Copan

(Found in Copan is a Double-headed Stone Dragon of the Mayans. Like the two heads, this scale encompasses two series, and by coincidence it is in the year of the dragon and the end of the Mayan Calender that is finding its first use.) It was first presented and dedicated to Billy Stiltner.

Musically the scale tends to cluster around notes in the vicinity ef 8 'equal' so can have a type of dual diminished quality making the scale easily useful for the dark and forboding without resolution.

This composition, A Molten Wind, based on a single chord with changing timbres, is one 19 minute example

Originally there were found two recurrent sequences that generated this scale with two proportional triads as chords or sonorities.

Hn = 2Hn-4 + Hn-1 [one can also express as ( 2A + D = E )

and

Hn = 4Hn-5 + 2Hn-3 ( 4A + 2C = F )

But two other recurrent sequences have now also been found bringing the number to 4.

the other two are

Hn = 8Hn-9 + 2Hn-2 ( 8A + 2H = J )

............example 116 x 8= 928

..........................................+

.....................2420 x 2 = 4840

................928 + 4840 = 5768

and

Hn = 16Hn-12 + 8Hn-5 ( 16A + 8H = M )

............example 116 x 16= 1856

..............................................+

............example 2420 x 8 = 19360

........................................= 21216

In this instance, I seeded the scale with harmonics as follows. I include the resultant scale step numbers (x.) in a 19 tone scale in order to facilitate details the reader might wish to pursue.

(14.) 116 seed beginning

(7.) 179

(0.) 276

(12.) 426 seed to first formula carried to here ( in example 116 times 2 + 426 = 658)

(5.) 658 seed to second formula carried to here ( in example 276 times 4 + 658 times 2 = 2420)

(17.) 1016

(10.) 1568

(3.) 2420 seed to both third and forth formulas ( in different octaves )

(15.) 3736

(8.) 5768

(1.) 8904

(13.) 13744

(6.) 21216

(18.) 32752

(11.) 50560

(4.) 78048

(16.) 120480

(9.) 185984

(2.) 287104

The scale converges on a generator 751.659 cents that would be used to construct a golden version of the scale.

Copan forms good Moments of Symmetries at (1, 3, 5) 8, 11, 19, 27, 35, 43, 51, 59, 67, 75, 83 etc.

copanchart

Here are examples of 8, 11, and 19 tone versions

The 8 tone is marked *, with the 11 requiring the addition of +

The complement to * is also an 11 and the complement to * and + 11 is another 8 tone

*=8 tone, add+ = 11 tone

[While the numerical series above starts with harmonic 116, choosing 276 as the starting point places many pitches closer to 12 ET where instrumentalist are used to playing to minimize fingering problems.]

0. [276]                0*cents

1. [8904]             14.05610468 +

2. [287104]          27.22733797

3. [2420]            158.7202505*

4. [78048]          172.2595032

5.   [658]            304.1031803*

6.   [21216]         317.208724

7.   [179]            450.3495846 *

8.   [5768]          462.397191 +

9.   [185984]       475.553437

10. [1568]          607.4224648 *

11. [50560]        620.6212635

12. [426]           751.4221961 *

13. [13744]        765.5878372

14. [116]          899.347846 *

15. [3736]         910.5051395 +

16. [120480]      923.8916313

17. [1016]        1056.192276 *

18. [32752]         1068.925116

19. [276]            1200*

Building proportional triads on the lowest tone of our series results in these examples in terms of scale step numbers

14. 12. 5.
14. 0. 17.
14. (+2 octaves ) 3. 8.
14. (+3 octaves ) 3. 6.

As an example in this 19 tone scale, only the following transpositions are possible.

2A + D = E
14.	12.	5.
7.	5.	17.
0.	17.	10.
12.	10.	3.
5.	3.	15.
17.	15.	8.
10.	8.	1.
3.	1.	13.
15.	13.	6.
8.	6.	18.
1.	18.	11.
13.	11.	4.
6.	4.	16.
18.	16.	9
11.	9	2.

4A + 2C = F
14.	0.	17.
7.	12.	10.
0.	5.	3.
12.	17.	15.
5.	10.	8.
17.	3.	1.
10.	15.	13.
3.	8.	6.
15.	1.	18.
8.	13.	11.
1.	6.	4.
13.	18.	16.
6.	11.	9
18.	4.	2.

8A + 2H = J
14.(+2 octaves )	3.	8.
7.	15.	1.
0.	8.	13.
12.	1.	6.
5.	13.	18.
17.	6.	11.
10.	18.	4.
3.	11.	16.
15.	4.	9
8.	16.	2.

16A + 8H = M
14.(+3 octaves )	3.	6.
7.	15.	18.
0.	8.	11.
12.	1.	4.
5.	13.	16.
17.	6.	9
10.	18.	2.

IN THE TABLES BELOW

3, 5 , 6 = 2A + D = E

2, 3, 5 = 4A + 2C = F

1, 2, 3 = 8A + 2H = J

1, 4, 5 = 16A + 8H = M

14.(+2 octaves )	3.	8.	3.	6.	18.
7.	15.	1.	15.	18.	11.
0.	8.	13.	8.	11.	4.
12.	1.	6.	1.	4.	16.
5.	13.	18.	13.	16.	9
17.	6.	11.	6.	9	2.

............................3................5.......6

...................2.......3................5

1.................2.......3

1..................................4.......5

THIS CHART SHOWS THE EXTENT OF THE

PRESENTS OF TRIADS WITH USING THE SPACING

FOUND ABOVE

		14.		12.	5.	+
		7.		5.	17.	+
	14.	0.	14.	17.	10.	+0
	7.	12.	7.	10.	3.	+1
	0.	5.	0.	3.	15.	+2
	12.	17.	12.	15.	8.	+3
	5.	10.	5.	8.	1.	+4
	17.	3.	17.	1.	13.	+5
	10.	15.	10.	13.	6.	+6
14.	3.	8.	3.	6.	18.	+7
7.	15.	1.	15.	18.	11.	+8
0.	8.	13.	8.	11.	4.	+9
12.	1.	6.	1.	4.	16.	+10
5.	13.	18.	13.	16.	9	+11
17.	6.	11.	6.	9	2.	+12
10.	18.	4.	18.	2.		+13
3.	11.	16.	11.
15.	4.	9	4.
8.	16.	2.	16.

.....................3................5.......6

............2.......3................5

1..........2.......3

1............................4.......5

Once I have looked at and listened to all of these, I pretty much compose by ear but don't regret being informed by looking at these simpest structures (they are just triads regardless of what the numbers say) within the tuning. Thanks if you made it this far.

14.	12.	5.
7.	5.	17.
0.	17.	10.
12.	10.	3.
5.	3.	15.
17.	15.	8.
10.	8.	1.
3.	1.	13.
15.	13.	6.
8.	6.	18.
1.	18.	11.
13.	11.	4.
6.	4.	16.
18.	16.	9
11.	9	2.

14.	0.	17.
7.	12.	10.
0.	5.	3.
12.	17.	15.
5.	10.	8.
17.	3.	1.
10.	15.	13.
3.	8.	6.
15.	1.	18.
8.	13.	11.
1.	6.	4.
13.	18.	16.
6.	11.	9
18.	4.	2.

		14.		12.	5.	+
		7.		5.	17.	+
	14.	0.	14.	17.	10.	+0
	7.	12.	7.	10.	3.	+1
	0.	5.	0.	3.	15.	+2
	12.	17.	12.	15.	8.	+3
	5.	10.	5.	8.	1.	+4
	17.	3.	17.	1.	13.	+5
	10.	15.	10.	13.	6.	+6
14.	3.	8.	3.	6.	18.	+7
7.	15.	1.	15.	18.	11.	+8
0.	8.	13.	8.	11.	4.	+9
12.	1.	6.	1.	4.	16.	+10
5.	13.	18.	13.	16.	9	+11
17.	6.	11.	6.	9	2.	+12
10.	18.	4.	18.	2.		+13
3.	11.	16.	11.
15.	4.	9	4.
8.	16.	2.	16.

14.	12.	5.
7.	5.	17.
0.	17.	10.
12.	10.	3.
5.	3.	15.
17.	15.	8.
10.	8.	1.
3.	1.	13.
15.	13.	6.
8.	6.	18.
1.	18.	11.
13.	11.	4.
6.	4.	16.
18.	16.	9
11.	9	2.

14.	0.	17.
7.	12.	10.
0.	5.	3.
12.	17.	15.
5.	10.	8.
17.	3.	1.
10.	15.	13.
3.	8.	6.
15.	1.	18.
8.	13.	11.
1.	6.	4.
13.	18.	16.
6.	11.	9
18.	4.	2.

		14.		12.	5.	+
		7.		5.	17.	+
	14.	0.	14.	17.	10.	+0
	7.	12.	7.	10.	3.	+1
	0.	5.	0.	3.	15.	+2
	12.	17.	12.	15.	8.	+3
	5.	10.	5.	8.	1.	+4
	17.	3.	17.	1.	13.	+5
	10.	15.	10.	13.	6.	+6
14.	3.	8.	3.	6.	18.	+7
7.	15.	1.	15.	18.	11.	+8
0.	8.	13.	8.	11.	4.	+9
12.	1.	6.	1.	4.	16.	+10
5.	13.	18.	13.	16.	9	+11
17.	6.	11.	6.	9	2.	+12
10.	18.	4.	18.	2.		+13
3.	11.	16.	11.
15.	4.	9	4.
8.	16.	2.	16.

14.	12.	5.
7.	5.	17.
0.	17.	10.
12.	10.	3.
5.	3.	15.
17.	15.	8.
10.	8.	1.
3.	1.	13.
15.	13.	6.
8.	6.	18.
1.	18.	11.
13.	11.	4.
6.	4.	16.
18.	16.	9
11.	9	2.

14.	0.	17.
7.	12.	10.
0.	5.	3.
12.	17.	15.
5.	10.	8.
17.	3.	1.
10.	15.	13.
3.	8.	6.
15.	1.	18.
8.	13.	11.
1.	6.	4.
13.	18.	16.
6.	11.	9
18.	4.	2.

		14.		12.	5.	+
		7.		5.	17.	+
	14.	0.	14.	17.	10.	+0
	7.	12.	7.	10.	3.	+1
	0.	5.	0.	3.	15.	+2
	12.	17.	12.	15.	8.	+3
	5.	10.	5.	8.	1.	+4
	17.	3.	17.	1.	13.	+5
	10.	15.	10.	13.	6.	+6
14.	3.	8.	3.	6.	18.	+7
7.	15.	1.	15.	18.	11.	+8
0.	8.	13.	8.	11.	4.	+9
12.	1.	6.	1.	4.	16.	+10
5.	13.	18.	13.	16.	9	+11
17.	6.	11.	6.	9	2.	+12
10.	18.	4.	18.	2.		+13
3.	11.	16.	11.
15.	4.	9	4.
8.	16.	2.	16.