The apical pentatonic @ 3x5

Set mapping pentatonic 3x5*

"w" is the non-specific (or generic) denomination of a whole step
"h" is the non-specific (or generic) denomination of a half step
"Es" is the h = -n Exterior set
"Ps" is the h = 0 Perimeter set
"Is" is the Interior set where n-tET > w � n-tad and h greater than or equal
to 1/n-tET
___________________________________________

0+w2+h3+w5+w7+h8*

0. (wh)w(wh)
2. [hw]w[hw]
3. w(wh)(wh)
5. (wh)(wh)w
6. [hw][hw]w
8. (wh)w(wh)

3 5
6 8 10
9 11 13 5x3
12 14�
3x5

3 1 4 2 5
8 6 9 7 10
13� 11� 14� 12� 15�

Es @ 1, 4, 2, 7, 5 and 10
Ps @ 3, 6, 9, 12, and 15
Is @ 8, 13, 11 and 14

w=2 h=1
0 2 3 5 7 8
0 300 450 750 1050 1200

w=3 h=2
0 3 5 8 11 13
0 277 462 738 1015 1200

w=4 h=3 0 4 7 11 15 18
w=5 h=4 0 5 9 14 19 23
w=6 h=5 0 6 11 17 23 28
w=7 h=6 0 7 13 20 27 33
...
w=3 h=1
0 3 4 7 10 11
0 327 436 764 1091 1200

w=4 h=2 0 4 6 10 14 16
w=5 h=3 0 5 8 13 18 21
w=6 h=4 0 6 10 16 22 26
w=7 h=5 0 7 12 19 26 31
...
w=4 h=1
0 4 5 9 13 14
0 343 429 771 1114 1200

w=5 h=2 0 5 7 12 17 19
w=6 h=3 0 6 9 15 21 24
w=7 h=4 0 7 11 18 25 29
w=8 h=5 0 8 13 21 29 34
...
w=5 h=1 0 5 6 11 16 17
w=6 h=2 0 6 8 14 20 22
w=7 h=3 0 7 10 17 24 27
w=8 h=4 0 8 12 20 28 32
...
w=6 h=1 0 6 7 13 19 20
w=7 h=2 0 7 9 16 23 25
w=8 h=3 0 8 11 19 27 30
w=9 h=4 0 9 13 22 31 35
...

Dan Stearns

*I started using this manner of set mapping in the hopes of constructing
internally consistent intertwining networks of compositional links between
all equidistant divisions of the octave�

**In a previous post (Set mapping the 9-tET pelog), I tried to use the 9-tET
example @ 1 3 1 1 3 to illustrate its position as a +2 Is of 2x5. Using the
principle of d � O � F (where d � O � F = n-tad), I would tend to see 9-tET
here (@ +2 Is of 2x5) as an inverted apical scale (the inverted w and h of F
out of d where d is 8).

Dan Stearns wrote:

> From: "Dan Stearns" <stearns@capecod.net>
>
> Set mapping pentatonic 3x5*
>
> "w" is the non-specific (or generic) denomination of a whole step
> "h" is the non-specific (or generic) denomination of a half step
> "Es" is the h = -n Exterior set
> "Ps" is the h = 0 Perimeter set
> "Is" is the Interior set where n-tET > w � n-tad and h greater than or equal
> to 1/n-tET
> ___________________________________________
>
> 0+w2+h3+w5+w7+h8*
>
> 0. (wh)w(wh)
> 2. [hw]w[hw]
> 3. w(wh)(wh)
> 5. (wh)(wh)w
> 6. [hw][hw]w
> 8. (wh)w(wh)

This is the pentatonic as a subset of a 7 tone scale with w=2steps and h=1st

>
> 3 5
> 6 8 10
> 9 11 13 5x3
> 12 14�
> 3x5

Not sure what this is except it resembles some navarro sequences. Don't
understand the following until *

>
> 3 1 4 2 5
> 8 6 9 7 10
> 13� 11� 14� 12� 15�
>
> Es @ 1, 4, 2, 7, 5 and 10
> Ps @ 3, 6, 9, 12, and 15
> Is @ 8, 13, 11 and 14
>
> w=2 h=1
> 0 2 3 5 7 8
> 0 300 450 750 1050 1200

It seems that what you are saying here is in the case of w=hx2 this is the
results in cents. The second line shows the scale against the necessary eight
steps and how it occurs.

>
> w=3 h=2
> 0 3 5 8 11 13
> 0 277 462 738 1015 1200

Here 2w=3h for instance. 13 steps needed. Etc. Still don't get the Es, Ps & Is
and what does it tell us

>
> w=4 h=3 0 4 7 11 15 18
> w=5 h=4 0 5 9 14 19 23
> w=6 h=5 0 6 11 17 23 28
> w=7 h=6 0 7 13 20 27 33
> ...
> w=3 h=1
> 0 3 4 7 10 11
> 0 327 436 764 1091 1200
>
> w=4 h=2 0 4 6 10 14 16
> w=5 h=3 0 5 8 13 18 21
> w=6 h=4 0 6 10 16 22 26
> w=7 h=5 0 7 12 19 26 31
> ...
> w=4 h=1
> 0 4 5 9 13 14
> 0 343 429 771 1114 1200
>
> w=5 h=2 0 5 7 12 17 19
> w=6 h=3 0 6 9 15 21 24
> w=7 h=4 0 7 11 18 25 29
> w=8 h=5 0 8 13 21 29 34
> ...
> w=5 h=1 0 5 6 11 16 17
> w=6 h=2 0 6 8 14 20 22
> w=7 h=3 0 7 10 17 24 27
> w=8 h=4 0 8 12 20 28 32
> ...
> w=6 h=1 0 6 7 13 19 20
> w=7 h=2 0 7 9 16 23 25
> w=8 h=3 0 8 11 19 27 30
> w=9 h=4 0 9 13 22 31 35
> ...
>
> Dan Stearns
>
> *I started using this manner of set mapping in the hopes of constructing
> internally consistent intertwining networks of compositional links between
> all equidistant divisions of the octave�
>
> **In a previous post (Set mapping the 9-tET pelog), I tried to use the 9-tET
> example @ 1 3 1 1 3 to illustrate its position as a +2 Is of 2x5. Using the
> principle of d � O � F (where d � O � F = n-tad), I would tend to see 9-tET
> here (@ +2 Is of 2x5) as an inverted apical scale (the inverted w and h of F
> out of d where d is 8).

-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com