Pub Date : 2021-04-12DOI: 10.1080/17459737.2021.1903591
D. Conklin
Sequential pattern mining in music is a central part of automated music analysis and music generation. This paper evaluates sequential pattern mining on a corpus of Mozarabic chant neume sequences that have been annotated by a musicologist with intra-opus patterns. Significant patterns are discovered in three settings: all closed patterns, maximal closed patterns, and minimal closed patterns. Each setting is evaluated against the annotated patterns using the measures of recall and precision. The results indicate that it is possible to retrieve all known patterns with an acceptable precision using significant closed pattern discovery.
{"title":"Mining contour sequences for significant closed patterns","authors":"D. Conklin","doi":"10.1080/17459737.2021.1903591","DOIUrl":"https://doi.org/10.1080/17459737.2021.1903591","url":null,"abstract":"Sequential pattern mining in music is a central part of automated music analysis and music generation. This paper evaluates sequential pattern mining on a corpus of Mozarabic chant neume sequences that have been annotated by a musicologist with intra-opus patterns. Significant patterns are discovered in three settings: all closed patterns, maximal closed patterns, and minimal closed patterns. Each setting is evaluated against the annotated patterns using the measures of recall and precision. The results indicate that it is possible to retrieve all known patterns with an acceptable precision using significant closed pattern discovery.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"29 1","pages":"112 - 124"},"PeriodicalIF":1.1,"publicationDate":"2021-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89421452","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 : 2021-04-05DOI: 10.1080/17459737.2021.1896811
A. Laaksonen, Kjell Lemström
We study the problem of identifying repetitions under transposition and time-warp invariances in polyphonic symbolic music. Using a novel onset-time-pair representation, we reduce the repeating pattern discovery problem to instances of the classical problem of finding the longest increasing subsequences. The resulting algorithm works in time where n is the number of notes in a musical work. We also study windowed variants of the problem where onset-time differences between notes are restricted, and show that they can also be solved in time using the algorithm.
{"title":"Discovering distorted repeating patterns in polyphonic music through longest increasing subsequences","authors":"A. Laaksonen, Kjell Lemström","doi":"10.1080/17459737.2021.1896811","DOIUrl":"https://doi.org/10.1080/17459737.2021.1896811","url":null,"abstract":"We study the problem of identifying repetitions under transposition and time-warp invariances in polyphonic symbolic music. Using a novel onset-time-pair representation, we reduce the repeating pattern discovery problem to instances of the classical problem of finding the longest increasing subsequences. The resulting algorithm works in time where n is the number of notes in a musical work. We also study windowed variants of the problem where onset-time differences between notes are restricted, and show that they can also be solved in time using the algorithm.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"82 1","pages":"99 - 111"},"PeriodicalIF":1.1,"publicationDate":"2021-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79306339","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 : 2021-04-05DOI: 10.1080/17459737.2021.1900436
Kerstin Neubarth, D. Conklin
In computational pattern discovery, pattern evaluation measures select or rank patterns according to their potential interestingness in a given analysis task. Many measures have been proposed to accommodate different pattern types and properties. This paper presents a method and case study employing measures for frequent, characteristic, associative, contrasting, dependent, and significant patterns to model pattern interestingness in a reference analysis, Frances Densmore's study of Teton Sioux songs. Results suggest that interesting changes from older to more recent Sioux songs according to Densmore's analysis are best captured by contrast, dependency, and significance measures.
{"title":"Modelling pattern interestingness in comparative music corpus analysis","authors":"Kerstin Neubarth, D. Conklin","doi":"10.1080/17459737.2021.1900436","DOIUrl":"https://doi.org/10.1080/17459737.2021.1900436","url":null,"abstract":"In computational pattern discovery, pattern evaluation measures select or rank patterns according to their potential interestingness in a given analysis task. Many measures have been proposed to accommodate different pattern types and properties. This paper presents a method and case study employing measures for frequent, characteristic, associative, contrasting, dependent, and significant patterns to model pattern interestingness in a reference analysis, Frances Densmore's study of Teton Sioux songs. Results suggest that interesting changes from older to more recent Sioux songs according to Densmore's analysis are best captured by contrast, dependency, and significance measures.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"32 1","pages":"154 - 167"},"PeriodicalIF":1.1,"publicationDate":"2021-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85102379","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 : 2021-03-11DOI: 10.1080/17459737.2022.2164626
M. Tran, Changbom Park, Jae-Hun Jung
Jeongganbo is a unique music representation invented by Sejong the Great. Contrary to the Western music notation, the pitch of each note is encrypted and the length is visualized directly in a matrix form. We use topological data analysis (TDA) to analyze the Korean music written in Jeongganbo for Suyeonjang, Songuyeo, and Taryong, those well-known pieces played among noble community. We define the nodes of each music with pitch and length and transform the music into a graph with the distance between the nodes defined as their adjacent occurrence rate. The graph homology is investigated by TDA. We identify cycles of each music and show how those cycles are interconnected. We found that the cycles of Suyeonjang and Songuyeo, categorized as a special type of cyclic music, frequently overlap each other in the music, while those of Taryong, which does not belong to the same class, appear only individually.
{"title":"Topological data analysis of Korean music in Jeongganbo: a cycle structure","authors":"M. Tran, Changbom Park, Jae-Hun Jung","doi":"10.1080/17459737.2022.2164626","DOIUrl":"https://doi.org/10.1080/17459737.2022.2164626","url":null,"abstract":"Jeongganbo is a unique music representation invented by Sejong the Great. Contrary to the Western music notation, the pitch of each note is encrypted and the length is visualized directly in a matrix form. We use topological data analysis (TDA) to analyze the Korean music written in Jeongganbo for Suyeonjang, Songuyeo, and Taryong, those well-known pieces played among noble community. We define the nodes of each music with pitch and length and transform the music into a graph with the distance between the nodes defined as their adjacent occurrence rate. The graph homology is investigated by TDA. We identify cycles of each music and show how those cycles are interconnected. We found that the cycles of Suyeonjang and Songuyeo, categorized as a special type of cyclic music, frequently overlap each other in the music, while those of Taryong, which does not belong to the same class, appear only individually.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"52 1","pages":"403 - 432"},"PeriodicalIF":1.1,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87148031","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 : 2021-03-11DOI: 10.1080/17459737.2021.1882600
D. Mukherjee
When the same sound is produced simultaneously with two different fundamental frequencies, auditory roughness is observed. If the first sound is fixed and the fundamental frequency of the second is varied continuously, auditory roughness also varies continuously. A vowel sound is distinguished by its spectral envelope – which is independent of the fundamental frequency. This is a motivation to define the metric space of timbres. Each timbre is associated with a dissonance function which has local minima at certain intervals of local consonance related to the timbre. This is related to the music-theoretical notion of consonant intervals and scales. For the subspace consisting of all timbres with an interval of local consonance at a chosen point β, the main theorem describes certain points on the boundary by the vanishing of one-sided derivatives of dissonance functions at β.
{"title":"Local minima of dissonance functions","authors":"D. Mukherjee","doi":"10.1080/17459737.2021.1882600","DOIUrl":"https://doi.org/10.1080/17459737.2021.1882600","url":null,"abstract":"When the same sound is produced simultaneously with two different fundamental frequencies, auditory roughness is observed. If the first sound is fixed and the fundamental frequency of the second is varied continuously, auditory roughness also varies continuously. A vowel sound is distinguished by its spectral envelope – which is independent of the fundamental frequency. This is a motivation to define the metric space of timbres. Each timbre is associated with a dissonance function which has local minima at certain intervals of local consonance related to the timbre. This is related to the music-theoretical notion of consonant intervals and scales. For the subspace consisting of all timbres with an interval of local consonance at a chosen point β, the main theorem describes certain points on the boundary by the vanishing of one-sided derivatives of dissonance functions at β.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"91 1","pages":"17 - 37"},"PeriodicalIF":1.1,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72565650","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 : 2021-03-11DOI: 10.1080/17459737.2021.1871789
T. Johnson
Deep Rhythms are normally constructed by choosing a length (l), and the difference (d) between one basic note and the next. If one begins at 0, and one wishes to construct rhythms in measures containing 8 notes, with 3 notes in each measure, and the difference between basic notes is 3, then l = 8, n = 3, one follows the cycle (0, 3, 6, 1, 4, 7, 2, 5, 0 . . . ) and the rhythms are (0, 3, 6), (1, 3, 6), (1,4,6) and so forth, as in the beginning measures of the music. Only (0,3,6) is given in Franck Jedrzejewski’s complete list of deep rhythms on the facing page, because this list includes only basic deep rhythms beginning with zero. But since we are dealing with a circle of 8, we can rotate around the cycle and find seven other deep rhythms, all of which are interesting to my ears. An infinite number of rhythms may be constructed in this way, but as the circles get larger, the rhythms get longer, and tend to follow repeating sequences in a boring way, so I just added a few more sections that I particularly liked and then stopped. The lower staff is simply accompaniment and should be more felt than heard. I find it easiest and most satisfying to repeat each rhythm four times before going on to the next, and to keep a steady tempo of about 120 quarter notes per minute. Since each variation is an independent little piece, one may select and order them however one wishes. I recommend outdoor performances, where the music becomes camouflaged, always blending well with the ambient sounds.
深节奏通常是通过选择一个长度(l)和一个基本音符与下一个基本音符之间的差异(d)来构建的。如果一个人从0开始，他希望在包含8个音符的小节中构建节奏，每个小节有3个音符，基本音符之间的差异是3，那么l = 8, n = 3，一个人遵循循环(0,3,6,1,4,7,2,5,0…)节奏是(0,3,6)(1,3,6)(1,4,6)等等，就像音乐的开始小节一样。在frank Jedrzejewski的完整的深层节奏列表中，只有(0,3,6)是给出的，因为这个列表只包括以0开头的基本深层节奏。但由于我们在处理一个8个的圆圈，我们可以围绕这个周期旋转，找到其他7个深节奏，所有这些对我的耳朵都很有趣。可以用这种方式构建无限多的节奏，但随着圆圈变大，节奏变长，并且倾向于以一种无聊的方式遵循重复序列，所以我只是添加了一些我特别喜欢的部分，然后停止。低五线谱只是伴奏，应该更多地感受而不是听到。我发现最简单和最令人满意的方法是在进行下一个节奏之前将每个节奏重复四次，并保持每分钟约120个四分音符的稳定节奏。因为每个变奏都是一个独立的小片段，你可以随心所欲地选择和排序它们。我推荐户外表演，在那里音乐变得隐蔽，总是与环境声音很好地融合在一起。
{"title":"Deep Rhythms VIIIWood block music*","authors":"T. Johnson","doi":"10.1080/17459737.2021.1871789","DOIUrl":"https://doi.org/10.1080/17459737.2021.1871789","url":null,"abstract":"Deep Rhythms are normally constructed by choosing a length (l), and the difference (d) between one basic note and the next. If one begins at 0, and one wishes to construct rhythms in measures containing 8 notes, with 3 notes in each measure, and the difference between basic notes is 3, then l = 8, n = 3, one follows the cycle (0, 3, 6, 1, 4, 7, 2, 5, 0 . . . ) and the rhythms are (0, 3, 6), (1, 3, 6), (1,4,6) and so forth, as in the beginning measures of the music. Only (0,3,6) is given in Franck Jedrzejewski’s complete list of deep rhythms on the facing page, because this list includes only basic deep rhythms beginning with zero. But since we are dealing with a circle of 8, we can rotate around the cycle and find seven other deep rhythms, all of which are interesting to my ears. An infinite number of rhythms may be constructed in this way, but as the circles get larger, the rhythms get longer, and tend to follow repeating sequences in a boring way, so I just added a few more sections that I particularly liked and then stopped. The lower staff is simply accompaniment and should be more felt than heard. I find it easiest and most satisfying to repeat each rhythm four times before going on to the next, and to keep a steady tempo of about 120 quarter notes per minute. Since each variation is an independent little piece, one may select and order them however one wishes. I recommend outdoor performances, where the music becomes camouflaged, always blending well with the ambient sounds.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"73 1","pages":"194 - 199"},"PeriodicalIF":1.1,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79219563","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 : 2021-02-23DOI: 10.1080/17459737.2022.2136775
J. Arias-Valero, O. A. Agust'in-Aquino, E. Lluis-Puebla
We re-create the essential results of a 1989 unpublished article by Mazzola and Muzzulini that contains the musicological aspects of a first-species counterpoint model. We include a summary of the mathematical counterpoint theory and several variations of the model that offer different perspectives on Mazzola's original principles.
{"title":"Musicological, computational, and conceptual aspects of first-species counterpoint theory","authors":"J. Arias-Valero, O. A. Agust'in-Aquino, E. Lluis-Puebla","doi":"10.1080/17459737.2022.2136775","DOIUrl":"https://doi.org/10.1080/17459737.2022.2136775","url":null,"abstract":"We re-create the essential results of a 1989 unpublished article by Mazzola and Muzzulini that contains the musicological aspects of a first-species counterpoint model. We include a summary of the mathematical counterpoint theory and several variations of the model that offer different perspectives on Mazzola's original principles.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"39 1","pages":"371 - 387"},"PeriodicalIF":1.1,"publicationDate":"2021-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79284806","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 : 2021-02-01DOI: 10.1080/17459737.2021.1871787
Maria Mannone
Mathematical music theory helps us investigate musical compositions in mathematical terms. Some hints can be extended towards the visual arts. Mathematical approaches can also help formalize a “translation” from the visual domain to the auditory one and vice versa. Thus, a visual artwork can be mathematically investigated, then translated into music. The final, refined musical rendition can be compared to the initial visual idea. Can an artistic idea be preserved through these changes of media? Can a non-trivial pattern be envisaged in an artwork, and then still be identified after the change of medium? Here, we consider a contemporary installation and an ensemble musical piece derived from it. We first mathematically investigate the installation, finding its patterns and structure, and then we compare them with structure and patterns of the musical composition. In particular, we apply two concepts of mathematical music theory, the Quantum GestART and the gestural similarity conjecture, to the analysis of Qwalala, realized for the Venice Biennale by Pae White, comparing it to its musical rendition in the homonymous piece for harp and ensemble composed by Federico Favali. Some sketches of generalizations follow, with the “Souvenir Theorem” and the “Art Conjecture.”
{"title":"A musical reading of a contemporary installation and back: mathematical investigations of patterns in Qwalala","authors":"Maria Mannone","doi":"10.1080/17459737.2021.1871787","DOIUrl":"https://doi.org/10.1080/17459737.2021.1871787","url":null,"abstract":"Mathematical music theory helps us investigate musical compositions in mathematical terms. Some hints can be extended towards the visual arts. Mathematical approaches can also help formalize a “translation” from the visual domain to the auditory one and vice versa. Thus, a visual artwork can be mathematically investigated, then translated into music. The final, refined musical rendition can be compared to the initial visual idea. Can an artistic idea be preserved through these changes of media? Can a non-trivial pattern be envisaged in an artwork, and then still be identified after the change of medium? Here, we consider a contemporary installation and an ensemble musical piece derived from it. We first mathematically investigate the installation, finding its patterns and structure, and then we compare them with structure and patterns of the musical composition. In particular, we apply two concepts of mathematical music theory, the Quantum GestART and the gestural similarity conjecture, to the analysis of Qwalala, realized for the Venice Biennale by Pae White, comparing it to its musical rendition in the homonymous piece for harp and ensemble composed by Federico Favali. Some sketches of generalizations follow, with the “Souvenir Theorem” and the “Art Conjecture.”","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"103 1","pages":"80 - 96"},"PeriodicalIF":1.1,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85847250","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 : 2021-01-19DOI: 10.1080/17459737.2021.1871788
B. Limkar, G. Chandekar
Operational Modal Analysis (OMA) of Tibetan singing bowl is performed to extract natural frequencies and mode shapes without measuring excitation data. It is kept free on a rigid surface, which is a common way of playing this musical instrument. OMA results are validated using Experimental Modal Analysis (EMA) and Numerical Methods using FEA. Numerical simulations using ANSYS® software establishes a benchmark for EMA results. The input and response data for 144 response points are collected using instrumented hammer and accelerometer, connected to a four-channel FFT analyser. A self-generated MATLAB® code processes the response signals for EMA and OMA. For natural frequencies, the absolute error lies within 6%, except for the first mode. For mode shapes, the Modal Assurance Criteria (MAC) value is more than 70%, except for the fourth mode. Thus, OMA is the best available method compared to the EMA and Numerical method using FEA for structural analysis under actual performance conditions.
{"title":"Structural dynamic analysis of a musical instrument: Tibetan singing bowl","authors":"B. Limkar, G. Chandekar","doi":"10.1080/17459737.2021.1871788","DOIUrl":"https://doi.org/10.1080/17459737.2021.1871788","url":null,"abstract":"Operational Modal Analysis (OMA) of Tibetan singing bowl is performed to extract natural frequencies and mode shapes without measuring excitation data. It is kept free on a rigid surface, which is a common way of playing this musical instrument. OMA results are validated using Experimental Modal Analysis (EMA) and Numerical Methods using FEA. Numerical simulations using ANSYS® software establishes a benchmark for EMA results. The input and response data for 144 response points are collected using instrumented hammer and accelerometer, connected to a four-channel FFT analyser. A self-generated MATLAB® code processes the response signals for EMA and OMA. For natural frequencies, the absolute error lies within 6%, except for the first mode. For mode shapes, the Modal Assurance Criteria (MAC) value is more than 70%, except for the fourth mode. Thus, OMA is the best available method compared to the EMA and Numerical method using FEA for structural analysis under actual performance conditions.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"53 Suppl 1 1","pages":"1 - 16"},"PeriodicalIF":1.1,"publicationDate":"2021-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77768155","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 : 2021-01-02DOI: 10.1080/17459737.2020.1716404
Reinhard Blutner, Peter beim Graben
Metaphors involving motion and forces are a source of inspiration for understanding tonal music and tonal harmonies since ancient times. Starting with the rise of quantum cognition, the modern interactional conception of forces as developed in gauge theory has recently entered the field of theoretical musicology. We develop a gauge model of tonal attraction based on SU(2) symmetry. This model comprises two earlier attempts, the phase model grounded on U(1) gauge symmetry, and the spatial deformation model derived from SO(2) gauge symmetry. In the neutral, force-free case both submodels agree and generate the same predictions as a simple qubit approach. However, there are several differences in the force-driven case. It is claimed that the deformation model gives a proper description of static tonal attraction. The full model combines the deformation model with the phase model through SU(2) gauge symmetry and unifies static and dynamic tonal attraction.
{"title":"Gauge models of musical forces","authors":"Reinhard Blutner, Peter beim Graben","doi":"10.1080/17459737.2020.1716404","DOIUrl":"https://doi.org/10.1080/17459737.2020.1716404","url":null,"abstract":"Metaphors involving motion and forces are a source of inspiration for understanding tonal music and tonal harmonies since ancient times. Starting with the rise of quantum cognition, the modern interactional conception of forces as developed in gauge theory has recently entered the field of theoretical musicology. We develop a gauge model of tonal attraction based on SU(2) symmetry. This model comprises two earlier attempts, the phase model grounded on U(1) gauge symmetry, and the spatial deformation model derived from SO(2) gauge symmetry. In the neutral, force-free case both submodels agree and generate the same predictions as a simple qubit approach. However, there are several differences in the force-driven case. It is claimed that the deformation model gives a proper description of static tonal attraction. The full model combines the deformation model with the phase model through SU(2) gauge symmetry and unifies static and dynamic tonal attraction.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"11 1","pages":"17 - 36"},"PeriodicalIF":1.1,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88653624","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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