Along with the Tonnetz, Hook lists other contributions of neo-Riemannian theory that include a “fresh” perspective on the concepts of consonance, dissonance, symmetry, and efficient voice leading in composition.24
Childs, however, remarks on a fundamental issue regarding the Tonnetz that has arisen in the last fifteen years or so: “the composers whose works seem best suited for neo-Riemannian analysis rarely limited their harmonic vocabulary to simple triads.”25
In approaching many of these composers’ works with the neoRiemannian tools currently at our disposal, theorists must often simplify chords that contain dissonances into triads.
This often involves disregarding the sevenths of major-minor seventh chords and the root of half-diminished chords.26
There is an inevitable loss with the simplification of seventh chords to triads; for example, one might choose to omit the seventh of a dominant seventh chord (scale degree 4 within the key) for the sake of showing the progression on a Tonnetz.
By doing so, the strong downward semitone “pull” from scale degree 4 to scale degree 3 – an important aspect of a V7 ! I progression – is left unaccounted for.27
そうすると、半音下(4度から3度へ)に解決する強い力を持つ7thの音(これはⅤ7 Ⅰという進行のとても重要な性質です！) が考慮されないことになります。
Jazz music, in particular, is often rich with seventh chords and this increase in cardinality enables more relations to be formed between chords.28
Adrian Childs highlights the need for “a transformational system for dominant and halfdiminished seventh chords which would allow all four pitches to participate in parsimonious voice leading.”29
In 1998, Childs and Edward Gollin both design three-dimensional (3- D) models to accommodate certain tetrachords.30
1998年にChilds と Edward Gollinは三次元のモデルを考案し、4和音に対応できるようにしました。
While Childs’ provides a more elaborate design, Gollin’s revamped 3-D Tonnetz (pictured in Figure 4) is more straightforward to navigate; therefore, it may be more useful for mapping short harmonic passages.
The figure shows a central tetrachord prism in the center, with the notes C, E, G, B-flat. From each of the prism’s six edges stems one other tetrachord that shares two of the same notes from the common edge.
This figure is particularly useful in demonstrating the inversional relationship between two tetrachords of the same set class but of a different mode with two notes in common.31
Gollin refers to this operation as the “S-transformation” which retains two pitches as common tones while the remaining two pitches move by half step in similar motion, shown in Figure 4.32
Not pictured below is another operation, “C-transformation,” which pertains to moving two of the four pitches a semitone in contrary motion.33
Childs explains that a seventh-chord model for mapping harmonies is “more powerful” in neo-Riemannian analysis due to its accurate tracking of all four voices, rather than the current limited system of triadic transformations.35
This type of Tonnetz is certainly innovative, though it is also quite limiting for the purpose of jazz analysis since the repertoire often features a variety of tetrachords types.
The major seventh chord (0158), diminished seventh chord (0369)and any 0258 tetrachord transposed from the CMm7 listed in Figure 4 requires an entirely new diagram.
This figure simply cannot accommodate a wide enough range of chords as it is.
As a result, I was required to transpose Gollin’s original framework by T5 so that it reflected the following excerpt of Charles Mingus’ ‘Fables of Faubles’ in Figure 5.
There is a Iiii iv relationship between FMm7 and Cø7 chords Figure 5.
The two chords invert (I) so that two points – the third (iii) note of the FMm7 , C, and the fourth (iv) note, E-flat – map onto one another.
These two inversion points are stationary pitches, shown in orange.
The other two pitches (G-flat and B-flat) begin as the colour red but transform through the Iiii iv relationship to become the pitches shown in yellow (A and F).
The three-dimensional model is limiting in that it does not accommodate near-transformations.
For example, the two moving voices must move by semitone; if one travels by semitone and the other by whole tone, the entire system is rendered ineffective.
Voice leading by whole tone often occurs in jazz and using only Gollin’s system to analyze seventh chords would severely limit its potential in the analysis of the genre as a whole.
Joseph Straus presents a more flexible analytical system to accommodate transformations – most commonly transposition and inversion – that do not quite fit the standard, rigid mould.38
These “fuzzy transformations,” originally presented by Ian Quinn (1997), entail examining each voice’s individual transformation between simultaneities and then selecting the best overall transformation to use.
This decision can be made by selecting what is essentially the median, mean, or mode (in the mathematical sense) of all the separate transformations.
According to Straus: “the connections created by such fuzzy transpositions [or inversions] may serve to link harmonies that would be judged as incomparable by traditional, crisp atonal set theory.”39
In this way, the “fuzzy transformation” is similar to a line of best fit for a graph, wherein the outliers are accounted for through the concept of offset.
The offset is calculated by adding the total number of semitones the outliers would need to shift up or down to be an exact (or “crisp”) transformation.40
In other words, one could measure the distance from each graphical outlier to the line of best fit and decipher the offset from the sum of these measurements.
‘One Note Samba’ contains various fuzzy inversions between a number of its seventh chords. For example, Figure 6 outlines a near-I i ii relationship in mm. 10-11 of the piece.41
In an ideal and exact transformation, this would be expressed by Gollin’s 3-D Tonnetz as an S-transformation; in actuality, only one chord is of the 0258 set class.
This then creates an Ii ii S-transformation with an offset of 1.
The offset is illustrated at the bottom of Figure 6 while the shaded outlier (E-flat) and expected transformational output (E-natural) appear on the staff
24. Ibid., 50. 25. Adrian P. Childs, “Moving beyond Neo-Riemannian Triads: Exploring a Transformational Model for Seventh Chords,” Journal of Music Theory 42, no. 2 (Autumn 1998): 181, http://jstor.org/stable/843872. The fact that these composers do not limit themselves to triads may imply that they are not, in fact, well-suited for standard neo-Riemannian analysis and the Tonnetz. I interpret this statement to mean that the triadic foundations of these composers’ harmonic progressions move parsimoniously and in a way that can be plotted on a Tonnetz; in this way, they are ideal for this type of analysis. However, in order to map the chords, one must overlook the seventh. The result of omitting the seventh is incomplete analysis, in that it can never accurately represent the music that is actually there. 26. Childs, “Moving beyond Neo-Riemannian Triads,” 182. Major-minor seventh chords (SC 0258) contain a SC 037 triad within them. For example, CMm7 contains the notes C-E-G-Bb, with the first three notes C-E-G creating a major triad. This is why the seventh (Bb) is often omitted in neo-Riemannian analysis. Half-diminished chords are also of the set class 0258, and they too contain a SC 037 triad: F#ø7 (F#-A-CE) becomes A minor (A-C-E) when the root (F#) is omitted in analysis. 27. This is not to say that a simplification such as this can never be justified; if the important motion occurs in the voices that are retained after the reduction, it can be a logical analytical choice. For that reason, I occasionally choose to reduce chords to their triadic foundation in my analyses. 28. Edward Gollin, “Some Aspects of Three-Dimensional ‘Tonnetze’,” Journal of Music Theory 42, no. 2 (Autumn 1998): 196, http://www.jstor.org/stable/843873. Cardinality refers to the number of elements in a set. In this case, it refers to the number of individual pitches in a chord: a C major chord has a cardinality of three (reflective of the notes C, E, and G), while a C Mm7 chord has a cardinality of four (C, E, G, and Bb). 29. Childs, “Moving beyond Neo-Riemannian Triads,” 185. 20. Gollin, “Some Aspects of Three-Dimensional ‘Tonnetze’,” 195. 31. There is an entirely different model for representing same-SC tetrachords with one note in common, also on page 201 of his article. While this, too, can easily lend itself to jazz analysis, I chose to omit it based on the relevance of Figure 4 to my argument, and the redundancies which would transpire. 32. Childs, “Moving beyond Neo-Riemannian Triads,” 185.