以感知不和諧度量測理論與DSP方法辯證中國古代曾侯乙編鐘的音階設計

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题名:	以感知不和諧度量測理論與DSP方法辯證中國古代曾侯乙編鐘的音階設計 The Use of Sensory Dissonance Study and Dsp Method for Authentication of the Scale Design of the Ancient Chnese Bells of Marquis Yi
作者:	劉宗慶翁志文
贡献者:	電機系
关键词:	非和諧式樂器的調音曾侯乙編鐘考古 ARMA 音訊合成
日期:	2017-2018
上传时间:	2018-03-02 11:12:17 (UTC+8)
摘要:	本計畫引用我們執行 102 年度計畫的結果(以聲音的感知不和諧度的量測理論，探究設計裝水玻璃管樂器(Glass Tube Gamelan)的音階)，在此次進一步考究於湖北出土之考古大發現-戰國時期曾侯乙編鐘的音階設計。裝水玻璃管樂器屬非和諧式樂器(Inharmonic Musical Instrument)的範疇，經由使用由湖北博物館致贈鴻禧美術館的一個仿古編鐘的初步測試，發覺以和諧度理論來說，亦歸類為非和諧式樂器，且為六或七個音階。為探究真實曾侯乙編鐘的音階設計，因此蒐集到的此古董編鐘的側錄原音(以中央C 之鐘音，鐘銘:宮; 出土號:中三8 正鼓音; 音高頻率:256.89 Hz)，以感知不和諧度量測理論重新計算音階。目前所得到各音階的頻率，卻與真實曾侯乙編鐘所側錄音的頻率並不相同，也就是說曾鐘的音階並不是最和諧的。此也如某些學者所研究的，以”fa”(音名F)為key 所形成類12 平均音律的調音，是為了便於各國(地)合奏所需之轉調方便的設計的猜測。為了要能進一步完備驗證我們理論所推論出的結果，除了將對所收集的真實編鐘的側錄原音，增加多個鐘音基音(key)的頻譜與和諧度分析，以求得不同key 下的虛擬音階來比對曾鐘; 並將使用電腦音訊合成的方法，根據我們所設計的音階製作一組虛擬的編鐘，以驗證7 音階的編鐘的設計是較佳的。(使用虛擬合成，此乃鑒於無法直接試做出實體的一組7 音階編鐘，它包含了每一次調音，即需重新鑄造，需投入相當大的資源，此為原始編鐘製造成功的可貴之處，而成為帝王的收藏)。在編鐘聲音合成的部分，先將取得的鐘音基音FFT 後得到的頻譜，選取其主要頻率的成分(共五個)，並將五個主要單頻(Principal Component、Mode 或partial) 取出(例:可以用Butterworth 數位帶通濾波器濾出各Partial)，並以ARMA 模型來合成每一單頻衰減信號，所以一個鐘音事實上包含了五個ARMA 模型所合成一個編鐘的聲音。根據編鐘實測的頻譜，其實這五個單頻，經微觀放大觀察，每一個單頻實為兩個非常相近的單頻所組成，即雙模式類單頻，這是鐘或鈴聲所特有的現象，因此會形成”差拍”的時域響應。頻率非常相近的兩個sine 波的雙模式的信號，若使用一般ARMA 模型，需使用數百甚至上千的階數，不實際。解決方法需使用FZ-ARMA (Frequency Zooming-Autoregressive Moving Average)來做。我們的初步分析，以FZ-ARMA(2,3)即可完成。依我們計算Sensory Dissonance Curve 來設計七個音階編鐘的音階，並做出兩個八度，每一音每由 5 個FZ-ARMA 加總來模擬。目前已由Matlab 的impz 與sound 指令來合成發聲測試過，聲音很像。此虛擬樂器完成時，可用來演奏由真實曾侯乙編鐘所錄製的同樣的古曲做比較，以驗證此新的虛擬7 聲編鐘是較佳的推斷。此比較的設計，可探究古代曾侯乙編鐘的音階設計之迷，將帶來考古音律學(調音)的分析的科學論據，更開啟有關中國古代歷史樂器，以和諧度量測理論的考古學研究之濫觴與討論。 On our research conducted in NSC102 project, we predicted that the famous archeological discovery-ancient Chinese chime-bells instrument from the Tomb of Marquis Yi of the Zeng State during 433 BC should have only 6 or 7-notes scale and should not have a 12-notes equal temperament scale in any range based on the Sensory Dissonance Curve study. To further authenticate our discovery, we had collected the original recorded bell sound and performed the preliminary Sensory Dissonance Curve analysis based on the Middle C bell sound which directs to the same conclusion. The scope of this project is to further investigate inside into more bell sounds and its relation to the scale design. This is then followed by using computer synthesis techniques to fabricate a set of virtual chime-bells where ARMA model is used to synthesize the instrument. From our preliminary study, the spectrum of sounding a Yi’s chime bell has five major peaks (principal components or partials). Band-pass filter could be used to extract each principal component. Therefore we need five parallel ARMA models added together to synthesize a bell sound. A closer look on each principal component shows that in fact each principal component has two peaks separated by 4 Hz. Beating of the partials is a perceptually important feature and must be incorporated properly in synthesis models in order to generate realistic sounding synthesis. It is known, however, that very closely located poles are often too demanding in practical estimation, requiring extremely high filter orders range from the impractical hundreds to a thousand that are impractical and end up to numerical problems. To solve this problem the FZ-ARMA is adopted where each partial is first modulated down to near the zero frequency and low-pass filtered, then is zoomed by decimation (reduce the sampling rate). The final signal is then modelled by Stieglitz-McBride (STMCB) Iteration ARMA. The resultant ARMA model then is back substituted the poles and zeros to obtain the true ARMA model. Our analysis shows that FZ-ARMA(2,3) can achieve the task. There are 7 sets of bell sound needed to be synthesized each with five FZ-ARMA models added together to become the virtual instrument. This instrument will be played and compared to the ancient chime bells. A CD performed by the Hubei Music and Dance Troupe using the real Marquis Yi’s chime bells will be used for the purpose. Virtual instrument synthesizes the same song for evaluation and followed by authentication of our prediction that 7 notes chime bell could be more consonant as music flows. Our method of tuning analysis and virtual instrument synthesis would bring us with the tuning study of the ancient Chinese musical instrument to a new era.
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