As a perceptual characteristic of sound, chroma has its roots in psychology, modeling human categorization of pitches. A chroma, essentially a pitch class under octave equivalence, stands distinct from the broader concept of a pitch class used in some music schools, which refers to a period within a construction (a potentially arbitrary selection for an equivalence relation).
Humans consider pitches in a 1:2 ratio as being equivalent, having the same "color". This aligns with the observed logarithmic perception of sound, highlighted by the timeless practice of functional music (the function of music refers to the ubiquitous tension/release nature of chord progressions like \(\text{V}_7 \to \text{I}\) ).
Chromas are the fractional part of the logarithm with base 2 of x. However, since we are talking about ratios, chromas are normalized to the octave space to express them as a ratio from 1:
\( \text{chroma}(x)=2^{\log_2(x)\mod 1} \) or \( \Xi(x) = x \mod 1:2 \)
So, the chroma of 3, 6, 12, 24, etc., is 1.5 or 3/2, the fifth.
Generalized Reduction (ratio/intervallic remainder) function:
\[ \mathop{ \Xi}_{[a,b]}(x) = x \bmod a:b = \frac xa \bmod \left[1,\frac ba\right] = \left(\frac ba \right)^{\left(log_{b/a}( x/a) \bmod 1\right) }\]
The term is not just convenient but accurate. Its precise definition and utility are often misunderstood due to misconceptions about synesthesia and holistic interpretations.
There is no "red pitch"; rather, by assigning a reference pitch to a color, one can map chromas to different ratios. It’s a relative property but with an exact correspondence. We are blind to an octave above the frequency of red, as the visible light spectrum spans precisely one octave in frequency.
If we select red as unison (1:1), the RGB set is in a 4:5:6 ratio, a major chord:
Red ~ 400 THz = f(1)
Green ~ 500 THz = f(Red * 5/4)
Blue ~ 600 THz = f(Red * 3/2)
With violet as root, the 4:5:6 chord wraps chromas, the color wheel is modular.
Violet ~ 700 THz = f(1)
Yellow ~ 466 THz = f(Violet * 5/4 mod Red)
Cyan ~ 525 THz = f(Violet * 3/2 mod Red)
(Note: The colors and frequencies presented are illustrative approximations. Determining the exact transition point between colors is analogous to a fundamental question in music theory: when does one pitch become the next? Functional music theory addresses this by emphasizing the importance of context over fuzzy interval boundaries. A more radical interpretation suggests that chord progressions, such as the high-entropy V7 to low-entropy I, are inherent musical structures independent of human perception.)
"Optical illusions" demonstrating how context affect chroma interpretation (the additive light synthesis works for far smaller resolutions than screens too), open and zoom!
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| Both squares are the same color; orange/red with yellow/light green, which are unidentifiable on the left. |
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| Both squares are the same light green. |
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| Both circles are red. |
To analyze any tuning system, it is more useful to understand its chroma content than any other aspect. A simple rule is that for finite generating sets, if the period is not in phase with the octave, the chromas are infinite, making these types of systems less useful for common music practice.
Consider the Bohlen-Pierce tuning, its equalized form named 13ed3 (13 equal divisions of 3, the tritave). Though it "has 13 classes," this is but a mirage, as the chromas are infinite. Also, tunings with uniform step size trivializes any equivalence relation, leaving no "fundamental domain."
With octave equivalence, you can join any jam by just trusting pitch classes with the same chromas. If the chord is an A, you can safely play it in any range. In a non-octave gathering, where every participant plays in a different, for example, tritave, while it may be a fun experiment, any intended functional music will be disrupted.
Non-octave tunings, with their siren call of possibilities, ultimately hinder collaborative music-making.
A set with infinite non repeated chromas are the prime numbers.
A set with infinite repeated chromas are the odd numbers.
A set with infinite identical chromas are the even numbers.
(DRAFT)
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