why is pattern broken?? [blackjack]

Here's a question mostly for Paul regarding blackjack chords...

Let's take, for example, the simple pattern P5-P4-P4, which Paul has
classified on one of his blackjack posts (I don't have the post
number handy...)

Using our "accepted" blackjack notation ]=1/4 tone up, >=1/6 tone up,
^=1/12 tone up, v=1/12 tone down, <= 1/6 tone down and [= 1/4 tone
down...

we have patterns of these P5-P4-P4 chords that go:

keyboard notation 72-tET notation quality
F#3 C#4 F#4 B4 A[ C> E[ Gb^ 4:5:6:7

G#3 D#4 G#4 C#5 Bb< D[ F< G> 4:5:6:7

Now, one can see that when, on the keyboad, these are transposed by a
keyboard "whole tone..." or a "secor" in blackjack, there is a
resulting 4:5:6:7 chord.

My question is why it is that this pattern of just tetrads is broken
after a string of eight of them?

After that, the next pattern is this:

A#4 F5 A#5 D#6 Gb^ B[ Db^ Ev dissonant

And, that one is labeled dissonant. Likewise the next one transposed
up a keyboard whole tone is also labeled "dissonant..."

And, going in the other direction *below* the F#3 C#4 F#4 B4 the
pattern is, apparently, also broken since:

E3 B3 E4 A4 G> C< D> F dissonant

is also labeled "dissonant."

So the pattern of just tetrads breaks off on that end, too.

Now, why is it that the pattern breaks and just tetrads don't run for
the remaining length of the keyboard?

I notice similar changes in the extremities... only they do different
things, in the other chord lists.

There has to be an easy explanation for this... I can sense it...

Thanks!

________ ________ _______
Joseph Pehrson

In-Reply-To: <9rijbl+jlah@eGroups.com>
Joseph, Blackjack isn't an equal temperament! You have 10 large and 11
small intervals to the octave. That means you can't transpose things
diatonically and expect them to be the same.

Graham

--- In tuning@y..., graham@m... wrote:

/tuning/topicId_29703.html#29710

> In-Reply-To: <9rijbl+jlah@e...>
> Joseph, Blackjack isn't an equal temperament! You have 10 large
and 11 small intervals to the octave. That means you can't transpose
things diatonically and expect them to be the same.
>
>
> Graham

Hi Graham!

Yes, that much I actually *did* know... so I wouldn't expect the same
results from a transposition by a "small step" of 33 cents, for
instance.

However, I was puzzled that when I transposed by a "secor" of 116
cents (a "whole tone" on the keyboard) the 4:5:6:7 tetrads were
maintained for EIGHT transpositions and then suddenly stopped.

I think I *may* have the answer, though. I remember now that
Blackjack has TWO different small intervals around the STARTING
pitch... (let's say C, for instance) and that's the only place where
they don't alternate.

My guess is that's what "throws it off" when the transpositions
extend over one "octave..."

Am I on the right track, Paul??

Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> However, I was puzzled that when I transposed by a "secor" of 116
> cents (a "whole tone" on the keyboard) the 4:5:6:7 tetrads were
> maintained for EIGHT transpositions and then suddenly stopped.

It's the same phenonmenon as trasposing by fifths in a meantone
system, and suddenly hitting the wolf. In a meantone system, you have
a pattern of 0,1,4 for triads--one fifth up gives the fifth, and four
fifths gives the third. If you have twelve notes tuned to meantone
fifths, you can transpose from 0,1,4 to 7,8,11 and the triads are all
fine; however another transposition and you hit the wolf third.

In miracle, you have 0,6,-7,-2 as the pattern for tetrads. Starting
from the third, this means 0 secors for the third, 5 for the 7/4, 7
for the octave, and 13 for the fifth. You can go from 0,5,7,13 all
the way to 7,12,14,20 with no problem, but 8,13,15,21 means you've
hit the wolf secor, and have a wolf fifth.

--- In tuning@y..., jpehrson@r... wrote:

> Now, why is it that the pattern breaks and just tetrads don't run
for
> the remaining length of the keyboard?

Because the scale is only a chain of 20 secors, reduced to within one
octave.

It's a bit like the case of the "wolf" in a 12-tone meantone, which
is a chain of 11 fifths, reduced to within one octave:

Bb D F 4:5:6
B Eb F# dissonant
C E G 4:5:6
C# F G# dissonant
D F# A 4:5:6
Eb G Bb 4:5:6
E G# B 4:5:6
F A C 4:5:6
F# Bb C# dissonant
G B D 4:5:6
G# C Eb dissonant
A C# E 4:5:6

More specifically, in the blackjack case, the pattern you noticed
would hold indefinitely up and down the keyboard if you used a non-
octave-repeating version of the blackjack scale:

. . . . 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 . . . .

But since, once an octave, you have two consecutive "2"s, the pattern
is broken -- usually resulting in a dissonant chord, but sometimes in
a new type of consonant chord.

And don't forget to consider all the inversions of the chords I've
listed for you!

--- In tuning@y..., genewardsmith@j... wrote:

/tuning/topicId_29703.html#29718

> --- In tuning@y..., jpehrson@r... wrote:
>
> > However, I was puzzled that when I transposed by a "secor" of 116
> > cents (a "whole tone" on the keyboard) the 4:5:6:7 tetrads were
> > maintained for EIGHT transpositions and then suddenly stopped.
>
> It's the same phenonmenon as trasposing by fifths in a meantone
> system, and suddenly hitting the wolf. In a meantone system, you
have
> a pattern of 0,1,4 for triads--one fifth up gives the fifth, and
four
> fifths gives the third. If you have twelve notes tuned to meantone
> fifths, you can transpose from 0,1,4 to 7,8,11 and the triads are
all fine; however another transposition and you hit the wolf third.
>

Hello Gene!

Thanks so much for your post, and I'm understanding the general
concept here.... However, according to your description isn't 7,8,11
a major triad beginning on Db?? I thought that was already a triad
that was considered "not good" in meantone... ??

_________ _______ _________
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> Hello Gene!
>
> Thanks so much for your post, and I'm understanding the general
> concept here.... However, according to your description isn't
7,8,11
> a major triad beginning on Db?? I thought that was already a triad
> that was considered "not good" in meantone... ??

Gene would be labelling the flatmost note as 0 -- that usually being
Ab or Eb. Thus 7,8,11 would be A major or E major, respectively.

--- In tuning@y..., jpehrson@r... wrote:

> Thanks so much for your post, and I'm understanding the general
> concept here.... However, according to your description isn't
7,8,11
> a major triad beginning on Db?? I thought that was already a triad
> that was considered "not good" in meantone... ??

If you start your circle of fifths at Eb, so that Eb is 0, you get:
Eb Bb F C G D A E B F# C# G#, so that everything from Eb major
to E major would be fine, but if you went down to Ab or up to B you
would be in trouble.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_29703.html#29725

>
> More specifically, in the blackjack case, the pattern you noticed
> would hold indefinitely up and down the keyboard if you used a non-
> octave-repeating version of the blackjack scale:
>
> . . . . 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 . . . .
>
> But since, once an octave, you have two consecutive "2"s, the
pattern is broken -- usually resulting in a dissonant chord, but
sometimes in a new type of consonant chord.
>

Thanks, Paul... So it looks as though my "guess" about the two "small
steps" surrounding the octave-equivalent pitch "throwing the pattern
off" was actually correct...

> And don't forget to consider all the inversions of the chords I've
> listed for you!

Gee... you're clarivoyant, Paul... I *was* forgetting about those to
a degree...

There's so much to work with here, it's incredible. I'm finding I
want to be quite *specific* with it, too... so even though this next
thing is entirely electronic, most of it I'm working out totally on
paper as well...

_________ ________ ______
Joseph Pehrson

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_29703.html#29754

> --- In tuning@y..., jpehrson@r... wrote:
>
> > Hello Gene!
> >
> > Thanks so much for your post, and I'm understanding the general
> > concept here.... However, according to your description isn't
> 7,8,11
> > a major triad beginning on Db?? I thought that was already a
triad
> > that was considered "not good" in meantone... ??
>
> Gene would be labelling the flatmost note as 0 -- that usually
being Ab or Eb. Thus 7,8,11 would be A major or E major,
respectively.

Oh.... Thanks!

_______ ______ _______
Joseph Pehrson