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In Is there anything special about A♭ in a Tonnetz?, the main diagram makes very clear that there is one adjacent move on the Tonnetz that, surprisingly, doesn't have a single-letter transformation associated with it. Here is the diagram:

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On the left side of the diagram, we clearly see that adjacent triangles that share two pitches (i.e., a side) can be understood as P, L, or R. Immediately above we see that, when two triangles share a single node, they can be understood as N or S.

But what about the third option, the PRL transformation that toggles between, e.g., FM and Cm? It seems odd that, in a theory so focused on smooth voice leading, a chord change that shares a common tone would not have a dedicated transformation. Is there a single-letter transformation for this switch?

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  • 1
    ...and if not, why?
    – Aaron
    Sep 4, 2022 at 17:45
  • @Aaron my guess is that it has something to do with the tritone between the thirds of the two chords (e.g. A and E flat).
    – phoog
    Sep 5, 2022 at 10:54

1 Answer 1

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In David Kopp's book Chromatic Transformations in Nineteenth-Century Music, he does introduce a single-letter transformation that could address this issue.

In Chapter 7 (see specifically pp. 170ff), Kopp introduces the F transformation. F stands for "fifth-change operation," and it connects two triads with different modes related by descending perfect fifth.

Beginning with a C-major triad as stated in the question, an F transformation changes the mode from major to minor and down a perfect fifth to F minor.

But there is a caveat—and a pretty big one in my view. The transformations cited in the question (P, L, R, S, N, and H) are involutions, meaning that applying the same transformation to the resulting triad returns you to the original triad. In other words, applying the P transformation to C major takes you to C minor, and applying that same P to C minor takes you right back to C major.

The F transformation, however, is not an involution, because applying F again to the resulting F-major triad from earlier takes us not back to C minor but instead to B♭ minor (recall that the definition of F connects two triads related by descending perfect fifth). However, Kopp does address this by creating the F-1 transformation, which would move by ascending fifth to get us back to C minor.

Perhaps this is one reason why—in addition to phoog's great comment that the chordal thirds are a tritone apart—this transformation is simply not mentioned in the same breath as the other six. Nevertheless, for the sake of completeness, I think it's worth having a single-letter transformation—and one that's an involution—for this system. The only question is what to call it...

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