Pub Date : 2023-07-24DOI: 10.1080/17459737.2023.2234126
C. Dettmann, Liam Taylor-West
,
，
{"title":"Algebraic tunings","authors":"C. Dettmann, Liam Taylor-West","doi":"10.1080/17459737.2023.2234126","DOIUrl":"https://doi.org/10.1080/17459737.2023.2234126","url":null,"abstract":",","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82441821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-17DOI: 10.1080/17459737.2023.2228546
Jeffrey R. Boland, Lane P. Hughston
Equal temperament, in which semitones are tuned in the irrational ratio of 21/12:1, is best seen as a serviceable compromise, sacrificing purity for flexibility. Just intonation, in which intervals are given by products of powers of 2, 3, and 5, is more natural, but of limited flexibility. We propose a new scheme in which ratios of Gaussian integers form the basis of an abstract tonal sys
{"title":"Mathematical foundations of complex tonality","authors":"Jeffrey R. Boland, Lane P. Hughston","doi":"10.1080/17459737.2023.2228546","DOIUrl":"https://doi.org/10.1080/17459737.2023.2228546","url":null,"abstract":"<p>Equal temperament, in which semitones are tuned in the irrational ratio of <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0001.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0001.gif\"}' src=\"//:0\"/><span></span></span><span><span style=\"color: inherit; display: none;\"></span><span data-mathml='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mn>2</mn><mrow><mn>1</mn><mrow><mo>/</mo></mrow><mn>12</mn></mrow></msup><mo>:</mo><mn>1</mn></math>' role=\"presentation\" style=\"position: relative;\" tabindex=\"0\"><nobr aria-hidden=\"true\"><span style=\"width: 3.951em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.322em; height: 0px; font-size: 118%;\"><span style=\"position: absolute; clip: rect(1.292em, 1003.25em, 2.504em, -1000em); top: -2.359em; left: 0em;\"><span><span><span style=\"display: inline-block; position: relative; width: 1.989em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.186em, 1000.45em, 4.141em, -1000em); top: -3.997em; left: 0em;\"><span style=\"font-family: MathJax_Main;\">2</span><span style=\"display: inline-block; width: 0px; height: 3.997em;\"></span></span><span style=\"position: absolute; top: -4.39em; left: 0.5em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">1</span><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">/</span></span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">12</span></span><span style=\"display: inline-block; width: 0px; height: 3.997em;\"></span></span></span></span><span style=\"font-family: MathJax_Main; padding-left: 0.278em;\">:</span><span style=\"font-family: MathJax_Main; padding-left: 0.278em;\">1</span></span><span style=\"display: inline-block; width: 0px; height: 2.359em;\"></span></span></span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.057em; border-left: 0px solid; width: 0px; height: 1.203em;\"></span></span></nobr><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mn>2</mn><mrow><mn>1</mn><mrow><mo>/</mo></mrow><mn>12</mn></mrow></msup><mo>:</mo><mn>1</mn></math></span></span><script type=\"math/mml\"><math><msup><mn>2</mn><mrow><mn>1</mn><mrow><mo>/</mo></mrow><mn>12</mn></mrow></msup><mo>:</mo><mn>1</mn></math></script></span>, is best seen as a serviceable compromise, sacrificing purity for flexibility. Just intonation, in which intervals are given by products of powers of 2, 3, and 5, is more natural, but of limited flexibility. We propose a new scheme in which ratios of Gaussian integers form the basis of an abstract tonal sys","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138540072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-04DOI: 10.1080/17459737.2023.2209793
F. Hazama
{"title":"Iterative method of construction for almost smooth rhythms","authors":"F. Hazama","doi":"10.1080/17459737.2023.2209793","DOIUrl":"https://doi.org/10.1080/17459737.2023.2209793","url":null,"abstract":"","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74521863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-31DOI: 10.1080/17459737.2023.2210346
Michael D. Fowler
{"title":"Composing Cagean silence","authors":"Michael D. Fowler","doi":"10.1080/17459737.2023.2210346","DOIUrl":"https://doi.org/10.1080/17459737.2023.2210346","url":null,"abstract":"","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90477961","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-29DOI: 10.1080/17459737.2023.2213471
Verónica Rumbo, Ernesto Mordecki, Martín Rocamora
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.
{"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":"https://doi.org/10.1080/17459737.2023.2213471","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":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135791949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-03DOI: 10.1080/17459737.2023.2197905
Mai Lan Tran, Dongjin Lee, Jae-Hun Jung
Common AI music composition algorithms train a machine by feeding a set of music pieces. This approach is a blackbox optimization, i.e. the underlying composition algorithm is, in general, unknown to users. In this paper, we present a method of machine composition that teaches a machine the compositional principles embedded in the music using the concept of overlap matrix. In (Tran Mai Lan, Changbom Park & Jae-Hun Jung (2023) Topological data analysis of Korean music in Jeongganbo: a cycle structure, Journal of Mathematics and Music, DOI: 10.1080/17459737.2022.2164626), a type of Korean music called Dodeuri music has been analysed using topological data analysis (TDA). To apply TDA, the music data is first reconstructed as a graph. Through TDA on the constructed graph, a unique set of cycles is found. The overlap matrix lets us visualize how those cycles are interconnected in music. We explain how we use the overlap matrix for machine composition. The overlap matrix is suitable for algorithmic composition and also provides seed music to train an artificial neural network.
常见的人工智能作曲算法通过输入一组音乐片段来训练机器。这种方法是一种黑盒优化，即底层的组合算法通常对用户来说是未知的。在本文中，我们提出了一种机器作曲的方法，该方法使用重叠矩阵的概念教机器嵌入音乐中的作曲原则。在(Tran Mai Lan, Changbom Park & Jae-Hun Jung, 2023)《Jeongganbo韩国音乐的拓扑数据分析:一个循环结构》，Journal of Mathematics and music, DOI: 10.1080/17459737.2022.2164626)中，使用拓扑数据分析(TDA)分析了一种称为Dodeuri音乐的韩国音乐类型。为了应用TDA，首先将音乐数据重构为图形。通过构造图上的TDA，找到了唯一的一组环。重叠矩阵让我们直观地看到这些循环在音乐中是如何相互联系的。我们解释了如何使用重叠矩阵进行机器组合。重叠矩阵不仅适用于算法合成，而且为训练人工神经网络提供了种子音乐。
{"title":"Machine composition of Korean music via topological data analysis and artificial neural network","authors":"Mai Lan Tran, Dongjin Lee, Jae-Hun Jung","doi":"10.1080/17459737.2023.2197905","DOIUrl":"https://doi.org/10.1080/17459737.2023.2197905","url":null,"abstract":"Common AI music composition algorithms train a machine by feeding a set of music pieces. This approach is a blackbox optimization, i.e. the underlying composition algorithm is, in general, unknown to users. In this paper, we present a method of machine composition that teaches a machine the compositional principles embedded in the music using the concept of overlap matrix. In (Tran Mai Lan, Changbom Park & Jae-Hun Jung (2023) Topological data analysis of Korean music in Jeongganbo: a cycle structure, Journal of Mathematics and Music, DOI: 10.1080/17459737.2022.2164626), a type of Korean music called Dodeuri music has been analysed using topological data analysis (TDA). To apply TDA, the music data is first reconstructed as a graph. Through TDA on the constructed graph, a unique set of cycles is found. The overlap matrix lets us visualize how those cycles are interconnected in music. We explain how we use the overlap matrix for machine composition. The overlap matrix is suitable for algorithmic composition and also provides seed music to train an artificial neural network.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134922204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-30DOI: 10.1080/17459737.2023.2194301
R. Cubarsi
{"title":"On the divisions of the octave in generalized Pythagorean scales and their bidimensional representation","authors":"R. Cubarsi","doi":"10.1080/17459737.2023.2194301","DOIUrl":"https://doi.org/10.1080/17459737.2023.2194301","url":null,"abstract":"","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73026419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-09DOI: 10.1080/17459737.2022.2164627
Roger Asensi Arranz, Daniel Harasim, T. Noll
The aim of this paper is to argue that complementation is an operation similarly fundamental to music theory as transposition and inversion. We focus on studying the chromatic complement mapping that translates diatonic seventh chords into 8-note scales which can also be interpreted as rhythmic beat patterns. Such complements of diatonic seventh chords are of particular importance since they correspond to the scales popularized by the Jazz theorist Barry Harris, as well as to rhythms used in African drum music and Steve Reich's Clapping Music. Our approach enables a systematic study of these scales and rhythms using established theories of efficient voice leading and generalized diatonic scales and chords, in particular the theory of second-order maximally even sets. The main contributions of this research are (1) to explicate the correspondence between voice leadings and rhythmic transformations, (2) to systematize the family of Barry Harris scales, and (3) to describe classes of voice leadings between chords of different cardinality that are invariant under complementation.
{"title":"Fishing for complements with chord, scale, and rhythm nets","authors":"Roger Asensi Arranz, Daniel Harasim, T. Noll","doi":"10.1080/17459737.2022.2164627","DOIUrl":"https://doi.org/10.1080/17459737.2022.2164627","url":null,"abstract":"The aim of this paper is to argue that complementation is an operation similarly fundamental to music theory as transposition and inversion. We focus on studying the chromatic complement mapping that translates diatonic seventh chords into 8-note scales which can also be interpreted as rhythmic beat patterns. Such complements of diatonic seventh chords are of particular importance since they correspond to the scales popularized by the Jazz theorist Barry Harris, as well as to rhythms used in African drum music and Steve Reich's Clapping Music. Our approach enables a systematic study of these scales and rhythms using established theories of efficient voice leading and generalized diatonic scales and chords, in particular the theory of second-order maximally even sets. The main contributions of this research are (1) to explicate the correspondence between voice leadings and rhythmic transformations, (2) to systematize the family of Barry Harris scales, and (3) to describe classes of voice leadings between chords of different cardinality that are invariant under complementation.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74995050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-20DOI: 10.1080/17459737.2023.2166136
H. Bedouelle
{"title":"Exhaustive chord progressions and their use in music composition","authors":"H. Bedouelle","doi":"10.1080/17459737.2023.2166136","DOIUrl":"https://doi.org/10.1080/17459737.2023.2166136","url":null,"abstract":"","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86806697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-09DOI: 10.1080/17459737.2022.2149869
Noah R. Fram
Prevailing theories of genre, derived primarily from literary and musical scholarship, differ in characteristics they ascribe to genre itself. Here, the temporally dynamic and culturally contingent nature of genre informs a computational framework that is reducible to extant theories of genre and connected to psychological theories of perceptual categorization. This framework, called genredynamics, interprets genres as perceptual categories in a space defined by aesthetic and sociocultural variables, and characterizes the behaviour and structure of genres using concepts from differential topology. Its existence demonstrates that disparate theoretical approaches to genre can be unified and implies that genre is best understood as both a psychological and musicological phenomenon. Classifications' temporal fluidity and incorporating sociocultural variables alongside sensory ones are necessary for this framework to be generalizable. Together, these theoretical results have broad implications for potential applications of genre theory, including the study of mental representations, social and cultural psychology, and cognition.
{"title":"Genredynamics: a perceptual calculus of genre","authors":"Noah R. Fram","doi":"10.1080/17459737.2022.2149869","DOIUrl":"https://doi.org/10.1080/17459737.2022.2149869","url":null,"abstract":"Prevailing theories of genre, derived primarily from literary and musical scholarship, differ in characteristics they ascribe to genre itself. Here, the temporally dynamic and culturally contingent nature of genre informs a computational framework that is reducible to extant theories of genre and connected to psychological theories of perceptual categorization. This framework, called genredynamics, interprets genres as perceptual categories in a space defined by aesthetic and sociocultural variables, and characterizes the behaviour and structure of genres using concepts from differential topology. Its existence demonstrates that disparate theoretical approaches to genre can be unified and implies that genre is best understood as both a psychological and musicological phenomenon. Classifications' temporal fluidity and incorporating sociocultural variables alongside sensory ones are necessary for this framework to be generalizable. Together, these theoretical results have broad implications for potential applications of genre theory, including the study of mental representations, social and cultural psychology, and cognition.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83123165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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