This paper shows how scale vectors (which can represent either pitch or rhythmic patterns) can be written as a linear combination of columns of scale matrices, thus decomposing the scale into musically relevant intervals. When the scales or rhythms have different cardinalities, they can be compared using a canonical form closely related to Lyndon words. The eigenvalues of the scale matrix are equal to the Fourier coefficients, which leads to a number of relationships between the scale vectors and the decompositions. Overcomplete dictionaries of frame elements can be used for more convincing representations by finding sparse decompositions, a technique that can also be applied to tiling problems. Scale matrices are related to familiar theoretical properties such as the interval function, Z-relation or homometry, all of which can be efficiently studied within this framework. In many cases, the determinant of the scale matrix is key: singular scale matrices correspond to Lewin's special cases, regular matrices allow a simple method of recovering the argument of an interval function and elicit unique decompositions, large determinant values correspond to flat interval distributions.View full textDownload full textKeywordscirculating matrices, scales, periodic rhythm, sum of scales, linear combination of scales, scale vector, scale matrix, Fourier, DFT, homometry, Z-relation, FLID, Lewin, tilingsRelated var addthis_config = { ui_cobrand: "Taylor & Francis Online", services_compact: "citeulike,netvibes,twitter,technorati,delicious,linkedin,facebook,stumbleupon,digg,google,more", pubid: "ra-4dff56cd6bb1830b" }; Add to shortlist Link Permalink http://dx.doi.org/10.1080/17459737.2011.640469
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机译：本文展示了如何将音阶向量（可以表示音高或节奏模式）写成音阶矩阵列的线性组合，从而将音阶分解为音乐上相关的区间。当音阶或节奏的基数不同时，可以使用与Lyndon单词密切相关的规范形式来比较它们。比例尺矩阵的特征值等于傅立叶系数，这导致比例尺矢量与分解之间存在许多关系。通过找到稀疏分解，可以将框架元素的不完整字典用于更具说服力的表示，该技术也可以应用于平铺问题。尺度矩阵与熟悉的理论属性（例如区间函数，Z关系或同构性）相关，所有这些都可以在此框架内有效地研究。在许多情况下，尺度矩阵的行列式是关键：奇异尺度矩阵对应于Lewin的特殊情况，规则矩阵允许使用简单的方法来恢复区间函数的参数并引发唯一分解，较大的行列式值对应于平坦区间分布。查看全文下载全文关键词循环矩阵，尺度，周期性节奏，尺度总和，尺度的线性组合，尺度向量，尺度矩阵，傅立叶，DFT，齐次性，Z关系，FLID，Lewin，平铺相关var addthis_config = {ui_cobrand：“ Taylor ＆Francis Online”，services_compact：“ citeulike，netvibes，twitter，technorati，delicious，linkedin，facebook，stumbleupon，digg，google，更多”，发布号：“ ra-4dff56cd6bb1830b”};添加到候选列表链接永久链接http://dx.doi.org/10.1080/17459737.2011.640469
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