# Element-by-element harmonic comparison

21 September 2026. Current corrected results; all distances are cents (100 cents = one equal-tempered semitone).

Each row uses all eligible neutral and singly-ionized strong lines for that element, not one representative line. Missing entries mean absent from the selected NIST handbook, not no spectrum in nature. All 118 element symbols are accounted for. The common correction was selected on training elements only. Hydrogen is descriptive because it was previously explored.

The final column is an exploratory best offset fitted separately to that same element's lines. It is NOT a validated prediction or a discovered physical property. Low-line-count optima can be essentially arbitrary. A shared factor would require agreement and transfer, which the held-out test did not establish.

| Z | Element | Catalog rows | Scored lines | Split | Mean absolute error | After shared factor | Own fitted offset (descriptive) |
|---:|---|---:|---:|---|---:|---:|---:|
| 1 | H — Hydrogen | 30 | 23 | previously_explored | 20.1416 | 21.7795 | 15.8085 |
| 2 | He — Helium | 138 | 125 | validation | 24.6787 | 23.6833 | 20.4412 |
| 3 | Li — Lithium | 87 | 82 | training | 25.9939 | 23.5452 | 25.7440 |
| 4 | Be — Beryllium | 45 | 43 | validation | 22.8969 | 29.7822 | -20.7856 |
| 5 | B — Boron | 37 | 37 | validation | 17.7404 | 26.0477 | 3.3032 |
| 6 | C — Carbon | 61 | 61 | training | 26.4090 | 25.4239 | -44.4087 |
| 7 | N — Nitrogen | 81 | 81 | validation | 24.5494 | 24.1678 | 15.7195 |
| 8 | O — Oxygen | 90 | 88 | validation | 19.5218 | 27.0767 | -2.8404 |
| 9 | F — Fluorine | 81 | 81 | validation | 24.1240 | 22.7899 | 13.2962 |
| 10 | Ne — Neon | 442 | 385 | validation | 25.0846 | 25.4151 | -28.3437 |
| 11 | Na — Sodium | 142 | 138 | training | 24.7030 | 25.1143 | 11.1257 |
| 12 | Mg — Magnesium | 106 | 105 | training | 19.6018 | 28.1463 | 0.9942 |
| 13 | Al — Aluminum | 53 | 53 | training | 20.7960 | 30.6792 | -18.2531 |
| 14 | Si — Silicon | 136 | 136 | validation | 25.4275 | 27.7716 | -27.4054 |
| 15 | P — Phosphorus | 67 | 67 | training | 25.1684 | 22.1596 | 28.5359 |
| 16 | S — Sulfur | 50 | 50 | validation | 25.0764 | 25.1626 | -7.6684 |
| 17 | Cl — Chlorine | 96 | 96 | training | 26.4161 | 21.6293 | 34.9781 |
| 18 | Ar — Argon | 313 | 308 | validation | 24.8335 | 25.4424 | -13.2841 |
| 19 | K — Potassium | 125 | 124 | validation | 24.3207 | 25.3317 | -0.5439 |
| 20 | Ca — Calcium | 84 | 75 | validation | 23.5503 | 26.6929 | -11.2160 |
| 21 | Sc — Scandium | 101 | 99 | validation | 30.2770 | 20.6386 | 41.6642 |
| 22 | Ti — Titanium | 264 | 264 | validation | 26.0133 | 25.8593 | -34.5965 |
| 23 | V — Vanadium | 217 | 217 | training | 22.3520 | 23.0342 | 12.5386 |
| 24 | Cr — Chromium | 257 | 257 | training | 27.2992 | 21.9172 | 37.1844 |
| 25 | Mn — Manganese | 130 | 130 | training | 28.9565 | 19.9388 | 35.2162 |
| 26 | Fe — Iron | 328 | 328 | validation | 25.8706 | 25.7894 | -38.8483 |
| 27 | Co — Cobalt | 109 | 109 | validation | 24.1079 | 22.1949 | 14.9651 |
| 28 | Ni — Nickel | 103 | 103 | validation | 23.5088 | 24.1335 | 16.7753 |
| 29 | Cu — Copper | 122 | 122 | validation | 23.8860 | 26.0664 | -6.0932 |
| 30 | Zn — Zinc | 32 | 32 | training | 21.6652 | 30.5618 | -25.7252 |
| 31 | Ga — Gallium | 33 | 33 | validation | 27.6360 | 25.0659 | -38.4016 |
| 32 | Ge — Germanium | 68 | 68 | training | 22.9527 | 27.4644 | -11.0375 |
| 33 | As — Arsenic | 43 | 43 | validation | 26.5797 | 26.3757 | -33.1090 |
| 34 | Se — Selenium | 49 | 45 | training | 25.3658 | 24.9803 | -37.9319 |
| 35 | Br — Bromine | 78 | 78 | training | 22.8774 | 28.1230 | -10.9282 |
| 36 | Kr — Krypton | 261 | 221 | training | 24.1477 | 24.3069 | 19.5254 |
| 37 | Rb — Rubidium | 102 | 102 | validation | 25.8916 | 24.2781 | 37.0571 |
| 38 | Sr — Strontium | 43 | 43 | training | 20.3305 | 23.4027 | 13.6481 |
| 39 | Y — Yttrium | 120 | 108 | training | 27.2341 | 23.3977 | 41.9580 |
| 40 | Zr — Zirconium | 199 | 199 | training | 27.8048 | 21.9728 | 39.2670 |
| 41 | Nb — Niobium | 190 | 190 | training | 25.8136 | 25.1099 | 45.8749 |
| 42 | Mo — Molybdenum | 205 | 205 | validation | 24.8433 | 24.2039 | 34.7772 |
| 43 | Tc — Technetium | 172 | 172 | validation | 25.2645 | 25.7838 | -36.3733 |
| 44 | Ru — Ruthenium | 180 | 180 | validation | 26.2000 | 24.8522 | -36.1463 |
| 45 | Rh — Rhodium | 144 | 144 | validation | 24.4285 | 25.1020 | -31.6120 |
| 46 | Pd — Palladium | 54 | 54 | validation | 20.9570 | 26.0364 | -2.1421 |
| 47 | Ag — Silver | 48 | 48 | validation | 23.9210 | 24.9504 | -3.1294 |
| 48 | Cd — Cadmium | 41 | 41 | validation | 25.4860 | 20.7465 | 27.0242 |
| 49 | In — Indium | 80 | 80 | validation | 21.6665 | 22.5149 | 13.8548 |
| 50 | Sn — Tin | 68 | 68 | training | 25.6518 | 23.4502 | 36.0990 |
| 51 | Sb — Antimony | 63 | 63 | validation | 25.2665 | 23.0217 | 28.0393 |
| 52 | Te — Tellurium | 89 | 86 | validation | 27.0789 | 22.7447 | 39.2177 |
| 53 | I — Iodine | 83 | 83 | validation | 25.3047 | 23.0810 | 31.7518 |
| 54 | Xe — Xenon | 302 | 262 | validation | 24.5256 | 25.0967 | 0.2043 |
| 55 | Cs — Cesium | 140 | 125 | training | 25.3692 | 23.1595 | 31.0265 |
| 56 | Ba — Barium | 45 | 43 | validation | 20.8000 | 25.8322 | 1.7005 |
| 57 | La — Lanthanum | 78 | 78 | training | 24.0733 | 24.3094 | 18.0356 |
| 58 | Ce — Cerium | 200 | 200 | validation | 25.8636 | 24.2005 | 42.4494 |
| 59 | Pr — Praseodymium | 196 | 196 | training | 25.9185 | 25.5689 | -40.5997 |
| 60 | Nd — Neodymium | 153 | 153 | validation | 22.0134 | 26.5461 | -1.2640 |
| 61 | Pm — Promethium | 152 | 152 | validation | 22.9164 | 27.3640 | -9.1210 |
| 62 | Sm — Samarium | 170 | 170 | training | 25.5591 | 26.9142 | -32.3576 |
| 63 | Eu — Europium | 144 | 144 | validation | 24.8448 | 24.6087 | 37.9377 |
| 64 | Gd — Gadolinium | 247 | 247 | training | 26.1051 | 23.8370 | 39.5174 |
| 65 | Tb — Terbium | 58 | 58 | training | 24.7009 | 23.0816 | 23.2156 |
| 66 | Dy — Dysprosium | 163 | 163 | training | 25.8359 | 26.6818 | -32.5481 |
| 67 | Ho — Holmium | 173 | 173 | validation | 23.1554 | 23.0483 | 9.7118 |
| 68 | Er — Erbium | 152 | 152 | training | 23.5787 | 25.4455 | 7.6992 |
| 69 | Tm — Thulium | 136 | 136 | training | 24.1053 | 25.7620 | -6.1271 |
| 70 | Yb — Ytterbium | 144 | 144 | training | 25.9333 | 24.8355 | -40.2526 |
| 71 | Lu — Lutetium | 60 | 60 | training | 23.1474 | 24.2451 | 13.8811 |
| 72 | Hf — Hafnium | 168 | 168 | validation | 24.8701 | 24.6000 | 17.7160 |
| 73 | Ta — Tantalum | 170 | 170 | training | 25.7755 | 24.1945 | 36.0911 |
| 74 | W — Tungsten | 199 | 199 | validation | 25.3219 | 25.4169 | -35.6741 |
| 75 | Re — Rhenium | 143 | 143 | training | 24.4293 | 25.5053 | 8.4465 |
| 76 | Os — Osmium | 174 | 169 | training | 25.6599 | 24.0866 | 37.7458 |
| 77 | Ir — Iridium | 120 | 120 | training | 27.0259 | 24.6468 | -49.2350 |
| 78 | Pt — Platinum | 407 | 395 | validation | 24.5207 | 24.9564 | -3.4878 |
| 79 | Au — Gold | 82 | 82 | training | 26.4261 | 25.0165 | -41.6254 |
| 80 | Hg — Mercury | 62 | 60 | validation | 25.7511 | 23.6245 | 13.1235 |
| 81 | Tl — Thallium | 45 | 45 | training | 27.2564 | 23.6885 | -49.7103 |
| 82 | Pb — Lead | 68 | 68 | training | 28.7717 | 21.3827 | 41.2873 |
| 83 | Bi — Bismuth | 60 | 60 | validation | 26.3007 | 23.0696 | 38.3935 |
| 84 | Po — Polonium | 12 | 11 | validation | 24.0172 | 34.1977 | -22.4346 |
| 85 | At — Astatine | 2 | 2 | validation | 32.1242 | 17.8758 | 22.2474 |
| 86 | Rn — Radon | 25 | 25 | validation | 25.3358 | 22.9918 | 13.0698 |
| 87 | Fr — Francium | 4 | 4 | validation | 31.3267 | 15.9761 | 21.9595 |
| 88 | Ra — Radium | 23 | 23 | training | 26.9617 | 20.8168 | 32.3860 |
| 89 | Ac — Actinium | 54 | 54 | training | 24.2509 | 26.2707 | -13.4235 |
| 90 | Th — Thorium | 287 | 282 | training | 25.2602 | 23.5174 | 26.1614 |
| 91 | Pa — Protactinium | 105 | 104 | training | 26.6694 | 24.1779 | 45.1034 |
| 92 | U — Uranium | 150 | 150 | training | 24.2998 | 25.9909 | -10.4230 |
| 93 | Np — Neptunium | 33 | 33 | validation | 22.8440 | 27.3013 | -12.0701 |
| 94 | Pu — Plutonium | 83 | 83 | training | 24.0801 | 25.5602 | -5.7193 |
| 95 | Am — Americium | 73 | 73 | validation | 24.0140 | 26.7800 | -23.8556 |
| 96 | Cm — Curium | 134 | 134 | validation | 26.2800 | 25.5078 | -44.1697 |
| 97 | Bk — Berkelium | 140 | 140 | validation | 23.4002 | 23.7593 | 10.9931 |
| 98 | Cf — Californium | 81 | 74 | validation | 24.1577 | 25.7977 | -26.0062 |
| 99 | Es — Einsteinium | 51 | 29 | training | 23.6659 | 31.9697 | -20.9989 |
| 100 | Fm | 0 | 0 | Not in catalog | — | — | — |
| 101 | Md | 0 | 0 | Not in catalog | — | — | — |
| 102 | No | 0 | 0 | Not in catalog | — | — | — |
| 103 | Lr | 0 | 0 | Not in catalog | — | — | — |
| 104 | Rf | 0 | 0 | Not in catalog | — | — | — |
| 105 | Db | 0 | 0 | Not in catalog | — | — | — |
| 106 | Sg | 0 | 0 | Not in catalog | — | — | — |
| 107 | Bh | 0 | 0 | Not in catalog | — | — | — |
| 108 | Hs | 0 | 0 | Not in catalog | — | — | — |
| 109 | Mt | 0 | 0 | Not in catalog | — | — | — |
| 110 | Ds | 0 | 0 | Not in catalog | — | — | — |
| 111 | Rg | 0 | 0 | Not in catalog | — | — | — |
| 112 | Cn | 0 | 0 | Not in catalog | — | — | — |
| 113 | Nh | 0 | 0 | Not in catalog | — | — | — |
| 114 | Fl | 0 | 0 | Not in catalog | — | — | — |
| 115 | Mc | 0 | 0 | Not in catalog | — | — | — |
| 116 | Lv | 0 | 0 | Not in catalog | — | — | — |
| 117 | Ts | 0 | 0 | Not in catalog | — | — | — |
| 118 | Og | 0 | 0 | Not in catalog | — | — | — |