十二音列樂曲的方陣與模型研究－以魏本和荀白克的樂曲為例

Studies on the Matrices and Models of Twelve-Tone Compositions-Examples from Webern and Schönberg's Music

郭美惠(Mei-Hui Guo) ； 王琛瑤(Chen-Yao Wang) ； 應廣儀(Guang-Yi Ying)

《中國統計學報》 45卷2期 (2007/06) Pp. 170-188

https://doi.org/10.29973/JCSA.200706.0004

摘要

本文介紹十二音列音樂所使用的十二音列方陣，並對魏本及荀白克的十二音列樂曲建立統計模型。本文所分析的樂曲包括魏本的第二十號到第三十一號作品和荀白克的第二十五號、第三十三a號以及第三十七號作品的第一樂章。樂曲資料的型態分為僅考慮十二音列序列的四個基本形式的四型資料以及將每個基本類型再分別細分為十二個子類型的四十八子型資料。在第一部份我們先介紹十二音列及十二音列方陣的建構方法以及樂曲資料的型式。在第二部份我們對樂曲的四型資料建立馬可夫模型及時間序列模型，藉此探討樂曲中十二音列四型資料的序列相關性及樂曲中所使用四型資料的實證分佈與模型平穩分佈的關係。最後我們對四十八子型資料建立自我迴歸及移動平均的時間序列模型，探討不同樂曲對十二音列序列安排的相關性。

關鍵字

十二音列 ； 馬可夫模型 ； 自我相關係數 ； 時間序列模型

並列摘要

In this study, we introduce the twelve-tone matrices of twelve-tone compositions and build statistical models for the twelve-tone musics of Webern and Schönberg. The music pieces analyzed here includes opus No.20, 21, …, 31 of Webern and opus No. 25, 33a and first movement of No.37 of Schönberg. Two types of data are considered in each music piece, the first type is the four-form data which contains the basic four row series and the other type is the forty-eight sub-form data which are the respective twelve sub-forms evolved from the four basic row forms. In this paper, we will first introduce the twelve-tone series, the methodology of building twelve-tone matrices and data forms of music pieces. Next, we will establish Markov models and time series models for the four-form data, based on the models we will study the autocorrelations of the twelve-tone four-form series data and the relation between the four-form frequencies used by the composers and the stationary distributions of models. Finally, we will build the autoregressive and moving average time series models for the forty-eight sub-form data, and discuss the autocorrelation structures of different twelve-tone music pieces.

並列關鍵字

Autocorrelation ； Markov model ； time series model ； twelve-tone series

參考文獻

Bailey, K.(1996).Webern Studies.Cambridge:Cambridge University Press.

Bailey, K.(2004).The Twelve-Note Music of Anton Webern: Old Forms in a New Languange.Cambridge:Cambridge University Press.

Beran, J.,Mazzola, G.(1999).Analyzing Musical Structure and Performance-A Statistical Approach.Statistical Science.14,47-49.

Brockwell, P. J.,Davis, R. A.(1996).Introduction to Time Series and Forecasting.New York:Springer-Verlag.

Conklin, S.(2003).Music Generation from Statistical Models.Proceedings of the AISB 2003 Symposium on Artifical Intelligence and Creativity in Arts and Sciences.(Proceedings of the AISB 2003 Symposium on Artifical Intelligence and Creativity in Arts and Sciences).: