论文标题
彩色立方体舞蹈中的（带有）自动形态：音乐和弦转换的互动工具
Composing (with) automorphisms in the colored Cube Dance: an interactive tool for musical chord transformation
论文作者
论文摘要
“彩色立方体舞蹈”是Douthett's和Steinbach的Cube Dance图的延伸，与在大型，次要和增强的三合会集中定义的二进制关系相关。这项贡献探讨了这种单型动作的自动形态群体，以此作为改变和弦进展的一种方式。我们表明，该自动形态组为7776，同构为$（{\ Mathbb {z} _3} _3}^4 \ rtimes d_8）\ rtimes（d_6 \ times \ times \ times \ mathbb {z} _2 _2 _2 _2）$。该小组的大小和复杂性使其变得笨拙：因此，我们通过基于常见的HTML/JavaScript框架为学生，音乐家和作曲家提供了一个互动工具，以探索这些自动形态，从而显示了这些技术在数学/音乐推广活动中的潜力。
The `colored Cube Dance' is an extension of Douthett's and Steinbach's Cube Dance graph, related to a monoid of binary relations defined on the set of major, minor, and augmented triads. This contribution explores the automorphism group of this monoid action, as a way to transform chord progressions. We show that this automorphism group is of order 7776 and is isomorphic to $({\mathbb{Z}_3}^4 \rtimes D_8) \rtimes (D_6 \times \mathbb{Z}_2)$. The size and complexity of this group makes it unwieldy: we therefore provide an interactive tool via a web interface based on common HTML/Javascript frameworks for students, musicians, and composers to explore these automorphisms, showing the potential of these technologies for math/music outreach activities.
