发表机构
Aalborg University（奥尔堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出用几何音高-时间表示匹配音乐模式，定义变换类和图案间变换图，精确刻画拉威尔作品中海顿主题的变换关系。
AI 中文摘要
我们回顾了使用点集表示音乐的概念，并论证了此类表示比顺序表示更适合于匹配无语音复调音乐（如键盘音乐）中的模式。诸如移调、倒影、缩减、增时和逆行等音乐变换，可以通过音高-时间表示中的几何变换来建模，这些变换将平移与平行于时间轴的缩放以及关于时间轴的反射相结合。我们识别出八种几何音高-时间表示类型，它们使用半音音高、莫菲音高、莫菲或色度来表示音高，并使用起始时间或中间时间来表时间。我们说明了这些表示类型如何使我们能够刻画不同类型的音乐变换。例如，通过使用中间时间而非起始时间，我们可以精确刻画某些逆行关系；通过使用莫菲和色度音高表示，我们可以刻画涉及八度位移和重复的变换。我们提出了变换类的概念，并考虑三个特定类别：$F_{\mathrm{2STR}}$、$F_{\mathrm{2STRMod7}}$和$F_{\mathrm{2STRMod12}}$。我们引入了模式集合$S$和变换类$F$的图案间变换图的概念。该图中的每个顶点表示$S$中的一个模式，并且当且仅当$P_1$可以通过$F$中的变换映射到$P_2$时，图中存在从模式$P_1$到模式$P_2$的边。我们借助此类图表明，拉威尔《海顿之名的小步舞曲》中海顿主题出现之间的音乐关系，可以在所考虑的音高-时间表示中，通过$F_{\mathrm{2STR}}$、$F_{\mathrm{2STRMod7}}$和$F_{\mathrm{2STRMod12}}$中的变换精确描述。
英文摘要
We review the notion of representing music using point sets and argue that such representations are better adapted than sequential representations for matching patterns in unvoiced, polyphonic music, such as keyboard music. Musical transformations such as transposition, inversion, diminution, augmentation and retrograde can be modelled by geometric transformations in pitch-time representations that combine translation with scaling parallel to and reflection in the time axis. We identify eight types of geometric pitch-time representation that use chromatic pitch, morphetic pitch, morph or chroma to represent pitch and either onset time or midtime to represent time. We illustrate how these types of representation allow us to characterise different types of musical transformation. For example, by using midtime instead of onset time, we can precisely characterise certain retrograde relationships; and by using morph and chroma pitch representations we can characterise transformations involving octave displacements and duplications. We present the concept of a transformation class and consider the three specific classes, $F_{\mathrm{2STR}}$, $F_{\mathrm{2STRMod7}}$ and $F_{\mathrm{2STRMod12}}$. We introduce the notion of an inter-pattern transformation graph for a set of patterns, $S$, and a transformation class, $F$. Each vertex in such a graph represents a pattern in $S$ and there is an edge in the graph from pattern $P_1$ to pattern $P_2$ if and only if $P_1$ can be mapped onto $P_2$ by a transformation in $F$. We show, with the aid of such graphs, that the musical relationships between the occurrences of the HAYDN theme in Ravel's Menuet sur le nom d'Haydn can be precisely described in terms of transformations in $F_{\mathrm{2STR}}$, $F_{\mathrm{2STRMod7}}$ and $F_{\mathrm{2STRMod12}}$ within the pitch-time representations considered.
Comments35 pages, 8 tables, 15 figures. Draft of first chapter of book currently in preparation for publication by Springer
