Abstract
We consider the mathematical properties of a new musical instrument referred to as a SphereHarmonic. The SphereHarmonic is designed based on the basic divisions of a temari, that is, a spherical surface with congruent triangles defined by great circles. Each triangle on the sphere corresponds to a tone, and each vertex, which is connected to multiple triangles, represents a chord composed of the corresponding tones. Different chords are created by different combinations of the triangles, which are obtained by rotating the hemispheres. We describe four types of SphereHar- monics characterized by their divisions: Sn, C6, C8, and C10. These four SphereHarmonics have different mathematical characteristics, including the numbers of triangles, the degrees of the verti- ces, and the numbers of axes. We summarize these properties and also calculate the combinations for the possible vertices.
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