BlackJack - Pentads, tetrads, triads, hexanies

Yes, as scales of around 20 notes with many consonances go, the 2 of {1 3 5
7 9 11} Eikosany is absolutely brilliant. I still remember how I felt when
I first understood it. Wilson is a genius, no doubt. But as Kraig says, the
Eikosany is essentially non-tonal and facilitates very different kinds of
harmony and melody to what Blackjack does. Blackjack has a tonal centre,
but still with plenty of opportunity for modulation.

The Eikosany gives all 11 limit consonances equal billing. Blackjack
favours the 7 limit (and the 9-limit sans 5's) (and the neutral, i.e.
11-limit sans 5's and 7's). So you actually get more of the more readily
recognised chords (but still exotic to most) and less of the _really_ exotic.

Paul Erlich wrote:

>For each of the above [15 categories of tetrad], the Eikosany has 1 otonal
tetrad, 1 utonal
>tetrad, and 2 hexanies.
>
>The blackjack scale is very different in that it has far more
>hexanies and tetrads in the 1-3-5-7 category. I invite Dave Keenan to
>tell us what it has in the other categories.

Here are the numbers of otonal and utonal tetrads in Blackjack, for of each
of the 15 possible kinds. e.g. The first line tells us that there are eight
1:3:5:7 (or in more familiar terms 4:5:6:7) otonal tetrads and eight
1/(7:6:5:4) utonal tetrads. Note that some are grouped according to what
larger chords they are subsets of.

1-3-5-7 8

1-3-7-9 7

1-3-9-11 6

1-3-7-11 4
1-7-9-11 4
3-7-9-11 4
as subsets of pentads
1-3-7-9-11 4

1-3-5-9 2
1-5-7-9 2
3-5-7-9 2
as subsets of pentads
1-3-5-7-9 2

1-5-7-11 0
1-5-9-11 0
3-5-7-11 0
3-5-9-11 0
1-3-5-11 0
5-7-9-11 0
as subsets of hexads
1-3-5-7-9-11 0

That's a total of 39 otonal and 39 utonal quasi-just tetrads in 21 notes.
Plus 6 otonal and 6 utonal pentads. Notice that there are no 5:11's in
Blackjack. You'd need to extend the chain to 23 notes before you would
obtain one of them (and hence a full 11-limit hexad).

On the subject of triads: Here are the 20 kinds and their quantities.

1-5-7 14
1-3-7 13
3-9-11 12
1-3-9 9
1-3-5 8
3-5-7 8
1-7-9 7
3-7-9 7
1-3-11 6
1-9-11 6
1-7-11 4
3-7-11 4
7-9-11 4
1-5-9 2
3-5-9 2
5-7-9 2
1-5-11 0
3-5-11 0
5-7-11 0
5-9-11 0

108 otonal and 108 utonal. But one mustn't forget there are also 15 neutral
triads (1 : 11/9 : 3).

You can see them all below. Just paste this diagram into a text editor and
insert spaces at the start of a line to move a pattern along the chain and
see all its possible positions. These show only the otonal chords (the most
consonant). To make the corresponding utonal for each of these, flip them
horizontally.

Degrees of 72-EDO arranged as a chain of 7-step (116.67c) generators.
2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70
+-------------------Blackjack-21-MOS-improper-CS------------+
+--------11 note MOS-improper-+
+-----10 note MOS-proper---+
1------11/9-------3
5--------------7-----1
7-----1-----------------3
3-----------------9-------11
1-----------------3-----------------9
5--------------7-----1-----------------3
7-----1-----------------3-----------------9
1-----------------3-----------------9-------11
7-----1-----------------3-----------------9-------11
5--------------7-----1-----------------3-----------------9
5--------------7-----1-----------------3-----------------9-------11
5*7---1*5------------1*73*5------------3*7---1*3
2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70

You will notice that the 10 note MOS has no tetrads or larger, but it has
triads.
3-9-11 1
1-3-7 2
1-5-7 3
neutral 4

That's 6 otonal, 6 utonal, and 4 neutral. 16 triads in 10-notes.

Working out hexany patterns on the chain is not so straighforward as otonal
chords. But the 1-3-5-7 hexany pattern is shown above, and we can see that
there are 6 of them in Blackjack.

Counting the other hexanies will have to wait for another time. Or for
someone else.

Regards,
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:

> Degrees of 72-EDO arranged as a chain of 7-step (116.67c)
generators.
> 2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70
> +-------------------Blackjack-21-MOS-improper-CS------------+
> +--------11 note MOS-improper-+
> +-----10 note MOS-proper---+
> 1------11/9-------3
> 5--------------7-----1
> 7-----1-----------------3
> 3-----------------9-------11
> 1-----------------3-----------------9
> 5--------------7-----1-----------------3
> 7-----1-----------------3-----------------9
> 1-----------------3-----------------9-------11
> 7-----1-----------------3-----------------9-------11
> 5--------------7-----1-----------------3-----------------9
> 5--------------7-----1-----------------3-----------------9-------11
> 5*7---1*5------------1*73*5------------3*7---1*3
> 2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70

Awesome, Dave! Just what we needed.
>
> You will notice that the 10 note MOS has no tetrads or larger, but
it has
> triads.
> 3-9-11 1
> 1-3-7 2
> 1-5-7 3
> neutral 4
>
> That's 6 otonal, 6 utonal, and 4 neutral. 16 triads in 10-notes.

This could be a place to start for composers. And the 11-note MOS,
which has one more of each of those, so 23 triads.

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> Yes, as scales of around 20 notes with many consonances go, the 2
of {1 3 5
> 7 9 11} Eikosany is absolutely brilliant. I still remember how I
felt when
> I first understood it. Wilson is a genius, no doubt. But as Kraig
says, the
> Eikosany is essentially non-tonal and facilitates very different
kinds of
> harmony and melody to what Blackjack does. Blackjack has a tonal
centre,

Well it has a center point in the lattice, but that doesn't mean it
will (or can) function as a tonal center.

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> > Yes, as scales of around 20 notes with many consonances go, the 2
> of {1 3 5
> > 7 9 11} Eikosany is absolutely brilliant. I still remember how I
> felt when
> > I first understood it. Wilson is a genius, no doubt. But as Kraig
> says, the
> > Eikosany is essentially non-tonal and facilitates very different
> kinds of
> > harmony and melody to what Blackjack does. Blackjack has a tonal
> centre,
>
> Well it has a center point in the lattice, but that doesn't mean it
> will (or can) function as a tonal center.

I guess I don't really know the definition of "tonal centre". But
ye, I just meant that Blackjack has some notes which are nearer the
center of the lattice and have more consonant relationships with other
notes, and some which are on the periphery. The eikosany notes are all
on the surface of a 5D hypersphere, although I would expect that a
1:3:5 would be favoured as a place of relative rest.

As far as being the 1 odentity in the most (and lowest limit)
otonalities, the notes 51 and 58 (of 72-tET) are equal winners in the
Blackjack scale (not note 0).

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., paul@s... wrote:

> I guess I don't really know the definition of "tonal centre". But
> ye, I just meant that Blackjack has some notes which are nearer the
> center of the lattice and have more consonant relationships with
other
> notes,

Oddly, the central point (point of symmetry) is _not_ the point with
the most consonant relationships to other notes.

> and some which are on the periphery. The eikosany notes are all
> on the surface of a 5D hypersphere, although I would expect that a
> 1:3:5 would be favoured as a place of relative rest.

I do see that potential bias.

--- In tuning@y..., paul@s... wrote:
> Oddly, the central point (point of symmetry) is _not_ the point with
> the most consonant relationships to other notes.

Which notes are?

I pointed out that 51 and 58 are most often odentity 1. But this is a
different question.

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., paul@s... wrote:
> > Oddly, the central point (point of symmetry) is _not_ the point
with
> > the most consonant relationships to other notes.
>
> Which notes are?

Just look at the lattice (and think 9-limit).