Blackjack chord progression

Blackjack chords demo1

I thought it would be interesting to demonstrate some of the
procedures I'm currently using with Blackjack harmony. As many of
you are aware, my earlier efforts were done by plotting the lattice
rather slavishly through common-tone harmonies.

Now, however, I'm trying to do this procedure more by ear, which is
how it should have been done in the first place! I notice, though,
that after my exposure to Blackjack harmonies I actually am using
common tone progressions, but the common tones are sometimes implied
and I use a greater variety of dissonant chords to contrast the
consonances.

I made a scan of the opening chords to my new woodwind quintet as an
illustration. I hope this comes up as nicely in your browser as it
does in mine. It works perfectly. Good luck!

It's here in the "files" section and is called chordsdemo.jpg:

/tuning/files/Pehrson/

Anyway, as you will see, the very first chord (from the bass) C<:F<:A
is an interesting dissonance with a 4:3 between the bottom notes and
an A which appears to be connected by a 7:6 (Paul or somebody, please
correct me if I'm wrong here…)

So, this is a chord that is part of a "parallelogram" (thank you,
Paul) that consists of F<:C<:D:A. This is a "seventhy" kind of
parallelogram that, essentially connects two of the basic "just"
tetrads.

Now, notice that the next chord substitutes a D for the F< in the
first chord making the "regular" triad: A:D:F#v. That becomes
a "resolution" of sorts.

Continuing on, rather than slavishly following the lattice, similar
substitutions take place as the D again RETURNS to an F<.

This F< becomes the 5:4 pitch of a triad C#v:F<:G#v. So, essentially
that alternation of pitches becomes a kind of "unifying factor"
without any slavish plotting.

And to continue to the fourth chord, note that the F< is maintained.
It becomes the root of a chord F<:Bb^:D. This is an interesting
chord because we have used the F< and the D so much before. The F<
and the Bb^ of this chord, if I am correct, are the far borders of
the hexany F<:C#v:G:Bb^:D:G#v yes? And, I believe F<:Bb^ outline a
ratio of 11 here. (Please correct me here if I am wrong).

Anyway, this chord is a quasi-dissonance on the utonal side, yes? It
resolves to E>:A>:Db^. Now, what makes this such a finalized
resolution to the ear? Quite frankly, I don't know:

This "resolution" is a little more tenacious, theoretically. From the
Bb^ to the E> is a 7:5, yes? (one of the cyan lines) and I'm
figuring there is an implied G making the transition between the
hexany and the tetrad that is part of the triad A>:E>: Db^.

Now maybe that theoretical logic is specious. However, the aural
result is clear, and there is definitely a strong resolution on the
final triad of this sequence.

I put an mp3 up of this sequence in the Tuning List Files forum. The
file is called, obviously, chordprogression.mp3:

/tuning/files/Pehrson/

If anybody has any other ideas as to why this resolution works so
well (if in fact somebody feels it does, I am more than open to
suggestion…)

I think it has something to do with an implied G and a motion down to
a tetrad a fifth below the prominent D (It's been prominent
throughout this progression).

Possibly??

Joseph

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> illustration. I hope this comes up as nicely in your browser as it
> does in mine.

oops!

It works perfectly. Good luck!
>
> It's here in the "files" section and is called chordsdemo.jpg:
>
> /tuning/files/Pehrson/
>
> Anyway, as you will see, the very first chord (from the bass)
C<:F<:A
> is an interesting dissonance with a 4:3 between the bottom notes
and
> an A which appears to be connected by a 7:6 (Paul or somebody,
please
> correct me if I'm wrong here…)

you're right, F<:A:C< is essentially a 1/(9:7:6) chord, aka
supermajor triad. it's a subset of the 9-limit utonality, so i'd
consider it somewhat dissonant, partch less so. but, taking the
octaves into consideration, in this position the 6:7 becomes a 7:24,
and the 7:9 becomes an 7:18, so the voicing is not helping the
potential consonance of these intervals along . . . probably just as
you intended.

> So, this is a chord that is part of a "parallelogram" (thank you,
> Paul) that consists of F<:C<:D:A. This is a "seventhy" kind of
> parallelogram that, essentially connects two of the basic "just"
> tetrads.

well, it's a basic "just" tetrad in itself (usually having D as
root), a so-called 9-limit ass:

http://x31eq.com/ass.htm

it can be expressed as 12:14:18:21, or, equally well, as 1/
(21:18:14:12). often called "septimal minor seventh" chord.

as you can see on that page, the other kind of 9-limit ass is simply
the "just minor seventh" chord.

> Now, notice that the next chord substitutes a D for the F< in the
> first chord making the "regular" triad: A:D:F#v. That becomes
> a "resolution" of sorts.
>
> Continuing on, rather than slavishly following the lattice, similar
> substitutions take place as the D again RETURNS to an F<.
>
> This F< becomes the 5:4 pitch of a triad C#v:F<:G#v. So,
essentially
> that alternation of pitches becomes a kind of "unifying factor"
> without any slavish plotting.

what's this "slavish" business?

> And to continue to the fourth chord, note that the F< is
maintained.
> It becomes the root of a chord F<:Bb^:D. This is an interesting
> chord because we have used the F< and the D so much before. The F<
> and the Bb^ of this chord, if I am correct, are the far borders of
> the hexany F<:C#v:G:Bb^:D:G#v yes? And, I believe F<:Bb^ outline a
> ratio of 11 here. (Please correct me here if I am wrong).

yes, F<:Bb^ approximates 8:11, F<:D approximates 7:24 (just like C<:A
in the first chord), Bb^:D approximates 2:5. three intervals that
wouldn't fit together in true just intonation. try D lower and Bb^
higher and see how you like the chord.

> Anyway, this chord is a quasi-dissonance on the utonal side, yes?

i'm not sure how it's utonal . . .

> It
> resolves to E>:A>:Db^.

> Now, what makes this such a finalized
> resolution to the ear? Quite frankly, I don't know

> I think it has something to do with an implied G

where/how are you hearing an implied G, exactly? as the "root" of
F<:Bb^:D? G:Bb^:D would be a conventional major triad, and G:D:F<
a 4:6:7, so that could make some sense . . .

> and a motion down to
> a tetrad a fifth
> below the prominent D (It's been prominent
> throughout this progression).

motion to a tetrad on G? i thought you were moving to A>:Db^:E>,
a "just major" triad on A> . . .

--- In tuning@yahoogroups.com, "wallyesterpaulrus"

/tuning/topicId_45231.html#45240

<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
>
> > illustration. I hope this comes up as nicely in your browser as
it does in mine.
>
> oops!
>

***That's too bad the scan wouldn't work, Paul. What did you get?
Something too large/small, or just an error message? It worked
easily for me both at home and at work when I click on the file... :(

Anyway, we really don't need it... :)

> you're right, F<:A:C< is essentially a 1/(9:7:6) chord, aka
> supermajor triad. it's a subset of the 9-limit utonality, so i'd
> consider it somewhat dissonant, partch less so. but, taking the
> octaves into consideration, in this position the 6:7 becomes a
7:24,
> and the 7:9 becomes an 7:18, so the voicing is not helping the
> potential consonance of these intervals along . . . probably just
as you intended.
>

***Yes, it's wanting a resolution...

> > So, this is a chord that is part of a "parallelogram" (thank
you,
> > Paul) that consists of F<:C<:D:A. This is a "seventhy" kind of
> > parallelogram that, essentially connects two of the basic "just"
> > tetrads.
>
> well, it's a basic "just" tetrad in itself (usually having D as
> root), a so-called 9-limit ass:
>

***GOT IT! Finally, I'm catching on to where the asses fit in the
lattice. (That sounds vaguely off color...)

> http://x31eq.com/ass.htm
>
> it can be expressed as 12:14:18:21, or, equally well, as 1/
> (21:18:14:12). often called "septimal minor seventh" chord.
>
> as you can see on that page, the other kind of 9-limit ass is
simply the "just minor seventh" chord.
>
> > Now, notice that the next chord substitutes a D for the F< in the
> > first chord making the "regular" triad: A:D:F#v. That becomes
> > a "resolution" of sorts.
> >
> > Continuing on, rather than slavishly following the lattice,
similar
> > substitutions take place as the D again RETURNS to an F<.
> >
> > This F< becomes the 5:4 pitch of a triad C#v:F<:G#v. So,
> essentially
> > that alternation of pitches becomes a kind of "unifying factor"
> > without any slavish plotting.
>
> what's this "slavish" business?
>

***Something that I actually took out of the post, and then it crept
back in when I had to repost again...

> > And to continue to the fourth chord, note that the F< is
> maintained.
> > It becomes the root of a chord F<:Bb^:D. This is an interesting
> > chord because we have used the F< and the D so much before. The
F<
> > and the Bb^ of this chord, if I am correct, are the far borders
of
> > the hexany F<:C#v:G:Bb^:D:G#v yes? And, I believe F<:Bb^ outline
a
> > ratio of 11 here. (Please correct me here if I am wrong).
>
> yes, F<:Bb^ approximates 8:11, F<:D approximates 7:24 (just like
C<:A
> in the first chord), Bb^:D approximates 2:5. three intervals that
> wouldn't fit together in true just intonation. try D lower and Bb^
> higher and see how you like the chord.
>
> > Anyway, this chord is a quasi-dissonance on the utonal side, yes?
>
> i'm not sure how it's utonal . . .
>

***Got it. That doesn't make sense since the hexany it's based on is
three dimensional and goes both in the otonal and utonal direction,
correct??

> > It
> > resolves to E>:A>:Db^.
>
> > Now, what makes this such a finalized
> > resolution to the ear? Quite frankly, I don't know
>
> > I think it has something to do with an implied G
>
> where/how are you hearing an implied G, exactly? as the "root" of
> F<:Bb^:D?

***That's what I was thinking as a possibility...

G:Bb^:D would be a conventional major triad,

***I believe you mean G:Bv:D, yes??

and G:D:F<
> a 4:6:7, so that could make some sense . . .
>
> > and a motion down to
> > a tetrad a fifth
> > below the prominent D (It's been prominent
> > throughout this progression).
>
> motion to a tetrad on G? i thought you were moving to A>:Db^:E>,
> a "just major" triad on A> . . .

***Right. I was just wondering how the F<:Bb^:D connected to
A>:Db^:E>. I think if you listen to the sound example you'll agree
it's a very pronounced "resolution" effect. There must be some
acoustic or psychological reason for that...

Thanks, Paul!

Joseph

hi Joe,

> From: "Joseph Pehrson" <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, July 02, 2003 7:25 PM
> Subject: [tuning] Re: Blackjack chord progression
>
>
> <snip>
>
> ... I was just wondering how the F<:Bb^:D connected
> to A>:Db^:E>. I think if you listen to the sound
> example you'll agree it's a very pronounced "resolution"
> effect. There must be some acoustic or psychological
> reason for that...

i got intrigued enough by this to do some analysis.

(to view it properly on the Yahoo web interface,
use "Expand Messages" mode)

---- 72edo ---- ------- cents -------- --- intervals ---
note Sims name from "C" above root cents ~ratios

D D 1400 933+(1/3)
> 383+(1/3) 5:4
Bb^ 1/12 high Bb 1016+(2/3) 550
> 550 11:8
F< 1/6 low F 466+(2/3) 0

E> 1/6 high E 1633+(1/3) 700
> 316+(2/3) 6:5
Db^ 1/12 high Db 1316+(2/3) 383+(1/3)
> 383+(1/3) 5:4
A> 1/6 high A 933+(1/3) 0

if the notes are actually arranged in the
chords from the bottom up as i have put them
in my table, which is how Joe described them,
then here is the voice-leading of the resolution
in cents:

cents

D + 233+(1/3) E>

Bb^ + 300 Db^

F< + 466+(2/3) A>

(but i'm not hearing it that way: it sounds to me
like the final chord is spelled E>:A<:Db^, i.e.,
2nd inversion.)

anyway, the first table shows that the second (final)
chord is very close to a JI major triad.

as for the first chord, F<:D is almost exactly a 12:7
ratio, and F<:Bb^ is almost exactly an 11:8, as you
(Joe) already mentioned. what's really interesting
is that a 12/7 is very nearly a 5:4 larger than 11:8.

the voice-leading table shows that F<:A> is a narrow
"wolf" 4th. but i think it's close enough to a 4:3
to give the effect of a "dominant --> tonic" root-movement.

if the final chord *is* the 2nd inversion, as i'm
hearing it, then this is a "parallel" chord progression
of the type "forbidden" in old-fashioned harmony books,
since two of the voices move the same distance in the
same direction. the third voice also moves the same
direction, but a very small distance. i suppose that
both of these qualities make it a strong resolution.

here, the voice-leading in all three voices moves less
than a semitone, as follows:

cents

D - (83+(1/3)) Db^

Bb^ - (83+(1/3)) A>

F< - (33+(1/3)) E>

in addition to the mathematical considerations, this
sounds to me very much like either a VI --> V or
ii7 --> V in D> minor, both of which are typical
strong progressions. ending with that progression
is not unlike a typical slow-middle-movement ending
of a Baroque piece, where the slow movement ends on V
to prepare for the I of the fast final movement.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

/tuning/topicId_45231.html#45249

> (but i'm not hearing it that way: it sounds to me
> like the final chord is spelled E>:A<:Db^, i.e.,
> 2nd inversion.)

***Hi Monz,

This is correct. The chords are parallel and in second inversion.
I'm sorry nobody seems to be able to pull up the scan of the harmony
that I put up here:

/tuning/files/Pehrson/

(chordsdemo.jpg)

It worked *fine* for me in a couple of browsers.

>
> anyway, the first table shows that the second (final)
> chord is very close to a JI major triad.

***Well, that would make sense, Monz, since it's one of the
Blackjack "just" triads! :)

>
> as for the first chord, F<:D is almost exactly a 12:7
> ratio, and F<:Bb^ is almost exactly an 11:8, as you
> (Joe) already mentioned. what's really interesting
> is that a 12/7 is very nearly a 5:4 larger than 11:8.
>
>
> the voice-leading table shows that F<:A> is a narrow
> "wolf" 4th. but i think it's close enough to a 4:3
> to give the effect of a "dominant --> tonic" root-movement.
>
>
>
> if the final chord *is* the 2nd inversion, as i'm
> hearing it, then this is a "parallel" chord progression
> of the type "forbidden" in old-fashioned harmony books,
> since two of the voices move the same distance in the
> same direction. the third voice also moves the same
> direction, but a very small distance. i suppose that
> both of these qualities make it a strong resolution.
>
> here, the voice-leading in all three voices moves less
> than a semitone, as follows:
>
>
> cents
>
> D - (83+(1/3)) Db^
>
> Bb^ - (83+(1/3)) A>
>
> F< - (33+(1/3)) E>
>
>
> in addition to the mathematical considerations, this
> sounds to me very much like either a VI --> V or
> ii7 --> V in D> minor, both of which are typical
> strong progressions. ending with that progression
> is not unlike a typical slow-middle-movement ending
> of a Baroque piece, where the slow movement ends on V
> to prepare for the I of the fast final movement.
>
>

***Well, actually, Monz these are interesting theories, but I think I
figured it out. In fact, it's not the *only* parallel chord in the
progression. They *all* are!

Here's a funny little observation on my part. Did you know that, if
one writes out the Blackjack notes, it really is possible to do
pretty "accurate" analysis in 12-equal?? This sounds like sacrilege,
but it really is possible. The deviations from 12-equal are really
small enough so that one still gets an overall aural picture related
to our "old" system.

Anyway, what we're hearing here in the last two chords is a
progression using the NEAPOLITAN... or a lowered ii chord. In
conjunction with the other parallel chords of the passage, it
creates a resolution like the traditional Neapolitan with its
descending stepwise motion...

J. Pehrson