Neo-riman /2




  • ネオリーマンセオリーの変形とそれを示した図について。
  • この図によってジャズによくあるコード進行を簡単に分析することができる。
  • ワンノートサンバ、イパネマの娘など。
  • Tymoczkoによって改変された図は、コードネームにフォーカスしており、よりジャズの分析に使用しやすい図である。


The fundamental venture of neo-Riemannian theory is to investigate transformational relationships among the twenty-four possible set class (SC) 037 triads – twelve major and twelve minor – in “algebraically elegant” and “musically suggestive” ways that can be visualized in various forms by the use of a graph called the Tonnetz (tone network).9

ネオリーマンセオリーが野心的なのは、24ある全てのトライアド−各12個あるメジャーとマイナーのトライアド− (Set Class で 037 と表されるトライアド) 同士の変形の可能性を調査した点です。しかも代数学的に洗練され、かつ音楽的な示唆に富んでおり、Tonnetz (tone network) と呼ばれる図式など様々な方法で視覚化しました。

The Tonnetz, or Cartesian plane, allows common-tone retention and harmonic motion to be plotted spatially.

Tonnetz もしくはデカルト平面 (訳注:二次元平面) 上に「共有する音」や「声部の動き」を示すことができます。

This is particularly important to some of the theoretical perspectives that emerged in response to nineteenth-century music: namely, that triadic proximity could be examined through the number of shared common tones rather than through rigid tonality and root relation.10

これは19世紀の音楽の分析をする上で、とても重要な観点です。三和音の空間的近さ(that triadic proximity)は、共有する音の数によって測定され、厳密な意味での調性やルートの関係からは影響をうけません。

The plane is designed so that each triangle represents a triad whose three points are representative of individual notes: a triangle with its point at the top is a major chord whereas one with its point facing down is a minor chord.


It is constructed using three axes, each representing a different interval from point to point (or note to note).

The horizontal axis is comprised of perfect fifth relationships (ex. D ! A ! E…); the axis travelling from bottom left to top right contains major third relationships (ex. Bb ! D ! F#…); and the axis from bottom right to top left shows minor third relationships (ex. Ab ! F ! D…).

この図には3方向の軸があります。そして進む方向によって異なった音程を示しています。横の軸はperfect 5thになっており、左下から右上への移動はMajor 3rd 、右下から左上への移動はminor 3rdとなっています。

One can then use the Tonnetz to map chords as they progress, as though there is a triangular object flipping along the various axes as the music transitions from harmony to harmony.11


Figure 1: A standard Cartesian plane with L, P, and R transformations labelled (adaptd from Cohn’s Figure 2).12

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The Tonnetz unites the concept of trichords in transformational theory with those in nineteenth-century harmonic theory through its combination of triads and transformations.13


The three primary transformations in neoRiemannian theory are mathematical operations – L, P, and R – that depict specific, pre-determined ways of transforming one chord into another.14


The Leittonwechsel relationship (L) takes the root of a major chord and moves it a semitone down to the leading tone, preserving the third and fifth; it preserves the root and third of a minor chord and shifts the fifth a semi-tone upward.15

The Leittonwechsel relationship (L) はメジャーコードのルートを半音下げ、導音へ移動させます。3度と5度はキープします。一方、マイナーコードにおいてはルートと3度はキープしたまま5度を半音あげます。

The Parallel (P) retains the root and fifth, moving the third down a semitone from major to minor or up a semitone from minor to major.

The Parallel (P)ではルートと5度をキープし、3度をメジャーコードであれば半音あげマイナーコードにし、マイナーコードであれば3度を半音上げてメジャーコードに変えます。

The Relative (R) moves the fifth of a major chord up a tone to become the root of its relative minor, or the root of the minor chord down a tone to become the fifth of its relative major.

The Relative (R)では、メジャーコードの五度を全音あげ、その音がルートとなるようなマイナーコードに変化させます。マイナーコードの場合はルートを全音下げ、その音が5thとなるようなメジャーコードに変化させます。

All of these transformations change the chord quality from major to minor or vice versa and can occur in either direction on the Tonnetz.


The Slide (S), a less commonly discussed transformation, maintains a stationary third while the root and fifth both slide up or down a semitone in similar motion.

The Slide (S)はあまり議論に上がらない変形ではありますが、3度を固定し、ルートと5度の両方を半音あげたり下げたりします。2音は同じ方向に動きます。

This changes the chord quality from major to minor or the reverse.16 When S is plotted on a Tonnetz, the triangle flips about a point on a horizontal axis.


Using Figure 1, the C minor chord would retain the note E-flat as a common tone while the notes C and G flip about the axis to become the notes C-flat and G-flat, respectively.


The Slide is a frequent transformation in jazz; it can be heard in Antonio Carlos Jobim’s ‘One Note Samba’ (1961) and ‘The Girl from Ipanema’(1963), shown in Figure 2.


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Since the Tonnetz is not suitable for seventh chords, as I will discuss further into the article, I have omitted the sevenths in my analysis.

Tonnetz はセブンスコードに対応していないので(後でこの点についてはより詳細に考察します)、セブンスコードの7thの音は今回の分析では省略します。

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While the above Tonnetz is a consistent method for modelling triadic transformations, Julian Hook presents a plane redesigned by Tymoczko that is more easily navigated in analysis.

上記のトネッツはよくできており3和音の変形を説明するのに適しているが、Julian HookはTymoczkoによってデザインされた別の図も提案しています。これはより簡単に変形を示すことができています。

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According to Hook, it successfully “relates the geometry of the spaces to the musical behaviour of the chords that inhabit them.”19


The entire plane is rotated and as a result, the perfect fifth relationship is no longer horizontal but is instead diagonal; the note names are converted to integers, faded to grey, and written in a smaller font;20


the chord names are embedded within each triangle; arrows now depict the transformational directions throughout the Tonnetz; and, finally, the transformations are colour-coded in the legend.


A number of these modifications make jazz analysis more straightforward – specifically, the inclusion of chord names (written in a larger font than the individual integers which construct the chords) and the addition of colour allow for greater ease when travelling through the diagram.

これらの変更によって、ジャズの分析をより直感的におこなうことができます。コードネーム(数字よりもコードは大きいサイズで書かれています) と色付きの矢印が書かれていることで、より簡単に変化を確認することができようになっています。

This ensures that the focus is on the triads and their transformations, rather than on individual pitches.


Figure 3 illustrates the visual proximity of chords and fluidity of the progression when using Tymoczko’s Tonnetz design. I then use his Tonnetz to show a phrase from ‘For Heaven’s Sake’ (1946), which exhibits a variety of single transformations as well as combinations. The chords are circled in orange on the Tonnetz and numbered in order of occurrence.21

図3はコードの空間的な近さとコード進行の円滑さを、Tymoczko の Tonnetz で示したものです。‘For Heaven’s Sake’を示しました。この曲には単一の変形や、変形の組み合わせによるコード進行が含まれています。各コードはオレンジ色のまるで囲まれて、現れる順番に数字が振られています。

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9. Julian Hook, “Exploring Musical Space,” Science 313, no. 49 (2006): 49, DOI: 10.1126/science.1129300.

10. Cohn, “Introduction to Neo-Riemannian Theory,” 174.

11. If one is to assume equally-tempered pitch classes, rather than just-intoned pitches, then Gollin (1998) clarifies that “the Tonnetz would be situated not in an infinite Cartesian plane but on the closed, unbounded surface of a […] hyper-torus in 4-dimensional space.” Whilethis is an important point in the study of neo-Riemannian theory, the 2-D Cartesian plane in Figure 1 is more appropriate and intuitive than the torus for deciphering small-scale transformations such as those in this article.

12. Cohn, “Introduction to Neo-Riemannian Theory,” 172.

13. Ibid.

14. Hook, “Exploring Musical Space,” 49. Lewin (among others) has discussed another transformation, “D”, as a valuable neo-Riemannian operation. I opt not to use it in my analyses for several reasons. First, it is not a true transformation, as a “V” chord can be the dominant of two different chords: “I” and “i”. By definition, a transformation can only yield one outcome. Second, it is not a contextual inversion like L, P, and R. Third, the D transformation can occur from the dominant to the tonic or in reverse and harmonically speaking, the functions of both are drastically different. Fourth, combining L and R in analyses usually presents a more accurate portrayal of what is occurring in the music – especially when travelling between two minor chords (which would rarely be tonally analyzed as dominant in function, anyway).

15. This can also be referred to as a Leading Tone relationship.

16. Lewin (1987) ascribed the label “SLIDE” to this operation, though Capuzzo (2004) refers to it as “P’”. There are in fact three different types of Slide transformations – one for each axis of the Tonnetz – though my analyses only pertain to one, which I will call “S”.

17. Antonio Carlos Jobim, “One Note Samba,” The Real Book, 5th ed., (1988): 331.

18. Antonio Carlos Jobim, “The Girl from Ipanema,” The Real Book, 5th ed., (1988): 171.

19. Hook, “Exploring Musical Space,” 49.

20. It is standard practice in certain types of music theory to represent the note C with the number 0, C# with 1, D with 2, and so forth.

21. Again, it was necessary to reduce the chords to triads due to the inherent limitations of the plane.

22. Elise Bretton, Sherman Edwards, and Donald Meyer, “For Heaven’s Sake,” The Real Book, 5th ed., (1988): 159.

23. Hook, “Exploring Musical Space,” 49.

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