本文針對Webern及Schoenberg的十二音列樂曲進行統計分析，所分析的樂曲包括Webern的第二十、二十一、二十二、二十三、二十四、二十五、二十六、二十七、二十八、二十九、三十及第三十一號作品和Schoenberg的第二十五、三十三a以及第三十七號作品的第一樂章。資料的型態分為僅考慮十二音列序列的四個基本形式的第一類資料以及將每個基本類型再分別細分為十二個子類型對應出十二音列方陣的第二類資料。本論文首先介紹十二音列，再針對十二音列序列探討兩個問題。第一部份是將第一類資料應用馬可夫模型進行統計分析，我們並利用樣本的自我相關係數（sample autocorrelation function） 及模型的自我相關係數（autocorrelation function of model）來探討馬可夫模型（Markov model）的配適程度。第二部份則是針對第二類資料建立時間序列的模型。我們利用樣本的自我相關函數（sample autocorrelation function）、偏自我相關函數(partial autocorrelation function）以及延伸的自我相關函數（extended autocorrelation function)來建立模型，並根據統計量AICC（corrected Akaike's information criterion）找出一個最佳的配適模型。最後利用樣本殘差值的自我相關係數及部分自我相關係數來探討模型的配適情形。

In the thesis, we study the data collected from twelve-note music of Webern and Schoenberg, including opus 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30 and opus 31 of Webern and opus 25, 33a and opus 37 of Schoenberg. The data consists of the following two kinds. The data of the first kind consists of the four basic forms of the twelve-tone music. And the data of the second kind consists of the twelve-tone derived from the matrix of the twelve-note music. We will introduce the twelve-note music first and then study two main topics about twelve-note music in this thesis. In the first part, we consider the Markov properties of the first kind data. We compare the sample autocorrelation function and autocorrelation function of the fitted model to determine the fitness of the Markovian model. In the second part, we build the time series model for the second kind data. Sample autocorrelation function、partial autocorrelation function and extended autocorrelation function are used to determine the orders of the models. The best model is selected based on the AICC. Finally, we check the fitness of the models using sample autocorrelation function and partial autocorrelation function of the residuals.

1 前言.........................1
2 十二音列的資料介紹...........3
2.1 十二音列...................3
2.2 資料來源...................6
2.3 基本統計量.................7
3 四種形式的統計模型..........11
3.1 馬可夫過程的介紹..........11
3.2 馬可夫模型................13
3.3 時間序列模型..............18
4 四十八個子類型的統計模型....20
4.1 統計模型的介紹............20
4.2 四十八個子類型的模型建立..24
4.3 四十八個子類型的模型分析..26
5 結論........................28
A 附圖........................32
B 附表........................56

1. K. Bailey(1991). The Twelve-Note Music of Anton Webern. Cambridge Uversity Press, Inc.

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

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

4. J. Lester(1989). Analytic Approach to Twentieth-Century Music. W. W. Norton & Company, Inc.

5. L. M. Liu, G. B. Hudak, G. E. P. Box, M. E. Muller, G. C. Taio(1986). The SCA Statistical System : Reference Manual for Forecasting and Time Series Analysis (Version 3). Scientific Computing Associates, Inc.

6. I. L. MacDonald and W. Zucchini(1997). Hidden Markov and Other Models for Discrete-valued Time Series. Chapman & Hall, Inc.

7. S. Milstein(1992). Arnold Schoenberg : notes, sets, forms. Cambridge University Press, Inc.

8. G. Perle(1991). Serial Composition and Atonality-An Introduction to the Music of Schoenberg, Berg, and Webern. University of California Press, Inc.

9. W.S. Wei(1990). Time Series Analysis-Univariate and Multivariate Methods. Addison-Wesley Publishing Company, Inc.

10. 林茂文(1992)，時間序列分析與預測，華泰書局。

11. 黃文璋(1995)，隨機過程，華泰書局。

12. 陳宏天(1996)，心電圖中P-R區間的統計分析與模型建立，國立中山大學

13. 應用數學系碩士班碩士論文。