{"title":"选择音乐节奏模型的最小描述长度","authors":"Verónica Rumbo, Ernesto Mordecki, Martín Rocamora","doi":"10.1080/17459737.2023.2213471","DOIUrl":null,"url":null,"abstract":"AbstractWe apply the minimum description length (MDL) methodology from information theory in order to select mathematical models for musical rhythm. We consider six models proposed by David Temperley and mathematically formalize them, which allows for an MDL analysis. As a consequence, we find that two of the models are not suitable to apply this methodology, as their codification strategy does not represent all possible rhythm sequences. A slight modification of Temperley's hierarchical model provides an improvement in codification performance, robust across all the four corpora considered in the experiments: three classical corpora commonly used in music studies and one of tango songs recently released. Our study confirms the usefulness of the MDL approach to solve the classical trade-off between model complexity and its ability to fit the data.Keywords: Musical rhythmprobabilistic modellinginformation theoryminimum description lengthmodel selection2010 Mathematics Subject Classification: 00A6568Q3062B102012 Computing Classification Scheme: Applied computing AcknowledgmentsThe authors would like to thank Dr. Ignacio Ramírez, Dr. Marcelo Fiori and Dr. Paola Bermolen for fruitful discussion about this work.Disclosure statementNo potential conflict of interest was reported by the author(s).Supplemental dataSupplemental data for this article can be accessed online at http://dx.doi.org/10.1080/17459737.2023.2213471.Correction StatementThis article has been republished with minor changes. These changes do not impact the academic content of the article.Notes1 Note that other metrical levels could be used as reference, such as the sixteenth-note level.2 This is not the only reason. The models discussed in the following also treat different time signatures differently.3 CitationMavromatis (2012) uses the term probabilistic musical grammar.4 As usual in information theory we use logx=log2x.5 To encode one element within a set of d=2ℓ a binary stings, we need to use ℓ=log2d=logd bits.6 In both cases the description length is proportional to logN. This fact was observed by CitationRissanen (1978).7 https://github.com/mrocamora/mdlfit8 Aesthetic preferences can result in a different rate of occurrence of different rhythmic patterns, see Figure 5.Additional informationFundingThis work was partially supported by funding agencies Agencia Nacional de Investigación e Innovación (ANII) and Comisión Sectorial de Investigación Científica (CSIC), UdelaR.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"2 1","pages":"0"},"PeriodicalIF":0.5000,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Minimum description length for selection of models of musical rhythm\",\"authors\":\"Verónica Rumbo, Ernesto Mordecki, Martín Rocamora\",\"doi\":\"10.1080/17459737.2023.2213471\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"AbstractWe apply the minimum description length (MDL) methodology from information theory in order to select mathematical models for musical rhythm. We consider six models proposed by David Temperley and mathematically formalize them, which allows for an MDL analysis. As a consequence, we find that two of the models are not suitable to apply this methodology, as their codification strategy does not represent all possible rhythm sequences. A slight modification of Temperley's hierarchical model provides an improvement in codification performance, robust across all the four corpora considered in the experiments: three classical corpora commonly used in music studies and one of tango songs recently released. Our study confirms the usefulness of the MDL approach to solve the classical trade-off between model complexity and its ability to fit the data.Keywords: Musical rhythmprobabilistic modellinginformation theoryminimum description lengthmodel selection2010 Mathematics Subject Classification: 00A6568Q3062B102012 Computing Classification Scheme: Applied computing AcknowledgmentsThe authors would like to thank Dr. Ignacio Ramírez, Dr. Marcelo Fiori and Dr. Paola Bermolen for fruitful discussion about this work.Disclosure statementNo potential conflict of interest was reported by the author(s).Supplemental dataSupplemental data for this article can be accessed online at http://dx.doi.org/10.1080/17459737.2023.2213471.Correction StatementThis article has been republished with minor changes. These changes do not impact the academic content of the article.Notes1 Note that other metrical levels could be used as reference, such as the sixteenth-note level.2 This is not the only reason. The models discussed in the following also treat different time signatures differently.3 CitationMavromatis (2012) uses the term probabilistic musical grammar.4 As usual in information theory we use logx=log2x.5 To encode one element within a set of d=2ℓ a binary stings, we need to use ℓ=log2d=logd bits.6 In both cases the description length is proportional to logN. This fact was observed by CitationRissanen (1978).7 https://github.com/mrocamora/mdlfit8 Aesthetic preferences can result in a different rate of occurrence of different rhythmic patterns, see Figure 5.Additional informationFundingThis work was partially supported by funding agencies Agencia Nacional de Investigación e Innovación (ANII) and Comisión Sectorial de Investigación Científica (CSIC), UdelaR.\",\"PeriodicalId\":50138,\"journal\":{\"name\":\"Journal of Mathematics and Music\",\"volume\":\"2 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2023-05-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematics and Music\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1080/17459737.2023.2213471\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematics and Music","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1080/17459737.2023.2213471","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
Minimum description length for selection of models of musical rhythm
AbstractWe apply the minimum description length (MDL) methodology from information theory in order to select mathematical models for musical rhythm. We consider six models proposed by David Temperley and mathematically formalize them, which allows for an MDL analysis. As a consequence, we find that two of the models are not suitable to apply this methodology, as their codification strategy does not represent all possible rhythm sequences. A slight modification of Temperley's hierarchical model provides an improvement in codification performance, robust across all the four corpora considered in the experiments: three classical corpora commonly used in music studies and one of tango songs recently released. Our study confirms the usefulness of the MDL approach to solve the classical trade-off between model complexity and its ability to fit the data.Keywords: Musical rhythmprobabilistic modellinginformation theoryminimum description lengthmodel selection2010 Mathematics Subject Classification: 00A6568Q3062B102012 Computing Classification Scheme: Applied computing AcknowledgmentsThe authors would like to thank Dr. Ignacio Ramírez, Dr. Marcelo Fiori and Dr. Paola Bermolen for fruitful discussion about this work.Disclosure statementNo potential conflict of interest was reported by the author(s).Supplemental dataSupplemental data for this article can be accessed online at http://dx.doi.org/10.1080/17459737.2023.2213471.Correction StatementThis article has been republished with minor changes. These changes do not impact the academic content of the article.Notes1 Note that other metrical levels could be used as reference, such as the sixteenth-note level.2 This is not the only reason. The models discussed in the following also treat different time signatures differently.3 CitationMavromatis (2012) uses the term probabilistic musical grammar.4 As usual in information theory we use logx=log2x.5 To encode one element within a set of d=2ℓ a binary stings, we need to use ℓ=log2d=logd bits.6 In both cases the description length is proportional to logN. This fact was observed by CitationRissanen (1978).7 https://github.com/mrocamora/mdlfit8 Aesthetic preferences can result in a different rate of occurrence of different rhythmic patterns, see Figure 5.Additional informationFundingThis work was partially supported by funding agencies Agencia Nacional de Investigación e Innovación (ANII) and Comisión Sectorial de Investigación Científica (CSIC), UdelaR.
期刊介绍:
Journal of Mathematics and Music aims to advance the use of mathematical modelling and computation in music theory. The Journal focuses on mathematical approaches to musical structures and processes, including mathematical investigations into music-theoretic or compositional issues as well as mathematically motivated analyses of musical works or performances. In consideration of the deep unsolved ontological and epistemological questions concerning knowledge about music, the Journal is open to a broad array of methodologies and topics, particularly those outside of established research fields such as acoustics, sound engineering, auditory perception, linguistics etc.