Yahoo Tuning Groups Ultimate Backup tuning Temperament in Bach's Well-Tempered Clavier

Dear Sergio,

thanks for your communication, and compliments for your dissertation, to
which I just gave a cursory glance.

Be sure that I will read it in detail as soon my present commitments allow,
and write to you - privately - my observations.

Best regards,

Claudio

Claudio Di Veroli

http://harps.braybaroque.ie/

_____

From: harpsichord@yahoogroups.com [mailto:harpsichord@yahoogroups.com] On
Behalf Of Sergio Martínez
Sent: 16 January 2012 14:15
To: bach_tunings@yahoogroups.com; bach-keyboard@yahoogroups.com;
clavichord@yahoogroups.com; tuning@yahoogroups.com;
harpsichord@yahoogroups.com
Subject: [harpsichord] Temperament in Bach's Well-Tempered Clavier

Hello,

I have uploaded a new paper in the "files" section of these forums:

Temperament in Bach's Well-Tempered Clavier. A historical survey and a new
evaluation according to dissonance theory

This is the abstract:
After a historical survey of temperament in Bach's Well-Tempered Clavier by
Johann Sebastian Bach, an analysis of the work has been made by applying a
number of historical good temperaments as well as some recent proposals. The
results obtained show that the global dissonance for all preludes and fugues
in major keys can be minimized using the Kirnberger II temperament. The
method of analysis used for this research is based on the mathematical
theories of sensory dissonance, which have been developed by authors such as
Hermann Ludwig Ferdinand von Helmholtz, Harry Partch, Reinier Plomp, Willem
J. M. Levelt and William A. Sethares.

Keywords:

Tuning. Temperament. Good temperament. Bach temperament. Sensory dissonance.
Timbre. Spectrum. Johann Sebastian Bach. The Well-Tempered Clavier.

Best wishes,

Sergio

[Non-text portions of this message have been removed]

--- In tuning@yahoogroups.com, Sergio Martínez <sergio.martinez.ruiz@...> wrote:
>
> Hello,
>
> I have uploaded a new paper in the "files" section of these forums:
> Temperament in Bach's Well-Tempered Clavier. A historical survey and > a new evaluation according to dissonance theory

Hi Sergio,
there in yours article on p.12 you quote Bach's son Carl-Phillip-Emanuel Bach [1753] recommendation in the german original:

§14:
"Durch Probierung der Quarten hat man den Vortheil,
daß man die nöthige Schwebung der Quinten deutlicher hören kann,
weil die Quarten ihrem Grund-Tone näher liegen als die Quinten"

inclusive yours own well done translation:

'The beats of the 5ths can be more easliy heard by probing 4ths,
an advantage the stems from the fact that the tones of the latter
lie closer together than 5ths.'

Here my personal translation of that sentence:

"By checking through the fourths one gains the advantage
that you can hear the necesarry beats of the fifths much more lucid,
because the fourths got nearer located to theirs fundamental-tone
than the fifths"

From that revealing infromation one can conclude,
that it is more reasonable to read Bach's [1722]
decorative ornament-scroll as an sequence of an dozen 4ths:

Pattern:

begin_x1-x2-x3__y1-y2-y3__z1~z2~z3~z4~z5~end from left to rigth

with an adequate allocation in stepping by 4ths,
for instance:

start=C-F-Bb-Eb__G#-C#-F#__B~?~E~?~A~?~D~?~G~?~C=end

or even more concrete realized with the 12 attributions
in note-names and theirs corresponing ratios:

start := C = 1/1 begin at unison
x1 := F = 4/3 quarte
x2 := Bb = 16/9 Pythagorean minor 7th
x3 := Eb = 32/27 { = D# } Pythagorean minor 3rd
_{ enharmonic exchange of note-names}_
y1 := G# = 128/81 { = Ab }
y2 := C# = 256/243 an Pythagorean limma upwards from C
y3 := F# = 1024/729 inverse tritone 729/512
_{last pure fourth of 4/3}_
z1 := B = 4096/2187 inverse apotome 2187/2048
729/728
z2 := E = 1024/819 = (5/4)*(4096/4095)
273/272
z3 := A = 256/153
289/288
z4 := D = 272/243
324/323
z5 := G = 256/153 = (3/2)*(512/513)
513/512
end = C = 1/1 our's loop returned back to the initial unison

that yields reverse in backward direction,
the desired cycle of downwards beating 5ths instead
of the initially upwards beating 4ths

C 512/513 G 323/324 D 288/289 A 272/273 E 728/729 B F# C# G# Eb Bb F C

It turns out as an 5-fold epimoric subdivision of the PC=3^12/2^19

Control:
3^12/2^19 = (513/512)(324/323)(289/288)(273/272)(729/728)

All higher primes than 2 or 3 do cancel out each others.

Arrange the above ratios in ascending order
apt to fit into the "scala"-file fromat,
including the individual seizes of the semitones among them:

! SpBach19lim.scl
!
Sparschuh's (2012) 19-limit Bach's decorative ornament tuning
12
!
!
256/243 ! C# = | 8 -5> ! initial limma
! * 17/16 = +| 4 0, 0 0 0, 0 1>
272/243 ! D = | 4 -3, 0 0 0, 0 1>
! * 18/17 = +| 1 2, 0 0 0, 0 -1>
32/27 ! Eb = | 5 -3>
! * 96/91 = +| 5 1, 0 -1 0, -1>
1024/819 ! E = | 10 -3, 0 -1 0, -1>
! * 273/256 = +| -8 1, 0 1 0, 1>
4/3 ! F = | 2 -1>
! * 256/243 = +| 8 -5>
1024/729 ! F# = | 10 -6>
! * 81/76 = +| -2 4, 0 0 0, 0 0 -1>
256/171 ! G = | 8 -3, 0 0 0, 0 0 -1>
! * 19/18 = +| -1 -2, 0 0 0, 0 0 1>
128/81 ! G# = | 7 -4>
! * 18/17 = +| 1 2, 0 0 0, 0 -1>
256/153 ! A = | 8 2, 0 0 0, 0 -1>
! * 17/16 = +| -4 0, 0 0 0, 0 1>
16/9 ! Bb = | -4 2>
256/243 ! = +| 8 -5>
4096/2187 ! B = | 12 -7>
! 2187/2048 = +|-11 7> ! finally apotome as C-major 'leading-tone'
2/1 ! C' = |1>
!
!

Quote:
http://en.wikipedia.org/wiki/Leading-tone
"For example, in the C major scale
(white keys on a piano, starting on C),
the leading note is the note B"

For memorizing consider the dozen semitones of the octave
in more concise representation:

C 256/243 C# 17/16 D 18/17 Eb 96/91 E 273/256 F
F 256/243 F# 81/76 G
G 19/18 G# 18/17 A 17/16 Bb 256/243 B 2187/2048 C'

Also attend the variance in deviations of the 3rds
relative against just 5/4

Amounts of graduation in the 3rds sharpening:

Ab-C: 81/80 = |-4 4,-1> Syntonic-Comma, hence ditone Ab-C = 81/64
Eb-G: 96/95 = |5 1, -1 0 0, 0 0 1>
Bb-F: 136/135 = |3 -3, -1 0 0, 0 1>
F-A : 256/255 = |8 -1, -1 0 0, 0 1>
C-E : 4096/4095 =|12 -2, -1 -1 0, -1> the smallest one of all 12
G-B : 1216/1215 =|6 -5, -1 0 0, 0 0 1>
D-F#: 256/255 = |8 -1, -1 0 0, 0 1> the same amount as already F-A
A-C#: 136/135 = |3 -3, -1 0 0, 0 1> same as Bb-F
E-G#: 91/90 = |-1 -2,-1 1 0,1>
B-D#: 81/80 as Ab-C
F#A#: 81/80
C#E#: 81/80
G#B#: 81/80 back again

That property of beating satisfies well Bach's demand,
as reported by his pupil Kirnberger:
"All 3rds sharp"
when tuning his masters harpsichord.

Here note in the remote keys do appear in the last four cases an
http://en.wikipedia.org/wiki/Ditone
s:
B-D#, F#-A# C#-F and finally Ab-C,
in postions where they do belong to correctly
according Werckmeister's concept of an 'well-temperament'.

Statement about my older proposals:
The above tuning replaces all my earlier attempts
in interpreting Bach's loops.

If a group member can find even better fitting ratios,
than my above ones, please present it here
for discussion as meassage too.

Conclusion:
Try it out yours own the above semi-tones

C
256/243=~90.224995673062911277566336313100947441113553845113637...cent
C#
17/16 =~104.95540950040728990487921297248522481352073881378482...cent
D
18/17 =~98.954592230367545584094252502274396210033839648169719...cent
Eb
96/91 =~92.601432626951896337948366951149159788421655922363488...cent
E
273/256 =~111.3085691038229391510250985236104612351329225395910..cent
F
256/243=~90.224995673062911277566336313100947441113553845113637...cent
F#
81/76 =~110.30698732924707842572362346131793481490007333006827...cent
G
19/18 =~93.603014401527757063249842013441686208654505131886272...cent
G#
18/17 =~98.954592230367545584094252502274396210033839648169719...cent
A
17/16 =~104.95540950040728990487921297248522481352073881378482...cent
Bb
256/243=~90.224995673062911277566336313100947441113553845113637...centB
2187/2048=~113.685006057711924211407129161658673582441024616840...centC'

Try to tune that within the accuracy
as precise as far yours ears able to apply that.

If that ratios were already used in coeval Baroque tuning-practice, that decision is left up to other experts in that field.

Nevertheless
have a lot of fun when playing Bach's works in that above tuning,
independent from the question, whether there is at the moment no answer if it is today brand-new invented or even not?
A.S.

Temperament in Bach's Well-Tempered Clavier

🔗Sergio Martínez <sergio.martinez.ruiz@...>

🔗genewardsmith <genewardsmith@...>

🔗Claudio Di Veroli <dvc@...>

🔗Andy <a_sparschuh@...>