Neo-Riemannian examples in music

The field of math/music can be sometimes abstract, and I always try to find examples to illustrate the notions I’m presenting. Recently, I had to give a course in transformational music theory, and thus to find notable examples in neo-Riemannian theory. Based on my previous posts about the T6 transformation (see here, and there), I decided to compile these examples in one single video: this allowed me to put cues showing exactly when the T6 transformation occurs, and to add the corresponding scene when dealing with opera or film music.

The T6 video was well-received during the course, so I decided I could actually do it for some other notable neo-Riemannian transformations. In the preparation of these videos, I relied heavily on the existing literature about neo-Riemannian analysis in music, opera, and more particularly film music. Here is a (non-exhaustive) list of references which I used:

  • Frank Lehman, Hollywood Harmony, Oxford University Press. The reference book on transformational music theory in film music.
  • Frank Lehman, Transformational Analysis and the Representation of Genius in Film Music, Music Theory Spectrum, Vol. 35 (1), 2013, pp. 1-22.
  • Scott Murphy, The Major Tritone Progression in Recent Hollywood Science-Fiction Films, Music Theory Online, available online.
  • Scott Murphy, Scoring Loss in Some Recent Popular Film and Television. Music Theory Spectrum (Fall 2014) 36 (2): 295-314.
  • Bribitzer-Stull, M. (2015). From Nibelheim to Hollywood: The associativity of harmonic progression. In Understanding the Leitmotif: From Wagner to Hollywood Film Music (pp. 131-156). Cambridge: Cambridge University Press.
  • Richard Cohn, Uncanny Resemblances: Tonal Signification in the Freudian Age, Journal of the American Musicological Society, Vol. 57, No. 2 (Summer 2004), pp. 285-324
  • Richard Cohn, Hexatonic Poles and the Uncanny in Parsifal, The Opera Quarterly, Volume 22, Issue 2, spring 2006, Pages 230–248.
  • Erik Heine, Chromatic Mediants and Narrative Context in Film, Music Analysis, 2018, 37: 103-132. https://doi.org/10.1111/musa.12106

And now, a quick presentation of the different videos, which have been posted on YouTube.

The T6 transformation

As said above, I have talked about this transformation on this blog before. It corresponds to the transposition of a major or a minor triad by six semitones, as shown below.

Major T6 transformationMinor T6 transformation

It combines many peculiarities, such as the tritone between the roots, the central inversion, and the parsimonious voice-leading of two tones, which have been used expressively in many pieces of music and film music. The video below shows a number of them.

Note that this list of examples is non-exhaustive. In fact, as I write this post, the movie Dune by Denis Villeneuve has been released, and the score by Hans Zimmer features a particularly interesting example of T6 transformation in the piece “Herald of the Change”.

In this piece, you can hear T6 transformations at 1’23” – 1’30” between a D major chord and a G# major chord. As in the previous discussions, the use of the T6 transformation makes sense, as it corresponds in the movie to the official transfer of power of the planet Arrakis from the Harkonnen House to the Atreides House (which are diametrically opposed), and to the subsequent travel of the Atreides family from Caladan to Arrakis (distant planets in space, water versus sand, etc….). Notice however that, although this is a major T6 transformation, it doesn’t sound at all like the examples in the compilation video above (compare with the Star Trek or Starship Troopers examples). This is because Hans Zimmer uses an octatonic scale, which allows major triads and T6 transformations, but gives at the same time a very dark mood to the music.

Speaking of major T6 transformations in otherwise dark moods, there is also a nice example of it in the score of 007: Skyfall by Thomas Newman. The track which is precisely named Skyfall can be heard in the movie when M and Bond are in Scotland, on their way to the Skyfall domain. There is a eery major T6 transformation between B major and F major, which you can hear in the video below at 3’07” (I’m not entirely sure, but it seems to me that the ‘eery’ effect is achieved by maintaining some of the pitches of B major over the chord of F major).

Notice how it happens right after Bond says “Storm’s coming”. By the way, Newman reuses it in the score of 007: Spectre, in the track named “The Eternal City”.

The SLIDE transformation

The neo-Riemannian SLIDE transformation is an involution which takes a minor chord of root X to the major chord of root one semitone below X, and vice-versa, as shown below.

Its use in film music is relatively recent, and is commonly associated with supernatural or magic properties. Here is a compilation of examples.

Then, about two months after I had completed this video, I discovered a striking example of SLIDE in Faure’s Requiem, which you can hear below at 24’46”. It is used in a repeated L-S-T7 series of transformations.

The ‘Tarnhelm’ LP transformation

The neo-Riemannian LP transformation is also called the ‘Tarnhelm’ transformation, as it features in Wagner’s Das Rheingold when Alberich uses the Tarnhelm (a magic helmet) to transform itself into a dragon. It is the transformations which takes a minor chord of root X to the minor chord whose root is eight semitones above X, as shown below.

It has been used to suggest (dark) magic, malefic objects or people, the most striking example being its use in the Imperial March in the score of Star Wars by John Williams. Here is a compilation video of examples.

The ‘Hexatonic Pole’ LPL transformation

The last compilation is about the neo-Riemannian LPL transformation, also named the ‘hexatonic pole’ transformation (sometimes notated ‘H’). It is the transformation which takes a major chord of root X to the minor chord whose root is eight semitones above X, as shown below.

It has been studied by Richard Cohn, who noted its peculiar properties and these were used to suggest the ‘uncanny’, often in magical or death-related settings. I was glad to find recently a particularly striking example of the ‘H’ transformation in the score of the last Harry Potter movie by Alexandre Desplat, which occurs precisely when Harry seizes the resurrection stone and see his deceased relatives. You can find this example, and many others, in the video below.

Conclusion

I hope these compilations of examples can prove useful to illustrate neo-Riemannian and transformational theories. Again, this is non-exhaustive and there will probably be many more examples to come in the future as music and movies are made. I’d be glad to know about them, so if you encounter any, don’t hesitate to contact me. As the neo-Riemannian group contains 24 elements, we have only covered a small fraction of the possible musical transformations in this group. I wonder if one could find musically relevant examples for all of these transformations…

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