245/1244)(933/932)(2097/2096)(393/392)(441/440)
> divsion into 7 epimoric subfactors.

How do you DO that - find such factorisations?
Probably a question for tuning-math, but a sketch
of the method might also be of interest to the
mathematically challenged o this list :-)

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.375 / Virus Database: 267.14.20/234 - Release Date: 18/1/06

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
Dears Yahya & Petr,
> > A_1 = 55,110,220,440 Hz start-frequency
> > E_1 = 41,82,164/165 lowest bass string 41Hz
> > B_3 = 123
> > F#4 = 369
> > C#6 = 1107
> > G#4 = 415,830,1660,3320/3321 neo-baroque Cammerthone a'=415Hz
> > Eb4 = 311,622,1244/1245
> > B_3 = 233,466,932/933
> > F_5 = 699
> > C_3 = 131,262,524,1048,2096/2097
> > G_1 = 49,98,196,392/393
> > D_3 = 147
> > A_4 = 440/441
>
> Impressive!
> > PC=531441/524288=3^12/2^19= "now the numerical value is correct
(165/164)(3321/3320)(1245/1244)(933/932)(2097/2096)(393/392)(441/440)
> > divsion into 7 epimoric subfactors.

> How do you DO that - find such factorisations?
Werckmeisters "Septenarius" of his"Musicalische Temperatur" 1691: P.73
did an similar subdivisin in string-lengths:

C 393/392,196,98,49 divide longest value by 3, 5th C>G 393/3=131
G 131/132,66,33
D 351/352,176,88,44,22,11
A 117
E 39
H 417/416,208,104,52,26,13 "H" german = brit: "B"
F# 279/278,139
C# 93
G# 495/496,248,124,62,31
D# 165
B 441/440,220,110,55 "B" german = brit: "Bb"
F 147
C 49

PC=531441/524288=
(393/392)(132/131)(351/352)(417/416)(279/278)(495/496)(441/440)
an 7-fold epimoric factorization.

C _ 1/1
C# 98/93
D _49/44
E _49/39
F _ 4/3 _ pure 4th
F#196/139
G 196/131
G# 49/31
A 196/117
Bb 98/55
B _49/26
C' _2/1

> Probably a question for tuning-math,
I.m.o:
Werckmeisters method of his #6 should be of general intrest.

> but a sketch
> of the method might also be of interest to the
> mathematically challenged o this list :-)
You can read W.s stringlengts reversre as pitch frequencies,
he used that backwards trick himself at other places too:

a 55,110,220,440 Hz begin, start here
e 165
b 31,62,124,248,496/495
f# 93
c# 139,278/279
g# 13,26,52,104,208,416/417
eb 39
bb 117
f 11,22,44,88,176,352/351 "some authors prefere here 175 instead 176
c 33,66,132/131
g 49,98,196,392/393 "from here on same steps as in my 440,41,415 ADO
d 147
a 440/441 circle closed back home on tuning-fork pitch 440Hz

same PC subdivision as in above stringlengths,
but here in reverse order backwards.

abs. vs. relative
264 __c___ 1/1
278 __c#_139/132
294 __d__147/132
312 __eb__39/33
330 __e___ 5/4 _ pure 3rd! E1=41.125Hz instead 41*8=328
352 __f___ 4/3 _ just 4th!
366 __f#__93/66
392 __g__ 49/33
416 __g#__52/33 " one Hz sharper than 415 the neo-baroquists A4
440 __a___ 5/3 _ also just pure 6th too!
468 __bb__39/22
496 __b__ 62/33
528 __c'__ 2/1

sounds great on my old piano!
I'm wondering & puzzeld however W. could predict our 440=11*5*2^3 ?
See look for further description at Werckmeister.
A.S.