Calculating Sign Vectors

The main tool introduced in this paper is the sign vector for voice leadings, which is calculated in musicMCT by the function vl_signvector(). Following the talk’s focus on Wagner’s Tristan und Isolde, let us study the voice leadings that resolve the Tristan chord in the Vorspiel’s first and third phrases (mm. 1-3 and 8-11). As in the paper, we’ll call those voice leadings \(\alpha\) and \(\beta\):

alpha <- c(-1, -3, -1, 3)
beta <- c(-1, -2, 1, 4)

In this model, voice leadings should be entered in ascending registral order. The first entry of a voice leading represents that bass’s motion; the last entry represents the highest voice’s motion.

We now can calculate the sign vectors for these two motions:

vl_signvector(alpha)
#>  [1] -1 -1 -1  1  1  1  0  1 -1  1  1  1 -1  1  1  1 -1  1  1  1  1
vl_signvector(beta)
#>  [1] -1 -1  0  1  1  1  1  1 -1  1  1  1 -1  1  1  1  1  1  1  1  1

Although it’s possible to inspect the sign vectors by hand, it’s more convenient to let R do calculations with them. For instance, one of the measures of similarity proposed in the paper involves counting how many positions in the sign vectors are different.

alpha_sv <- vl_signvector(alpha)
beta_sv <- vl_signvector(beta)
ab_difference <- alpha_sv - beta_sv

which(ab_difference != 0)
#> [1]  3  7 17

This shows that the two sign vectors differ in only three positions: the 3rd, 7th, and 17th entries of the sign vector.

We might then want to know what specific contrapuntal behaviors those differences correspond to. For this, we need to know what hyperplane arrangement was used to compute the sign vectors. Musically, this is equivalent to knowing what specific questions we asked about the voice leadings. The default hyperplane arrangement, for voice leadings, is what I call the “infrared” hyperplane arrangement (see make_infrared_ineqmat()). We can get information about the specific hyperplanes where \(\alpha\) and \(\beta\) differ with the following.

tetrachord_arrangement <- make_infrared_ineqmat(4)
tetrachord_arrangement[c(3, 7, 17), ]
#>           [,1]      [,2]     [,3] [,4] [,5]
#> [1,]  0.000000 -1.000000 2.000000   -1    0
#> [2,] -2.000000  0.000000 2.000000    0   -1
#> [3,] -2.666667  1.333333 1.333333    0   -1

Each row of the matrix above corresponds to one hyperplane in the arrangement, reflecting some musical quality on which the voice leadings differ. The columns (except for the last column) correspond to the voices in the progression, so that the first column corresponds to the bass, the second column to the tenor, and so on.

Thus, the first row of the matrix above shows that one hyperplane where \(\alpha\) and \(\beta\) differ has to do with the relative motions of tenor, alto, and soprano. This row has a 0 in the first column, indicating that the motion of the bass doesn’t affect this musical quality at all. The third column has a 2, which indicates that the behavior of the alto is especially important for this property. Ultimately, this hyperplane asks whether the tenor or soprano bunches up with the alto more. In \(\alpha\), the alto ends up moving closer to the tenor, whereas in \(\beta\), the soprano and tenor move away from the alto by equal amounts.

The second row of the matrix only has entries for the first and third columns, so this hyperplane only cares about the relative motion of the bass and alto. Do the voices get farther apart? In \(\alpha\) the answer is “no,” so the sign vector for \(\alpha\) has a \(0\) in the corresponding position for the sign vector. In \(\beta\), the bass and alto do get rather apart, so the sign vector for \(\beta\) has a \(1\) here:

vl_signvector(alpha)[7]
#> [1] 0
vl_signvector(beta)[7]
#> [1] 1

Which hyperplane arrangement to choose?

The voice-leading properties discussed in the “Duality” paper are the ones captured by the “infrared” hyperplane arrangement. These have to do with (1) how voices move relative to each other, and (2) how pairs of duets compare to each other, measuring how much voices “bunch up” with one another. However, you might not necessarily always want to ask exactly those questions.

For instance, you might not be interested in the “bunching up” behavior described by questions of type (2). If you only want to know about the two-voice contrapuntal motions of (1), these are described by what I call the “white” hyperplane arrangement. The white arrangements themselves are created by the function make_white_ineqmat(), but they can be specified in calls to other functions directly. For instance, we can compute the sign vectors for \(\alpha\) and \(\beta\) using only the white hyperplanes as follows:

vl_signvector(alpha, ineqmat="white")
#> [1] -1  0  1  1  1  1
vl_signvector(beta, ineqmat="white")
#> [1] -1  1  1  1  1  1

The parameter in musicMCT functions that specifies which hyperplane arrangement to use is the ineqmat parameter as in the code block above.

Instead of asking about relative motions as in the white arrangements, you might simply want to know whether each voice moves up or down. Those properties are capture by the “black” arrangements (make_black_ineqmat()):

vl_signvector(alpha, ineqmat="black")
#> [1] -1 -1 -1  1
vl_signvector(beta, ineqmat="black")
#> [1] -1 -1  1  1

The black signvector for \(\alpha\) starts with three \(-1\)s because the bass, tenor, and alto all descend. The soprano ascends, so the last entry of \(\alpha\)’s sign vector is a positive 1. (If a voice doesn’t move, its corresponding entry in the black signvector would be \(0\).) Since the alto voice in \(\beta\) does ascend, the third entry of \(\beta\)’s black sign vector is \(1\).

Another hyperplane arrangement that might be useful for exploring voice leadings is the “ultraviolet” arrangement, which is interested in perfectly balanced contrary motion. For instance, one hyperplane of the ultraviolet arrangement reads out a \(0\) if the bass and soprano move in opposite directions by the same amount. Let’s look at \(\alpha\) and \(\beta\) through this lens:

vl_signvector(alpha, ineqmat="ultraviolet")
#> [1] -1 -1  1 -1  0  1
vl_signvector(beta, ineqmat="ultraviolet")
#> [1] -1  0  1 -1  1  1

Each has one zero in its sign vector, which means that each has exactly one pair of voices moving in balanced contrary motion. In \(\alpha\), that pair is the tenor and soprano, whose voice exchange makes each line move by 3 semitones in opposite directions. In \(\beta\), that pair is the bass and alto, which wedge apart semitonally.

It’s possible to mix and match hyperplane arrangements, if you are interested in multiple sets of questions. For instance, you might want to know about the absolute motions of individual voices (the black hyperplanes) as well as the relative motions of voice pairs (the white hyperplanes). For some combinations of hyperplane arrangements, musicMCT has pre-defined names for them. For instance, the combination of black and white arrangements is called the gray arrangement:

vl_signvector(alpha, ineqmat="gray")
#>  [1] -1  0  1  1  1  1 -1 -1 -1  1
vl_signvector(beta, ineqmat="gray")
#>  [1] -1  1  1  1  1  1 -1 -1  1  1

In other cases, the specific combination you want may not be pre-programmed. In this case, you can define your own arrangement by manually creating its matrix. Here, for instance, is how you could find the sign vector for infrared and ultraviolet arrangements at the same time:

tetrachord_infrared <- make_infrared_ineqmat(4)
tetrachord_ultraviolet <- make_ultraviolet_ineqmat(4)
ir_uv_spectrum <- rbind(tetrachord_infrared, tetrachord_ultraviolet)

vl_signvector(alpha, ineqmat=ir_uv_spectrum)
#>  [1] -1 -1 -1  1  1  1  0  1 -1  1  1  1 -1  1  1  1 -1  1  1  1  1 -1 -1  1 -1
#> [26]  0  1
vl_signvector(beta, ineqmat=ir_uv_spectrum)
#>  [1] -1 -1  0  1  1  1  1  1 -1  1  1  1 -1  1  1  1  1  1  1  1  1 -1  0  1 -1
#> [26]  1  1

Notice that, in this case where I have defined a hyperplane arrangement manually with a matrix, the ineqmat parameter takes a variable name without quotation marks (ir_uv_spectrum) unlike in the earlier examples, which named a general category of arrangements (such as "white" or "black") as a string in quotation marks.

Other structural properties

The central claim of “Chord-Voice Leading Duality” is that the tools musicMCT offers for chords and scales are also useful for voice leadings. It’s therefore very much worth trying to use other functions, like howfree() and whichsvzeroes().

In this case, it’s important to know that musicMCT defaults to a scale-oriented coordinate system, and that to get correct results from functions other than vl_signvector() we need to convert voice leadings into the right coordinates.

That is, musicMCT is designed to accept scales defined as pitch numbers. For instance, a C major triad would be represented as \((0, 4, 7)\). As “Duality” describes, the right coordinate system for treating chords and voice leadings as dual is the one in which perfectly even scales and voice leadings are represented as the origin. In this coordinate system, the C major triad should be represented as \((0, 0, -1)\), since it differs from the perfectly even trichord \((0, 4, 8)\) by just a semitone in the top voice.

The package has functions that convert between these coordinate systems:

c_major <- c(0, 4, 7)
cmaj_displacement <- c(0, 0, -1)

coord_to_edo(c_major)
#> [1]  0  0 -1
coord_from_edo(cmaj_displacement)
#> [1] 0 4 7

Because most functions in musicMCT expect inputs in the pitch-centric coordinate system, voice leadings need to be converted to those coordinates with coord_from_edo(). For instance, we might be interested in how many independent ways \(\beta\) can vary without losing its qualitative identity. The right answer here is \(2\), but we won’t get that if we forget to use the right coordinate system:

howfree(beta, ineqmat="infrared")
#> [1] 1
howfree(coord_from_edo(beta), ineqmat="infrared")
#> [1] 2

In the above example, not too that we had to explicitly specify that howfree() should use the infrared hyperplane arrangement. Each function has an arrangement type that it defaults to; while vl_signvector() defaults to the infrared arrangements, most other functions default to the “Modal Color Theory” arrangements introduced in the Journal of Music Theory article of the same name. (The MCT arrangements are essentially a subset of the full infrared arrangements.)

For an introduction to other functions that might be useful for exploring voice leadings, I encourage you to explore the package’s other vignettes, as well as its documentation for individual functions.

Last updated: 1 November 2026