Pub Date : 2022-04-05DOI: 10.1080/17459737.2022.2049383
B. Fugiel
I propose a quantum-like approach to the description of melody perception where classic intervals that constitute a melody are replaced by acoustical qubits, i.e. two-level acoustic systems, using Shepard tones for this purpose. Each of such qubits is considered to be a superposition of two intervals, ascending and descending, that form an octave when put together. Any melody perception can thus be treated analogously to a sequence of quantum measurements. Because of an acoustical collapse, analogous to the wave function reduction in quantum mechanics, just a single interval, ascending or descending, can be heard each time. Different melodies generated by the same sequence of acoustical qubits can be then perceived.
{"title":"Quantum-like melody perception","authors":"B. Fugiel","doi":"10.1080/17459737.2022.2049383","DOIUrl":"https://doi.org/10.1080/17459737.2022.2049383","url":null,"abstract":"I propose a quantum-like approach to the description of melody perception where classic intervals that constitute a melody are replaced by acoustical qubits, i.e. two-level acoustic systems, using Shepard tones for this purpose. Each of such qubits is considered to be a superposition of two intervals, ascending and descending, that form an octave when put together. Any melody perception can thus be treated analogously to a sequence of quantum measurements. Because of an acoustical collapse, analogous to the wave function reduction in quantum mechanics, just a single interval, ascending or descending, can be heard each time. Different melodies generated by the same sequence of acoustical qubits can be then perceived.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"64 1","pages":"319 - 331"},"PeriodicalIF":1.1,"publicationDate":"2022-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85787825","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 : 2022-03-17DOI: 10.1080/17459737.2022.2044927
Fabian C. Moss, M. Neuwirth, M. Rohrmeier
In this study, we determine the fundamental role of the line of fifths for the organization of tonal material by applying dimensionality reduction to a large historical corpus of pitch-class counts (ca. 1360–1940). We observe a historically growing trend in the exploitation of the fifths range, i.e. the size of segments that pitch-class distributions cover on the line of fifths. Moreover, we introduce the novel concept of pitch-class (co-)evolution, which traces the changing co-occurrence of pitch classes over time and likewise reaffirms the centrality of this linear tonal space from a historical angle, allowing us also to distinguish between historical periods in terms of the usage of pitch classes.
{"title":"The line of fifths and the co-evolution of tonal pitch-classes","authors":"Fabian C. Moss, M. Neuwirth, M. Rohrmeier","doi":"10.1080/17459737.2022.2044927","DOIUrl":"https://doi.org/10.1080/17459737.2022.2044927","url":null,"abstract":"In this study, we determine the fundamental role of the line of fifths for the organization of tonal material by applying dimensionality reduction to a large historical corpus of pitch-class counts (ca. 1360–1940). We observe a historically growing trend in the exploitation of the fifths range, i.e. the size of segments that pitch-class distributions cover on the line of fifths. Moreover, we introduce the novel concept of pitch-class (co-)evolution, which traces the changing co-occurrence of pitch classes over time and likewise reaffirms the centrality of this linear tonal space from a historical angle, allowing us also to distinguish between historical periods in terms of the usage of pitch classes.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"94 1","pages":"173 - 197"},"PeriodicalIF":1.1,"publicationDate":"2022-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80270596","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 : 2022-02-19DOI: 10.1080/17459737.2022.2025625
Francisco Gómez-Martín
This is both a personal and academic review of Godfried Toussaint's The Geometry of Musical Rhythm.
这是对哥德弗里德·杜桑的《音乐节奏的几何》的个人和学术评论。
{"title":"A review of Godfried Toussaint's The Geometry of Musical Rhythm","authors":"Francisco Gómez-Martín","doi":"10.1080/17459737.2022.2025625","DOIUrl":"https://doi.org/10.1080/17459737.2022.2025625","url":null,"abstract":"This is both a personal and academic review of Godfried Toussaint's The Geometry of Musical Rhythm.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"142 1","pages":"239 - 247"},"PeriodicalIF":1.1,"publicationDate":"2022-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76620374","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 : 2022-01-10DOI: 10.1080/17459737.2021.2008035
Dmitri Tymoczko
Musicians often operate with a hierarchy of scale-like collections, each embedded within the next, and with transposition and inversion available at every level. A particularly common technique is to counteract a transformation at one level with an analogous transformation in the intrinsic scale consisting of a chord’s own notes.
{"title":"Hierarchical set theory","authors":"Dmitri Tymoczko","doi":"10.1080/17459737.2021.2008035","DOIUrl":"https://doi.org/10.1080/17459737.2021.2008035","url":null,"abstract":"Musicians often operate with a hierarchy of scale-like collections, each embedded within the next, and with transposition and inversion available at every level. A particularly common technique is to counteract a transformation at one level with an analogous transformation in the intrinsic scale consisting of a chord’s own notes.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"13 1","pages":"282 - 290"},"PeriodicalIF":1.1,"publicationDate":"2022-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79226101","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 : 2021-12-09DOI: 10.1080/17459737.2021.2002446
Olof Misgeld, A. Holzapfel, Petter Kallioinen, Sven Ahlbäck
Some triple-beat forms in Scandinavian Folk Music are characterized by non-isochronous beat durations: asymmetric beats. Theorists of folk music have suggested that the variability of rhythmic figures and asymmetric metre are fundamental to these forms. The aim of this study is to obtain a deeper understanding of the relationship between melodic structure and asymmetric metre by analysing semi-automatically annotated performances. Our study considers archive and contemporary recordings of fiddlers' different versions of the same musical pieces: polska tunes in a local Swedish tradition. Results show that asymmetric beat patterns are consistent between performances and that they correspond with structural features of rhythmic figures, such as the note density within beats. The present study goes beyond previous work by exploring the use of a state-of-the-art automatic music notation tool in a corpus study of Swedish traditional music, and by employing statistical methods for a comparative analysis of performances across different players.
{"title":"The melodic beat: exploring asymmetry in polska performance","authors":"Olof Misgeld, A. Holzapfel, Petter Kallioinen, Sven Ahlbäck","doi":"10.1080/17459737.2021.2002446","DOIUrl":"https://doi.org/10.1080/17459737.2021.2002446","url":null,"abstract":"Some triple-beat forms in Scandinavian Folk Music are characterized by non-isochronous beat durations: asymmetric beats. Theorists of folk music have suggested that the variability of rhythmic figures and asymmetric metre are fundamental to these forms. The aim of this study is to obtain a deeper understanding of the relationship between melodic structure and asymmetric metre by analysing semi-automatically annotated performances. Our study considers archive and contemporary recordings of fiddlers' different versions of the same musical pieces: polska tunes in a local Swedish tradition. Results show that asymmetric beat patterns are consistent between performances and that they correspond with structural features of rhythmic figures, such as the note density within beats. The present study goes beyond previous work by exploring the use of a state-of-the-art automatic music notation tool in a corpus study of Swedish traditional music, and by employing statistical methods for a comparative analysis of performances across different players.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"45 1","pages":"138 - 159"},"PeriodicalIF":1.1,"publicationDate":"2021-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88397763","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 : 2021-12-07DOI: 10.1080/17459737.2021.2001699
Daniele Ghisi
“Barberpole” tempo illusions are a family of auditory illusions based on the synchronization of faded rhythmic streams playing at different rates, often manufacturing experiences of seemingly eternal acceleration or deceleration. The forefather of all such illusions, based on layers whose rates are powers of two apart (“octaves”), was studied by Jean-Claude Risset in the late seventies and is now known as Risset rhythm. This article provides a mathematical framework for barberpole tempo illusions, generalizing Risset rhythms for arbitrary numbers of subdivisions, non-integer proportions, arbitrary rate modulation, and increasingly accelerating tempi. Furthermore, this article describes a new illusion of eternal rallentando/accelerando based on the full harmonic spectrum of rates. This construction shows that Risset rhythms are related to barberpole variable-rate polyrhythms. A notable application of the study of divisional structures that barberpole illusions underpin is the construction of bistable auditory figures (accelerating or decelerating depending on the stream being focused).
{"title":"Barberpole tempo illusions","authors":"Daniele Ghisi","doi":"10.1080/17459737.2021.2001699","DOIUrl":"https://doi.org/10.1080/17459737.2021.2001699","url":null,"abstract":"“Barberpole” tempo illusions are a family of auditory illusions based on the synchronization of faded rhythmic streams playing at different rates, often manufacturing experiences of seemingly eternal acceleration or deceleration. The forefather of all such illusions, based on layers whose rates are powers of two apart (“octaves”), was studied by Jean-Claude Risset in the late seventies and is now known as Risset rhythm. This article provides a mathematical framework for barberpole tempo illusions, generalizing Risset rhythms for arbitrary numbers of subdivisions, non-integer proportions, arbitrary rate modulation, and increasingly accelerating tempi. Furthermore, this article describes a new illusion of eternal rallentando/accelerando based on the full harmonic spectrum of rates. This construction shows that Risset rhythms are related to barberpole variable-rate polyrhythms. A notable application of the study of divisional structures that barberpole illusions underpin is the construction of bistable auditory figures (accelerating or decelerating depending on the stream being focused).","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"70 1","pages":"266 - 281"},"PeriodicalIF":1.1,"publicationDate":"2021-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87187677","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 : 2021-12-06DOI: 10.1080/17459737.2021.2002956
Tiasa Mondol, Daniel G. Brown
We apply Context-free Grammars (CFG) to measure the structural information content of a symbolic music string. CFGs are appropriate to this domain because they highlight hierarchical patterns, and their dictionary of rules can be used for compression. We adapt this approach to estimate the conditional Kolmogorov complexity of a string with a concise CFG of another string. Thus, a related string may be compressed with the production rules for the first string. We then define an information distance between two symbolic music strings, and show that this measure can separate genres, composers and musical styles. Next, we adapt our approach to a model-selection problem, expressing the model as a CFG with restricted size, generated from a set of representative strings. We show that a well-generated CFG for a composer identifies characteristic patterns that can significantly compress other pieces from the same composer, while not being useful on pieces from different composers. We identify further opportunities of this approach, including using CFGs for generating new music in the style of a composer.
{"title":"Grammar-based compression and its use in symbolic music analysis","authors":"Tiasa Mondol, Daniel G. Brown","doi":"10.1080/17459737.2021.2002956","DOIUrl":"https://doi.org/10.1080/17459737.2021.2002956","url":null,"abstract":"We apply Context-free Grammars (CFG) to measure the structural information content of a symbolic music string. CFGs are appropriate to this domain because they highlight hierarchical patterns, and their dictionary of rules can be used for compression. We adapt this approach to estimate the conditional Kolmogorov complexity of a string with a concise CFG of another string. Thus, a related string may be compressed with the production rules for the first string. We then define an information distance between two symbolic music strings, and show that this measure can separate genres, composers and musical styles. Next, we adapt our approach to a model-selection problem, expressing the model as a CFG with restricted size, generated from a set of representative strings. We show that a well-generated CFG for a composer identifies characteristic patterns that can significantly compress other pieces from the same composer, while not being useful on pieces from different composers. We identify further opportunities of this approach, including using CFGs for generating new music in the style of a composer.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"88 1","pages":"133 - 150"},"PeriodicalIF":1.1,"publicationDate":"2021-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83815427","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 : 2021-09-18DOI: 10.1080/17459737.2021.1969599
M. Buongiorno Nardelli
I introduce the concept of dynamical score networks for the representation and analysis of tonal compositions: a score is interpreted as a dynamical network where every chord is a node and each progression links successive chords. This network can be viewed as a time series of a non-stationary signal, and as such, it can be partitioned for the automatic identification of tonal regions using time series analysis and change point detection without relying on comparisons with pre-determined reference sets or extensive corpora. I demonstrate that the essential features of tonal harmony, centricity, referentiality, directedness and hierarchy, emerge naturally from the network topology and its scale-free properties. Finally, solving for the minimal length path through a route optimization algorithm on these graphs provides an abstraction of harmonic sequences that can be generalized for the conception of generative models of tonal compositional design.
{"title":"Tonal harmony and the topology of dynamical score networks","authors":"M. Buongiorno Nardelli","doi":"10.1080/17459737.2021.1969599","DOIUrl":"https://doi.org/10.1080/17459737.2021.1969599","url":null,"abstract":"I introduce the concept of dynamical score networks for the representation and analysis of tonal compositions: a score is interpreted as a dynamical network where every chord is a node and each progression links successive chords. This network can be viewed as a time series of a non-stationary signal, and as such, it can be partitioned for the automatic identification of tonal regions using time series analysis and change point detection without relying on comparisons with pre-determined reference sets or extensive corpora. I demonstrate that the essential features of tonal harmony, centricity, referentiality, directedness and hierarchy, emerge naturally from the network topology and its scale-free properties. Finally, solving for the minimal length path through a route optimization algorithm on these graphs provides an abstraction of harmonic sequences that can be generalized for the conception of generative models of tonal compositional design.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"23 1","pages":"198 - 212"},"PeriodicalIF":1.1,"publicationDate":"2021-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87072573","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 : 2021-07-08DOI: 10.1080/17459737.2023.2180812
Gennaro Auricchio, L. Ferrarini, Greta Lanzarotto
In this paper, we propose an integer linear programming model whose solutions are the aperiodic rhythms tiling with a given rhythm A. We show how it can be used to define an iterative algorithm that, given a period n, finds all the rhythms which tile with a given rhythm A and also to efficiently check the necessity of the Coven-Meyerowitz condition (T2). To conclude, we run several experiments to validate the time efficiency of the model.
{"title":"An integer linear programming model for tilings","authors":"Gennaro Auricchio, L. Ferrarini, Greta Lanzarotto","doi":"10.1080/17459737.2023.2180812","DOIUrl":"https://doi.org/10.1080/17459737.2023.2180812","url":null,"abstract":"In this paper, we propose an integer linear programming model whose solutions are the aperiodic rhythms tiling with a given rhythm A. We show how it can be used to define an iterative algorithm that, given a period n, finds all the rhythms which tile with a given rhythm A and also to efficiently check the necessity of the Coven-Meyerowitz condition (T2). To conclude, we run several experiments to validate the time efficiency of the model.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"89 1","pages":"514 - 530"},"PeriodicalIF":1.1,"publicationDate":"2021-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81134031","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 : 2021-07-05DOI: 10.1080/17459737.2021.1927214
J. Besada
In 2003, Spanish composer José Manuel López López (b. Madrid 1956) wrote his Estudio II sobre la modulación métrica for four percussionists, commissioned by the Portuguese ensemble Drumming. As the title of the percussion quartet reveals, Elliott Carter’s metric modulation was a key concept for López López during the compositional process. Nevertheless, his theoretical and aesthetic position up to this time was rather close to Karlheinz Stockhausen and to the French spectral composers. López López conceived a chart of tempi in order to reconcile the metric modulation with some ideas he borrowed from Stockhausen, in particular from his famous article “ . . . wie die Zeit vergeht . . . .” In the columns, López López calculated a series of metric modulations among basic notes – from a whole note to a 16th-note – and their dotted counterparts. Conversely, for each row, given a starting value, the cells were filled by iteratively multiplying this value by 12 √ 2, which is the frequency ratio of an equally tempered semitone. López López lastly rounded the obtained values to the nearest integer, giving rise in each row to a series of “chromatic tempi.” This chart puts in evidence a double analogy for understanding tempo relationships as tuning ones. On the one hand, the vertical rational proportions are equivalent to those of the partials of the harmonic series, which are concomitants of just intonation. On the other hand, the horizontal irrational proportions reflect those of the chromatic scale in the equal temperament. Notice that, although not present within the table, López López took also into account subdivisions based on tuplets for his metric modulations, as evident in several calculations below the charted values. This device is a significant milestone in López López’s creative development. Estudio II sobre la modulación métrica is an unpitched score, but this temporal conception was also present in the next piece he composed: Entrance-Exit. For this kind of pianistic tombeau for his dear friend Fausto Romitelli, López López paired tempi with spectral fundamentals by means of these methods. Next, López López implemented and refined his table via Excel, a platform allowing him to remap values as fast as possible. This choice also led him to choose new frequency ratios beyond 12 √ 2 – and therefore related to further tuning systems – for analogous purposes.
2003年，西班牙作曲家何塞•曼努埃尔López López(出生于1956年的马德里)受葡萄牙Drumming乐团委托，为四名打击乐手创作了他的作品《Estudio II sobre la modulación msamtrica》。正如打击乐四重奏的标题所示，Elliott Carter的韵律调制是López López在作曲过程中的一个关键概念。然而，他的理论和美学立场，直到这个时候是相当接近卡尔海因茨·施托克豪森和法国谱作曲家。López López构思了一个节奏图，以便与他从斯托克豪森那里借来的一些想法调和，特别是从他著名的文章“…”在这些列中，López López计算了一系列基本音符之间的韵律调制——从全音到16音——以及它们的虚线对应。相反，对于给定起始值的每行，通过将该值迭代地乘以12√2(即同等调质半音的频率比)来填充单元格。López López最后将得到的值四舍五入到最接近的整数，每行产生一系列“半音节拍”。这张图表证明了将节奏关系理解为调音关系的双重类比。一方面，纵向有理比例相当于谐音级数的偏分比例，它们是正音的伴音。另一方面，水平的不合理比例反映了同等气质中半音音阶的不合理比例。请注意，虽然没有出现在表中，但López López也考虑了基于tuplets的度量调制的细分，这在图表值下面的几个计算中很明显。这个装置是López López创意发展的一个重要里程碑。Estudio II sobre la modulación m录影带是一首无音调的乐谱，但这种时间概念也出现在他创作的下一首作品中:入口-出口。对于这种钢琴墓为他亲爱的朋友福斯托·罗米特利，López López配对的速度和频谱基础通过这些方法。接下来，López López通过Excel实现并改进了他的表，这个平台允许他尽可能快地重新映射值。这个选择也导致他选择新的频率比超过12√2 -因此与进一步的调谐系统有关-用于类似的目的。
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