Visualizing Inferred Fundamentals in the 3D Harmony Structure

[jimishol]

Although the arguments about the [missing fundamental](https://en.wikipedia.org/wiki/Missing_fundamental) were essential for discovering the three‑dimensional structure, the fundamentals were not shown visually on the screen. Like the surfaces (which can be toggled off by pressing s), there is subjectivity about whether their visualization actually helps musical analysis. While watching MIDI playback of classical pieces I often wondered what the fundamental frequency of certain chords might be, so I added an inferred‑fundamental visualization to the project.

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What I added and where to look

• Maxima script implementing the original computation:
  asset_pipeline/missing_fountamental.mac
  https://github.com/jimishol/cholidean-harmony-structure/blob/main/asset_pipeline/missing_fountamental.mac

• Lua implementation used by the renderer:
  src/systems/fountamental.lua
  https://github.com/jimishol/cholidean-harmony-structure/blob/main/src/systems/fountamental.lua

• Test script for the Lua module (now in asset_pipeline):
  asset_pipeline/test_fountamental.lua
  https://github.com/jimishol/cholidean-harmony-structure/blob/main/asset_pipeline/test_fountamental.lua

• Visualization control constant:
  src/constants.lua (see mfState)
  https://github.com/jimishol/cholidean-harmony-structure/blob/main/src/constants.lua

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How to run

• Run the Maxima script:

maxima -b asset_pipeline/missing_fountamental.mac

• Test the Lua module:

lua asset_pipeline/test_fountamental.lua

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Implementation details

• Computation
  The algorithm estimates an inferred fundamental using a heuristic based on optimal division by two‑digit integers. A detected fundamental is accepted only if it is at or above C0, implemented as:

local has_fundamental = fundamental_midi >= 12

• Visualization
  Inferred fundamentals are visualized by increasing the size of joints and/or labels. The behavior is controlled by the mfState constant in src/constants.lua:

  • mfState = 0 — fundamentals are not visualized (the has_fundamental guard still applies).
  • mfState = 1 — joints are enlarged (imprecise detection; requires calibration of label_distance).
  • mfState = 2 — labels are enlarged (default).
  • mfState = 3 — both labels and joints are enlarged (visually heavy; may obscure other information and requires label_distance tuning).

• Interpretation
  Enlarged, glowing elements represent fundamentals that are actually sounding; non‑glowing elements indicate inferred fundamentals that are not physically present but are perceived to exist. For example, the interval E–G can give the impression of a C fundamental, and some tuners will interpret that chord as a C.

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Caveats and open questions

• I am not certain whether estimating fundamentals by the simple quotient of two‑digit integers is accepted practice among music theorists.
• The missing fundamental concept is perceptual and context dependent; the heuristic here is exploratory.
• I welcome suggestions for alternative detection methods and feedback on whether the visualization helps or distracts in harmonic analysis.