在彩色立方体舞蹈中创作自同构：音乐和弦转换的交互式工具
Composing (with) automorphisms in the colored Cube Dance: an interactive tool for musical chord transformation
摘要：
“彩色立方体舞蹈”是杜塞特和斯坦巴赫的立方体舞蹈图的扩展，与定义在大、小和增强三和弦集合上的二元关系的幺半群有关。这一贡献探讨了自同构群的幺半群行动，作为一种方式来转换和弦进行。我们证明了这个自同构群的阶为7776，并且同构于$({\mathbb{Z}_3}^4 \rtimes D_8) \rtimes (D_6 \times \mathbb{Z}_2)$。这个群体的规模和复杂性使得它笨拙：因此，我们通过基于常见HTML/JavaScript框架的Web界面为学生，音乐家和作曲家提供了一个交互式工具，以探索这些自同构，显示这些技术在数学/音乐推广活动中的潜力。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
