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首页> 外文期刊>Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance >Algorithmic and computational approaches to pure-tone approximations of equal-tempered musical scales
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Algorithmic and computational approaches to pure-tone approximations of equal-tempered musical scales

机译：等温音阶纯音近似的算法和计算方法

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We develop formalism for generating pure-tone systems that best approximate the modulation/transposition properties of equal-tempered scales. We define six measurements to determine the closeness of scales generated by pure intervals to equal-tempered scales. Two measures apply to scales generated by single intervals and rely heavily on continued fraction analysis. We show that generated scales can be optimized to preserve the pure-tone character of the scale by compromising the modulation/transposition properties by a least amount. The other four measures are generalizations that apply to scales generated by multiple intervals. These measures are directly applied to individual scales and numerically compared. We apply this formalism to pure intervals of import in antiquity and show that this approach yields the historically important just 12-tone system as an optimum scale. Finally, we show how this formalism can be applied to the analysis of non-standard scales.View full textDownload full textKeywordsmusical scales, spectra, spectral variation, continued fractions, computational algorithms, equal temperament, just temperament, well-tempered scales2000 Mathematics Subject Classifications00A65Related 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/10.1080/17459737.2011.641765
机译：我们开发用于生成纯音系统的形式主义，该纯音系统最近似于等温音阶的调制/转置特性。我们定义了六个测量值，以确定纯间隔到等温刻度所产生的刻度的接近程度。两种度量适用于单个间隔生成的标度，并且严重依赖于连续的分数分析。我们表明可以通过最小程度地调制/转置特性来优化生成的音阶，以保留音阶的纯音特征。其他四个度量是适用于多个间隔生成的标度的概括。这些度量直接应用于单个标度并进行数值比较。我们将这种形式主义应用于古代的纯净进口间隔，并证明这种方法产生了具有历史意义的仅12音系统作为最佳比例。最后，我们展示了这种形式主义可以如何应用于非标准音阶的分析。 var addthis_config = {ui_cobrand：“泰勒和弗朗西斯在线”，servicescompact：“ 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/10.1080/17459737.2011.641765

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