whichsvzeroes: Which interval-comparison equalities does a scale satisfy?

In musicMCT: Analyze the Structure of Musical Scales

Which interval-comparison equalities does a scale satisfy?

Description

As "Modal Color Theory" (p. 26) describes, one useful measure of a scale's regularity is the number of zeroes in its sign vector. This indicates how many hyperplanes a scale lies on, a geometrical fact whose musical interpretation is, roughly speaking, how many times two generic intervals equal each other in specific size. (I say only "roughly speaking" because one hyperplane usually represents multiple comparisons: see Appendix 1.1.) Scales with a great degree of symmetry or other forms of regularity such as well-formedness tend to be on a very high number of hyperplanes compared to all sets of a given cardinality.

musicMCT offers two convenience functions that return pertinent information from signvector(). countsvzeroes returns this count of the number of sign-vector zeroes, while whichsvzeroes gives a list of the specific hyperplanes the scale lines on (numbered according to their position on the given ineqmat). The specific information in whichsvzeroes can be useful because it determines the "flat" of the hyperplane arrangement that the scale lies on, which is a more general kind of scalar structure than color (as determined by the entire sign vector).

Usage

whichsvzeroes(set, ineqmat = NULL, edo = 12, rounder = 10)

countsvzeroes(set, ineqmat = NULL, edo = 12, rounder = 10)

Arguments

set	Numeric vector of pitch-classes in the set
ineqmat	Specifies which hyperplane arrangement to consider. By default (or by explicitly entering "mct") it supplies the standard "Modal Color Theory" arrangements of getineqmat(), but can be set to strings "white," "black", "gray", "roth", "infrared", "pastel", "rosy", "infrared", or "anaglyph", giving the ineqmats of make_white_ineqmat(), make_black_ineqmat(), make_gray_ineqmat(), make_roth_ineqmat(), make_infrared_ineqmat(), make_pastel_ineqmat(), make_rosy_ineqmat(), make_infrared_ineqmat(), or make_anaglyph_ineqmat(). For other arrangements, this parameter accepts explicit matrices.
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.

Value

Single numeric value for countsvzeroes and a numeric vector for whichsvzeroes

Examples

# Sort 12edo heptachords by how many sign vector zeroes they have (from high to low)
heptas12 <- unique(apply(utils::combn(12, 7), 2, primeform), MARGIN=2)
heptas12_svzeroes <- apply(heptas12, 2, countsvzeroes)
colnames(heptas12) <- apply(heptas12, 2, fortenum)
heptas12[, order(heptas12_svzeroes, decreasing=TRUE)]

# Multiple hexachords on the same flat but of different colors
hex1 <- c(0, 2, 4, 5, 7, 9)
hex2 <- convert(c(0, 1, 2, 4, 5, 6), 9, 12)
hex3 <- convert(c(0, 3, 6, 8, 11, 14), 15, 12)
hex_words <- rbind(asword(hex1), asword(hex2), asword(hex3))
rownames(hex_words) <- c("hex1", "hex2", "hex3")
c(colornum(hex1), colornum(hex2), colornum(hex3))
whichsvzeroes(hex1)
whichsvzeroes(hex2)
whichsvzeroes(hex3)
hex_words

musicMCT documentation built on June 21, 2026, 9:06 a.m.