论文标题
音乐色调通过欧拉n翼霍普夫奇异性的分叉控制通过
Musical tone coloring via bifurcation control of Eulerian n-tuple Hopf singularities
论文作者
论文摘要
音乐中声音的内在本质是它们定性类型的演变,而在数学中,我们通过分叉来解释每个定性变化。 HOPF分叉是生成任意频率信号的重要场所。因此,通过分叉控制理论对音乐声音的调查是长期的和自然的贡献。在本文中,我们通过光谱和时间信封的动态建模来解决声音的音调着色。音符的多个领先的谐波部分(Modulo A听力速度阈值)归因于具有N-Tuple Hopf奇异性的Eulerian差异系统。然后,通过对差分系统的分叉控制,在一组连续的时间间隔内模拟了时间信封的定性演变。对于一个例子,我们提出的方法应用于从钢琴和小提琴获得的音频C#4文件上。傅立叶分析用于生成振幅光谱向量。然后,我们将每个振幅光谱向量与欧拉流动不变叶相关联。分叉控制足以准确构建音符的所需光谱和振幅信封。这些对应于涉及欧拉差分系统的Clifford Toral歧管的丰富分叉场景。为了减少技术,我们采用了几种还原技术,并使用一个分叉参数。我们展示了不同有序的基本分叉集,例如干草叉和(双重)马鞍节点分叉与由钢琴或小提琴演奏的定性时间信封变化相关联。在小提琴演奏的C#4的时间包络分叉中观察到完整的磁滞类型周期。
An intrinsic essence of sounds in music is the evolution of their qualitative types while in mathematics we interpret each qualitative change by a bifurcation. Hopf bifurcation is an important venue to generate a signal with an arbitrary frequency. Hence, the investigations of musical sounds via bifurcation control theory are long-overdue and natural contributions. In this paper, we address the tone coloring of sounds by dynamical modeling of spectral and temporal envelopes. Multiple number of leading harmonic partials of a note (modulo a hearing sound velocity threshold) are attributed into an Eulerian differential system with n-tuple Hopf singularity. The qualitative evolution of the temporal envelop is then simulated over a set of consecutive time-intervals via bifurcation control of the differential system. For an instance, our proposed approach is applied on audio C#4 files obtained from piano and violin. Fourier analysis is used to generate the amplitude spectral vectors. Then, we associate each amplitude spectral vector with an Eulerian flow-invariant leaf. Bifurcation control suffices to accurately construct the desired spectral and amplitude envelopes of musical notes. These correspond with a rich bifurcation scenarios involving Clifford toral manifolds for the Eulerian differential system. In order to reduce the technicalities, we employ several reduction techniques and use one bifurcation parameter. We show how different ordered sets of elementary bifurcations such as pitchfork and (double) saddle-node bifurcations are associated with the qualitative temporal envelop changes of a C]4 played by either a piano or a violin. A complete hysteresis type cycle is observed within the temporal envelop bifurcations of the C#4 played by violin.