musicMCT (version 0.2.0)

eps: The brightness ratio

Description

Section 3.3 of "Modal Color Theory" describes a "brightness ratio" which characterizes the modes of a scale in terms of how well "sum brightness" acts as a proxy for "voice-leading brightness." Scales with a brightness ratio less than 1 are pretty well behaved from this perspective, while ones with a brightness ratio greater than 1 are poorly behaved. When the brightness ratio is 0, sum brightness and voice-leading brightness give exactly the same results. (This can happen for sets on two extremes: those like the diatonic scale which are well formed and those like the Weitzmann scales, which differ from "white" in only one scale degree.)

I wish I had come up with a more descriptive name than "brightness ratio" for this property, because it's not really a ratio of brightness in the sense you might expect (i.e. "this scale is 20% bright"). Rather, it's a ratio of two brightness-related properties, delta and eps. "Modal Color Theory" (p. 20) offers definitions of these. Delta is "the largest sum difference between (voice-leading) incomparable modes," with value 0 by definition if all of the modes are comparable. ("This, in a sense, is a measure of how badly voice-leading brightness breaks down from the perspective of sum brightness.") Epsilon "represents the smallest sum difference between non-identical but comparable modes." This is harder to give an intuitive gloss on, but my attempt in "MCT" was "Essentially, epsilon measures the finest distinction that voice-leading brightness is capable of parsing."

The brightness ratio (ratio) itself is simply delta divided by epsilon.

Usage

eps(set, edo = 12, rounder = 10)
delta(set, edo = 12, rounder = 10)
ratio(set, edo = 12, rounder = 10)

Value

Single non-negative numeric value

Arguments

set

Numeric vector of pitch-classes in the set

edo

Number of unit steps in an octave. Defaults to 12.

rounder

Numeric (expected integer), defaults to 10: number of decimal places to round to when testing for equality.

Examples

harmonic_minor <- c(0, 2, 3, 5, 7, 8, 11)
hypersaturated_harmonic_minor <- saturate(2, harmonic_minor)
c(delta(harmonic_minor), eps(harmonic_minor))
c(delta(hypersaturated_harmonic_minor), eps(hypersaturated_harmonic_minor))

# Delta and epsilon depend on the precise scale, but ratio() is constant on a hue
ratio(harmonic_minor)
ratio(hypersaturated_harmonic_minor)

#### Sort all 12tet heptachords by brightness ratio
heptas12 <- unique(apply(combn(12, 7), 2, primeform),MARGIN=2)
hepta_ratios <- apply(heptas12, 2, ratio)
sorted_heptas <- heptas12[, order(hepta_ratios)]
colnames(sorted_heptas) <- apply(sorted_heptas, 2, fortenum)
sorted_heptas

#### Compare evenness to ratio for 12tet hetpachords
plot(apply(heptas12, 2, evenness), hepta_ratios, xlab="Evenness", ylab="Brightness Ratio")