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 "white," "roth", "pastel," or "rosy", giving theineqmats ofmake_white_ineqmat(),make_roth_ineqmat(),make_pastel_ineqmat(), andmake_rosy_ineqmat(). For other arrangements, the desired inequality matrix can be entered directly.- 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
# 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)]
#> 7-1 7-35 7-34 7-33 7-32 7-2 7-3 7-30 7-5 7-7 7-37 7-22 7-31 7-4 7-6 7-13
#> [1,] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
#> [2,] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
#> [3,] 2 3 3 2 3 2 2 2 2 2 3 2 3 2 2 2
#> [4,] 3 5 4 4 4 3 3 4 3 3 4 5 4 3 3 4
#> [5,] 4 6 6 6 6 4 4 6 5 6 5 6 6 4 4 5
#> [6,] 5 8 8 8 8 5 5 8 6 7 7 8 7 6 7 6
#> [7,] 6 10 10 10 9 7 8 9 7 8 8 9 9 7 8 8
#> 7-38 7-21 7-29 7-20 7-8 7-12 7-11 7-19 7-17 7-27 7-25 7-15 7-28 7-24 7-26
#> [1,] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
#> [2,] 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1
#> [3,] 2 2 2 2 3 2 3 2 2 2 3 2 3 2 3
#> [4,] 4 4 4 5 4 3 4 3 4 4 4 4 5 3 4
#> [5,] 5 5 6 6 5 4 5 6 5 5 6 6 6 5 5
#> [6,] 7 8 7 7 6 7 6 7 6 7 7 7 7 7 7
#> [7,] 8 9 9 9 8 9 8 9 9 9 9 8 9 9 9
#> 7-9 7-10 7-36 7-16 7-14 7-18 7-23
#> [1,] 0 0 0 0 0 0 0
#> [2,] 1 1 1 1 1 1 2
#> [3,] 2 2 2 2 2 4 3
#> [4,] 3 3 3 3 3 5 4
#> [5,] 4 4 5 5 5 6 5
#> [6,] 6 6 6 6 7 7 7
#> [7,] 8 9 8 9 8 9 9
# 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))
#> NULL
whichsvzeroes(hex1)
#> [1] 1 4 5 7 8 10 17
whichsvzeroes(hex2)
#> [1] 1 4 5 7 8 10 17
whichsvzeroes(hex3)
#> [1] 1 4 5 7 8 10 17
hex_words
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> hex1 2 2 1 2 2 3
#> hex2 1 1 2 1 1 3
#> hex3 3 3 2 3 3 1