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