Frequency gives a precise way to describe temporal repetition in a light field. Colour names a perceptual and measurement problem that usually involves an entire spectrum and an observer. Connecting the two is useful; treating them as identical hides how much an observation can leave unresolved.
From period to frequency
For a monochromatic wave, frequency f and period T satisfy f = 1/T. In vacuum, wavelength λ satisfies c = λf. Thus a vacuum wavelength of 550 nanometres corresponds to about 5.451 × 1014 cycles per second and a period of about 1.835 femtoseconds.
These numbers describe an ideal monochromatic component. A short pulse or ordinary illumination generally contains a range of frequencies. Assigning one frequency can be a useful approximation or a reference value, but its meaning should be stated.
The wavelength also depends on the medium. At a stationary interface in a time-independent system, the temporal frequency is preserved while the phase velocity and wavelength can change. The relation λ = c/f applies to vacuum; replacing c with an appropriate phase velocity is necessary inside a material.
A spectrum has more than three numbers
A spectrometer estimates a distribution over wavelength or frequency. A colour measurement often reduces that distribution to a few weighted responses. In standard colorimetry, tristimulus values are calculated using specified colour-matching functions and viewing assumptions.
The CIE publishes the 1931 standard observer functions and documents how such functions enter tristimulus calculations. Those standard functions are a defined observational description. They should not be confused with three narrow filters that each detect only one wavelength.
An explicit ambiguity
Consider an invented instrument that receives light in four spectral bins. Let their nonnegative intensities be s₁, s₂, s₃ and s₄. Give it three output channels:
r₁ = s₁ + s₂, r₂ = s₂ + s₃, r₃ = s₃ + s₄.
The spectrum (2, 2, 2, 2) produces (4, 4, 4). The different spectrum (1, 3, 1, 3) produces the same output. Their difference, (−1, 1, −1, 1), lies in the measurement map's kernel: adding it changes the spectrum but leaves all three recorded values unchanged.
This is a teaching example, not a model fitted to human vision or actual CIE data. It demonstrates the mathematical possibility behind a familiar issue: different spectra can match under a specified colour measurement. Such matches are called metameric when they concern colour stimuli under the relevant observer and conditions.
Add a fourth measurement that reads s₁ alone. The two example spectra now give different results. The additional channel resolves this particular ambiguity because it is sensitive to the previously invisible difference. Merely duplicating r₁ would add another recorded number without adding independent information.
Measurement and experience
Colour appearance depends on more than a wavelength label. The illuminating spectrum, an object's wavelength-dependent response and viewing conditions all matter. A specified standard observer makes comparison reproducible within its intended conditions; it does not erase every difference between individual observers or visual contexts.
A camera's three channels also need calibration before they can stand for a standard colour measurement. Three arbitrary numbers are not automatically CIE tristimulus values. Displaying those numbers then introduces the display's own characteristics and the conditions under which the image is viewed.
Connect the readout to the world
The distinction between a spectrum and its recorded channels connects directly to three-dimensional readout and memory for prediction. A lower-dimensional description can be adequate for one task while losing distinctions needed for another.
In SMM/DTT, a proposed map from inner generation to observable completion must likewise state which distinctions the readout preserves. Our three-channel example does not derive three-dimensional space or explain colour experience from the Single Monad. It makes a narrower requirement concrete: identify the map, its kernel and the new experiment that would reveal a hidden difference.
Sources and further reading
- ISO/CIE 11664-3:2019: CIE tristimulus values.
- BIPM: the SI metre. The frequency example uses the exact SI value of c.
- What an image records; what a proof proves about a model; Duality of Time: book contents.