# Hydrogen, musical octaves, and alpha: checked results

21 September 2026 · Standalone exploratory study HARMONIC-ALPHA-01

## Outcome

Hydrogen's red H-alpha reference line is remarkably close to G-sharp after dividing its optical frequency by 2^40: **415.34437 Hz, only +0.16537 cents from G-sharp4**. But the seven-line pattern does not establish a special musical law: **23.5747% of common tuning shifts match the complete pattern at least as well as A4=440 Hz** under the explicitly chosen reference comparison.

The illustrative expression 4*pi^3 + pi^2 + pi comes close to alpha's inverse but is not equal to its reference value. Neither finding supplies a derivation from higher-dimensional geometry. No master theory or released engine was modified.

## Fixed mapping and data provenance

We used seven consecutive Balmer transitions, n=3 through 9 to n=2. For the resolved n=3 and n=4 multiplets, we selected the component with the highest listed intensity, before calculating the musical score. These are [NIST reference wavelengths](https://physics.nist.gov/PhysRefData/Handbook/Tables/hydrogentable2.htm), not seven new experimental measurements. NIST attributes n=3,4,5 to Mohr and Kotochigova's unpublished calculations and the remaining entries to Reader et al.'s 1980 compilation; see [source notes](https://physics.nist.gov/PhysRefData/Handbook/Tables/hydrogentable7.htm).

The source lists these wavelengths in air. We converted to vacuum using the Peck-Reeder standard-air formula, following [NIST's conversion convention](https://physics.nist.gov/PhysRefData/ASD/Html/lineshelp.html). This corrects a medium convention; it does not reconstruct experimental temperature, pressure or humidity. No wavelength uncertainties are supplied in the selected table, so the calculation is not an experimental-sigma test.

The fixed rule was:

\[
f_{\rm audible}=\frac{c}{\lambda_{\rm vacuum}2^{40}},\qquad
m=69+12\log_2\left(\frac{f_{\rm audible}}{440\ {\rm Hz}}\right).
\]

The nearest integer m identifies a note in twelve-tone equal temperament. The signed error is 100 times the difference from that integer. One semitone is 100 cents; one octave is 1200 cents. This is a mathematical sonification rule, not a claim that light is acoustic vibration.

| Transition | Air wavelength (nm) | Derived vacuum wavelength (nm) | Divided frequency (Hz) | Nearest note | Error (cents) |
|---|---:|---:|---:|---|---:|
| H-alpha, 3→2 | 656.28518 | 656.46647 | 415.34437 | G-sharp4 | +0.16537 |
| H-beta, 4→2 | 486.13615 | 486.27195 | 560.71434 | C-sharp5 | +19.71492 |
| H-gamma, 5→2 | 434.0462 | 434.16823 | 628.00462 | D-sharp5 | +15.92598 |
| H-delta, 6→2 | 410.174 | 410.28976 | 664.55388 | E5 | +13.85919 |
| H-epsilon, 7→2 | 397.0072 | 397.11952 | 686.59343 | F5 | −29.65695 |
| H-zeta, 8→2 | 388.9049 | 389.01510 | 700.89734 | F5 | +6.03956 |
| H-eta, 9→2 | 383.5384 | 383.64721 | 710.70413 | F5 | +30.09466 |

Displayed extra calculation digits do not imply corresponding physical accuracy. The mean absolute error is **16.49381 cents**; **2 of 7** entries are within the fixed 10-cent descriptive tolerance.

## Is the alignment unusual?

Assigning the nearest note guarantees every frequency a residual of at most 50 cents. Therefore “it corresponds to a note” alone is not informative.

For the comparison we shifted the whole note grid by a single offset u, uniform over one semitone, without moving hydrogen's lines relative to one another. The statistic was the mean absolute nearest-note error over all seven lines. This preserves the correlations and all frequency ratios instead of treating the lines as independent random draws.

The score is piecewise linear in u. Integrating the lengths of its below-threshold intervals gives **0.2357465138** for the fraction at least as well aligned as the fixed A440 grid. Independently checking 100,000 midpoint offsets gives **23,575/100,000 = 0.23575**. The mean score across all offsets is 25 cents.

This is a chosen random-tuning reference, **not the probability that the theory is false**, and not a universal chance model for atomic spectra. The line selection and mapping are exploratory choices made with familiar data. Saving the protocol before computing its score is a local freeze, not external preregistration or fresh held-out validation.

There is also a useful exact statement: replacing 40 with any integer k shifts m by 12(40−k), an integer number of octaves. It cannot change the nearest-note error in cents. Searching different integer octave counts cannot improve this score.

## A physical spectral benchmark

The nonrelativistic reduced-mass Coulomb model predicts

\[
f_{n\to2}=\frac{cR_\infty}{1+m_e/m_p}
\left(\frac14-\frac1{n^2}\right).
\]

Using [CODATA 2022 constants](https://physics.nist.gov/cuu/Constants/Table/allascii.txt), without fitting these seven entries, gives a mean absolute frequency discrepancy of **0.01463 cents**, with maximum absolute discrepancy **11.04 parts per million**. This is an established gross-structure benchmark; it omits fine structure, Lamb shifts and higher corrections. It is not a fully independent experimental validation of quantum theory: the constants and several reference entries already incorporate spectroscopy or theoretical calculations.

The musical score and physical discrepancy do not have identical statistical meanings: the musical procedure chooses the nearest note for every line, whereas the physical expression predicts each transition from its n. We do not assign a likelihood ratio between them.

## Alpha: approximate resemblance versus equality

The low-energy CODATA 2022 reference is

\[
\alpha^{-1}=137.035999177\pm0.000000021
\]

where the quoted uncertainty is one standard uncertainty. Alpha itself is dimensionless, so it does not depend on choosing seconds, metres or a musical tuning convention. That makes it a legitimate target for a physical derivation, but not a license to equate any nearby expression with it.

| Proposed inverse-alpha expression | Calculated value | Error relative to reference | Error (ppm) |
|---|---:|---:|---:|
| 137 exactly | 137 | −0.035999177 | −262.69869 |
| 4*pi^3 + pi^2 + pi | 137.0363037758784 | +0.0003045988784 | +2.22277 |

The pi expression differs by roughly **14,505 reference standard uncertainties**. That is a discrepancy scale, not a complete statistical model or a claim of a “14,505-sigma discovery.” It is an already-known illustrative numerical approximation, not a new result derived from the user's theory. We did not search a catalog of expressions for a best fit. The rounded shorthand alpha approximately 1/137 remains useful; the assertion of exact equality fails this check.

## Verification and reproduction

**Ten unit tests passed.** A separately written audit, importing none of the main implementation, reproduced the material calculations using 60-digit Decimal arithmetic, bisection for the air conversion, natural logarithms, and Machin's formula for pi. It checked the common-offset result by enumeration rather than piecewise integration.

This is numerical verification, not a second experiment or independent source data.

Run with Python 3.10 or later; no third-party packages or network required:

```sh
python3 -m unittest -v test_checks.py
python3 run_checks.py
python3 independent_audit.py
```

The saved `RESULTS.json` and `AUDIT.json` retain full arithmetic output; their extra digits are not claims of physical precision. `--output NEW_NAME.json` writes a new file and refuses to overwrite an existing result. `PROTOCOL.json` is hashed into both records.

## Next test

Keep this mapping frozen and compare hydrogen with deuterium isotope shifts, specifying the transitions and score before retrieving the next dataset. Require predictions of the shifts, not just fresh nearest-note labels. The reduced-mass model provides a baseline; any proposed geometric alternative must state a different, quantitative prediction to be distinguishable. Deriving alpha needs an explicit model equation first, not another fitted constant.

Conclusion: one beautiful near-match is real arithmetic. The proposed general musical law and geometric derivation of alpha are not established. These results do not establish sound-to-light conversion, extra physical dimensions, or a theological identification.