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Plato’s ‘Lambda’ Diagram from the Timaeus

On a Class of Locally Symmetric Sequences: The Right Infinite Word Λ θ Norman Carey CUNY Graduate Center ncarey@gc.cuny.edu. Mathematics and Computation in Music IRCAM : Paris, June 15-17, 2011 Paper Session 1: Word and scale theory I. Plato’s ‘Lambda’ Diagram from the Timaeus. 1. 2.

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Plato’s ‘Lambda’ Diagram from the Timaeus

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  1. On a Class ofLocally Symmetric Sequences:The Right Infinite Word ΛθNorman CareyCUNY Graduate Centerncarey@gc.cuny.edu Mathematics and Computation in Music IRCAM: Paris, June 15-17, 2011Paper Session 1: Word and scale theory I

  2. Plato’s ‘Lambda’ Diagramfrom the Timaeus 1 2 3 4 9 8 27

  3. The axes of the Lambda diagram are rotated and values are filled in to become…

  4. The Nicomachus Triangle 81 162 324 5184 648 1296 2592 54 432 864 1728 27 216 108 36 18 9 72 144 288 576 24 6 12 48 3 96 192 32 64 2 4 8 16 1 …which can be ordered…

  5. 1 1 RATIO: LETTERS:

  6. 12 2 1 RATIO: 1:2 LETTERS: a

  7. 1 2 3 3 2 1 RATIO: 1:22:3 LETTERS: a b

  8. 1 2 3 4 3 2 4 1 RATIO: 1:22:3 3:4 LETTERS: a b c

  9. 1 2 3 4 6 6 3 2 4 1 RATIO: 1:22:3 3:4 2:3 LETTERS: a b c b

  10. 1 2 3 4 6 8 6 3 2 4 8 1 RATIO: 1:22:3 3:4 2:3 3:4 LETTERS: a b c b c

  11. 1 2 3 4 6 8 9 9 6 3 2 4 8 1 RATIO: 1:22:3 3:4 2:3 3:4 8:9 LETTERS: a b c b c d

  12. 1 2 3 4 6 8 9 12 9 6 12 3 2 4 8 1 RATIO: 1:22:3 3:4 2:3 3:4 8:9 3:4 LETTERS: a b c b c d c

  13. 1 2 3 4 6 8 9 12 16 9 6 12 3 2 4 8 16 1 RATIO: 1:22:3 3:4 2:3 3:4 8:9 3:4 3:4 LETTERS: a b c b c d c c

  14. 1 2 3 4 6 8 9 12 16 18 18 9 6 12 3 2 4 8 16 1 RATIO: 1:22:3 3:4 2:3 3:4 8:9 3:4 3:4 8:9 LETTERS: a b c b c d c c d

  15. 1 2 3 4 6 8 9 12 16 18 24 18 9 24 6 12 3 2 4 8 16 1 RATIO: 1:22:3 3:4 2:3 3:4 8:9 3:4 3:4 8:9 3:4 LETTERS: a b c b c d c c d c

  16. 1 2 3 4 6 8 9 12 16 18 24 27 27 18 9 24 6 12 3 2 4 8 16 1 RATIO: 1:22:3 3:4 2:3 3:4 8:9 3:4 3:4 8:9 3:4 8:9 LETTERS: a b c b c d c c d c d

  17. 1 2 3 4 6 8 9 12 16 18 24 27 32 27 18 9 24 6 12 3 32 2 4 8 16 1 RATIO: 1:22:3 3:4 2:3 3:4 8:9 3:4 3:4 8:9 3:4 8:9 27:32 LETTERS: a b c b c d c c d c d e

  18. Lambda diagram: logarithmic representation 0 1 ϑ = Log2(3) = 1.5849625… 2 2ϑ= 3.169925… 3 3ϑ = 4.7548875…

  19. The ordering in the Nicomachus Triangle remains the same when values are replaced by logarithms

  20. 0, 1 1 0 VECTOR: (+1,0) LETTERS: a

  21. 0, 1, 1.58 1 0 VECTOR: (+1,0)(-1,+1) LETTERS: a b

  22. 0, 1, 1.58, 2 1 2 0 VECTOR: (+1,0)(-1,+1) (+2,-1) LETTERS: a b c

  23. 0, 1, 1.58, 2, 2.58 +1 1 2 0 VECTOR: (+1,0)(-1,+1) (+2,-1) (-1,+1) LETTERS: a b c b

  24. 0, 1, 1.58, 2, 2.58, 3 +1 1 2 3 0 VECTOR: (+1,0)(-1,+1) (+2,-1) (-1,+1) (+2,-1) LETTERS: a b c b c

  25. 0, 1, 1.58, 2, 2.58, 3, 3.17 2 +1 1 2 3 0 VECTOR: (+1,0)(-1,+1) (+2,-1) (-1,+1) (+2,-1) (-3,2) LETTERS: a b c b c d

  26. 0, 1, 1.58, 2, 2.58, 3, 3.17, 3.58 2 +1 +2 1 2 3 0 VECTOR: (+1,0)(-1,+1) (+2,-1) (-1,+1) (+2,-1) (-3,2) (+2,-1) LETTERS: a b c b c d c

  27. 0, 1, 1.58, 2, 2.58, 3, 3.17, 3.58, 4 2 +1 +2 1 2 3 4 0 VECTOR: (+1,0)(-1,+1) (+2,-1) (-1,+1) (+2,-1) (-3,2) (+2,-1) (+2,-1) LETTERS: a b c b c d c c

  28. 0, 1, 1.58, 2, 2.58, 3, 3.17, 3.58, 4, 4.17 2 2 +1 +1 +2 1 2 3 4 0 VECTOR: (+1,0)(-1,+1) (+2,-1) (-1,+1) (+2,-1) (-3,2) (+2,-1) (+2,-1) (-3,2) LETTERS: a b c b c d c c d

  29. 0, 1, 1.58, 2, 2.58, 3, 3.17, 3.58, 4, 4.17, 4.58 2 2 +1 +3 +1 +2 1 2 3 4 0 VECTOR: (+1,0)(-1,+1) (+2,-1) (-1,+1) (+2,-1) (-3,2) (+2,-1) (+2,-1) (-3,2) (+2,-1) LETTERS: a b c b c d c c d c

  30. 0, 1, 1.58, 2, 2.58, 3, 3.17, 3.58, 4, 4.17, 4.58, 4.76 3 2 2 +1 +3 +1 +2 1 2 3 4 0 VECTOR: (+1,0)(-1,+1) (+2,-1) (-1,+1) (+2,-1) (-3,2) (+2,-1) (+2,-1) (-3,2) (+2,-1) (-3,2) LETTERS: a b c b c d c c d c d

  31. 0, 1, 1.58, 2, 2.58, 3, 3.17, 3.58, 4, 4.17, 4.58, 4.76, 5 3 2 2 +1 +3 +1 +2 5 1 2 3 4 0 VECTOR: (+1,0)(-1,+1) (+2,-1) (-1,+1) (+2,-1) (-3,2) (+2,-1) (+2,-1) (-3,2) (+2,-1) (-3,2) (+5,-3) LETTERS: a b c b c d c c d c d e

  32. Introducing the Lambda word • Λϑ = abcbcdccdcde… • The Lambda word generated by ϑ = Log2(3) • Our standard “test case”

  33. Introducing the Lambda word • Λϑ = abcbcdccdcde… • The Lambda word generated by ϑ = Log2(3) • Our standard “test case” • Λθ = abcb… • The Lambda word generated by θ where θ is irrational; 1 < θ < 2

  34. Forming the Word

  35. Forming the Word

  36. Forming the Word

  37. Forming the Word

  38. Differences and encodings

  39. Differences and encodings

  40. Differences and encodings

  41. The first hundred letters of Λϑ

  42. The first hundred letters of Λϑ

  43. A sampling of palindromes in Λϑ

  44. Properties of the Lambda Word • A Lambda word is a right infinite word on an infinite alphabet.

  45. Properties of the Lambda Word • A Lambda word is a right infinite word on an infinite alphabet. • Each letter appears a calculably finite number of times.

  46. Properties of the Lambda Word • A Lambda word is a right infinite word on an infinite alphabet. • Each letter appears a calculably finite number of times. • Regions are encoded as central words.

  47. Properties of the Lambda Word • A Lambda word is a right infinite word on an infinite alphabet. • Each letter appears a calculably finite number of times. • Regions are encoded as central words. • There exists, for any Lambda word, a unique (non-trivial) factorization into palindromes.

  48. Properties of the Lambda Word • A Lambda word is a right infinite word on an infinite alphabet. • Each letter appears a calculably finite number of times. • Regions are encoded as central words. • There exists, for any Lambda word, a unique (non-trivial) factorization into palindromes. • Between any two successive appearances of the same letter appears a palindrome.

  49. A Lambda word is a right infinite word on an infinite alphabet.

  50. A Lambda word is a right infinite word on an infinite alphabet. Becauseθis irrational, the number of such approximations is infinite, thus the alphabet that encodes them is infinite.

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