Quotes

"I don't believe in the golden section. At least, I don't believe that it is the single formal principle for our sense of beauty; rather it is one of the highest among many, among countless many." - Arnold Schoenberg, Harmonielehre

"The systematic organization of a musical composition within a pre-determined time span by means of the lower numbers of the Fibonacci and Lucas series, singly or in combination, is common practice indeed. It seems that the more profound the composer, the stricter is his application of these proportions in the musical structure." - Hugo Norden, Proportions in Music, Fibonacci Quarterly

"Schillinger, inspired by the 'organic' nature of the golden mean, suggests deriving melodic lines from the Fibonacci series (and also from other summation series). He freely makes octave transpositions, so that the proportional and additive properties are lost, as is the Fibonacci source of the resulting melody..., and, if the resulting lines are in fact aesthetically pleasing, I suspect that the Fibonacci series is not the reason." - Jonathan Kramer, The Fibonacci Series in Twentieth-Century Music, Journal of Music Theory

"Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel." - Johannes Kepler, Harmonices Mundi

"The Golden Ratio manifests in the whole of creation. Take the ratio of the length of a man and the height of his navel. The ratio of the sides of the Great Temple. The ratio between the long and short sides of a pentagram. Why is this? Because the ratio of the Whole to the Greater is the ratio of the Greater to the Lesser." - Pythagoras of Samos

Sequences

Fibonacci Numbers

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 4807526976, 7778742049, 12586269025, 20365011074, 32951280099, 53316291173, 86267571272, 139583862445, 225851433717, 365435296162, 591286729879, 956722026041, 1548008755920, 2504730781961, 4052739537881, 6557470319842, 10610209857723, 17167680177565, 27777890035288, 44945570212853, 72723460248141, 117669030460994, 190392490709135, 308061521170129, 498454011879264, 806515533049393, 1304969544928657, 2111485077978050, 3416454622906707, 5527939700884757, 8944394323791464, 14472334024676221, 23416728348467685, 37889062373143906, 61305790721611591, 99194853094755497, 160500643816367088, 259695496911122585, 420196140727489673, 679891637638612258, 1100087778366101931, ...

Lucas Numbers

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196, 12752043, 20633239, 33385282, 54018521, 87403803, 141422324, 228826127, 370248451, 599074578, 969323029, 1568397607, 2537720636, 4106118243, 6643838879, 10749957122, 17393796001, 28143753123, 45537549124, 73681302247, 119218851371, 192900153618, 312119004989, 505019158607, 817138163596, 1322157322203, 2139295485799, 3461452808002, 5600748293801, 9062201101803, 14662949395604, 23725150497407, 38388099893011, 62113250390418, 100501350283429, 162614600673847, 263115950957276, 425730551631123, 688846502588399, 1114577054219522, 1803423556807921, 2918000611027443, 4721424167835364, 7639424778862807, 12360848946698171, 20000273725560978, 32361122672259149, 52361396397820127, 84722519070079276, 137083915467899403, 221806434537978679, 358890350005878082, 580696784543856761, 939587134549734843, 1520283919093591604, 2459871053643326447, ...

Vertical Para-Fibonacci Sequence

0, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 4, 0, 5, 3, 2, 6, 1, 7, 4, 0, 8, 5, 3, 9, 2, 10, 6, 1, 11, 7, 4, 12, 0, 13, 8, 5, 14, 3, 15, 9, 2, 16, 10, 6, 17, 1, 18, 11, 7, 19, 4, 20, 12, 0, 21, 13, 8, 22, 5, 23, 14, 3, 24, 15, 9, 25, 2, 26, 16, 10, 27, 6, 28, 17, 1, 29, 18, 11, 30, 7, 31, 19, 4, 32, 20, 12, 33, 0, 34, 21, 13, 35, 8, 36, 22, 5, 37, 23, 14, 38, 3, 39, 24, 15, 40, 9, 41, 25, 2, 42, 26, 16, 43, 10, 44, 27, 6, 45, 28, 17, 46, 1, 47, 29, 18, 48, 11, 49, 30, 7, 50, 31, 19, 51, 4, 52, 32, 20, 53, 12, 54, 33, 0, 55, 34, 21, 56, 13, 57, 35, 8, 58, 36, 22, 59, 5, 60, 37, 23, 61, 14, 62, 38, 3, 63, 39, 24, 64, 15, 65, 40, 9, 66, 41, 25, 67, 2, 68, 42, 26, 69, 16, 70, 43, 10, 71, 44, 27, 72, 6, 73, 45, 28, 74, 17, 75, 46, 1, 76, 47, 29, 77, 18, 78, 48, 11, 79, 49, 30, 80, 7, 81, 50, 31, 82, 19, 83, 51, 4, 84, 52, 32, 85, 20, 86, 53, 12, 87, 54, 33, 88, 0, ...

Horizontal Para-Fibonacci Sequence

1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 6, 1, 2, 3, 1, 4, 1, 2, 7, 1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 8, 1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 6, 1, 2, 3, 1, 4, 1, 2, 9, 1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 6, 1, 2, 3, 1, 4, 1, 2, 7, 1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 10, 1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 6, 1, 2, 3, 1, 4, 1, 2, 7, 1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 8, 1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 6, 1, 2, 3, 1, 4, 1, 2, 11, 1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 6, 1, 2, 3, 1, 4, 1, 2, 7, 1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 8, 1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 6, 1, 2, 3, 1, 4, 1, 2, 9, 1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 6, 1, 2, 3, 1, 4, 1, 2, 7, 1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 12, ...

Beatty Sequences; n(phi), n(phi2)

1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45, 46, 48, 50, 51, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 92, 93, 95, 97, 98, 100, 101, 103, 105, 106, 108, 110, 111, 113, 114, 116, 118, 119, 121, 122, 124, 126, 127, 129, 131, 132, 134, 135, 137, 139, 140, 142, 144, 145, 147, 148, 150, 152, 153, 155, 156, 158, 160, 161, 163, 165, 166, 168, 169, 171, 173, 174, 176, 177, 179, 181, 182, 184, 186, 187, 189, 190, 192, 194, 195, 197, 199, 200, 202, 203, 205, 207, 208, 210, 211, 213, 215, 216, 218, 220, 221, 223, 224, 226, 228, 229, 231, 232, ...

2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, 49, 52, 54, 57, 60, 62, 65, 68, 70, 73, 75, 78, 81, 83, 86, 89, 91, 94, 96, 99, 102, 104, 107, 109, 112, 115, 117, 120, 123, 125, 128, 130, 133, 136, 138, 140, 141, 143, 146, 149, 151, 154, 157, 159, 162, 164, 167, 170, 172, 175, 178, 180, 183, 185, 188, 191, 193, 196, 198, 201, 204, 206, 209, 212, 214, 217, 219, 222, 225, 227, 230, 233, ...

Golden String (Rabbit Sequence)

1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, ...


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