A Geometry of Musicー対訳

A Geometry of Music
:Harmony and Counterpoint in the Extended Common Practice
(Oxford Studies in Music Theory) by Dmitri Tymoczko


The word “tonal” is contested territory. Some writers use it restrictively, to describe only the Western art music of the eighteenth and nineteenth centuries. For them, more recent music is “post-tonal”—a catch-all term including everything from Arvo Pärt’s consonances to the organized sonic assaults of Varèse and Xenakis. This way of categorizing music makes it seem as if Pärt, Varèse, and Xenakis are clearly and obviously of a kind, resembling one another more than any of them resembles earlier composers.


“Tonal” can also be used expansively. Here, the term describes not just eighteenth-and nineteenth-century Western art music, but rock, folk, jazz, impressionism, minimalism, medieval and Renaissance music, and a good deal of non-Western music besides. “Tonality” in this sense is almost synonymous with “non-atonality”—a double negative, most naturally understood in contrast to music that was deliberately written to contrast with it. The expansive usage accords with the intuition that Schubert, the Beatles, and Pärt share musical preoccupations that are not shared by composers such as Varèse, Xenakis, and Cage. But it also raises awkward questions. What musical feature or features lead us to consider works to be tonal? Is “tonality” a single property, or does it have several components? And how does tonality manifest itself across the broad spectrum of Western and non-Western styles? Faced with these questions, contemporary music theory stares at its feet in awkward silence.


The purpose of this book is to provide general categories for discussing music that is neither classically tonal nor completely atonal. This, in my view, includes some of the most fascinating music of the twentieth century, from impressionism to postminimalism. It also includes some of the most mysterious and alluring music of Chopin, Liszt, and Wagner—music that is as beloved by audiences as it is recalcitrant to analytical scrutiny. My goal is to try to develop a set of theoretical tools that will help us think about these sophisticated tonal styles, which are in some ways freer and less rule-bound than either eighteenth-century classical music or twentieth-century modernism. More specifically, I will argue that five features are present in a wide range of genres, Western and non-Western, past and present, and that they jointly contribute to a sense of tonality.


1. Conjunct melodic motion.
Melodies tend to move by short distances from note to note.

2. Acoustic consonance.
Consonant harmonies are preferred to dissonant harmonies, and tend to be used at points of musical stability.

3. Harmonic consistency.
The harmonies in a passage of music, whatever they may be, tend to be structurally similar to one another.

4. Limited macroharmony.
I use the term “macroharmony” to refer to the total collection of notes heard over moderate spans of musical time. Tonal music tends to use relatively small macroharmonies, often involving five to eight notes.
5. Centricity.
Over moderate spans of musical time, one note is heard as being more prominent than the others, appearing more frequently and serving as a goal of musical motion.



3.「和音の一貫性」ー 進行中のハーモニーは、それがどうゆう種類のものであれ、構造的に互いに似ている傾向にある。
4.「限定されたマクロハーモニー」 ーこのマクロハーモニーという造語を、ある程度の長さの音楽的時間内に使われる音の集合を指し示す用語として使用する。 調性音楽においては比較的小さな規模マクロハーモニーが使用される。大抵は5から8音である。

5. 「中心性」ー ある程度の長さの音楽的時間内に、特定の音が他の音よりも特出した存在として感じられ、音楽上の動きの最終地点としての役割を果たす。

The aim of this book is to investigate the ways composers can use these five features to produce interesting musical effects. This project has empirical, theoretical, and historical components. Empirically, we might ask how each of the five features contributes to listeners’ perceptions of tonality: which is the most influential, and are there any interesting interactions between them? For instance, is harmonic consistency more important in the context of some scales than others? Theoretically, we might ask how the various features can in principle be combined. Is it the case, for example, that diatonic music necessarily involves a tonic? Conversely, is chromatic music necessarily non-centric? Finally, we can ask historical questions about how different Western styles have combined these five tonal ingredients—treating the features as determining a space of possibilities, and investigating the ways composers have explored that space.

 本書の目的は、これら5つの特徴を使い、音楽的に面白い効果を生み出す方法を研究することだ。本研究は、実証的、理論的、そして歴史的研究からなる複合体である。実証的には、どのように五つの特徴が 聴取者の調性の知覚に寄与しているのかを問うていく。5つの特徴のうち、何が一番影響力のある要素なのか、またこれら5つの特徴間に興味深い相互作用はあるのだろうか?例えば、「和音の一貫性」が、他の要素よりも、あるスケールの中の文脈では重要であったりするのだろうか。理論的な側面では、どの特徴が原理的に結びついているのだろうか?例えば、調性のある音楽に主和音は必ず必要なのだろうか?対して半音階的音楽には、中心があってはいけないのだろうか?最後に歴史的な問いであるが、異なる西洋音楽のスタイルは、五つの特徴をどのように組み合わせた結果なのだろうか?この五つの特徴が音楽的な空間の可能性を規定する要素と考えて、作曲家がこの音楽の空間を探索してきた方法を調査する。

This book is primarily concerned with the theoretical and historical questions. I am a musician, not a scientist, and although I will sometimes touch on perceptual issues, I will largely leave empirical psychology to the professionals. Instead, I will ask how composers have combined and might combine the five features. Part I of the book develops theoretical tools for thinking about the five features. Part II uses these tools to argue for a broader, more continuous conception of the Western musical tradition. Rather than focusing narrowly on the eighteenth and nineteenth centuries (the so-called “common practice period”), I attempt to identify an “extended common practice” stretching from the beginning of Western counterpoint to the music of recent decades. The point is to retell the history of Western music in such a way that the tonal styles of the last century—including jazz, rock, and minimalism—emerge as vibrant and interesting successors to the tonal music of earlier periods.

 本書の第一の関心は、理論上の問と歴史的な問にある。私は音楽家であって科学者ではないので、ー時的に知覚上の問題にふれることはあるにせよー、大方の実験心理学的領域は専門家に譲ることとする。そのかわりに、作曲家が五つの特徴をどのように組み合わせたのかを探っていく。第一章では、五つの特徴を分析する理論的手段を開発する。第二章では、これらの手段を使い、西洋音楽の伝統のなかに広範にそして連続したコンセプトが存在していることを証明していく。 18〜19世紀といういわゆるよく知られたクラシック作品のある狭い時期に注目するのではなく、 対位法の始まりからここ数十年まで拡張されたより普遍的な音楽をテーマに探求していく。ポイントは、21世紀の調性音楽ージャズやロックやミニマリズムを含むーは、クラシックの古典的な調性音楽を受け継ぐ形で、力強く現れたのだと、西洋音楽史を再構築することである。

My central conclusion is that the five features impose much stronger constraints than we would intuitively expect. For example, if we want to combine conjunct melodic motion and harmonic consistency, then we have only a few options, the most important of which involve acoustically consonant sonorities. And if we want to combine harmonic consistency with limited macroharmony, then we are led to a collection of very familiar scales and modes. Thus the materials of tonal music are, in Richard Cohn’s apt description, “overdetermined,” in the sense that they are special or distinctive for multiple different reasons.2 This suggests that when we look closely, we should find important similarities between different tonal styles: since there are only a few ways to combine the five features, different composers—from before Palestrina to after Bill Evans—will necessarily use the same basic techniques. In the second part of this book I make good on this claim, tracing common practices that connect the earliest examples of Western counterpoint to music of the very recent past.

 5つの特徴が私たちの想像以上に音楽を束縛している、というのが私の主要な結論である。例えば、「旋律の順次進行性」と「一貫した和音」の規則を組み合わせて従った場合、私たちにはほとんど選択肢がない。またもしマクロハーモニーと一貫したハーモニーを採用したならば、私たちはよく知られた旋法を使用することになる。そうゆうわけで、調性音楽の素材は、リチャード・カーンの適切な表現を借りれば、『過剰に決定』 されているのだ。これは、様々なスタイルの調性音楽の中に、重要な共通点があることを示唆している。五つの特徴を組み合わせる方法は限られているため、パレスチナ様式の作曲家からビルエヴァンスまで、同じテクニックを使うことになった。第二章では、初期の西洋音楽と本当に最近の音楽を結びつけるよく知られた作品を実際に検討しながら、この点について特に議論していく。

Let’s consider the five features in more detail, with an eye toward understanding why they might be so widespread throughout Western and non-Western music. A preference for conjunct melodic motion likely derives from the features of the auditory system that create a three-dimensional “auditory scene.”3 An eardrum, in effect, is a one-dimensional system that can only move back and forth. From this meager input our brains create a vivid three-dimensional sonic space consisting of individually localized sounds: the phone ringing in front of you, the honk of a car horn outside your window, and the sound of a droning music theorist off to your right. To accomplish this dazzling transfiguration, the brain relies on a number of computational tricks, one of which is to group sonic events that are nearby in pitch.4 Thus, a sequence like Figure 1.1.1a tends to be heard as belonging to a single sound source, whereas Figure 1.1.1b creates the impression of multiple sources. In this sense, small melodic steps are intrinsic to the very notion of “melody.” Figure 1.1.1 Small movements sound melodic (a), while large registral leaps create the impression of multiple melodies (b).


 5つの特徴について、より詳細にみていくこととする。なぜこの特徴が西洋と非西洋の広範な範囲にわたって存在しているのだろうか。「旋律の順次進行性」はおそらく『三次元の聴覚世界』をつくりだすシステムに由来している。 鼓膜は実際には単に前後に動くだけの、1次元のシステムでしかない。しかし脳は、この小さな刺激から鮮明な三次元の音響空間を作り出すのだ。ここではそれぞれのサウンドは個別の処理がなされている。電話があなたの目の前で鳴り、車の警笛が窓の外で、そしてドローンミュージックがあなたの右側に聞こえてくる。このように聞こえるようにするため、脳は沢山の計算上のトリックを使っている。その一つは、音高の近い音をひとまとめに認識することだ。結果として、図1.1.1aのような譜例は単一の音源から発せられた音と認識され、反対に1.1.1bは複数の音源からなるように聞こえる。この意味では旋律が小さい動きをとることは、まさに旋律の概念に生来的な要素なのだ。
(図1.1.1) ー(小さい動き(a)は単一の一塊の旋律として聞こえ、跳躍の大きい動き(b)は複数の旋律に聞こえる。)

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