- --- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
> > Here's a tally of the 19-limit consonances.

Hi Carl, Tom & all others,

> >

> > 19/18 1

> > 18/17 4

> > 17/16 2

> > 19/17 0

> > 9/8 2

> > 19/16 5

> > 24/19 2

> > 4/3 4

> > 17/12 2

> > ---------

> > 22

> (a,b,c,b,a,b,1,1,b,1,1,b) where a=6137/6144, b=512/513, c=4617/4624.

yours ... 19,17,3 approachs work similar as my older attempts in:

http://launch.groups.yahoo.com/group/clavichord/message/8737

.

That proposal intends to offer an arguable

http://en.wikipedia.org/wiki/Historically_informed_performance

tuning on modern replicas of

http://en.wikipedia.org/wiki/Michael_Mietke

"Clavessin"

"He delivered a harpsichord to the court at Cöthen in 1719 on the

recommendation of Johann Sebastian Bach, which was probably the

instrument for which Bach composed Brandenburg concerto no.5 as a

show-piece."

The scholary literature offers some further details

about Bachs's journey to Berlin for buying there the instrument.

Even the bill is still preserved.

Dated: Cöthen, March 1st 1719,

It lists $130 expenses altogehter.

Reprint in:

Bach-Dokumente, Kassel 1969, Vol.II p.73-74, #95

Translation into english by A.Mendel, Bach reader p.431,490

But when Bach moved in 1723 to Leipzig

that piano remained back in Coethen at least until 1784,

recorded among the court's inventory:

"Specifickation derer Fürstlichen Instrumenten in der

Musikalienkammer"

lists it on March, 8th, 1784 as "defect".

The problem:

Today, nobody knows any more how Bach had tuned that instrument.

But HIPerformers need for Mietke-replicas

adequate modern so called "Bach"-tunings,

preferably in the coeval "Cammerthone"

pitch of a4=(~405+-5)cps

as known from Mietke's others instruments too.

Hence a modern "Bach"-tuning should contain that specification,

in order to represent the state of the art in the research of the

last ~10 years:

New proposal for absolute pitches located at the frequencies:

243 middle_c'

256

272 d'

304

304 e'

312 f'

342

364 g'

392

406 a' cps or Hz

432

456 b'

486 c"

as again obtained from Bach's 1722(1723?) WTC autograph,

when reading the 'squiggle' pattern layout as an instruction

how to temper 5ths the corresponding to the "squiggles"

http://en.wikipedia.org/wiki/Well_Tempered_Clavier

Here an older photo of JSB's "decorative-ornaments" that i do prefer:

http://www.strukturbildung.de/Andreas.Sparschuh/Bach_Handschrift.jpg

That doodle at the top can be interpreted symbolic as:

Concise: start~2~2~2~1-1-1~3-3-3-3-3-3=end

when understood as an cylce 12 times 5ths

Or when labeld with note-names:

start=C~2G~2D~2A~2E-1B-1F#-1C#~3G#-3D#-3A#-3F-3C=end

Or even when expanded into full detailed pitches:

start=C 243 := 3^5 obtained from the last 5 ternary 'squiggles'

~2~ G 91 182 364 728 (<729 :=3^6)

~2~ D 68 136 272 (<273 := 3*G)

~2~ A 203 (<204 := 3*68)

~1- E 19 38 76 152 304 608 (<609 := 3*A) ; 3 single 'squiggles'

-1- B 57

-1~ F# 171

~3- C# 1...512 (<513 := 3*F#) ; the concluding 5 triply 'squiggles'

-3- G# 3

-3- D# 9 := 3^2

-3- A# 27 := 3^3

-3- F 81 := 3^4

-3. C=end 243 := 3^5

That's in today's modern terms of the:

http://www.xs4all.nl/~huygensf/scala/scl_format.html

!Sparschuh_proposal_Mietke.scl

!absolute pitches from the middle_c' on:

c'243#256 d'272#288 e'304 f'324#342 g'364#384 a'406#432 b'456

!

12

!

256/243 ! ~90.22...(cents)

272/243 ! ~195.81... D

32/27 ! ~294.13...

304/243 ! ~387.74... E (5/4)*(1216/1215)

4/3 ! ~498.04... F

38/27 ! ~591.65...

364/243 ! ~699.58... G (3/2)*(728/729)

128/81 ! ~792.18...

406/243 ! ~888.63... A4=406cps ~ Coeval Berlin, Coethen Cammer-thone

16/9 ! ~996.09...

152/81 ! ~1089.69... B

2/1

Attend,

that tuning contains some consecutive partials

out of the overtone-series inbetween the 4 semitones:

16 : 17 : 18 : 19 == C#1 : D1 : Eb1 : E1

in reference to:

http://en.wikipedia.org/wiki/Johann_Gottfried_Walther

specification in his:

"He wrote a handbook for the young Duke with the title Praecepta der

musicalischen Composition, 1708. It remained handwritten until Peter

Benary's edition (Leipzig, 1955)."

in german:

http://de.wikipedia.org/wiki/Johann_Gottfried_Walther

's

"Praecepta der musicalischen Composition, Weimar 1708;

Neu hrsg. von Peter Benary in:

Jenaer Beiträge zur Musikforschung. Band 2,

Breitkopf & Härtel, Leipzig 1955 "

bye

A.S. - I wrote:

> With 7 pure 5ths, the number is 68:

Heh, not quite. (P P P b P P a P P b c a) also has 68

> (p p b p p a p p b p c a) a=(-3 1 1) b=(-2 0 -1) c=(3 -2 0)

>

> Each of these scales uniquely hits the maximum number of

> 19-limit consonances in its category.

intervals. And despite having one 32/27 and one non-17-limit

tritone, its pattern of M3s is more equal, and its key-color

pattern is more sinusoidal on the circle of fifths.

!

(P P P b P P a P P b c a) a=(-3 1 1) b=(-2 0 -1) c=(3 -2 0)

12

!

19/18

323/288

19/16

64/51

4/3

24/17

3/2

19/12

57/34

16/9

32/17

2/1

!

As for 4 different nonpure 5ths... with 5 pure 5ths and

searching to radius 5, we get 19-limit intervals = 54:

(p p b c p p a p b c d a)

a=(-3 1 1) b=(-1 -1 -3) c=(0 1 2) d=(3 -2 0)

With 6 pure 5ths and radius 5, we get 60:

(p p p b p p p b c b d a)

a=(-3 1 1) b=(-2 0 -1) c=(0 1 2) d=(3 -2 0)

With 7 pure 5ths and radius 9, we get 64:

(p p b p p a p p b p c d)

a=(-3 1 1) b=(-2 0 -1) c=(-2 3 6) d=(2 -4 -5)

With this last, I think I broke Scala. 309237645312/206806264579

shows up in cents even though it's rational in the .scl file.

-Carl