: What do the terms "pitch relationships" and "vertical dimension" signify in relation to harmony? I am trying to learn more about the neurological basis of harmony perception. However, I am having a bit of trouble understanding

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What do the terms "pitch relationships" and "vertical dimension" signify in relation to harmony?
I am trying to learn more about the neurological basis of harmony perception. However, I am having a bit of trouble understanding the musical technical terminology being used by the authors in a chapter of a book entitled "Neurobiology of Harmony Perception" by Tramo et al. in The Cognitive Neuroscience of Music.

In particular, they say

In this paper, we present neurophysiological, neurological, and psychoacoustic evidence to support our contentions that (1) pitch relationships among tones in the vertical dimension influence consonance perception and (2) consonance cannot be explained solely by the absence of roughness. (pg 130)

This claim seemed hard to interpret for me without defining the ideas of "pitch relationships" and "vertical dimension."

Here, I think they define "vertical dimension"

We restrict our consideration of harmony to basic tenets articulated
by Piston,2 among others. Harmony has a vertical dimension and a
horizontal dimension. The vertical dimension encompasses the
relationships among simultaneous notes. By convention, note refers to
a pitch in the musical scale, and harmonic interval refers to two
notes sounded simultaneously. (pg 128).

By "pitch relationships", I think this paragraphs clarifies what it means

In summary, the temporal fine structure of the perfect fifth and fourth contains representations of the two notes constituting the interval, plus harmonically related bass notes that are implied by the interval. In music, these bass notes support the deep structure of harmony. Parncutt demonstrated experimentally that listeners associate major triads with pitches that are harmonically related to note F?s, including the fundamental bass, plus
the pitches of note F?s actually in the stimulus. These pitches cannot be accounted for simply on the basis of combination tones. Houtsma and Goldstein showed that musicians can use missing F? pitches to identify melodic intervals (major and minor seconds and thirds), even when two upper harmonics are presented
separately (dichotically) to each ear. (134-135)

Also this may be of help for interpreting

We interpret our findings and the results of previous psychoacoustic experiments as evidence in favour of the hypothesis that harmony in the vertical dimension, like harmony in the horizontal dimension, is principally a function of the pitch relationships among tones, with roughness playing a secondary role. In light of these observations, and in view of the likelihood that cognitive representations of pitch hierarchies influence harmony perception in the vertical dimension, we urge that the terms sensory consonance and sensory dissonance be reconsidered. (pg 147)

Thus, if I am interpreting their claim accurately, I think they mean the following

a chord’s consonance depends on both (1) the ratios between the notes and (2) the absence of roughness

MY QUESTION

Am I interpreting their claim correctly? What do the terms "vertical dimension" and "pitch relationship" signify in this context?

ASIDE

Regarding the term roughness, the following passage may help define it:

Figure 9.6C shows Plomp and Levelt’s idealized plot of the relationship between consonance and critical bandwidth (the latter is defined here by loudness summation). Note that the curve reaches an asymptote near the end of the x axis, at about one critical bandwidth. Thus a critical band account of consonance as the absence of roughness cannot apply to pure-tone intervals that are wider than a minor third or so. (pg 142).

So I assume they mean "roughness" in the sense of Plomp and Levelt. See this answer for more information about newer versions related to their models.

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