PUZZLE #4452: Wave Superposition Cipher (diff 6)
A sampled audio waveform containing 5 superimposed sine waves. Perform spectral analysis (FFT) to find frequency peaks, compute semitone intervals between adjacent peaks, and map each interval to a letter (4 letters).
DATA
| Difficulty |
6
|
| Sample Rate |
4000
|
| Duration |
0.25
|
| Num Frequencies |
5
|
| Waveform |
0.6399, -0.1124, -1.7977, 0.2486, -0.608, -1.6232, 0.6571, -0.0368, -0.2757, 1.705, 0.8328, 0.5874, 1.5525, 0.2692, -0.2472, -0.0532, -1.2079, -1.3034, -1.0991, -1.2034, -0.4413, -0.1164, 0.5483, 1.3526, 0.8117, 1.3055, 1.3197, -0.1302, 0.3547, -0.1188, -1.6721, -0.3889, -0.5799, -1.7977, 0.4295, 0.0636, -0.9417, 1.5015, 0.5424, -0.5859, 1.7843, 0.2796, -0.7458, 1.6595, -0.2972, -1.0026, 1.1364, -1.0714, -1.4608, 0.6688, -1.4543, -1.1777, 1.0003, -0.7123, 0.148, 2.0177, 0.2406, 0.8939, 1.8294, -0.3935, -0.086, 0.0321, -1.7313, -1.1706, -0.8817, -1.5301, -0.2316, 0.2796, 0.3739, 1.2655, 1.2369, 1.1994, 0.98, 0.1635, 0.2065, -0.5272, -1.2733, -0.4585, -1.144, -1.1452, 0.4121, -0.4707, -0.1846, 1.408, -0.1426, 0.1941, 1.6403, -0.4422, 0.1579, 1.4391, -1.0441, -0.1499, 0.8924, -1.7974, -0.5463, 0.3652, -2.0538, -0.1786, 0.819, -1.2418, 1.1631, 1.7166, -0.2219, 1.7237, 1.4658, -0.8394, 0.5856, -0.2946, -2.1183, -0.563, -1.0094, -1.7555, 0.3136, 0.2015, 0.2164, 1.6702, 1.0946, 0.8676, 1.1916, 0.1271, -0.2012, -0.4528, -1.05, -0.774, -0.9968, -0.7497, 0.0307, -0.4211, 0.2599, 1.0155, -0.1919, 0.8072, 1.0977, -0.4828, 0.7925, 0.733, -1.0392, 0.5705, 0.1767, -1.7989, 0.2311, -0.5031, -2.0424, 0.5791, -0.0269, -1.0093, 1.9265, 0.954, -0.0263, 2.3548, 0.5422, -0.6766, 1.0875, -1.2, -2.0367, -0.0254, -1.6987, -1.4169, 0.8092, -0.342, 0.4963, 2.0928, 0.6255, 0.8979, 1.4698, -0.2998, -0.222, -0.2748, -1.3101, -0.8638, -0.6584, -0.8668, -0.0947, -0.102, 0.3071, 0.6303, 0.2368, 0.8421, 0.5739, -0.0858, 0.9242, 0.1545, -0.554, 0.6705, -0.6264, -1.2044, 0.4154, -1.2355, -1.3069, 0.7319, -0.8891, -0.2613, 2.0266, 0.1017, 0.7566, 2.3759, -0.4295, 0.0369, 0.9909, -2.08, -1.2289, -0.079, -2.484, -0.5929, 0.8724, -0.902, 1.2293, 2.0072, 0.0687, 1.5688, 1.2283, -0.7889, 0.202, -0.3272, -1.7043, -0.3737, -0.6353, -1.1262, 0.2295, 0.0235, 0.0223, 0.8033, 0.4253, 0.5177, 0.5919, 0.2237, 0.5845, 0.0178, -0.0845, 0.3432, -0.8211, -0.6267, -0.0488, -1.4798, -0.6072, 0.3381, -1.0004, 0.5898, 1.4711, -0.1088, 1.6255, 1.6949, -0.676, 0.8508, 0.1945, -2.2936, -0.2854, -0.8703, -2.5397, 0.3477, 0.1578, -0.9073, 2.1277, 1.291, 0.007, 2.2203, 0.4581, -0.9589, 0.8604, -0.9865, -1.6611, 0.2098, -1.2038, -0.9861, 0.7168, -0.3762, -0.0152, 1.1398, 0.1131, 0.3986, 0.9275, 0.0859, 0.4257, 0.314, -0.1831, -0.0408, -0.6253, -0.5974, -0.538, -1.2003, -0.3923, -0.216, -0.5975, 0.9791, 0.9305, 0.2386, 1.9967, 0.9522, -0.3297, 1.1077, -0.7477, -1.912, 0.0035, -1.7381, -1.8439, 0.8328, -0.6863, -0.2075, 2.4375, 0.3365, 0.6747, 2.3361, -0.5092, -0.3108, 0.9243, -1.8462, -1.0023, 0.2818, -1.8883, -0.433, 0.8975, -0.961, 0.5876, 1.2278, -0.4385, 0.8085, 0.972, -0.3465, 0.6721, 0.434, -0.5154, 0.1152, -0.4584, -0.8978, -0.5379, -0.934, -0.5974, -0.2555, -0.1478, 0.8051, 0.7811, 0.8521, 1.7297, 0.5413, 0.2795, 0.725, -1.2502, -1.1677, -0.3812, -2.1575, -1.0272, 0.4696, -1.1149, 0.8122, 1.997, -0.1006, 1.5256, 1.7841, -0.941, 0.5731, 0.2959, -2.2455, -0.1339, -0.31, -2.0727, 0.6005, 0.2711, -1.1396, 1.3632, 0.6288, -0.6301, 1.509, 0.5201, -0.5617, 1.2415, 0.019, -0.6732, 0.5572, -0.7416, -1.0632, -0.1781, -1.0964, -0.7397, 0.0744, -0.1205, 0.5841, 1.0209, 1.0212, 1.3883, 0.6573, 0.4601, 0.2149, -1.1384, -0.8489, -0.9369, -1.9597, -0.4487, -0.1222, -0.7566, 1.3933, 1.2676, 0.2003, 2.0792, 0.8975, -0.6993, 1.0796, -0.7145, -1.9357, 0.4025, -1.1718, -1.6311, 1.0492, -0.6074, -0.6958, 1.8208, -0.3483, -0.0895, 1.8925, -0.381, 0.0234, 1.6065, -0.7906, -0.2634, 0.7848, -1.4412, -0.7145, -0.0384, -1.6241, -0.4814, 0.3814, -0.4357, 0.7966, 1.3355, 0.6875, 1.3757, 0.9084, 0.1911, -0.0863, -0.8714, -1.1059, -1.2352, -1.5422, -0.575, -0.4554, -0.1448, 1.4238, 0.7227, 0.8337, 1.9316, 0.255, -0.0449, 0.8649, -1.3625, -1.1107, 0.2734, -1.7932, -0.6644, 0.8176, -1.271, 0.2763, 1.4452, -0.8851, 0.7797, 1.5419, -0.9528, 0.911, 1.2338, -1.2522, 0.5845, 0.3638, -1.9059, 0.0404, -0.3465, -1.982, 0.2874, 0.1009, -0.722, 1.4986, 1.1618, 0.4839, 1.7618, 0.7093, -0.2285, 0.2105, -0.9725, -1.3508, -1.0675, -1.3494, -0.7008, -0.2329, 0.1904, 1.104, 0.8903, 1.1715, 1.4555, 0.1387, 0.4046, 0.2806, -1.3822, -0.4692, -0.3738, -1.7992, 0.0303, 0.1526, -1.2099, 1.0116, 0.7031, -0.8099, 1.5336, 0.6955, -0.7633, 1.6199, 0.2506, -1.0508, 1.2107, -0.6253, -1.6781, 0.6182, -1.2552, -1.6874, 0.7829, -0.7063, -0.2998, 1.9866, 0.3816, 0.7607, 2.1318, 0.006, -0.0462, 0.4621, -1.5283, -1.2829, -0.7605, -1.7024, -0.674, 0.1527, 0.0148, 1.0169, 1.2321, 1.0866, 1.1465, 0.3901, 0.2374, -0.1595, -1.0474, -0.442, -0.7945, -1.272, 0.1653, -0.3984, -0.5985, 1.1108, -0.037, -0.1555, 1.6673, -0.106, -0.1128, 1.6861, -0.5482, -0.284, 1.2305, -1.4758, -0.869, 0.5409, -2.0806, -0.7474, 0.7228, -1.4021, 0.5411, 1.9259, -0.1629, 1.5073, 2.0552, -0.7017, 0.5687, 0.1743, -2.1892, -0.6992, -0.8779, -2.08, 0.0058, 0.1639, -0.2316, 1.5028, 1.2163, 0.7103, 1.3113, 0.3947, -0.1263, -0.1038, -0.9225, -0.7165, -0.8764, -0.8906, -0.1628, -0.5133, -0.0794, 0.7211, -0.1463, 0.4334, 1.194, -0.2859, 0.6616, 1.0818, -0.7183, 0.5086, 0.5486, -1.6146, -0.0611, -0.2331, -2.2327, 0.109, 0.0571, -1.3996, 1.4699, 1.2405, -0.2562, 2.3735, 1.0445, -0.7034, 1.2789, -0.5977, -2.0866, -0.0385, -1.5216, -1.9399, 0.6599, -0.3782, -0.1465, 2.0493, 0.7244, 0.7466, 1.6994, -0.0034, -0.2175, 0.1571, -1.1609, -0.8134, -0.4907, -0.9973, -0.2793, -0.1472, -0.0642, 0.4254, 0.1959, 0.5958, 0.754, 0.0055, 0.8794, 0.5356, -0.4051, 0.7454, -0.1428, -1.2366, 0.1837, -1.0658, -1.6791, 0.4129, -0.7656, -0.7255, 1.8208, 0.3469, 0.4847, 2.6014, 0.0893, -0.1047, 1.4364, -1.5955, -1.4345, 0.0864, -2.4011, -1.16, 0.8034, -1.0417, 0.7238, 2.146, -0.0103, 1.3229, 1.6844, -0.6701, 0.2435, 0.1835, -1.6065, -0.4745, -0.4364, -1.3163, -0.0427, -0.0048, -0.3462, 0.5664, 0.3889, 0.2604, 0.6875, 0.2868, 0.5849, 0.4321, 0.018, 0.4508, -0.477, -0.7961, -0.1012, -1.3606, -1.0402, 0.1316, -1.0927, 0.1507, 1.5064, 0.0128, 1.3349, 2.1316, -0.2904, 0.7752, 0.7479, -2.1166, -0.5811, -0.5004, -2.7149, -0.1286, 0.2626, -1.2954, 1.6695, 1.5041, -0.2177, 2.1175, 0.9973, -0.8649, 0.9587, -0.4271, -1.8446, 0.2633, -0.9108, -1.4257, 0.5633, -0.3559, -0.467, 1.0478, 0.0994, 0.1061, 1.0668, 0.1836, 0.4437, 0.6785, 0.0216, 0.1308, -0.3187, -0.642, -0.5063, -1.1886, -0.7727, -0.3094, -0.8295, 0.5454, 0.9686, 0.306, 1.8422, 1.3654, -0.0876, 1.1562, -0.1564, -1.8328, -0.1589, -1.46, -2.2839, 0.4045, -0.5784, -0.7213, 2.1717, 0.5564, 0.2878, 2.4459, -0.0433, -0.4461, 1.2389, -1.349, -1.2754, 0.5041, -1.6958, -0.8209, 0.8681, -1.0356, 0.0338, 1.2104, -0.5339, 0.5783, 1.2922, -0.2804, 0.6742, 0.8812, -0.4251, 0.1961, -0.1287, -0.923, -0.5393, -0.9602, -1.0047, -0.4697, -0.3882, 0.45, 0.7529, 0.7658, 1.7103, 0.9422, 0.3845, 0.9367, -0.7535, -1.2416, -0.3291, -1.9728, -1.4616, 0.2862, -1.1041, 0.254, 1.9238, -0.0954, 1.213, 2.0848, -0.661, 0.432, 0.8179, -2.0022, -0.3466, 0.1083, -2.178, 0.101, 0.4294, -1.4357, 0.9359, 0.8125, -0.9039, 1.3352, 0.8023, -0.6551, 1.356, 0.5026, -0.723, 0.7524, -0.4404, -1.2529, -0.2082, -1.1164, -1.2093, -0.1242, -0.3789, 0.2374, 1.1004, 0.9499, 1.3526, 1.0166, 0.6395, 0.4845, -0.7883, -0.9158, -0.8314, -1.9278, -0.9031, -0.2, -0.9772, 0.8817, 1.3097, 0.0585, 1.8779, 1.2768, -0.6361, 1.0666, -0.0612, -1.8645, 0.4031, -0.7866, -1.8898, 0.8686, -0.4283, -1.2038, 1.5298, -0.1452, -0.5515, 1.8304, 0.0029, -0.2822, 1.8693, -0.2338, -0.3515, 1.1465, -1.1348, -1.0429, 0.1173, -1.6676, -0.9675, 0.2646, -0.6639, 0.4516, 1.3907, 0.6695, 1.3879, 1.2448, 0.2766, 0.321, -0.5687, -1.0423, -1.0685, -1.5334, -0.7953, -0.5557, -0.4152, 1.0104, 0.8777, 0.5536, 1.869, 0.5547, -0.059, 1.076, -0.8368, -1.1405, 0.4236, -1.5184, -1.0739, 0.8283, -1.2592, -0.3153, 1.4365, -0.8914, 0.3259, 1.7835, -0.7313, 0.6262, 1.694, -1.031, 0.4506, 0.8959, -1.7895, -0.2827, -0.1112, -2.1672, -0.2122, 0.1261, -1.1303, 1.1531, 1.2418, 0.2693, 1.8391, 1.107, -0.0835, 0.4921, -0.6317, -1.414, -0.8793, -1.4492, -1.0681, -0.3599, -0.1567, 0.7355, 0.9304, 1.0189, 1.4681, 0.4919, 0.3705, 0.5542, -1.0206, -0.5552, -0.0749, -1.6348, -0.2936, 0.236, -1.271, 0.5461, 0.7468, -0.9506, 1.1653, 0.9842, -0.867, 1.51, 0.8344, -1.0208, 1.2158, -0.0135, -1.8205, 0.5144, -0.919, -2.0713, 0.559, -0.5341, -0.8685, 1.8662, 0.5521, 0.4454, 2.3802, 0.4652, -0.046, 0.8866, -1.2178, -1.4574, -0.5867, -1.7768, -1.0887, 0.066, -0.3156, 0.7598, 1.1882, 0.9471, 1.2208, 0.6917, 0.3223, 0.1177, -0.7494, -0.4433, -0.6433, -1.2591, -0.0774, -0.312, -0.8361, 0.7846
|
| Instructions |
This is a sampled audio waveform created by summing pure sine waves. Each sine wave has a rationally-related frequency (consonant musical interval). To decode:
1. Compute FFT of the waveform to find frequency peaks
2. Apply a Hann window before FFT to reduce spectral leakage
3. Sort peaks by frequency (low to high)
4. For each adjacent pair, compute: semitones = 12 * log2(f2/f1)
5. Round semitones to the nearest integer
6. Map each semitone count to a letter using the table
Concatenate letters in order.
WARNING: Noise floor is present (0.03). Filter frequencies with magnitude below the noise floor estimate.
|
| Semitone To Letter |
{
"0": "A",
"2": "B",
"3": "C",
"4": "D",
"5": "E",
"7": "F",
"8": "G",
"9": "H",
"10": "I",
"11": "J",
"12": "K",
"14": "L",
"15": "M",
"16": "N",
"17": "O",
"19": "P",
"21": "Q",
"22": "R",
"24": "S",
"26": "T",
"28": "U",
"29": "V",
"31": "W",
"33": "X",
"35": "Y",
"36": "Z"
}
|
| Hann Window Formula |
w[i] = 0.5 * (1 - cos(2*pi*i/(n-1)))
|
| Fft Tip |
Sample rate = 4000 Hz (Nyquist = 2000 Hz). All signal frequencies are between 80 Hz and 1900 Hz.
|
| Hint |
5 frequencies → 4 semitone intervals → 4 letters. Apply Hann window, FFT, find peaks, compute log2 ratios.
|
| Answer Format |
single lowercase letter
|
author's note: Pool fill: wave-superposition diff 6
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