Difference between revisions of "User:Jon Awbrey/GRAPHICS"

Latest revision as of 18:52, 7 January 2011

Animations

Riffs 1 to 60

Rotes 1 to 60

Image Gallery

Reduction 6:1

Cacti

Image	Scale
	117 px ↓ 20 px
	117 px ↓ 20 px

Riffs

Integer	Riff	Scale
1
2		117 px ↓ 20 px
3		240 px ↓ 40 px
4		233 px ↓ 40 px
5		384 px ↓ 65 px
6		384 px ↓ 65 px
7		377 px ↓ 65 px
8		384 px ↓ 65 px
9		240 px ↓ 40 px
10		528 px ↓ 90 px
11		528 px ↓ 90 px
12		384 px ↓ 65 px
13		384 px ↓ 65 px
14		521 px ↓ 90 px
15		528 px ↓ 90 px
16		377 px ↓ 65 px
17		521 px ↓ 90 px
18		384 px ↓ 65 px
19		528 px ↓ 90 px
20		528 px ↓ 90 px
21		521 px ↓ 90 px
22		672 px ↓ 115 px
23		384 px ↓ 65 px
24		672 px ↓ 115 px
25		384 px ↓ 65 px
26		528 px ↓ 90 px
27		384 px ↓ 65 px
28		521 px ↓ 90 px
29		528 px ↓ 90 px
30		672 px ↓ 115 px
31		672 px ↓ 115 px
32		528 px ↓ 90 px
33		672 px ↓ 115 px
34		665 px ↓ 115 px
35		521 px ↓ 90 px
36		384 px ↓ 65 px
37		384 px ↓ 65 px
38		672 px ↓ 115 px
39		672 px ↓ 115 px
40		816 px ↓ 135 px
41		528 px ↓ 90 px
42		665 px ↓ 115 px
43		521 px ↓ 90 px
44		672 px ↓ 115 px
45		528 px ↓ 90 px
46		528 px ↓ 90 px
47		528 px ↓ 90 px
48		384 px ↓ 65 px
49		377 px ↓ 65 px
50		528 px ↓ 90 px
51		665 px ↓ 115 px
52		528 px ↓ 90 px
53		521 px ↓ 90 px
54		528 px ↓ 90 px
55		672 px ↓ 115 px
56		809 px ↓ 135 px
57		672 px ↓ 115 px
58		672 px ↓ 115 px
59		665 px ↓ 115 px
60		672 px ↓ 115 px
64		384 px ↓ 65 px
81		377 px ↓ 65 px
105		665 px ↓ 115 px
128		521 px ↓ 90 px
165		816 px ↓ 135 px
256		528 px ↓ 90 px
512		384 px ↓ 65 px
2010		1104 px ↓ 185 px
2500		528 px ↓ 90 px
65536		521 px ↓ 90 px
802701		1248 px ↓ 210 px
123456789		1296 px ↓ 220 px

Rotes

Integer	Rote	Scale
1		117 px ↓ 20 px
2		233 px ↓ 40 px
3		233 px ↓ 40 px
4		377 px ↓ 65 px
5		233 px ↓ 40 px
6		473 px ↓ 80 px
7		377 px ↓ 65 px
8		377 px ↓ 65 px
9		473 px ↓ 80 px
10		473 px ↓ 80 px
11		233 px ↓ 40 px
12		617 px ↓ 105 px
13		473 px ↓ 80 px
14		617 px ↓ 105 px
15		473 px ↓ 80 px
16		521 px ↓ 90 px
17		377 px ↓ 65 px
18		713 px ↓ 120 px
19		377 px ↓ 65 px
20		617 px ↓ 105 px
21		617 px ↓ 105 px
22		473 px ↓ 80 px
23		473 px ↓ 80 px
24		617 px ↓ 105 px
25		473 px ↓ 80 px
26		713 px ↓ 120 px
27		473 px ↓ 80 px
28		761 px ↓ 130 px
29		473 px ↓ 80 px
30		713 px ↓ 120 px
31		233 px ↓ 40 px
32		377 px ↓ 65 px
33		473 px ↓ 80 px
34		617 px ↓ 105 px
35		617 px ↓ 105 px
36		857 px ↓ 145 px
37		617 px ↓ 105 px
38		617 px ↓ 105 px
39		713 px ↓ 120 px
40		617 px ↓ 105 px
41		473 px ↓ 80 px
42		857 px ↓ 145 px
43		617 px ↓ 105 px
44		617 px ↓ 105 px
45		713 px ↓ 120 px
46		713 px ↓ 120 px
47		473 px ↓ 80 px
48		617 px ↓ 105 px
49		473 px ↓ 80 px
50		713 px ↓ 120 px
51		617 px ↓ 105 px
52		857 px ↓ 145 px
53		521 px ↓ 90 px
54		713 px ↓ 120 px
55		473 px ↓ 80 px
56		761 px ↓ 130 px
57		617 px ↓ 105 px
58		713 px ↓ 120 px
59		377 px ↓ 65 px
60		905 px ↓ 155 px
64		617 px ↓ 105 px
81		617 px ↓ 105 px
105		857 px ↓ 145 px
128		521 px ↓ 90 px
165		713 px ↓ 120 px
256		521 px ↓ 90 px
512		617 px ↓ 105 px
2010		1145 px ↓ 190 px
2500		1001 px ↓ 170 px
65536		665 px ↓ 115 px
802701		1961 px ↓ 330 px
123456789		2048 px ↓ 345 px

Reduction 10:1

Cacti

Image	Scale
	117 px ↓ 12 px
	117 px ↓ 12 px

Riffs

Integer	Riff	Scale
1
2		117 px ↓ 12 px
3		240 px ↓ 24 px
4		233 px ↓ 24 px
5		384 px ↓ 38 px
6		384 px ↓ 38 px
7		377 px ↓ 38 px
8		384 px ↓ 38 px
9		240 px ↓ 24 px
10		528 px ↓ 54 px
11		528 px ↓ 54 px
12		384 px ↓ 38 px
13		384 px ↓ 38 px
14		521 px ↓ 54 px
15		528 px ↓ 54 px
16		377 px ↓ 38 px
17		521 px ↓ 54 px
18		384 px ↓ 38 px
19		528 px ↓ 54 px
20		528 px ↓ 54 px
21		521 px ↓ 54 px
22		672 px ↓ 68 px
23		384 px ↓ 38 px
24		672 px ↓ 68 px
25		384 px ↓ 38 px
26		528 px ↓ 54 px
27		384 px ↓ 38 px
28		521 px ↓ 54 px
29		528 px ↓ 54 px
30		672 px ↓ 68 px
31		672 px ↓ 68 px
32		528 px ↓ 54 px
33		672 px ↓ 68 px
34		665 px ↓ 68 px
35		521 px ↓ 54 px
36		384 px ↓ 38 px
37		384 px ↓ 38 px
38		672 px ↓ 68 px
39		672 px ↓ 68 px
40		816 px ↓ 82 px
41		528 px ↓ 54 px
42		665 px ↓ 68 px
43		521 px ↓ 54 px
44		672 px ↓ 68 px
45		528 px ↓ 54 px
46		528 px ↓ 54 px
47		528 px ↓ 54 px
48		384 px ↓ 38 px
49		377 px ↓ 38 px
50		528 px ↓ 54 px
51		665 px ↓ 68 px
52		528 px ↓ 54 px
53		521 px ↓ 54 px
54		528 px ↓ 54 px
55		672 px ↓ 68 px
56		809 px ↓ 82 px
57		672 px ↓ 68 px
58		672 px ↓ 68 px
59		665 px ↓ 68 px
60		672 px ↓ 68 px
64		384 px ↓ 38 px
81		377 px ↓ 38 px
105		665 px ↓ 68 px
128		521 px ↓ 54 px
165		816 px ↓ 82 px
256		528 px ↓ 54 px
512		384 px ↓ 38 px
2010		1104 px ↓ 110 px
2500		528 px ↓ 54 px
65536		521 px ↓ 54 px
802701		1248 px ↓ 125 px
123456789		1296 px ↓ 130 px

Rotes

Integer	Rote	Scale
1		117 px ↓ 12 px
2		233 px ↓ 24 px
3		233 px ↓ 24 px
4		377 px ↓ 38 px
5		233 px ↓ 24 px
6		473 px ↓ 48 px
7		377 px ↓ 38 px
8		377 px ↓ 38 px
9		473 px ↓ 48 px
10		473 px ↓ 48 px
11		233 px ↓ 24 px
12		617 px ↓ 62 px
13		473 px ↓ 48 px
14		617 px ↓ 62 px
15		473 px ↓ 48 px
16		521 px ↓ 54 px
17		377 px ↓ 38 px
18		713 px ↓ 72 px
19		377 px ↓ 38 px
20		617 px ↓ 62 px
21		617 px ↓ 62 px
22		473 px ↓ 48 px
23		473 px ↓ 48 px
24		617 px ↓ 62 px
25		473 px ↓ 48 px
26		713 px ↓ 72 px
27		473 px ↓ 48 px
28		761 px ↓ 76 px
29		473 px ↓ 48 px
30		713 px ↓ 72 px
31		233 px ↓ 24 px
32		377 px ↓ 38 px
33		473 px ↓ 48 px
34		617 px ↓ 62 px
35		617 px ↓ 62 px
36		857 px ↓ 86 px
37		617 px ↓ 62 px
38		617 px ↓ 62 px
39		713 px ↓ 72 px
40		617 px ↓ 62 px
41		473 px ↓ 48 px
42		857 px ↓ 86 px
43		617 px ↓ 62 px
44		617 px ↓ 62 px
45		713 px ↓ 72 px
46		713 px ↓ 72 px
47		473 px ↓ 48 px
48		617 px ↓ 62 px
49		473 px ↓ 48 px
50		713 px ↓ 72 px
51		617 px ↓ 62 px
52		857 px ↓ 86 px
53		521 px ↓ 54 px
54		713 px ↓ 72 px
55		473 px ↓ 48 px
56		761 px ↓ 76 px
57		617 px ↓ 62 px
58		713 px ↓ 72 px
59		377 px ↓ 38 px
60		905 px ↓ 90 px
64		617 px ↓ 62 px
81		617 px ↓ 62 px
105		857 px ↓ 86 px
128		521 px ↓ 54 px
165		713 px ↓ 72 px
256		521 px ↓ 54 px
512		617 px ↓ 62 px
2010		1145 px ↓ 115 px
2500		1001 px ↓ 100 px
65536		665 px ↓ 68 px
802701		1961 px ↓ 196 px
123456789		2048 px ↓ 205 px

Riffs in Numerical Order

\(\text{Riffs in Numerical Order}\!\)
\(1\!\) \(\begin{array}{l} \varnothing \\ 1 \end{array}\)	\(\text{p}\!\) \(\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\)	\(\text{p}_\text{p}\!\) \(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\)	\(\text{p}^\text{p}\!\) \(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\)	\(\text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\)
\(\text{p} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}\)	\(\text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\)	\(\text{p}^{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\)	\(\text{p}_\text{p}^\text{p}\!\) \(\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\)	\(\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\)
\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\)	\(\text{p}^\text{p} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\)	\(\text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\)	\(\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}\)	\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\)
\(\text{p}^{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\)	\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\)	\(\text{p} \text{p}_\text{p}^\text{p}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\)	\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\)	\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\)
\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\)	\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\)	\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\)	\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\)	\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\)
\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\)	\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\)	\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\)	\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\)	\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\)
\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\) \(\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\)	\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\)	\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\)	\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\)	\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\)
\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\)	\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\)	\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}\)	\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\)	\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\)
\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\) \(\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\)	\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\)	\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 14\!:\!1 \\ 43 \end{array}\)	\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\)	\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}\)
\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\)	\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\)	\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\)	\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\) \(\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\)	\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\)
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\)	\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\)	\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\)	\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\)	\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\)
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\)	\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\)	\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\)	\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\) \(\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\)	\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\)

Rotes in Numerical Order

\(1\!\)

\(\begin{array}{l} \varnothing \\ 1 \end{array}\)

\(\text{p}\!\)

\(\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\)

\(\text{p}_\text{p}\!\)

\(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\)

\(\text{p}^\text{p}\!\)

\(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\)

\(\text{p}_{\text{p}_\text{p}}\!\)

\(\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\)

\(\text{p} \text{p}_\text{p}\!\)

\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}\)

\(\text{p}_{\text{p}^\text{p}}\!\)

\(\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\)

\(\text{p}^{\text{p}_\text{p}}\!\)

\(\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\)

\(\text{p}_\text{p}^\text{p}\!\)

\(\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\)

\(\text{p} \text{p}_{\text{p}_\text{p}}\!\)

\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\)

\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)

\(\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\)

\(\text{p}^\text{p} \text{p}_\text{p}\!\)

\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\)

\(\text{p}_{\text{p} \text{p}_\text{p}}\!\)

\(\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\)

\(\text{p} \text{p}_{\text{p}^\text{p}}\!\)

\(\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}\)

\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)

\(\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\)

\(\text{p}^{\text{p}^\text{p}}\!\)

\(\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\)

\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)

\(\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\)

\(\text{p} \text{p}_\text{p}^\text{p}\!\)

\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\)

\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)

\(\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\)

\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\)

\(\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\)

\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\)

\(\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\)

\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)

\(\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\)

\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\)

\(\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\)

\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\)

\(\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\)

\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\)

\(\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\)

\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)

\(\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\)

\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\)

\(\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\)

\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\)

\(\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\)

\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\)

\(\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\)

\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)

\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\)

\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\)

\(\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\)

\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\)

\(\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\)

\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)

\(\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\)

\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)

\(\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\)

\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\)

\(\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\)

\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\)

\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\)

\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\)

\(\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\)

\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)

\(\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}\)

\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)

\(\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\)

\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\)

\(\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\)

\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\)

\(\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\)

\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\)

\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\)

\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\)

\(\begin{array}{l} 14\!:\!1 \\ 43 \end{array}\)

\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)

\(\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\)

\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\)

\(\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}\)

\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\)

\(\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\)

\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\)

\(\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\)

\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\)

\(\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\)

\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\)

\(\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\)

\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\)

\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\)

\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)

\(\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\)

\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)

\(\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\)

\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\)

\(\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\)

\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\)

\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\)

\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)

\(\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\)

\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\)

\(\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\)

\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)

\(\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\)

\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\)

\(\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\)

\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\)

\(\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\)

\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)

\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\)

Miscellaneous Examples (Reduction 8:1)

\(\text{Integers, Riffs, Rotes}\!\)

\(\text{Integer}\!\)

\(\text{Riff}\!\)

\(\text{Rote}\!\)

\(1\!\)
\(2\!\)
\(3\!\)
\(4\!\)
\(2010\!\)
\(2011\!\)
\(2500\!\)
\(802701\!\)
\(123456789\!\)

Cactus Graphs

Hi Res

	117 px → 20 px
	117 px → 20 px
	117 px → 20 px
	117 px → 20 px
	290 px → 50 px
	386 px → 65 px
	204 px → 35 px
	348 px → 60 px
	386 px → 65 px
	386 px → 65 px
	386 px → 65 px
	386 px → 65 px
	386 px → 65 px
	386 px → 65 px
	386 px → 65 px
	386 px → 65 px
	530 px → 90 px
	530 px → 90 px

Lo Res

Differential Logic

ASCII Graphics

Series 1

o-------------------------------------------------o
|                                                 |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /                 o                 \      |
|     /                 /%\                 \     |
|    /                 /%%%\                 \    |
|   o                 o%%%%%o                 o   |
|   |                 |%%%%%|                 |   |
|   |        P        |%%%%%|        Q        |   |
|   |                 |%%%%%|                 |   |
|   o                 o%%%%%o                 o   |
|    \                 \%%%/                 /    |
|     \                 \%/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
|  f =                  p q                       |
o-------------------------------------------------o
Figure 22-a.  Conjunction pq : X -> B

o-------------------------------------------------o
|                                                 |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /        P        o        Q        \      |
|     /                 /%\                 \     |
|    /                 /%%%\                 \    |
|   o                 o.->-.o                 o   |
|   |    p(q)(dp)dq   |%\%/%|  (p)q dp(dq)    |   |
|   | o---------------|->o<-|---------------o |   |
|   |                 |%%^%%|                 |   |
|   o                 o%%|%%o                 o   |
|    \                 \%|%/                 /    |
|     \                 \|/                 /     |
|      \                 o                 /      |
|       \               /|\               /       |
|        o-------------o | o-------------o        |
|                        |                        |
|                        |                        |
|                        |                        |
|                        o                        |
|                  (p)(q) dp dq                   |
|                                                 |
o-------------------------------------------------o
|  f =                  p q                       |
o-------------------------------------------------o
|                                                 |
| Ef =              p  q   (dp)(dq)               |
|                                                 |
|           +       p (q)  (dp) dq                |
|                                                 |
|           +      (p) q    dp (dq)               |
|                                                 |
|           +      (p)(q)   dp  dq                |
|                                                 |
o-------------------------------------------------o
Figure 22-b.  Enlargement E[pq] : EX -> B

o-------------------------------------------------o
|                                                 |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /        P        o        Q        \      |
|     /                 /%\                 \     |
|    /                 /%%%\                 \    |
|   o                 o%%%%%o                 o   |
|   |       (dp)dq    |%%%%%|    dp(dq)       |   |
|   | o<--------------|->o<-|-------------->o |   |
|   |                 |%%^%%|                 |   |
|   o                 o%%|%%o                 o   |
|    \                 \%|%/                 /    |
|     \                 \|/                 /     |
|      \                 o                 /      |
|       \               /|\               /       |
|        o-------------o | o-------------o        |
|                        |                        |
|                        |                        |
|                        v                        |
|                        o                        |
|                      dp dq                      |
|                                                 |
o-------------------------------------------------o
|  f =                  p q                       |
o-------------------------------------------------o
|                                                 |
| Df =              p  q  ((dp)(dq))              |
|                                                 |
|           +       p (q)  (dp) dq                |
|                                                 |
|           +      (p) q    dp (dq)               |
|                                                 |
|           +      (p)(q)   dp  dq                |
|                                                 |
o-------------------------------------------------o
Figure 22-c.  Difference D[pq] : EX -> B

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o                                    |
|           /                     \                                   |
|          /                       \                                  |
|         /                         \                                 |
|        /                           \                                |
|       /                             \                               |
|      /                               \                              |
|     /                                 \                             |
|    o                                   o                            |
|    |                                   |                            |
|    |                                   |                            |
|    |                                   |                            |
|    |                 G                 |                            |
|    |                                   |                            |
|    |                                   |                            |
|    |                                   |                            |
|    o                                   o                            |
|     \                                 /                             |
|      \                               /                              |
|       \                           T /                               |
|        \             o<------------/-------------o                  |
|         \                         /                                 |
|          \                       /                                  |
|           \                     /                                   |
|            o-------------------o                                    |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 23.  Elements of a Cybernetic System

Series 2

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       /%\                       \         |
|        /                       /%%%\                       \        |
|       /                       /%%%%%\                       \       |
|      /                       /%%%%%%%\                       \      |
|     /                       /%%%%%%%%%\                       \     |
|    o                       o%%%%%%%%%%%o                       o    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |          P            |%%%%%%%%%%%|            Q          |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    o                       o%%%%%%%%%%%o                       o    |
|     \                       \%%%%%%%%%/                       /     |
|      \                       \%%%%%%%/                       /      |
|       \                       \%%%%%/                       /       |
|        \                       \%%%/                       /        |
|         \                       \%/                       /         |
|          \                       o                       /          |
|           \                     / \                     /           |
|            o-------------------o   o-------------------o            |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 24-1.  Proposition pq : X -> B

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (dp) (dq) o                       o    |
|    |                       |  o-->--o  |                       |    |
|    |                       |   \   /   |                       |    |
|    |             (dp) dq   |    \ /    |   dp (dq)             |    |
|    |          o<-----------------o----------------->o          |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               dp | dq                               |
|                                  |                                  |
|                                  v                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 24-2.  Tacit Extension !e![pq] : EX -> B

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (dp) (dq) o                       o    |
|    |                       |  o-->--o  |                       |    |
|    |                       |   \   /   |                       |    |
|    |             (dp) dq   |    \ /    |   dp (dq)             |    |
|    |          o----------------->o<-----------------o          |    |
|    |                       |     ^     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               dp | dq                               |
|                                  |                                  |
|                                  |                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 25-1.  Enlargement E[pq] : EX -> B

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o           o                       o    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |             (dp) dq   |           |   dp (dq)             |    |
|    |          o<---------------->o<---------------->o          |    |
|    |                       |     ^     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               dp | dq                               |
|                                  |                                  |
|                                  v                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 25-2.  Difference Map D[pq] : EX -> B

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /   o   \                       \      |
|     /                       /   ^ ^   \                       \     |
|    o                       o   /   \   o                       o    |
|    |                       |  /     \  |                       |    |
|    |                       | /       \ |                       |    |
|    |                       |/         \|                       |    |
|    |                   (dp)/ dq     dp \(dq)                   |    |
|    |                      /|           |\                      |    |
|    |                     / |           | \                     |    |
|    |                    /  |           |  \                    |    |
|    o                   /   o           o   \                   o    |
|     \                 v     \  dp dq  /     v                 /     |
|      \               o<--------------------->o               /      |
|       \                       \     /                       /       |
|        \                       \   /                       /        |
|         \                       \ /                       /         |
|          \                       o                       /          |
|           \                     / \                     /           |
|            o-------------------o   o-------------------o            |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 26-1.  Differential or Tangent d[pq] : EX -> B

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o           o                       o    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |                       |   dp dq   |                       |    |
|    |            o<------------------------------->o            |    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |                       |     o     |                       |    |
|    o                       o     ^     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                    dp | dq                    /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                                  |                                  |
|                                  |                                  |
|                                  v                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 26-2.  Remainder r[pq] : EX -> B

JPEG Graphics

Series 1

\(\text{Figure 22-a. Conjunction}~ pq : X \to \mathbb{B}\)

\(\text{Figure 22-b. Enlargement}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{E}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\)

\(\text{Figure 22-c. Difference}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{D}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \end{array}\)

Series 2

\(\text{Figure 24-1. Proposition}~ pq : X \to \mathbb{B}\)

\(\text{Figure 24-2. Tacit Extension}~ \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \varepsilon (pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\)

\(\text{Figure 25-1. Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{E}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\)

\(\text{Figure 25-2. Difference Map}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{D}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \end{array}\)

\(\text{Figure 26-1. Tangent Map}~ \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{d}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{,} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & 0 \end{array}\)

\(\text{Figure 26-2. Remainder Map}~ \operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{r}(pq) & = & p & \cdot & q & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}p ~ \operatorname{d}q \end{array}\)

Propositional Equation Reasoning Systems

Analysis of contingent propositions

(26)

	(27)
\(\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}\)

	(28)
\(\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}\)

(29)

Equation 1 : Proof 1

(30)

Equation 1 : Proof 2

Single Image Version

(31)

Serial Image Version

(31)

(33)

Equation 1 : Proof 3

Variant 1

start
cast p
dom
can
empty
can
cast q
dom
can
dom
spike
can
cast r
can
empty
spike
can

Variant 2

start
cast p
dom
can
empty
can
cast q
can
dom
can
spike
can
cast r
can
empty
spike
can

Variant 1

o-----------------------------------------------------------o
| Equation E_1.  Proof 3.                                   |
o-----------------------------------------------------------o
|                                                        1  |
|               q o   o r   q o r                           |
|                 |   |       |                             |
|               p o   o p   p o                             |
|                  \ /        |                             |
|                   o---------o                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        o                                  |
|                        |                                  |
|                        |                                  |
|                        |                                  |
|                        |                                  |
|                        @                                  |
|                                                           |
o==================================< CAST "p" >=============o
|                                                        2  |
|          q   r    q r    q   r    qr                      |
|          o   o     o     o o o o   o o                    |
|          |   |     |     |/  |/    |/                     |
|          o   o     o     o   o     o                      |
|           \ /      |      \ /      |                      |
|            o-------o       o-------o                      |
|             \     /         \     /                       |
|              \   /           \   /                        |
|               \ /             \ /                         |
|                o               o                          |
|                |               |                          |
|                |               |                          |
|                |               |                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Domination >===========o
|                                                        3  |
|          q   r    q r                                     |
|          o   o     o       o   o     o                    |
|          |   |     |      /   /     /                     |
|          o   o     o     o   o     o                      |
|           \ /      |      \ /      |                      |
|            o-------o       o-------o                      |
|             \     /         \     /                       |
|              \   /           \   /                        |
|               \ /             \ /                         |
|                o               o                          |
|                |               |                          |
|                |               |                          |
|                |               |                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                        4  |
|          q   r    q r                                     |
|          o   o     o                                      |
|          |   |     |                                      |
|          o   o     o                                      |
|           \ /      |                                      |
|            o-------o       o-------o                      |
|             \     /         \     /                       |
|              \   /           \   /                        |
|               \ /             \ /                         |
|                o               o                          |
|                |               |                          |
|                |               |                          |
|                |               |                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Emptiness >============o
|                                                        5  |
|          q   r    q r                                     |
|          o   o     o                                      |
|          |   |     |                                      |
|          o   o     o                                      |
|           \ /      |                                      |
|            o-------o           o                          |
|             \     /            |                          |
|              \   /             |                          |
|               \ /              |                          |
|                o               o                          |
|                |               |                          |
|                |               |                          |
|                |               |                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                        6  |
|          q   r    q r                                     |
|          o   o     o                                      |
|          |   |     |                                      |
|          o   o     o                                      |
|           \ /      |                                      |
|            o-------o                                      |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|                o                                          |
|                |                                          |
|                |                                          |
|                |                                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< CAST "q" >=============o
|                                                        7  |
|                      o         o                          |
|          r     r     |   r     |                          |
|      o   o     o     o   o     o r                        |
|      |   |     |     |   |     |                          |
|      o   o     o     o   o     o                          |
|       \ /      |      \ /      |                          |
|        o-------o       o-------o                          |
|         \     /         \     /                           |
|          \   /           \   /                            |
|           \ /             \ /                             |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Domination >===========o
|                                                        8  |
|                      o         o                          |
|          r     r     |   r     |                          |
|      o   o     o     o   o     o                          |
|      |   |     |     |   |     |                          |
|      o   o     o     o   o     o                          |
|       \ /      |      \ /      |                          |
|        o-------o       o-------o                          |
|         \     /         \     /                           |
|          \   /           \   /                            |
|           \ /             \ /                             |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                        9  |
|          r     r         r                                |
|          o     o         o                                |
|          |     |         |                                |
|          o     o     o   o     o                          |
|         /      |      \ /      |                          |
|        o-------o       o-------o                          |
|         \     /         \     /                           |
|          \   /           \   /                            |
|           \ /             \ /                             |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Domination >===========o
|                                                       10  |
|          r     r                                          |
|          o     o                                          |
|          |     |                                          |
|          o     o     o         o                          |
|         /      |      \        |                          |
|        o-------o       o-------o                          |
|         \     /         \     /                           |
|          \   /           \   /                            |
|           \ /             \ /                             |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Spike >================o
|                                                       11  |
|          r     r                                          |
|          o     o                                          |
|          |     |                                          |
|          o     o                                          |
|         /      |                                          |
|        o-------o           o                              |
|         \     /            |                              |
|          \   /             |                              |
|           \ /              |                              |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                       12  |
|          r     r                                          |
|          o     o                                          |
|          |     |                                          |
|          o     o                                          |
|         /      |                                          |
|        o-------o                                          |
|         \     /                                           |
|          \   /                                            |
|           \ /                                             |
|            o                                              |
|            |                                              |
|            |                                              |
|            |                                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< CAST "r" >=============o
|                                                       13  |
|                      o     o                              |
|                      |     |                              |
|      o     o         o     o                              |
|      |     |         |     |                              |
|      o     o         o     o                              |
|     /      |        /      |                              |
|    o-------o       o-------o                              |
|     \     /         \     /                               |
|      \   /           \   /                                |
|       \ /             \ /                                 |
|        o               o                                  |
|        |               |                                  |
|        |               |                                  |
|        |               |                                  |
|      r o---------------o---o r                            |
|         \             /                                   |
|          \           /                                    |
|           \         /                                     |
|            \       /                                      |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|              q o-------o---o q                            |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                       14  |
|                      o     o                              |
|                     /      |                              |
|    o-------o       o-------o                              |
|     \     /         \     /                               |
|      \   /           \   /                                |
|       \ /             \ /                                 |
|        o               o                                  |
|        |               |                                  |
|        |               |                                  |
|        |               |                                  |
|      r o---------------o---o r                            |
|         \             /                                   |
|          \           /                                    |
|           \         /                                     |
|            \       /                                      |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|              q o-------o---o q                            |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Emptiness & Spike >====o
|                                                       15  |
|        o               o                              16  |
|        |               |                                  |
|        |               |                                  |
|        |               |                                  |
|        o               o                                  |
|        |               |                                  |
|        |               |                                  |
|        |               |                                  |
|      r o---------------o---o r                            |
|         \             /                                   |
|          \           /                                    |
|           \         /                                     |
|            \       /                                      |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|              q o-------o---o q                            |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                       17  |
|          r o-------o---o r                                |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|              q o-------o---o q                            |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< QED >==================o

(40)

Variant 2

(40)

Praeclarum Theorema : Proof by CAST

(23)

@@ Line 1: / Line 1: @@
+==Animations==
+===Riffs 1 to 60===
+{| align="center"
+| [[Image:Animation Riff 60 x 0.16.gif]]
+|}
+===Rotes 1 to 60===
+{| align="center"
+| [[Image:Animation Rote 60 x 0.16.gif]]
+|}
 ==Image Gallery==
-{| align="center" border="1" cellpadding="10" style="text-align:center; width:90%"
+===Reduction 6:1===
+====Cacti====
+{| align="center" border="1" cellpadding="20" style="text-align:center; width:90%"
 |-
 | width="50%" | '''Image'''
 | width="50%" | '''Scale'''
 |-
-| height="100px" | [[Image:Rooted Node Big.jpg|20px]]
+| [[Image:Rooted Node Big.jpg|20px]] || 117 px <br> &darr; <br> 20 px
-| 117 px &rarr; 20 px
+|-
+| [[Image:Rooted Edge Big.jpg|20px]] || 117 px <br> &darr; <br> 20 px
+|}
+====Riffs====
+{| align="center" border="1" cellpadding="20" style="text-align:center; width:90%"
+|-
+| width="15%" | '''Integer'''
+| width="70%" | '''Riff'''
+| width="15%" | '''Scale'''
+|-
+| 1 || &nbsp; || &nbsp;
+|-
+| 2 || [[Image:Riff 2 Big.jpg|20px]] || 117 px <br> &darr; <br> 20 px
+|-
+| 3 || [[Image:Riff 3 Big.jpg|40px]] || 240 px <br> &darr; <br> 40 px
+|-
+| 4 || [[Image:Riff 4 Big.jpg|40px]] || 233 px <br> &darr; <br> 40 px
+|-
+| 5 || [[Image:Riff 5 Big.jpg|65px]] || 384 px <br> &darr; <br> 65 px
+|-
+| 6 || [[Image:Riff 6 Big.jpg|65px]] || 384 px <br> &darr; <br> 65 px
+|-
+| 7 || [[Image:Riff 7 Big.jpg|65px]] || 377 px <br> &darr; <br> 65 px
+|-
+| 8 || [[Image:Riff 8 Big.jpg|65px]] || 384 px <br> &darr; <br> 65 px
+|-
+| 9 || [[Image:Riff 9 Big.jpg|40px]] || 240 px <br> &darr; <br> 40 px
+|-
+| 10 || [[Image:Riff 10 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 11 || [[Image:Riff 11 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 12 || [[Image:Riff 12 Big.jpg|65px]] || 384 px <br> &darr; <br> 65 px
+|-
+| 13 || [[Image:Riff 13 Big.jpg|65px]] || 384 px <br> &darr; <br> 65 px
+|-
+| 14 || [[Image:Riff 14 Big.jpg|90px]] || 521 px <br> &darr; <br> 90 px
+|-
+| 15 || [[Image:Riff 15 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 16 || [[Image:Riff 16 Big.jpg|65px]] || 377 px <br> &darr; <br> 65 px
+|-
+| 17 || [[Image:Riff 17 Big.jpg|90px]] || 521 px <br> &darr; <br> 90 px
+|-
+| 18 || [[Image:Riff 18 Big.jpg|65px]] || 384 px <br> &darr; <br> 65 px
+|-
+| 19 || [[Image:Riff 19 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 20 || [[Image:Riff 20 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 21 || [[Image:Riff 21 Big.jpg|90px]] || 521 px <br> &darr; <br> 90 px
+|-
+| 22 || [[Image:Riff 22 Big.jpg|115px]] || 672 px <br> &darr; <br> 115 px
+|-
+| 23 || [[Image:Riff 23 Big.jpg|65px]] || 384 px <br> &darr; <br> 65 px
+|-
+| 24 || [[Image:Riff 24 Big.jpg|115px]] || 672 px <br> &darr; <br> 115 px
+|-
+| 25 || [[Image:Riff 25 Big.jpg|65px]] || 384 px <br> &darr; <br> 65 px
+|-
+| 26 || [[Image:Riff 26 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 27 || [[Image:Riff 27 Big.jpg|65px]] || 384 px <br> &darr; <br> 65 px
+|-
+| 28 || [[Image:Riff 28 Big.jpg|90px]] || 521 px <br> &darr; <br> 90 px
+|-
+| 29 || [[Image:Riff 29 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 30 || [[Image:Riff 30 Big.jpg|115px]] || 672 px <br> &darr; <br> 115 px
+|-
+| 31 || [[Image:Riff 31 Big.jpg|115px]] || 672 px <br> &darr; <br> 115 px
+|-
+| 32 || [[Image:Riff 32 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 33 || [[Image:Riff 33 Big.jpg|115px]] || 672 px <br> &darr; <br> 115 px
+|-
+| 34 || [[Image:Riff 34 Big.jpg|115px]] || 665 px <br> &darr; <br> 115 px
+|-
+| 35 || [[Image:Riff 35 Big.jpg|90px]] || 521 px <br> &darr; <br> 90 px
+|-
+| 36 || [[Image:Riff 36 Big.jpg|65px]] || 384 px <br> &darr; <br> 65 px
+|-
+| 37 || [[Image:Riff 37 Big.jpg|65px]] || 384 px <br> &darr; <br> 65 px
+|-
+| 38 || [[Image:Riff 38 Big.jpg|115px]] || 672 px <br> &darr; <br> 115 px
+|-
+| 39 || [[Image:Riff 39 Big.jpg|115px]] || 672 px <br> &darr; <br> 115 px
+|-
+| 40 || [[Image:Riff 40 Big.jpg|135px]] || 816 px <br> &darr; <br> 135 px
+|-
+| 41 || [[Image:Riff 41 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 42 || [[Image:Riff 42 Big.jpg|115px]] || 665 px <br> &darr; <br> 115 px
+|-
+| 43 || [[Image:Riff 43 Big.jpg|90px]] || 521 px <br> &darr; <br> 90 px
+|-
+| 44 || [[Image:Riff 44 Big.jpg|115px]] || 672 px <br> &darr; <br> 115 px
+|-
+| 45 || [[Image:Riff 45 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 46 || [[Image:Riff 46 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 47 || [[Image:Riff 47 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 48 || [[Image:Riff 48 Big.jpg|65px]] || 384 px <br> &darr; <br> 65 px
+|-
+| 49 || [[Image:Riff 49 Big.jpg|65px]] || 377 px <br> &darr; <br> 65 px
+|-
+| 50 || [[Image:Riff 50 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 51 || [[Image:Riff 51 Big.jpg|115px]] || 665 px <br> &darr; <br> 115 px
+|-
+| 52 || [[Image:Riff 52 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 53 || [[Image:Riff 53 Big.jpg|90px]] || 521 px <br> &darr; <br> 90 px
+|-
+| 54 || [[Image:Riff 54 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 55 || [[Image:Riff 55 Big.jpg|115px]] || 672 px <br> &darr; <br> 115 px
+|-
+| 56 || [[Image:Riff 56 Big.jpg|135px]] || 809 px <br> &darr; <br> 135 px
+|-
+| 57 || [[Image:Riff 57 Big.jpg|115px]] || 672 px <br> &darr; <br> 115 px
+|-
+| 58 || [[Image:Riff 58 Big.jpg|115px]] || 672 px <br> &darr; <br> 115 px
+|-
+| 59 || [[Image:Riff 59 Big.jpg|115px]] || 665 px <br> &darr; <br> 115 px
+|-
+| 60 || [[Image:Riff 60 Big.jpg|115px]] || 672 px <br> &darr; <br> 115 px
+|-
+| 64 || [[Image:Riff 64 Big.jpg|65px]] || 384 px <br> &darr; <br> 65 px
+|-
+| 81 || [[Image:Riff 81 Big.jpg|65px]] || 377 px <br> &darr; <br> 65 px
+|-
+| 105 || [[Image:Riff 105 Big.jpg|115px]] || 665 px <br> &darr; <br> 115 px
+|-
+| 128 || [[Image:Riff 128 Big.jpg|90px]] || 521 px <br> &darr; <br> 90 px
+|-
+| 165 || [[Image:Riff 165 Big.jpg|135px]] || 816 px <br> &darr; <br> 135 px
+|-
+| 256 || [[Image:Riff 256 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 512 || [[Image:Riff 512 Big.jpg|65px]] || 384 px <br> &darr; <br> 65 px
+|-
+| 2010 || [[Image:Riff 2010 Big.jpg|185px]] || 1104 px <br> &darr; <br> 185 px
+|-
+| 2500 || [[Image:Riff 2500 Big.jpg|90px]] || 528 px <br> &darr; <br> 90 px
+|-
+| 65536 || [[Image:Riff 65536 Big.jpg|90px]] || 521 px <br> &darr; <br> 90 px
+|-
+| 802701 || [[Image:Riff 802701 Big.jpg|210px]] || 1248 px <br> &darr; <br> 210 px
+|-
+| 123456789 || [[Image:Riff 123456789 Big.jpg|220px]] || 1296 px <br> &darr; <br> 220 px
+|}
+====Rotes====
+{| align="center" border="1" cellpadding="20" style="text-align:center; width:90%"
+|-
+| width="15%" | '''Integer'''
+| width="70%" | '''Rote'''
+| width="15%" | '''Scale'''
+|-
+| 1 || [[Image:Rote 1 Big.jpg|20px]] || 117 px <br> &darr; <br> 20 px
+|-
+| 2 || [[Image:Rote 2 Big.jpg|40px]] || 233 px <br> &darr; <br> 40 px
+|-
+| 3 || [[Image:Rote 3 Big.jpg|40px]] || 233 px <br> &darr; <br> 40 px
+|-
+| 4 || [[Image:Rote 4 Big.jpg|65px]] || 377 px <br> &darr; <br> 65 px
+|-
+| 5 || [[Image:Rote 5 Big.jpg|40px]] || 233 px <br> &darr; <br> 40 px
+|-
+| 6 || [[Image:Rote 6 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 7 || [[Image:Rote 7 Big.jpg|65px]] || 377 px <br> &darr; <br> 65 px
+|-
+| 8 || [[Image:Rote 8 Big.jpg|65px]] || 377 px <br> &darr; <br> 65 px
+|-
+| 9 || [[Image:Rote 9 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 10 || [[Image:Rote 10 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 11 || [[Image:Rote 11 Big.jpg|40px]] || 233 px <br> &darr; <br> 40 px
+|-
+| 12 || [[Image:Rote 12 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 13 || [[Image:Rote 13 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 14 || [[Image:Rote 14 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 15 || [[Image:Rote 15 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 16 || [[Image:Rote 16 Big.jpg|90px]] || 521 px <br> &darr; <br> 90 px
+|-
+| 17 || [[Image:Rote 17 Big.jpg|65px]] || 377 px <br> &darr; <br> 65 px
+|-
+| 18 || [[Image:Rote 18 Big.jpg|120px]] || 713 px <br> &darr; <br> 120 px
+|-
+| 19 || [[Image:Rote 19 Big.jpg|65px]] || 377 px <br> &darr; <br> 65 px
+|-
+| 20 || [[Image:Rote 20 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 21 || [[Image:Rote 21 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 22 || [[Image:Rote 22 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 23 || [[Image:Rote 23 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 24 || [[Image:Rote 24 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 25 || [[Image:Rote 25 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 26 || [[Image:Rote 26 Big.jpg|120px]] || 713 px <br> &darr; <br> 120 px
+|-
+| 27 || [[Image:Rote 27 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 28 || [[Image:Rote 28 Big.jpg|130px]] || 761 px <br> &darr; <br> 130 px
+|-
+| 29 || [[Image:Rote 29 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 30 || [[Image:Rote 30 Big.jpg|120px]] || 713 px <br> &darr; <br> 120 px
+|-
+| 31 || [[Image:Rote 31 Big.jpg|40px]] || 233 px <br> &darr; <br> 40 px
+|-
+| 32 || [[Image:Rote 32 Big.jpg|65px]] || 377 px <br> &darr; <br> 65 px
+|-
+| 33 || [[Image:Rote 33 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 34 || [[Image:Rote 34 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 35 || [[Image:Rote 35 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 36 || [[Image:Rote 36 Big.jpg|145px]] || 857 px <br> &darr; <br> 145 px
+|-
+| 37 || [[Image:Rote 37 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 38 || [[Image:Rote 38 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 39 || [[Image:Rote 39 Big.jpg|120px]] || 713 px <br> &darr; <br> 120 px
+|-
+| 40 || [[Image:Rote 40 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 41 || [[Image:Rote 41 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 42 || [[Image:Rote 42 Big.jpg|145px]] || 857 px <br> &darr; <br> 145 px
+|-
+| 43 || [[Image:Rote 43 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 44 || [[Image:Rote 44 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 45 || [[Image:Rote 45 Big.jpg|120px]] || 713 px <br> &darr; <br> 120 px
+|-
+| 46 || [[Image:Rote 46 Big.jpg|120px]] || 713 px <br> &darr; <br> 120 px
+|-
+| 47 || [[Image:Rote 47 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 48 || [[Image:Rote 48 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 49 || [[Image:Rote 49 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 50 || [[Image:Rote 50 Big.jpg|120px]] || 713 px <br> &darr; <br> 120 px
+|-
+| 51 || [[Image:Rote 51 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 52 || [[Image:Rote 52 Big.jpg|145px]] || 857 px <br> &darr; <br> 145 px
+|-
+| 53 || [[Image:Rote 53 Big.jpg|90px]] || 521 px <br> &darr; <br> 90 px
+|-
+| 54 || [[Image:Rote 54 Big.jpg|120px]] || 713 px <br> &darr; <br> 120 px
+|-
+| 55 || [[Image:Rote 55 Big.jpg|80px]] || 473 px <br> &darr; <br> 80 px
+|-
+| 56 || [[Image:Rote 56 Big.jpg|130px]] || 761 px <br> &darr; <br> 130 px
+|-
+| 57 || [[Image:Rote 57 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 58 || [[Image:Rote 58 Big.jpg|120px]] || 713 px <br> &darr; <br> 120 px
+|-
+| 59 || [[Image:Rote 59 Big.jpg|65px]] || 377 px <br> &darr; <br> 65 px
+|-
+| 60 || [[Image:Rote 60 Big.jpg|155px]] || 905 px <br> &darr; <br> 155 px
+|-
+| 64 || [[Image:Rote 64 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 81 || [[Image:Rote 81 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 105 || [[Image:Rote 105 Big.jpg|145px]] || 857 px <br> &darr; <br> 145 px
+|-
+| 128 || [[Image:Rote 128 Big.jpg|90px]] || 521 px <br> &darr; <br> 90 px
+|-
+| 165 || [[Image:Rote 165 Big.jpg|120px]] || 713 px <br> &darr; <br> 120 px
+|-
+| 256 || [[Image:Rote 256 Big.jpg|90px]] || 521 px <br> &darr; <br> 90 px
+|-
+| 512 || [[Image:Rote 512 Big.jpg|105px]] || 617 px <br> &darr; <br> 105 px
+|-
+| 2010 || [[Image:Rote 2010 Big.jpg|190px]] || 1145 px <br> &darr; <br> 190 px
+|-
+| 2500 || [[Image:Rote 2500 Big.jpg|170px]] || 1001 px <br> &darr; <br> 170 px
+|-
+| 65536 || [[Image:Rote 65536 Big.jpg|115px]] || 665 px <br> &darr; <br> 115 px
+|-
+| 802701 || [[Image:Rote 802701 Big.jpg|330px]] || 1961 px <br> &darr; <br> 330 px
+|-
+| 123456789 || [[Image:Rote 123456789 Big.jpg|345px]] || 2048 px <br> &darr; <br> 345 px
+|}
+===Reduction 10:1===
+====Cacti====
+{| align="center" border="1" cellpadding="20" style="text-align:center; width:90%"
+|-
+| width="50%" | '''Image'''
+| width="50%" | '''Scale'''
+|-
+| [[Image:Rooted Node Big.jpg|12px]] || 117 px <br> &darr; <br> 12 px
+|-
+| [[Image:Rooted Edge Big.jpg|12px]] || 117 px <br> &darr; <br> 12 px
+|}
+====Riffs====
+{| align="center" border="1" cellpadding="20" style="text-align:center; width:90%"
+|-
+| width="15%" | '''Integer'''
+| width="70%" | '''Riff'''
+| width="15%" | '''Scale'''
+|-
+| 1 || &nbsp; || &nbsp;
+|-
+| 2 || [[Image:Riff 2 Big.jpg|12px]] || 117 px <br> &darr; <br> 12 px
+|-
+| 3 || [[Image:Riff 3 Big.jpg|24px]] || 240 px <br> &darr; <br> 24 px
+|-
+| 4 || [[Image:Riff 4 Big.jpg|24px]] || 233 px <br> &darr; <br> 24 px
+|-
+| 5 || [[Image:Riff 5 Big.jpg|38px]] || 384 px <br> &darr; <br> 38 px
+|-
+| 6 || [[Image:Riff 6 Big.jpg|38px]] || 384 px <br> &darr; <br> 38 px
+|-
+| 7 || [[Image:Riff 7 Big.jpg|38px]] || 377 px <br> &darr; <br> 38 px
+|-
+| 8 || [[Image:Riff 8 Big.jpg|38px]] || 384 px <br> &darr; <br> 38 px
+|-
+| 9 || [[Image:Riff 9 Big.jpg|24px]] || 240 px <br> &darr; <br> 24 px
+|-
+| 10 || [[Image:Riff 10 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 11 || [[Image:Riff 11 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 12 || [[Image:Riff 12 Big.jpg|38px]] || 384 px <br> &darr; <br> 38 px
+|-
+| 13 || [[Image:Riff 13 Big.jpg|38px]] || 384 px <br> &darr; <br> 38 px
+|-
+| 14 || [[Image:Riff 14 Big.jpg|54px]] || 521 px <br> &darr; <br> 54 px
+|-
+| 15 || [[Image:Riff 15 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 16 || [[Image:Riff 16 Big.jpg|38px]] || 377 px <br> &darr; <br> 38 px
+|-
+| 17 || [[Image:Riff 17 Big.jpg|54px]] || 521 px <br> &darr; <br> 54 px
+|-
+| 18 || [[Image:Riff 18 Big.jpg|38px]] || 384 px <br> &darr; <br> 38 px
+|-
+| 19 || [[Image:Riff 19 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 20 || [[Image:Riff 20 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 21 || [[Image:Riff 21 Big.jpg|54px]] || 521 px <br> &darr; <br> 54 px
+|-
+| 22 || [[Image:Riff 22 Big.jpg|68px]] || 672 px <br> &darr; <br> 68 px
+|-
+| 23 || [[Image:Riff 23 Big.jpg|38px]] || 384 px <br> &darr; <br> 38 px
+|-
+| 24 || [[Image:Riff 24 Big.jpg|68px]] || 672 px <br> &darr; <br> 68 px
+|-
+| 25 || [[Image:Riff 25 Big.jpg|38px]] || 384 px <br> &darr; <br> 38 px
+|-
+| 26 || [[Image:Riff 26 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 27 || [[Image:Riff 27 Big.jpg|38px]] || 384 px <br> &darr; <br> 38 px
+|-
+| 28 || [[Image:Riff 28 Big.jpg|54px]] || 521 px <br> &darr; <br> 54 px
+|-
+| 29 || [[Image:Riff 29 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 30 || [[Image:Riff 30 Big.jpg|68px]] || 672 px <br> &darr; <br> 68 px
+|-
+| 31 || [[Image:Riff 31 Big.jpg|68px]] || 672 px <br> &darr; <br> 68 px
+|-
+| 32 || [[Image:Riff 32 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 33 || [[Image:Riff 33 Big.jpg|68px]] || 672 px <br> &darr; <br> 68 px
+|-
+| 34 || [[Image:Riff 34 Big.jpg|68px]] || 665 px <br> &darr; <br> 68 px
+|-
+| 35 || [[Image:Riff 35 Big.jpg|54px]] || 521 px <br> &darr; <br> 54 px
+|-
+| 36 || [[Image:Riff 36 Big.jpg|38px]] || 384 px <br> &darr; <br> 38 px
+|-
+| 37 || [[Image:Riff 37 Big.jpg|38px]] || 384 px <br> &darr; <br> 38 px
+|-
+| 38 || [[Image:Riff 38 Big.jpg|68px]] || 672 px <br> &darr; <br> 68 px
+|-
+| 39 || [[Image:Riff 39 Big.jpg|68px]] || 672 px <br> &darr; <br> 68 px
+|-
+| 40 || [[Image:Riff 40 Big.jpg|82px]] || 816 px <br> &darr; <br> 82 px
+|-
+| 41 || [[Image:Riff 41 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 42 || [[Image:Riff 42 Big.jpg|68px]] || 665 px <br> &darr; <br> 68 px
+|-
+| 43 || [[Image:Riff 43 Big.jpg|54px]] || 521 px <br> &darr; <br> 54 px
+|-
+| 44 || [[Image:Riff 44 Big.jpg|68px]] || 672 px <br> &darr; <br> 68 px
+|-
+| 45 || [[Image:Riff 45 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 46 || [[Image:Riff 46 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 47 || [[Image:Riff 47 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 48 || [[Image:Riff 48 Big.jpg|38px]] || 384 px <br> &darr; <br> 38 px
+|-
+| 49 || [[Image:Riff 49 Big.jpg|38px]] || 377 px <br> &darr; <br> 38 px
+|-
+| 50 || [[Image:Riff 50 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 51 || [[Image:Riff 51 Big.jpg|68px]] || 665 px <br> &darr; <br> 68 px
+|-
+| 52 || [[Image:Riff 52 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 53 || [[Image:Riff 53 Big.jpg|54px]] || 521 px <br> &darr; <br> 54 px
+|-
+| 54 || [[Image:Riff 54 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 55 || [[Image:Riff 55 Big.jpg|68px]] || 672 px <br> &darr; <br> 68 px
+|-
+| 56 || [[Image:Riff 56 Big.jpg|82px]] || 809 px <br> &darr; <br> 82 px
+|-
+| 57 || [[Image:Riff 57 Big.jpg|68px]] || 672 px <br> &darr; <br> 68 px
+|-
+| 58 || [[Image:Riff 58 Big.jpg|68px]] || 672 px <br> &darr; <br> 68 px
+|-
+| 59 || [[Image:Riff 59 Big.jpg|68px]] || 665 px <br> &darr; <br> 68 px
+|-
+| 60 || [[Image:Riff 60 Big.jpg|68px]] || 672 px <br> &darr; <br> 68 px
+|-
+| 64 || [[Image:Riff 64 Big.jpg|38px]] || 384 px <br> &darr; <br> 38 px
+|-
+| 81 || [[Image:Riff 81 Big.jpg|38px]] || 377 px <br> &darr; <br> 38 px
+|-
+| 105 || [[Image:Riff 105 Big.jpg|68px]] || 665 px <br> &darr; <br> 68 px
+|-
+| 128 || [[Image:Riff 128 Big.jpg|54px]] || 521 px <br> &darr; <br> 54 px
+|-
+| 165 || [[Image:Riff 165 Big.jpg|82px]] || 816 px <br> &darr; <br> 82 px
+|-
+| 256 || [[Image:Riff 256 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 512 || [[Image:Riff 512 Big.jpg|38px]] || 384 px <br> &darr; <br> 38 px
+|-
+| 2010 || [[Image:Riff 2010 Big.jpg|110px]] || 1104 px <br> &darr; <br> 110 px
+|-
+| 2500 || [[Image:Riff 2500 Big.jpg|54px]] || 528 px <br> &darr; <br> 54 px
+|-
+| 65536 || [[Image:Riff 65536 Big.jpg|54px]] || 521 px <br> &darr; <br> 54 px
+|-
+| 802701 || [[Image:Riff 802701 Big.jpg|125px]] || 1248 px <br> &darr; <br> 125 px
+|-
+| 123456789 || [[Image:Riff 123456789 Big.jpg|130px]] || 1296 px <br> &darr; <br> 130 px
+|}
+====Rotes====
+{| align="center" border="1" cellpadding="20" style="text-align:center; width:90%"
+|-
+| width="15%" | '''Integer'''
+| width="70%" | '''Rote'''
+| width="15%" | '''Scale'''
+|-
+| 1 || [[Image:Rote 1 Big.jpg|12px]] || 117 px <br> &darr; <br> 12 px
+|-
+| 2 || [[Image:Rote 2 Big.jpg|24px]] || 233 px <br> &darr; <br> 24 px
+|-
+| 3 || [[Image:Rote 3 Big.jpg|24px]] || 233 px <br> &darr; <br> 24 px
+|-
+| 4 || [[Image:Rote 4 Big.jpg|38px]] || 377 px <br> &darr; <br> 38 px
+|-
+| 5 || [[Image:Rote 5 Big.jpg|24px]] || 233 px <br> &darr; <br> 24 px
+|-
+| 6 || [[Image:Rote 6 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
+|-
+| 7 || [[Image:Rote 7 Big.jpg|38px]] || 377 px <br> &darr; <br> 38 px
+|-
+| 8 || [[Image:Rote 8 Big.jpg|38px]] || 377 px <br> &darr; <br> 38 px
+|-
+| 9 || [[Image:Rote 9 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
+|-
+| 10 || [[Image:Rote 10 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
+|-
+| 11 || [[Image:Rote 11 Big.jpg|24px]] || 233 px <br> &darr; <br> 24 px
+|-
+| 12 || [[Image:Rote 12 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 13 || [[Image:Rote 13 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
+|-
+| 14 || [[Image:Rote 14 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 15 || [[Image:Rote 15 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
+|-
+| 16 || [[Image:Rote 16 Big.jpg|54px]] || 521 px <br> &darr; <br> 54 px
+|-
+| 17 || [[Image:Rote 17 Big.jpg|38px]] || 377 px <br> &darr; <br> 38 px
+|-
+| 18 || [[Image:Rote 18 Big.jpg|72px]] || 713 px <br> &darr; <br> 72 px
+|-
+| 19 || [[Image:Rote 19 Big.jpg|38px]] || 377 px <br> &darr; <br> 38 px
+|-
+| 20 || [[Image:Rote 20 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 21 || [[Image:Rote 21 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 22 || [[Image:Rote 22 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
+|-
+| 23 || [[Image:Rote 23 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
+|-
+| 24 || [[Image:Rote 24 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 25 || [[Image:Rote 25 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
+|-
+| 26 || [[Image:Rote 26 Big.jpg|72px]] || 713 px <br> &darr; <br> 72 px
+|-
+| 27 || [[Image:Rote 27 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
+|-
+| 28 || [[Image:Rote 28 Big.jpg|76px]] || 761 px <br> &darr; <br> 76 px
+|-
+| 29 || [[Image:Rote 29 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
+|-
+| 30 || [[Image:Rote 30 Big.jpg|72px]] || 713 px <br> &darr; <br> 72 px
+|-
+| 31 || [[Image:Rote 31 Big.jpg|24px]] || 233 px <br> &darr; <br> 24 px
+|-
+| 32 || [[Image:Rote 32 Big.jpg|38px]] || 377 px <br> &darr; <br> 38 px
+|-
+| 33 || [[Image:Rote 33 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
+|-
+| 34 || [[Image:Rote 34 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 35 || [[Image:Rote 35 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 36 || [[Image:Rote 36 Big.jpg|86px]] || 857 px <br> &darr; <br> 86 px
+|-
+| 37 || [[Image:Rote 37 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 38 || [[Image:Rote 38 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 39 || [[Image:Rote 39 Big.jpg|72px]] || 713 px <br> &darr; <br> 72 px
+|-
+| 40 || [[Image:Rote 40 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 41 || [[Image:Rote 41 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
+|-
+| 42 || [[Image:Rote 42 Big.jpg|86px]] || 857 px <br> &darr; <br> 86 px
+|-
+| 43 || [[Image:Rote 43 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 44 || [[Image:Rote 44 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 45 || [[Image:Rote 45 Big.jpg|72px]] || 713 px <br> &darr; <br> 72 px
 |-
-| height="100px" | [[Image:Rooted Edge Big.jpg|20px]]
+| 46 || [[Image:Rote 46 Big.jpg|72px]] || 713 px <br> &darr; <br> 72 px
-| 117 px &rarr; 20 px
 |-
-| height="100px" | [[Image:Rote 2 Big.jpg|40px]]
+| 47 || [[Image:Rote 47 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
-| 233 px &rarr; 40 px
 |-
-| height="150px" | [[Image:Rote 3 Big.jpg|40px]]
+| 48 || [[Image:Rote 48 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
-| 233 px &rarr; 40 px
 |-
-| height="150px" | [[Image:Rote 4 Big.jpg|65px]]
+| 49 || [[Image:Rote 49 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
-| 377 px &rarr; 65 px
 |-
-| height="200px" | [[Image:Rote 5 Big.jpg|40px]]
+| 50 || [[Image:Rote 50 Big.jpg|72px]] || 713 px <br> &darr; <br> 72 px
-| 233 px &rarr; 40 px
 |-
-| height="150px" | [[Image:Rote 6 Big.jpg|80px]]
+| 51 || [[Image:Rote 51 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
-| 473 px &rarr; 80 px
 |-
-| height="200px" | [[Image:Rote 7 Big.jpg|65px]]
+| 52 || [[Image:Rote 52 Big.jpg|86px]] || 857 px <br> &darr; <br> 86 px
-| 377 px &rarr; 65 px
 |-
-| height="200px" | [[Image:Rote 8 Big.jpg|65px]]
+| 53 || [[Image:Rote 53 Big.jpg|54px]] || 521 px <br> &darr; <br> 54 px
-| 377 px &rarr; 65 px
 |-
-| height="150px" | [[Image:Rote 9 Big.jpg|80px]]
+| 54 || [[Image:Rote 54 Big.jpg|72px]] || 713 px <br> &darr; <br> 72 px
-| 473 px &rarr; 80 px
 |-
-| height="200px" | [[Image:Rote 10 Big.jpg|80px]]
+| 55 || [[Image:Rote 55 Big.jpg|48px]] || 473 px <br> &darr; <br> 48 px
-| 473 px &rarr; 80 px
 |-
-| height="250px" | [[Image:Rote 11 Big.jpg|40px]]
+| 56 || [[Image:Rote 56 Big.jpg|76px]] || 761 px <br> &darr; <br> 76 px
-| 233 px &rarr; 40 px
+|-
+| 57 || [[Image:Rote 57 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 58 || [[Image:Rote 58 Big.jpg|72px]] || 713 px <br> &darr; <br> 72 px
+|-
+| 59 || [[Image:Rote 59 Big.jpg|38px]] || 377 px <br> &darr; <br> 38 px
+|-
+| 60 || [[Image:Rote 60 Big.jpg|90px]] || 905 px <br> &darr; <br> 90 px
+|-
+| 64 || [[Image:Rote 64 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 81 || [[Image:Rote 81 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 105 || [[Image:Rote 105 Big.jpg|86px]] || 857 px <br> &darr; <br> 86 px
+|-
+| 128 || [[Image:Rote 128 Big.jpg|54px]] || 521 px <br> &darr; <br> 54 px
+|-
+| 165 || [[Image:Rote 165 Big.jpg|72px]] || 713 px <br> &darr; <br> 72 px
+|-
+| 256 || [[Image:Rote 256 Big.jpg|54px]] || 521 px <br> &darr; <br> 54 px
+|-
+| 512 || [[Image:Rote 512 Big.jpg|62px]] || 617 px <br> &darr; <br> 62 px
+|-
+| 2010 || [[Image:Rote 2010 Big.jpg|115px]] || 1145 px <br> &darr; <br> 115 px
+|-
+| 2500 || [[Image:Rote 2500 Big.jpg|100px]] || 1001 px <br> &darr; <br> 100 px
+|-
+| 65536 || [[Image:Rote 65536 Big.jpg|68px]] || 665 px <br> &darr; <br> 68 px
+|-
+| 802701 || [[Image:Rote 802701 Big.jpg|196px]] || 1961 px <br> &darr; <br> 196 px
+|-
+| 123456789 || [[Image:Rote 123456789 Big.jpg|205px]] || 2048 px <br> &darr; <br> 205 px
 |}
-==Riffs & Rotes==
+==Riffs in Numerical Order==
 {| align="center" border="1" cellpadding="10"
+|+ style="height:25px" | <math>\text{Riffs in Numerical Order}\!</math>
 | valign="bottom" |
-<p>[[Image:Rooted Node Big.jpg|20px]]</p><br>
+<p>&nbsp;</p><br>
+<p><math>1\!</math></p><br>
+<p><math>\begin{array}{l} \varnothing \\ 1 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 2 Big.jpg|20px]]</p><br>
+<p><math>\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 \\ 2 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 3 Big.jpg|40px]]</p><br>
+<p><math>\text{p}_\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 4 Big.jpg|40px]]</p><br>
+<p><math>\text{p}^\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 5 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 3\!:\!1 \\ 5 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 6 Big.jpg|65px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 7 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 4\!:\!1 \\ 7 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 8 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!3 \\ 8 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 9 Big.jpg|40px]]</p><br>
+<p><math>\text{p}_\text{p}^\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!2 \\ 9 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 10 Big.jpg|90px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 11 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 5\!:\!1 \\ 11 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 12 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 13 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 6\!:\!1 \\ 13 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 14 Big.jpg|90px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 15 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 16 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!4 \\ 16 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 17 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 7\!:\!1 \\ 17 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 18 Big.jpg|65px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 19 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 8\!:\!1 \\ 19 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 20 Big.jpg|90px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 21 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 22 Big.jpg|115px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 23 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 9\!:\!1 \\ 23 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 24 Big.jpg|115px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 25 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 3\!:\!2 \\ 25 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 26 Big.jpg|90px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 27 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!3 \\ 27 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 28 Big.jpg|90px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 29 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 10\!:\!1 \\ 29 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 30 Big.jpg|115px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 31 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!</math></p><br>
+<p><math>\begin{array}{l} 11\!:\!1 \\ 31 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 32 Big.jpg|90px]]</p><br>
+<p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!5 \\ 32 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 33 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 34 Big.jpg|115px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 35 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 36 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 37 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 12\!:\!1 \\ 37 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 38 Big.jpg|115px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 39 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 40 Big.jpg|135px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 41 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 13\!:\!1 \\ 41 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 42 Big.jpg|115px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 43 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 14\!:\!1 \\ 43 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 44 Big.jpg|115px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 45 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 46 Big.jpg|90px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 47 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 15\!:\!1 \\ 47 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 48 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 49 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 4\!:\!2 \\ 49 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 50 Big.jpg|90px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 51 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 52 Big.jpg|90px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 53 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 16\!:\!1 \\ 53 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 54 Big.jpg|90px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 55 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 56 Big.jpg|135px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 57 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 58 Big.jpg|115px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 59 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!</math></p><br>
+<p><math>\begin{array}{l} 17\!:\!1 \\ 59 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 60 Big.jpg|115px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}</math></p>
+|}
+==Rotes in Numerical Order==
+{| align="center" border="1" cellpadding="6"
+| valign="bottom" |
+<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
+<p><math>1\!</math></p><br>
 <p><math>\begin{array}{l} \varnothing \\ 1 \end{array}</math></p>
 | valign="bottom" |
 <p>[[Image:Rote 2 Big.jpg|40px]]</p><br>
+<p><math>\text{p}\!</math></p><br>
 <p><math>\begin{array}{l} 1\!:\!1 \\ 2 \end{array}</math></p>
 | valign="bottom" |
 <p>[[Image:Rote 3 Big.jpg|40px]]</p><br>
+<p><math>\text{p}_\text{p}\!</math></p><br>
 <p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p>
 | valign="bottom" |
 <p>[[Image:Rote 4 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^\text{p}\!</math></p><br>
 <p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p>
 | valign="bottom" |
 <p>[[Image:Rote 5 Big.jpg|40px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}}\!</math></p><br>
 <p><math>\begin{array}{l} 3\!:\!1 \\ 5 \end{array}</math></p>
 |-
 | valign="bottom" |
 <p>[[Image:Rote 6 Big.jpg|80px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p}\!</math></p><br>
 <p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}</math></p>
 | valign="bottom" |
 <p>[[Image:Rote 7 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}^\text{p}}\!</math></p><br>
 <p><math>\begin{array}{l} 4\!:\!1 \\ 7 \end{array}</math></p>
 | valign="bottom" |
 <p>[[Image:Rote 8 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}}\!</math></p><br>
 <p><math>\begin{array}{l} 1\!:\!3 \\ 8 \end{array}</math></p>
 | valign="bottom" |
 <p>[[Image:Rote 9 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_\text{p}^\text{p}\!</math></p><br>
 <p><math>\begin{array}{l} 2\!:\!2 \\ 9 \end{array}</math></p>
 | valign="bottom" |
 <p>[[Image:Rote 10 Big.jpg|80px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
 <p><math>\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 11 Big.jpg|40px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 5\!:\!1 \\ 11 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 12 Big.jpg|105px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 13 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 6\!:\!1 \\ 13 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 14 Big.jpg|105px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 15 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 16 Big.jpg|90px]]</p><br>
+<p><math>\text{p}^{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!4 \\ 16 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 17 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 7\!:\!1 \\ 17 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 18 Big.jpg|120px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 19 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 8\!:\!1 \\ 19 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 20 Big.jpg|105px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 21 Big.jpg|105px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 22 Big.jpg|80px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 23 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 9\!:\!1 \\ 23 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 24 Big.jpg|105px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 25 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 3\!:\!2 \\ 25 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 26 Big.jpg|120px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 27 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!3 \\ 27 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 28 Big.jpg|130px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 29 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 10\!:\!1 \\ 29 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 30 Big.jpg|120px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 31 Big.jpg|40px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!</math></p><br>
+<p><math>\begin{array}{l} 11\!:\!1 \\ 31 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 32 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!5 \\ 32 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 33 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 34 Big.jpg|105px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 35 Big.jpg|105px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 36 Big.jpg|145px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 37 Big.jpg|105px]]</p><br>
+<p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 12\!:\!1 \\ 37 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 38 Big.jpg|105px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 39 Big.jpg|120px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 40 Big.jpg|105px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 41 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 13\!:\!1 \\ 41 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 42 Big.jpg|145px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 43 Big.jpg|105px]]</p><br>
+<p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 14\!:\!1 \\ 43 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 44 Big.jpg|105px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 45 Big.jpg|120px]]</p><br>
+<p><math>\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 46 Big.jpg|120px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 47 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 15\!:\!1 \\ 47 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 48 Big.jpg|105px]]</p><br>
+<p><math>\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 49 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 4\!:\!2 \\ 49 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 50 Big.jpg|120px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 51 Big.jpg|105px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 52 Big.jpg|145px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 53 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 16\!:\!1 \\ 53 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 54 Big.jpg|120px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 55 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 56 Big.jpg|130px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 57 Big.jpg|105px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 58 Big.jpg|120px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 59 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!</math></p><br>
+<p><math>\begin{array}{l} 17\!:\!1 \\ 59 \end{array}</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 60 Big.jpg|155px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}</math></p>
 |}
-<br>
+==Miscellaneous Examples (Reduction 8:1)==
-{| align="center" border="1" cellpadding="10" style="text-align:center; width:90%"
+{| align="center" border="1" width="96%"
+|+ style="height:24px" | <math>\text{Integers, Riffs, Rotes}\!</math>
+|- style="height:50px; background:#f0f0ff"
+|
+{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
+| width="10%" | <math>\text{Integer}\!</math>
+| width="45%" | <math>\text{Riff}\!</math>
+| width="45%" | <math>\text{Rote}\!</math>
+|}
 |-
-| width="50%" | '''Image'''
+|
-| width="50%" | '''Scale'''
+{| cellpadding="12" style="text-align:center; width:100%"
+| width="10%" | <math>1\!</math>
+| width="45%" | &nbsp;
+| width="45%" | [[Image:Rote 1 Big.jpg|15px]]
 |-
-| height="100px" | [[Image:Rooted Node Big.jpg|20px]]
+| <math>2\!</math>
-| 117 px &rarr; 20 px
+| [[Image:Riff 2 Big.jpg|15px]]
+| [[Image:Rote 2 Big.jpg|30px]]
 |-
-| height="100px" | [[Image:Rote 2 Big.jpg|40px]]
+| <math>3\!</math>
-| 233 px &rarr; 40 px
+| [[Image:Riff 3 Big.jpg|30px]]
+| [[Image:Rote 3 Big.jpg|30px]]
 |-
-| height="150px" | [[Image:Rote 3 Big.jpg|40px]]
+| <math>4\!</math>
-| 233 px &rarr; 40 px
+| [[Image:Riff 4 Big.jpg|30px]]
+| [[Image:Rote 4 Big.jpg|48px]]
 |-
-| height="150px" | [[Image:Rote 4 Big.jpg|65px]]
+| <math>2010\!</math>
-| 377 px &rarr; 65 px
+| [[Image:Riff 2010 Big.jpg|138px]]
+| [[Image:Rote 2010 Big.jpg|144px]]
 |-
-| height="200px" | [[Image:Rote 5 Big.jpg|40px]]
+| <math>2011\!</math>
-| 233 px &rarr; 40 px
+| [[Image:Riff 2011 Big.jpg|84px]]
+| [[Image:Rote 2011 Big.jpg|120px]]
 |-
-| height="150px" | [[Image:Rote 6 Big.jpg|80px]]
+| <math>2500\!</math>
-| 473 px &rarr; 80 px
+| [[Image:Riff 2500 Big.jpg|66px]]
+| [[Image:Rote 2500 Big.jpg|125px]]
 |-
-| height="200px" | [[Image:Rote 7 Big.jpg|65px]]
+| <math>802701\!</math>
-| 377 px &rarr; 65 px
+| [[Image:Riff 802701 Big.jpg|156px]]
+| [[Image:Rote 802701 Big.jpg|245px]]
 |-
-| height="200px" | [[Image:Rote 8 Big.jpg|65px]]
+| <math>123456789\!</math>
-| 377 px &rarr; 65 px
+| [[Image:Riff 123456789 Big.jpg|162px]]
-|-
+| [[Image:Rote 123456789 Big.jpg|256px]]
-| height="150px" | [[Image:Rote 9 Big.jpg|80px]]
+|}
-| 473 px &rarr; 80 px
-|-
-| height="200px" | [[Image:Rote 10 Big.jpg|80px]]
-| 473 px &rarr; 80 px
 |}