Euclidean

Euclidean rhythm categories

Euclidean rhythms are listed here by the necklace length and characterized as Even, Deep, Irregular or by the number of consecutive beats as Single, Double, Triple, Quad, Quintuple and rest as Dense.

Categories Even, Deep and Irregular (Aksak) are described in The Distance Geometry of Music (Etymologies are also mostly derived from the same source). Even rhythms have all offbeats evenly spaced without any consecutive beats. Some rhythms can also be categorized as deep (Erdos or Winograd), using fancy mathematical proofs.

For a rhythm to be Winograd-deep, all the distances must appear a unique number of times. In order to be an Erdos-deep, it is only required that any distance that appears must have a unique multiplicity. In Erdos-deep rhythms have points equally spaced around the necklace and all Winograd-deep rhytms are also Erdos deep but also always have exactly one consecutive beat in the rhythm necklace. Irregular rhythms (also known as Aksak have a combination of unequal beats and have sort of ethnic feel.

Rest of the euclidean rhythms fall into the categories characterized by number of consecutive beats. Single is used for uneven rhythms that dont qualify as irreqular, but also doesnt have any consecutive beats. Double is used for rhythms that have pair of onbeats in a row. Mixed Double is used for mixed pairs and single beats. Triple is used for three consecutive beats and so on ... Consecutive beats larger than 5 (Quintuple) are just marked as Dense

2

Even: 1 2

Euclid	Binary	Category	Type	Etymylogy
<1,2>	10	Deep	Erdos
<2,2>	11	Dense

4

Even: 1 3 Odd: 2

Euclid	Binary	Category	Type	Etymylogy
<1,3>	100	Even - Deep	Erdos
<2,3>	101	Deep	Winograd	West Africa, Latin America, Nubia, Northern Canada
<3,3>	111	Dense

4

Even: 1 2 4 Odd: 3

Euclid	Binary	Category	Type	Etymylogy
<1,4>	1000	Deep	Erdos	Classic
<2,4>	1010	Even - Deep	Erdos
<3,4>	1011	Triple		Brazil, Bali, Colombia, Greece, Spain, Persia, Trinidad
<4,4>	1111	Dense

5

Even: 1 5 Odd: 2 3 4

Euclid	Binary	Category	Type	Etymylogy
<1,5>	10000	Deep	Erdos
<2,5>	10010	Irregular	Authentic Aksak	Classical music, Jazz, Greece, Macedonia, Namibia, Persia, Rwanda
<3,5>	10110	Deep	Winograd	Korean, Rumanian, Persia
<4,5>	10111	Dense		Greece
<5,5>	11111	Dense

6

Even: 1 2 3 6 Odd: 4 5

Euclid	Binary	Category	Type	Etymylogy
<1,6>	100000	Deep	Erdos
<2,6>	100100	Deep	Erdos
<3,6>	101010	Even - Deep	Erdos
<4,6>	101101	Double
<5,6>	101111	Dense		Arab
<6,6>	111111	Dense

7

Even: 1 7 Odd: 2 3 4 5 6

Euclid	Binary	Category	Type	Etymylogy
<1,7>	1000000	Deep	Erdos
<2,7>	1000100	Deep	Erdos
<3,7>	1001010	Irregular	Authentic Aksak	Bulgaria, Greece, Sudan, Turkestan
<4,7>	1011010	Deep	Winograd	Bulgaria
<5,7>	1011101	Mixed Triple		Arab
<6,7>	1011111	Dense		Greek
<7,7>	1111111	Dense

8

Even: 1 2 4 8 Odd: 3 5 6 7

Euclid	Binary	Category	Type	Etymylogy
<1,8>	10000000	Deep	Erdos
<2,8>	10001000	Even - Deep	Erdos
<3,8>	10010100	Irregular	Pseudo Aksak	Balinese, Central Africa, Greece, India, Latin America, West Africa, Sudan, Cuba
<4,8>	10101010	Even
<5,8>	10110101	Mixed Double		Egypt, Korea, Latin America, West Africa, Cuba
<6,8>	10111011	Triple
<7,8>	10111111	Dense		Libya, Tuareng
<8,8>	11111111	Dense

9

Even: 1 3 9 Odd: 2 4 5 6 7 8

Euclid	Binary	Category	Type	Etymylogy
<1,9>	100000000	Deep	Erdos
<2,9>	100001000	Deep	Erdos
<3,9>	100100100	Even - Deep	Erdos
<4,9>	100101010	Deep - Irregular	Erdos - Quasi Aksak	Greece, Macedonia, Turkey, Zaïre
<5,9>	101101010	Deep	Winograd	Arab
<6,9>	101101101	Double
<7,9>	101111011	Dense		Greece
<8,9>	101111111	Dense
<9,9>	111111111	Dense

10

Even: 1 2 5 10 Odd: 3 4 6 7 8 9

Euclid	Binary	Category	Type	Etymylogy
<1,10>	1000000000	Deep	Erdos
<2,10>	1000010000	Even - Deep	Erdos
<3,10>	1000100100	Deep	Erdos	India
<4,10>	1001010010	Irregular	Pseudo Aksak
<5,10>	1010101010	Even
<6,10>	1011010110	Mixed Double
<7,10>	1011101101	Mixed Triple		Turkey
<8,10>	1011110111	Dense
<9,10>	1011111111	Dense
<10,10>	1111111111	Dense

11

Even: 1 11 Odd: 2 3 4 5 6 7 8 9 10

Euclid	Binary	Category	Type	Etymylogy
<1,11>	10000000000	Deep	Erdos
<2,11>	10000010000	Deep	Erdos
<3,11>	10001001000	Deep	Erdos	North India
<4,11>	10010100100	Irregular	Authentic Aksak	Southern India, Serbia
<5,11>	10010101010	Irregular	Authentic Aksak	Classical music, Bulgaria, Northern India, Serbia
<6,11>	10110101010	Deep	Winograd
<7,11>	10110101101	Mixed Double
<8,11>	10111011011	Mixed Triple
<9,11>	10111110111	Dense
<10,11>	10111111111	Dense
<11,11>	11111111111	Dense

12

Even: 1 2 3 4 6 12 Odd: 5 7 8 9 10 11

Euclid	Binary	Category	Type	Etymylogy
<1,12>	100000000000	Deep	Erdos
<2,12>	100000100000	Even - Deep	Erdos	Africa
<3,12>	100010001000	Even - Deep	Erdos	Africa
<4,12>	100100100100	Even		African
<5,12>	100101010010	Irregular	Pseudo Aksak	Spain, Macedonia, South Africa
<6,12>	101010101010	Even		Africa
<7,12>	101101010110	Mixed Double		West Africa, Ghana, Cuba
<8,12>	101101101101	Double		Guinea
<9,12>	101110111011	Triple
<10,12>	101111101111	Dense
<11,12>	101111111111	Dense		Oman
<12,12>	111111111111	Dense

13

Even: 1 13 Odd: 2 3 4 5 6 7 8 9 10 11 12

Euclid	Binary	Category	Type	Etymylogy
<1,13>	1000000000000	Deep	Erdos
<2,13>	1000000100000	Deep	Erdos
<3,13>	1000010001000	Deep	Erdos
<4,13>	1000100100100	Deep	Erdos
<5,13>	1001010010010	Irregular	Authentic Aksak	Macedonia
<6,13>	1001010101010	Irregular	Authentic Aksak	Macedonia
<7,13>	1011010101010	Deep	Winograd
<8,13>	1011010110110	Mixed Double
<9,13>	1011101101101	Mixed Triple
<10,13>	1011110111011	Triple
<11,13>	1011111101111	Dense
<12,13>	1011111111111	Dense
<13,13>	1111111111111	Dense

14

Even: 1 2 7 14 Odd: 3 4 5 6 8 9 10 11 12 13

Euclid	Binary	Category	Type	Etymylogy
<1,14>	10000000000000	Deep	Erdos
<2,14>	10000001000000	Even - Deep	Erdos
<3,14>	10000100010000	Deep	Erdos	North India
<4,14>	10001001000100	Single
<5,14>	10010100100100	Irregular	Pseudo Aksak
<6,14>	10010101001010	Irregular	Pseudo Aksak
<7,14>	10101010101010	Even
<8,14>	10110101011010	Mixed Double
<9,14>	10110101101101	Mixed Double		Algeria
<10,14>	10111011011101	Triple
<11,14>	10111101110111	Mixed Quad
<12,14>	10111111011111	Dense
<13,14>	10111111111111	Dense
<14,14>	11111111111111	Dense

15

Even: 1 3 5 15 Odd: 2 4 6 7 8 9 10 11 12 13 14

Euclid	Binary	Category	Type	Etymylogy
<1,15>	100000000000000	Deep	Erdos
<2,15>	100000001000000	Deep	Erdos
<3,15>	100001000010000	Even - Deep	Erdos	India
<4,15>	100010010001000	Deep	Erdos	North India
<5,15>	100100100100100	Even
<6,15>	100101001010010	Irregular	Quasi Aksak
<7,15>	100101010101010	Deep - Irregular	Erdos - Quasi Aksak	Bulgaria
<8,15>	101101010101010	Deep	Winograd
<9,15>	101101011010110	Mixed Double
<10,15>	101101101101101	Double
<11,15>	101110110111011	Triple
<12,15>	101111011110111	Quad
<13,15>	101111111011111	Dense
<14,15>	101111111111111	Dense
<15,15>	111111111111111	Dense

16

Even: 1 2 4 8 16 Odd: 3 5 6 7 9 10 11 12 13 14 15

Euclid	Binary	Category	Type	Etymylogy
<1,16>	1000000000000000	Deep	Erdos
<2,16>	1000000010000000	Even - Deep	Erdos
<3,16>	1000001000010000	Deep	Erdos
<4,16>	1000100010001000	Even
<5,16>	1000100100100100	Deep	Erdos	Brazilian, West Africa, Rap, Rumba, Son, Gahu
<6,16>	1001010010010100	Irregular	Pseudo Aksak
<7,16>	1001010101001010	Irregular	Pseudo Aksak	Brazilian, Macedonian, West Africa
<8,16>	1010101010101010	Even
<9,16>	1011010101011010	Mixed Double		West and Central African, and Brazilia
<10,16>	1011010110110101	Mixed Double
<11,16>	1011101101101101	Mixed Triple
<12,16>	1011101110111011	Triple
<13,16>	1011111011110111	Dense
<14,16>	1011111110111111	Dense
<15,16>	1011111111111111	Dense
<16,16>	1111111111111111	Dense

17

Even: 1 17 Odd: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Euclid	Binary	Category	Type	Etymylogy
<1,17>	10000000000000000	Deep	Erdos
<2,17>	10000000010000000	Deep	Erdos
<3,17>	10000010000100000	Deep	Erdos
<4,17>	10000100010001000	Deep	Erdos
<5,17>	10001001001000100	Deep	Erdos
<6,17>	10010100100100100	Irregular	Authentic Aksak
<7,17>	10010101001010010	Irregular	Authentic Aksak	Macedonia
<8,17>	10010101010101010	Irregular	Authentic Aksak	Bulgaria
<9,17>	10110101010101010	Deep	Winograd
<10,17>	10110101011010110	Mixed Double
<11,17>	10110101101101101	Mixed Double
<12,17>	10111011011011101	Mixed Triple
<13,17>	10111101110111011	Mixed Quad
<14,17>	10111110111101111	Dense
<15,17>	10111111110111111	Dense
<16,17>	10111111111111111	Dense
<17,17>	11111111111111111	Dense

18

Even: 1 2 3 6 9 18 Odd: 4 5 7 8 10 11 12 13 14 15 16 17

Euclid	Binary	Category	Type	Etymylogy
<1,18>	100000000000000000	Deep	Erdos
<2,18>	100000000100000000	Even - Deep	Erdos
<3,18>	100000100000100000	Even - Deep	Erdos
<4,18>	100001000100001000	Single
<5,18>	100010010001000100	Deep	Erdos
<6,18>	100100100100100100	Even
<7,18>	100101001001010010	Irregular	Pseudo Aksak	Bulgaria
<8,18>	100101010100101010	Irregular	Pseudo Aksak
<9,18>	101010101010101010	Even
<10,18>	101101010101101010	Mixed Double
<11,18>	101101011011010110	Mixed Double
<12,18>	101101101101101101	Double
<13,18>	101110110111011101	Mixed Triple
<14,18>	101111011101111011	Mixed Quad
<15,18>	101111101111101111	Quintuple
<16,18>	101111111101111111	Dense
<17,18>	101111111111111111	Dense
<18,18>	111111111111111111	Dense

19

Even: 1 19 Odd: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Euclid	Binary	Category	Type	Etymylogy
<1,19>	1000000000000000000	Deep	Erdos
<2,19>	1000000000100000000	Deep	Erdos
<3,19>	1000000100000100000	Deep	Erdos
<4,19>	1000010001000010000	Deep	Erdos
<5,19>	1000100100010001000	Deep	Erdos
<6,19>	1000100100100100100	Deep	Erdos
<7,19>	1001010010010010100	Irregular	Authentic Aksak
<8,19>	1001010100101001010	Irregular	Authentic Aksak	Bulgaria
<9,19>	1001010101010101010	Irregular	Authentic Aksak
<10,19>	1011010101010101010	Deep	Winograd
<11,19>	1011010101101011010	Mixed Double
<12,19>	1011010110110110101	Mixed Double
<13,19>	1011101101101101101	Double
<14,19>	1011101101110111011	Mixed Triple
<15,19>	1011110111011110111	Mixed Quad
<16,19>	1011111101111101111	Dense
<17,19>	1011111111101111111	Dense
<18,19>	1011111111111111111	Dense
<19,19>	1111111111111111111	Dense

20

Even: 1 2 4 5 10 20 Odd: 3 6 7 8 9 11 12 13 14 15 16 17 18 19

Euclid	Binary	Category	Type	Etymylogy
<1,20>	10000000000000000000	Deep	Erdos
<2,20>	10000000001000000000	Deep	Erdos
<3,20>	10000001000001000000	Deep	Erdos	India
<4,20>	10000100001000010000	Even
<5,20>	10001000100010001000	Even
<6,20>	10001001001000100100	Even
<7,20>	10010100100100100100	Irregular	Pseudo Aksak
<8,20>	10010100101001010010	Irregular	Pseudo Aksak
<9,20>	10010101010100101010	Irregular	Pseudo Aksak
<10,20>	10101010101010101010	Even
<11,20>	10110101010101101010	Mixed Double		West and Central Africa
<12,20>	10110101101011010110	Mixed Double
<13,20>	10110101101101101101	Mixed Double
<14,20>	10111011011011101101	Mixed Triple
<15,20>	10111011101110111011	Triple
<16,20>	10111101111011110111	Quad
<17,20>	10111111011111011111	Dense
<18,20>	10111111111011111111	Dense
<19,20>	10111111111111111111	Dense
<20,20>	11111111111111111111	Dense

21

Even: 1 3 7 21 Odd: 2 4 5 6 8 9 10 11 12 13 14 15 16 17 18 19 20

Euclid	Binary	Category	Type	Etymylogy
<1,21>	100000000000000000000	Deep	Erdos
<2,21>	100000000001000000000	Deep	Erdos
<3,21>	100000010000001000000	Even - Deep	Erdos
<4,21>	100000100001000010000	Deep	Erdos
<5,21>	100001000100010001000	Deep	Erdos
<6,21>	100010010001001000100	Irregular
<7,21>	100100100100100100100	Even
<8,21>	100101001001010010100	Deep - Irregular	Erdos - Quasi Aksak
<9,21>	100101010010101001010	Irregular	Quasi Aksak
<10,21>	100101010101010101010	Deep - Irregular	Erdos - Quasi Aksak
<11,21>	101101010101010101010	Deep	Winograd
<12,21>	101101010110101011010	Mixed Double
<13,21>	101101011011010110101	Mixed Double
<14,21>	101101101101101101101	Double
<15,21>	101110110111011011101	Mixed Triple
<16,21>	101111011101110111011	Triple
<17,21>	101111101111011110111	Dense
<18,21>	101111110111111011111	Dense
<19,21>	101111111111011111111	Dense
<20,21>	101111111111111111111	Dense
<21,21>	111111111111111111111	Dense

22

Even: 1 2 11 22 Odd: 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21

Euclid	Binary	Category	Type	Etymylogy
<1,22>	1000000000000000000000	Deep	Erdos
<2,22>	1000000000010000000000	Even - Deep	Erdos
<3,22>	1000000010000001000000	Deep	Erdos
<4,22>	1000001000010000010000	Single
<5,22>	1000010001000100001000	Deep	Erdos
<6,22>	1000100100010001001000	Even
<7,22>	1000100100100100100100	Deep	Erdos
<8,22>	1001010010010010100100	Irregular	Pseudo Aksak
<9,22>	1001010100101001010010	Irregular	Pseudo Aksak	Bulgaria
<10,22>	1001010101010010101010	Irregular	Pseudo Aksak
<11,22>	1010101010101010101010	Even
<12,22>	1011010101010110101010	Mixed Double
<13,22>	1011010101101011010110	Mixed Double
<14,22>	1011010110110110101101	Mixed Double
<15,22>	1011101101101101101101	Mixed Triple
<16,22>	1011101101110111011011	Mixed Triple
<17,22>	1011110111011101111011	Mixed Quad
<18,22>	1011111011110111110111	Dense
<19,22>	1011111110111111011111	Dense
<20,22>	1011111111110111111111	Dense
<21,22>	1011111111111111111111	Dense
<22,22>	1111111111111111111111	Dense

23

Even: 1 23 Odd: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Euclid	Binary	Category	Type	Etymylogy
<1,23>	10000000000000000000000	Deep	Erdos
<2,23>	10000000000010000000000	Deep	Erdos
<3,23>	10000000100000010000000	Deep	Erdos
<4,23>	10000010000100000100000	Deep	Erdos
<5,23>	10000100010000100001000	Deep	Erdos
<6,23>	10001001000100010001000	Deep	Erdos
<7,23>	10001001001001000100100	Deep	Erdos
<8,23>	10010100100100100100100	Irregular	Authentic Aksak
<9,23>	10010100100101001010010	Irregular	Authentic Aksak	Bulgaria
<10,23>	10010101010010101001010	Irregular	Authentic Aksak
<11,23>	10010101010101010101010	Irregular	Authentic Aksak
<12,23>	10110101010101010101010	Deep	Winograd
<13,23>	10110101010110101011010	Mixed Double
<14,23>	10110101101101011010110	Mixed Double
<15,23>	10110101101101101101101	Double
<16,23>	10111011011011011101101	Mixed Triple
<17,23>	10111011011101110111011	Mixed Triple
<18,23>	10111101110111101111011	Mixed Quad
<19,23>	10111110111101111101111	Dense
<20,23>	10111111101111110111111	Dense
<21,23>	10111111111110111111111	Dense
<22,23>	10111111111111111111111	Dense
<23,23>	11111111111111111111111	Dense

24

Even: 1 2 3 4 6 8 12 24 Odd: 5 7 9 10 11 13 14 15 16 17 18 19 20 21 22 23

Euclid	Binary	Category	Type	Etymylogy
<1,24>	100000000000000000000000	Deep	Erdos
<2,24>	100000000000100000000000	Even - Deep	Erdos
<3,24>	100000001000000010000000	Deep	Erdos
<4,24>	100000100000100000100000	Even
<5,24>	100001000100001000010000	Deep	Erdos
<6,24>	100010001000100010001000	Even
<7,24>	100010010010001001000100	Deep	Erdos
<8,24>	100100100100100100100100	Even
<9,24>	100101001001010010010100	Irregular	Pseudo Aksak	Central Africa
<10,24>	100101010010100101010010	Irregular	Pseudo Aksak
<11,24>	100101010101010010101010	Irregular	Pseudo Aksak	Central African and Bulgaria
<12,24>	101010101010101010101010	Even
<13,24>	101101010101010110101010	Mixed Double		West and Central Africa
<14,24>	101101010110101101010110	Mixed Double
<15,24>	101101011011010110110101	Mixed Double
<16,24>	101101101101101101101101	Double
<17,24>	101110110110111011011101	Mixed Triple
<18,24>	101110111011101110111011	Triple
<19,24>	101111011101111011110111	Mixed Quad
<20,24>	101111101111101111101111	Quintuple
<21,24>	101111111011111110111111	Dense
<22,24>	101111111111101111111111	Dense
<23,24>	101111111111111111111111	Dense
<24,24>	111111111111111111111111	Dense

25

Even: 1 5 25 Odd: 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Euclid	Binary	Category	Type	Etymylogy
<1,25>	1000000000000000000000000	Deep	Erdos
<2,25>	1000000000000100000000000	Deep	Erdos
<3,25>	1000000001000000010000000	Deep	Erdos
<4,25>	1000000100000100000100000	Deep	Erdos
<5,25>	1000010000100001000010000	Even
<6,25>	1000010001000100010001000	Deep	Erdos
<7,25>	1000100100010001001000100	Deep	Erdos
<8,25>	1000100100100100100100100	Deep	Erdos
<9,25>	1001010010010010010100100	Deep - Irregular	Erdos - Quasi Aksak
<10,25>	1001010010100101001010010	Irregular	Quasi Aksak
<11,25>	1001010101001010100101010	Deep - Irregular	Erdos - Quasi Aksak
<12,25>	1001010101010101010101010	Deep - Irregular	Erdos - Quasi Aksak
<13,25>	1011010101010101010101010	Deep	Winograd
<14,25>	1011010101011010101101010	Mixed Double
<15,25>	1011010110101101011010110	Mixed Double
<16,25>	1011010110110110110101101	Mixed Double
<17,25>	1011101101101101101101101	Double
<18,25>	1011101101110111011011101	Mixed Triple
<19,25>	1011110111011101110111011	Triple
<20,25>	1011110111101111011110111	Quad
<21,25>	1011111101111101111101111	Quintuple
<22,25>	1011111111011111110111111	Dense
<23,25>	1011111111111101111111111	Dense
<24,25>	1011111111111111111111111	Dense
<25,25>	1111111111111111111111111	Dense

26

Even: 1 2 13 26 Odd: 3 4 5 6 7 8 9 10 11 12 14 15 16 17 18 19 20 21 22 23 24 25

Euclid	Binary	Category	Type	Etymylogy
<1,26>	10000000000000000000000000	Deep	Erdos
<2,26>	10000000000001000000000000	Even - Deep	Erdos
<3,26>	10000000010000000100000000	Deep	Erdos
<4,26>	10000001000001000000100000	Single
<5,26>	10000010000100001000010000	Deep	Erdos
<6,26>	10000100010001000010001000	Even
<7,26>	10001001000100010001001000	Deep	Erdos
<8,26>	10001001001001000100100100	Single
<9,26>	10010100100100100100100100	Irregular	Pseudo Aksak
<10,26>	10010100100101001010010010	Irregular	Pseudo Aksak
<11,26>	10010101001010010101001010	Irregular	Pseudo Aksak
<12,26>	10010101010101001010101010	Irregular	Pseudo Aksak
<13,26>	10101010101010101010101010	Even
<14,26>	10110101010101011010101010	Mixed Double
<15,26>	10110101011010110101011010	Mixed Double
<16,26>	10110101101101011010110110	Mixed Double
<17,26>	10110101101101101101101101	Mixed Double
<18,26>	10111011011011011101101101	Mixed Triple
<19,26>	10111011011101110111011011	Mixed Triple
<20,26>	10111101110111011110111011	Mixed Quad
<21,26>	10111110111101111011110111	Mixed Quintuple
<22,26>	10111111011111011111101111	Quintuple
<23,26>	10111111110111111101111111	Dense
<24,26>	10111111111111011111111111	Dense
<25,26>	10111111111111111111111111	Dense
<26,26>	11111111111111111111111111	Dense

27

Even: 1 3 9 27 Odd: 2 4 5 6 7 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Euclid	Binary	Category	Type	Etymylogy
<1,27>	100000000000000000000000000	Deep	Erdos
<2,27>	100000000000001000000000000	Deep	Erdos
<3,27>	100000000100000000100000000	Even - Deep	Erdos
<4,27>	100000010000010000001000000	Deep	Erdos
<5,27>	100000100001000010000010000	Deep	Erdos
<6,27>	100001000100001000100001000	Even
<7,27>	100010010001000100010001000	Deep	Erdos
<8,27>	100010010010001001000100100	Deep	Erdos
<9,27>	100100100100100100100100100	Even
<10,27>	100101001001001010010010100	Deep - Irregular	Erdos - Quasi Aksak
<11,27>	100101010010100101001010010	Deep - Irregular	Erdos - Quasi Aksak
<12,27>	100101010100101010100101010	Irregular	Quasi Aksak
<13,27>	100101010101010101010101010	Deep - Irregular	Erdos - Quasi Aksak
<14,27>	101101010101010101010101010	Deep	Winograd
<15,27>	101101010101101010101101010	Mixed Double
<16,27>	101101010110101101011010110	Mixed Double
<17,27>	101101011011011010110110101	Mixed Double
<18,27>	101101101101101101101101101	Double
<19,27>	101110110110111011011101101	Mixed Triple
<20,27>	101110110111011101110111011	Mixed Triple
<21,27>	101111011101111011101111011	Mixed Quad
<22,27>	101111101111011110111110111	Dense
<23,27>	101111110111110111111011111	Dense
<24,27>	101111111101111111101111111	Dense
<25,27>	101111111111111011111111111	Dense
<26,27>	101111111111111111111111111	Dense
<27,27>	111111111111111111111111111	Dense

28

Even: 1 2 4 7 14 28 Odd: 3 5 6 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24 25 26 27

Euclid	Binary	Category	Type	Etymylogy
<1,28>	1000000000000000000000000000	Deep	Erdos
<2,28>	1000000000000010000000000000	Even - Deep	Erdos
<3,28>	1000000000100000000100000000	Deep	Erdos
<4,28>	1000000100000010000001000000	Even
<5,28>	1000001000010000010000010000	Deep	Erdos
<6,28>	1000010001000010000100010000	Single
<7,28>	1000100010001000100010001000	Even
<8,28>	1000100100010010001001000100	Single
<9,28>	1000100100100100100100100100	Deep	Erdos
<10,28>	1001010010010010010100100100	Irregular	Pseudo Aksak
<11,28>	1001010010010100101001010010	Irregular	Pseudo Aksak
<12,28>	1001010100101010010101001010	Irregular	Pseudo Aksak
<13,28>	1001010101010101001010101010	Irregular	Pseudo Aksak
<14,28>	1010101010101010101010101010	Even
<15,28>	1011010101010101011010101010	Mixed Double
<16,28>	1011010101101010110101011010	Mixed Double
<17,28>	1011010110110101101011010110	Mixed Double
<18,28>	1011010110110110110101101101	Mixed Double
<19,28>	1011101101101101101101101101	Mixed Triple
<20,28>	1011101101110110111011011101	Mixed Triple
<21,28>	1011101110111011101110111011	Triple
<22,28>	1011110111011110111101110111	Mixed Quad
<23,28>	1011111011110111110111110111	Dense
<24,28>	1011111101111110111111011111	Dense
<25,28>	1011111111101111111101111111	Dense
<26,28>	1011111111111110111111111111	Dense
<27,28>	1011111111111111111111111111	Dense
<28,28>	1111111111111111111111111111	Dense

29

Even: 1 29 Odd: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Euclid	Binary	Category	Type	Etymylogy
<1,29>	10000000000000000000000000000	Deep	Erdos
<2,29>	10000000000000010000000000000	Deep	Erdos
<3,29>	10000000001000000001000000000	Deep	Erdos
<4,29>	10000000100000010000001000000	Deep	Erdos
<5,29>	10000010000100000100000100000	Deep	Erdos
<6,29>	10000100010000100001000010000	Deep	Erdos
<7,29>	10000100010001000100010001000	Deep	Erdos
<8,29>	10001001000100010010001001000	Deep	Erdos
<9,29>	10001001001001001000100100100	Deep	Erdos
<10,29>	10010100100100100100100100100	Irregular	Authentic Aksak
<11,29>	10010100100101001010010010100	Irregular	Authentic Aksak
<12,29>	10010101001010010100101010010	Irregular	Authentic Aksak
<13,29>	10010101010100101010100101010	Irregular	Authentic Aksak
<14,29>	10010101010101010101010101010	Irregular	Authentic Aksak
<15,29>	10110101010101010101010101010	Deep	Winograd
<16,29>	10110101010101101010101101010	Mixed Double
<17,29>	10110101011010110101101010110	Mixed Double
<18,29>	10110101101101011010110110101	Mixed Double
<19,29>	10110101101101101101101101101	Double
<20,29>	10111011011011011011101101101	Mixed Triple
<21,29>	10111011011101110110111011011	Mixed Triple
<22,29>	10111101110111011101110111011	Mixed Quad
<23,29>	10111101110111101111011110111	Mixed Quad
<24,29>	10111110111101111101111101111	Dense
<25,29>	10111111101111110111111011111	Dense
<26,29>	10111111111011111111011111111	Dense
<27,29>	10111111111111110111111111111	Dense
<28,29>	10111111111111111111111111111	Dense
<29,29>	11111111111111111111111111111	Dense

30

Even: 1 2 3 5 6 10 15 30 Odd: 4 7 8 9 11 12 13 14 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Euclid	Binary	Category	Type	Etymylogy
<1,30>	100000000000000000000000000000	Deep	Erdos
<2,30>	100000000000000100000000000000	Even - Deep	Erdos
<3,30>	100000000010000000001000000000	Even - Deep	Erdos
<4,30>	100000001000000100000001000000	Single
<5,30>	100000100000100000100000100000	Single
<6,30>	100001000010000100001000010000	Even
<7,30>	100001000100010001000010001000	Deep	Erdos
<8,30>	100010010001000100010010001000	Single
<9,30>	100010010010001001001000100100	Single
<10,30>	100100100100100100100100100100	Even
<11,30>	100101001001001010010010100100	Irregular	Pseudo Aksak
<12,30>	100101001010010100101001010010	Irregular	Pseudo Aksak
<13,30>	100101010100101010010101001010	Irregular	Pseudo Aksak
<14,30>	100101010101010100101010101010	Irregular	Pseudo Aksak
<15,30>	101010101010101010101010101010	Even
<16,30>	101101010101010101101010101010	Mixed Double
<17,30>	101101010101101010110101011010	Mixed Double
<18,30>	101101011010110101101011010110	Mixed Double
<19,30>	101101011011011010110110101101	Mixed Double
<20,30>	101101101101101101101101101101	Double
<21,30>	101110110110111011011011101101	Mixed Triple
<22,30>	101110110111011101110110111011	Mixed Triple
<23,30>	101111011101110111011110111011	Mixed Quad
<24,30>	101111011110111101111011110111	Quad
<25,30>	101111101111101111101111101111	Quintuple
<26,30>	101111111011111101111111011111	Dense
<27,30>	101111111110111111111011111111	Dense
<28,30>	101111111111111101111111111111	Dense
<29,30>	101111111111111111111111111111	Dense
<30,30>	111111111111111111111111111111	Dense

31

Even: 1 31 Odd: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Euclid	Binary	Category	Type	Etymylogy
<1,31>	1000000000000000000000000000000	Deep	Erdos
<2,31>	1000000000000000100000000000000	Deep	Erdos
<3,31>	1000000000010000000001000000000	Deep	Erdos
<4,31>	1000000010000001000000010000000	Deep	Erdos
<5,31>	1000000100000100000100000100000	Deep	Erdos
<6,31>	1000001000010000100001000010000	Deep	Erdos
<7,31>	1000010001000100001000100001000	Deep	Erdos
<8,31>	1000100100010001000100010001000	Deep	Erdos
<9,31>	1000100100100010010001001000100	Deep	Erdos
<10,31>	1000100100100100100100100100100	Deep	Erdos
<11,31>	1001010010010010010010100100100	Irregular	Authentic Aksak
<12,31>	1001010010010100101001010010010	Irregular	Authentic Aksak
<13,31>	1001010100101001010100101010010	Irregular	Authentic Aksak
<14,31>	1001010101010010101010010101010	Irregular	Authentic Aksak
<15,31>	1001010101010101010101010101010	Irregular	Authentic Aksak
<16,31>	1011010101010101010101010101010	Deep	Winograd
<17,31>	1011010101010110101010110101010	Mixed Double
<18,31>	1011010101101011010101101010110	Mixed Double
<19,31>	1011010110110101101011010110110	Mixed Double
<20,31>	1011010110110110110110101101101	Mixed Double
<21,31>	1011101101101101101101101101101	Mixed Triple
<22,31>	1011101101101110110111011011101	Mixed Triple
<23,31>	1011101101110111011101110111011	Mixed Triple
<24,31>	1011110111011101111011101111011	Mixed Quad
<25,31>	1011111011110111101111011110111	Dense
<26,31>	1011111101111101111101111101111	Dense
<27,31>	1011111110111111011111110111111	Dense
<28,31>	1011111111110111111111011111111	Dense
<29,31>	1011111111111111101111111111111	Dense
<30,31>	1011111111111111111111111111111	Dense
<31,31>	1111111111111111111111111111111	Dense

32

Even: 1 2 4 8 16 32 Odd: 3 5 6 7 9 10 11 12 13 14 15 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Euclid	Binary	Category	Type	Etymylogy
<1,32>	10000000000000000000000000000000	Deep	Erdos
<2,32>	10000000000000001000000000000000	Even - Deep	Erdos
<3,32>	10000000000100000000010000000000	Deep	Erdos
<4,32>	10000000100000001000000010000000	Even
<5,32>	10000001000001000001000000100000	Deep	Erdos
<6,32>	10000010000100001000001000010000	Single
<7,32>	10000100010000100001000100001000	Deep	Erdos
<8,32>	10001000100010001000100010001000	Even
<9,32>	10001001000100010010001001000100	Deep	Erdos
<10,32>	10001001001001001000100100100100	Single
<11,32>	10010100100100100100100100100100	Irregular	Pseudo Aksak
<12,32>	10010100100101001001010010010100	Irregular	Pseudo Aksak
<13,32>	10010101001010010100101001010010	Irregular	Pseudo Aksak
<14,32>	10010101010010101001010101001010	Irregular	Pseudo Aksak
<15,32>	10010101010101010100101010101010	Irregular	Pseudo Aksak
<16,32>	10101010101010101010101010101010	Even
<17,32>	10110101010101010101101010101010	Mixed Double
<18,32>	10110101010110101011010101011010	Mixed Double
<19,32>	10110101011010110101101011010110	Mixed Double
<20,32>	10110101101101011011010110110101	Mixed Double
<21,32>	10110101101101101101101101101101	Mixed Double
<22,32>	10111011011011011011101101101101	Mixed Triple
<23,32>	10111011011101110110111011011101	Mixed Triple
<24,32>	10111011101110111011101110111011	Triple
<25,32>	10111101110111101111011101111011	Mixed Quad
<26,32>	10111110111101111011111011110111	Dense
<27,32>	10111111011111011111011111101111	Dense
<28,32>	10111111101111111011111110111111	Dense
<29,32>	10111111111101111111110111111111	Dense
<30,32>	10111111111111111011111111111111	Dense
<31,32>	10111111111111111111111111111111	Dense
<32,32>	11111111111111111111111111111111	Dense