Re: J.S. Bach: Badinerie from Suite in B Minor (golden meantone)

Danny Wier uploaded the MIDI file badinerie.mid this morning, and I
went and grabbed it. Sounds very nice! The harpsichord part is choppy
throughout much of the piece, but I'm becoming used to it.

For fun, I peeked at the tuning. Was very surprised to find the circle
of fifths going from C to E#! The piece is in B minor.

But Danny is right. I ran it thru my COFT analysis, and got the same
basic tuning (cents deviation from 12-tET):

Danny JdL's COFT
------ ----------
C 0.00 +14.22
G -3.78 +13.57
D -7.57 +11.42
A -11.35 +10.75
E -15.14 +2.39
B -18.92 -1.57
F# -22.71 -2.49
C# -26.49 -3.97
G# -30.27 -8.72
D# -34.06 -7.54
A# -37.87 -11.76
E# -41.65 -16.30

I haven't adjusted for the absolute differences, but the change in
tuning across the circle of fifths tells the story. His is even (.267
comma meantone); mine is of course jagged, reflecting the particular
intervals in the piece, and with a somewhat narrower overall range.
(what is the definition of "golden meantone" again?).

I show adaptive 5-limit pain as not much less than COFT pain (22534 vs.
29652), which is not much less than 1/4 comma meantone C to E# (36033).
So this is one of those pieces that is extremely well suited to
meantone, just as Danny has done it.

Thanks, Danny!

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> Danny Wier uploaded the MIDI file badinerie.mid this morning, and I
> went and grabbed it. Sounds very nice! The harpsichord part is
choppy
> throughout much of the piece, but I'm becoming used to it.

........

...His is even (.267
> comma meantone); mine is of course jagged, reflecting the particular
> intervals in the piece, and with a somewhat narrower overall range.
> (what is the definition of "golden meantone" again?).
>
Bob Wendell:
I dont' know what the conceptual definition is, John, but if its
practical definition is 0.26666....-comma meantone, this lies
slightly under Woolhouse's optimized 7/26-comma (0.26923...), which
means the fifths are all of 0.055 cents flatter.

It is also only 0.41 cents flatter than 2/7-comma, which I think I
remember was Alessandro Scarlatti's favorite? All three lie in this
0.41 cent range of difference. Not terribly significant, I would
venture. ("Please don't hit for saying that, whoever you are out
there," he said, flinching.)

[I wrote:]
>>(what is the definition of "golden meantone" again?).

[Danny wrote:]
>I promise, this is my last P.S. to this post! Golden meantone is
>based on a diatonic scale where the ratio of major seconds to minor
>seconds are 1.6180:1.0000 -- the first number is called the golden
>mean and is defined as 0.5 + half the square root of 5. That number
>is the limit of the ratio of two adjacent numbers in the Fibonacci
>sequence 1 1 2 3 5 8 13 21 34 55 89 144...

>I actually came up with this temperament independently before I found
>out who did it first. I nicknamed it "infinite-tone equal
>temperament" since I wanted to have some idea of what equal
>temperaments might sound like beyond 19 and 31 in that class (that
>is, 50-tone, 81-tone, 131-tone, 212-tone etc.)

I get it.

n-tET fifth maj 2nd min 2nd ratio
----- ----- ------- ------- ------
19 11 3 2 1.500
31 18 5 3 1.667
50 29 8 5 1.600
81 47 13 8 1.625
131 76 21 13 1.615
212 123 34 21 1.619
(etc.)

At the higher ET's, the fifth picks are no longer the closest to 1.5,
but for purposes of approaching the golden ratio, the series continues.
So, since an octave is 5 whole steps plus two small steps, if a small
step has S cents, and R is the ratio (logarithmic) between the two step
sizes,

2 * S + 5 * R * S = 1200 cents

S * (2 + 5*R) = 1200 cents

S = 1200 cents / (2 + 5*R)

If R == (1 + sqrt(5))/2 =~ 1.618, then

S = 1200 cents / 10.090 = 118.928 cents;
R*S = 192.429 cents

A fifth is 696.214 cents.

[Danny:]
>17-tone, 22-tone and 53-tone work differently by the way.

Yeah, for starters they aren't meantone. Are they involved in a similar
game?

Thanks for the clarification!

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> Yeah, for starters they aren't meantone. Are they involved in a
similar
> game?

Sure -- any MOS scale can be similarly identified with one of
the "archetypes" on the Scale Tree and thus lead to an endless
sequence of approximating ETs, involving always the same Golden ratio
(phi). I think two years ago there was a lot of discussion on this
here, involving Jason Yust, Dave Keenan, and Kraig Grady (especially
in showing us Wilson's work).