Pub Date : 2021-06-17DOI: 10.1080/17459737.2021.1933631
Jay Schweig, Aurian Kutner
We discuss and enumerate pc-sets that are both not contained in any diatonic collection and are minimal with respect to this property, and we generalize this idea to other collections. We also consider related simplicial complexes and examine how some of their geometric properties reflect qualities of the associated pc-sets.
{"title":"Minimally non-diatonic pc-sets","authors":"Jay Schweig, Aurian Kutner","doi":"10.1080/17459737.2021.1933631","DOIUrl":"https://doi.org/10.1080/17459737.2021.1933631","url":null,"abstract":"We discuss and enumerate pc-sets that are both not contained in any diatonic collection and are minimal with respect to this property, and we generalize this idea to other collections. We also consider related simplicial complexes and examine how some of their geometric properties reflect qualities of the associated pc-sets.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"61 1","pages":"38 - 45"},"PeriodicalIF":1.1,"publicationDate":"2021-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80480256","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 : 2021-06-06DOI: 10.1080/17459737.2021.1924303
F. Hazama
The present article introduces the notion of smoothness of rhythm and proposes a unified method that transforms an arbitrary rhythm into a smooth one. The method employs a self-map Rav, discrete average map, on the space of rhythms of arbitrary length with a fixed number of onsets. It is shown that, for any rhythm in the space, the iterations become eventually periodic, and that the final cycle consists only of smooth rhythms. The discrete average map leads naturally to a finite directed graph, which visualizes the realm of smooth rhythms in the whole world of rhythms. This article has an Online Supplement, in which we give detailed proof of the main result.
{"title":"Iterative method of construction for smooth rhythms","authors":"F. Hazama","doi":"10.1080/17459737.2021.1924303","DOIUrl":"https://doi.org/10.1080/17459737.2021.1924303","url":null,"abstract":"The present article introduces the notion of smoothness of rhythm and proposes a unified method that transforms an arbitrary rhythm into a smooth one. The method employs a self-map Rav, discrete average map, on the space of rhythms of arbitrary length with a fixed number of onsets. It is shown that, for any rhythm in the space, the iterations become eventually periodic, and that the final cycle consists only of smooth rhythms. The discrete average map leads naturally to a finite directed graph, which visualizes the realm of smooth rhythms in the whole world of rhythms. This article has an Online Supplement, in which we give detailed proof of the main result.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"21 1","pages":"216 - 235"},"PeriodicalIF":1.1,"publicationDate":"2021-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74584225","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 : 2021-06-02DOI: 10.1080/17459737.2021.1928311
Luka Marohnić
In this paper, we propose a model for idealized diatonic and chromatic voice leadings between seventh chords and study its computational aspects. The model provides a mathematical formalization of the concept of tonal seventh chord realizations including augmented sixth chords. Certain contrapuntal aspects, such as preparation and resolution of certain dissonant intervals, are taken into account. Possible applications of the model are discussed using a graph-theoretic approach. In particular, we present an algorithm for generating concrete voicings from sequences of seventh-chord symbols.
{"title":"A model for tonal progressions of seventh chords","authors":"Luka Marohnić","doi":"10.1080/17459737.2021.1928311","DOIUrl":"https://doi.org/10.1080/17459737.2021.1928311","url":null,"abstract":"In this paper, we propose a model for idealized diatonic and chromatic voice leadings between seventh chords and study its computational aspects. The model provides a mathematical formalization of the concept of tonal seventh chord realizations including augmented sixth chords. Certain contrapuntal aspects, such as preparation and resolution of certain dissonant intervals, are taken into account. Possible applications of the model are discussed using a graph-theoretic approach. In particular, we present an algorithm for generating concrete voicings from sequences of seventh-chord symbols.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"47 1","pages":"78 - 99"},"PeriodicalIF":1.1,"publicationDate":"2021-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90574724","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 : 2021-06-01DOI: 10.1080/17459737.2021.1924304
I. Nemoto, M. Kawakatsu
Musical chords play an essential role, especially in Western music, and the corresponding consonance–dissonance contrasts and emotional continua are still the targets of investigation in psychoacoustics and neurophysiology. In the present study, we define the simplicity of frequency ratios and simplicity of period ratios for the constituents of a chord and propose that those measures should be used as the two coordinate axes in the plane for plotting chords. In this plane, the major and minor chords are reflections of each other with respect to the diagonal . The other chord pair in the same relationship is the major–minor seventh (dominant seventh) and the half-diminished seventh chords. The other triads and seventh chords do not make such pairs of different categories of chords. It was also found that at least for triads, the sum and the difference seem to correspond to subjective consonance and the melancholic/sad emotional ratings, respectively. Implications of these findings are discussed. The proposed simple presentation may help interpret and model psychoacoustic and neurophysiological results on musical chords.
{"title":"A two-dimensional representation of musical chords using the simplicity of frequency and period ratios as coordinates","authors":"I. Nemoto, M. Kawakatsu","doi":"10.1080/17459737.2021.1924304","DOIUrl":"https://doi.org/10.1080/17459737.2021.1924304","url":null,"abstract":"Musical chords play an essential role, especially in Western music, and the corresponding consonance–dissonance contrasts and emotional continua are still the targets of investigation in psychoacoustics and neurophysiology. In the present study, we define the simplicity of frequency ratios and simplicity of period ratios for the constituents of a chord and propose that those measures should be used as the two coordinate axes in the plane for plotting chords. In this plane, the major and minor chords are reflections of each other with respect to the diagonal . The other chord pair in the same relationship is the major–minor seventh (dominant seventh) and the half-diminished seventh chords. The other triads and seventh chords do not make such pairs of different categories of chords. It was also found that at least for triads, the sum and the difference seem to correspond to subjective consonance and the melancholic/sad emotional ratings, respectively. Implications of these findings are discussed. The proposed simple presentation may help interpret and model psychoacoustic and neurophysiological results on musical chords.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"2 1","pages":"65 - 77"},"PeriodicalIF":1.1,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83150055","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 : 2021-05-04DOI: 10.1080/17459737.2021.1923843
C. White
The computational approach of autocorrelation relies on recurrent patterns within a musical signal to identify and analyze the meter of musical passages. This paper suggests that the autocorrelation process can act as a computational proxy for the act of period extraction, a crucial aspect of the cognition of musical meter, by identifying periodicities with which similar events tend to occur within a musical signal. Three analytical vignettes highlight three aspects of the identified patterns: (1) that the similarities between manifestations of the same patterns are often inexact, (2) that these patterns have ambiguous boundaries, and (3) that many more patterns exist on the musical surface than contribute to the passage's notated/felt meter, each of which overlaps with observations from music theory and behavioral research. An Online Supplement at chriswmwhite.com/autocorrelation contains accompanying data.
{"title":"Some observations on autocorrelated patterns within computational meter identification","authors":"C. White","doi":"10.1080/17459737.2021.1923843","DOIUrl":"https://doi.org/10.1080/17459737.2021.1923843","url":null,"abstract":"The computational approach of autocorrelation relies on recurrent patterns within a musical signal to identify and analyze the meter of musical passages. This paper suggests that the autocorrelation process can act as a computational proxy for the act of period extraction, a crucial aspect of the cognition of musical meter, by identifying periodicities with which similar events tend to occur within a musical signal. Three analytical vignettes highlight three aspects of the identified patterns: (1) that the similarities between manifestations of the same patterns are often inexact, (2) that these patterns have ambiguous boundaries, and (3) that many more patterns exist on the musical surface than contribute to the passage's notated/felt meter, each of which overlaps with observations from music theory and behavioral research. An Online Supplement at chriswmwhite.com/autocorrelation contains accompanying data.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"13 1","pages":"181 - 193"},"PeriodicalIF":1.1,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88444178","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 : 2021-05-04DOI: 10.1080/17459737.2021.1925762
David R. W. Sears, D. Forrest
Many musical traditions – from Western art, to popular and commercial – organize pitch phenomena around a referential pitch class (or tonic) and feature triads and seventh chords. As a result, triadic progressions associated with one tradition sometimes resurface in others. How, then, are we to distinguish between the conventional harmonic patterns that span several time periods, and the characteristic idioms that delimit a single period? This essay presents a comparative study of triadic progressions in four data sets comprised of expert harmonic annotations: Annotated Beethoven Corpus (ABC), Theme and Variation Encodings with Roman Numerals (TAVERN), Rolling Stone-200 (RS-200), and McGill Billboard (Billboard). Using methods for counting, filtering, and ranking multichord expressions, we reveal conventional and characteristic progressions and examine broad trends over time. We also include an accompanying standalone application that allows users to adjust various stages of the model pipeline and export the data for further exploration and analysis.
{"title":"Triadic patterns across classical and popular music corpora: stylistic conventions, or characteristic idioms?","authors":"David R. W. Sears, D. Forrest","doi":"10.1080/17459737.2021.1925762","DOIUrl":"https://doi.org/10.1080/17459737.2021.1925762","url":null,"abstract":"Many musical traditions – from Western art, to popular and commercial – organize pitch phenomena around a referential pitch class (or tonic) and feature triads and seventh chords. As a result, triadic progressions associated with one tradition sometimes resurface in others. How, then, are we to distinguish between the conventional harmonic patterns that span several time periods, and the characteristic idioms that delimit a single period? This essay presents a comparative study of triadic progressions in four data sets comprised of expert harmonic annotations: Annotated Beethoven Corpus (ABC), Theme and Variation Encodings with Roman Numerals (TAVERN), Rolling Stone-200 (RS-200), and McGill Billboard (Billboard). Using methods for counting, filtering, and ranking multichord expressions, we reveal conventional and characteristic progressions and examine broad trends over time. We also include an accompanying standalone application that allows users to adjust various stages of the model pipeline and export the data for further exploration and analysis.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"91 1","pages":"140 - 153"},"PeriodicalIF":1.1,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85665951","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 : 2021-05-04DOI: 10.1080/17459737.2021.1943026
Darian Tomašević, S. Wells, I. Ren, A. Volk, Matevž Pesek
The study of inter-annotator agreement in musical pattern annotations has gained increased attention over the past few years. While expert annotations are often taken as the reference for evaluating pattern discovery algorithms, relying on just one reference is not usually sufficient to capture the complex musical relations between patterns. In this paper, we address the potential of digital annotation tools to enable large-scale annotations of musical patterns, by comparing datasets gathered with two recently developed digital tools. We investigate the influence of the tools and different annotator backgrounds on the annotation process by performing inter-annotator agreement analysis and feature-based analysis on the annotated patterns. We discuss implications for further adaptation of annotation tools, and the potential for deriving reference data from such rich annotation datasets for the evaluation of automatic pattern discovery algorithms in the future.
{"title":"Exploring annotations for musical pattern discovery gathered with digital annotation tools","authors":"Darian Tomašević, S. Wells, I. Ren, A. Volk, Matevž Pesek","doi":"10.1080/17459737.2021.1943026","DOIUrl":"https://doi.org/10.1080/17459737.2021.1943026","url":null,"abstract":"The study of inter-annotator agreement in musical pattern annotations has gained increased attention over the past few years. While expert annotations are often taken as the reference for evaluating pattern discovery algorithms, relying on just one reference is not usually sufficient to capture the complex musical relations between patterns. In this paper, we address the potential of digital annotation tools to enable large-scale annotations of musical patterns, by comparing datasets gathered with two recently developed digital tools. We investigate the influence of the tools and different annotator backgrounds on the annotation process by performing inter-annotator agreement analysis and feature-based analysis on the annotated patterns. We discuss implications for further adaptation of annotation tools, and the potential for deriving reference data from such rich annotation datasets for the evaluation of automatic pattern discovery algorithms in the future.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"31 1","pages":"194 - 207"},"PeriodicalIF":1.1,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76047054","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 : 2021-05-04DOI: 10.1080/17459737.2021.1917010
Thomas Nuttall, M. Casado, Andrés Ferraro, D. Conklin, Rafael Caro Repetto
Here we present a computational approach to identifying melodic patterns in a dataset of 145 MusicXML scores with the aim of contributing to centonization theory in the Moroccan tradition of Arab-Andalusian Music – a theory in development by expert performer and researcher of this tradition, Amin Chaachoo. Central to his work is the definition of a set of characteristic patterns, or centos, for each ṭab‘, or melodic mode. We apply three methods: TF-IDF, Maximally General Distinctive Patterns (MGDP) and the Structure Induction Algorithm (SIA) to identify characteristic patterns at the level of ṭab‘. A substantial number of the centos proposed by Chaachoo are identified and new melodic patterns are retrieved. A discussion with Chaachoo about the obtained results promoted the elicitation of other categories of recurrent patterns in the tradition different from the centos, contributing to a deeper musicological knowledge of the tradition.
{"title":"A computational exploration of melodic patterns in Arab-Andalusian music","authors":"Thomas Nuttall, M. Casado, Andrés Ferraro, D. Conklin, Rafael Caro Repetto","doi":"10.1080/17459737.2021.1917010","DOIUrl":"https://doi.org/10.1080/17459737.2021.1917010","url":null,"abstract":"Here we present a computational approach to identifying melodic patterns in a dataset of 145 MusicXML scores with the aim of contributing to centonization theory in the Moroccan tradition of Arab-Andalusian Music – a theory in development by expert performer and researcher of this tradition, Amin Chaachoo. Central to his work is the definition of a set of characteristic patterns, or centos, for each ṭab‘, or melodic mode. We apply three methods: TF-IDF, Maximally General Distinctive Patterns (MGDP) and the Structure Induction Algorithm (SIA) to identify characteristic patterns at the level of ṭab‘. A substantial number of the centos proposed by Chaachoo are identified and new melodic patterns are retrieved. A discussion with Chaachoo about the obtained results promoted the elicitation of other categories of recurrent patterns in the tradition different from the centos, contributing to a deeper musicological knowledge of the tradition.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"31 1","pages":"168 - 180"},"PeriodicalIF":1.1,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90889934","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 : 2021-05-04DOI: 10.1080/17459737.2021.1923844
L. Nuño
Parsimonious transformations are common patterns in different musical styles and eras. In some cases, they can be represented on the Tonnetz, Cube Dance, Power Towers, or the central region of an orbifold, mainly when they only include the most even trichords and tetrachords. In this paper, two novel graphs, called Cyclopes, are presented, which include more than double the number of chord types in previously published graphs, thus allowing to represent a larger musical repertoire in a practical way. Apart from parsimonious transformations, they are also especially suitable for representing trichords a major third apart, tetrachords a minor third apart, and the cadences V7–I(m) and II –V7–I(m) with major or minor tonic chords. Therefore, they allow to clearly visualize the relationship among the corresponding chords and better understand those patterns, as well as being efficient mnemonic resources, all of which make them useful tools both for music analysis and composition.
简约变换是不同音乐风格和时代的常见模式。在某些情况下，它们可以表现在Tonnetz, Cube Dance, Power Towers或orbitold的中心区域，主要是当它们只包括最均匀的三和弦和四和弦时。在本文中，提出了两种新的曲线图，称为Cyclopes，其中包含的和弦类型数量是以前发表的曲线图的两倍多，从而允许以实用的方式表示更大的音乐曲目。除了简洁的转换外，它们还特别适用于表示大调三度的三和弦，小调三度的四和弦，以及带有大调或小调主音和弦的V7-I (m)和II -V7-I (m)的节奏。因此，它们可以清晰地可视化相应和弦之间的关系，更好地理解这些模式，同时也是有效的记忆资源，所有这些都使它们成为音乐分析和作曲的有用工具。
{"title":"Parsimonious graphs for the most common trichords and tetrachords","authors":"L. Nuño","doi":"10.1080/17459737.2021.1923844","DOIUrl":"https://doi.org/10.1080/17459737.2021.1923844","url":null,"abstract":"Parsimonious transformations are common patterns in different musical styles and eras. In some cases, they can be represented on the Tonnetz, Cube Dance, Power Towers, or the central region of an orbifold, mainly when they only include the most even trichords and tetrachords. In this paper, two novel graphs, called Cyclopes, are presented, which include more than double the number of chord types in previously published graphs, thus allowing to represent a larger musical repertoire in a practical way. Apart from parsimonious transformations, they are also especially suitable for representing trichords a major third apart, tetrachords a minor third apart, and the cadences V7–I(m) and II –V7–I(m) with major or minor tonic chords. Therefore, they allow to clearly visualize the relationship among the corresponding chords and better understand those patterns, as well as being efficient mnemonic resources, all of which make them useful tools both for music analysis and composition.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"7 1","pages":"125 - 139"},"PeriodicalIF":1.1,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84935860","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 : 2021-05-04DOI: 10.1080/17459737.2021.1947404
D. Conklin
Pattern in music, referring to the discovery, representation, selection, and interpretation of repeated structures within single pieces (intra-opus) or corpora (inter-opus), is a central part of music analysis, musical style and genre, improvisation, music perception, and composition. This special issue of the Journal of Mathematics and Music presents a diverse selection of papers on the topic of pattern in music from computational and mathematical perspectives. The following overview will introduce the papers considering three facets: representation, discovery, and evaluation and interpretation.
{"title":"Pattern in music","authors":"D. Conklin","doi":"10.1080/17459737.2021.1947404","DOIUrl":"https://doi.org/10.1080/17459737.2021.1947404","url":null,"abstract":"Pattern in music, referring to the discovery, representation, selection, and interpretation of repeated structures within single pieces (intra-opus) or corpora (inter-opus), is a central part of music analysis, musical style and genre, improvisation, music perception, and composition. This special issue of the Journal of Mathematics and Music presents a diverse selection of papers on the topic of pattern in music from computational and mathematical perspectives. The following overview will introduce the papers considering three facets: representation, discovery, and evaluation and interpretation.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"33 1","pages":"95 - 98"},"PeriodicalIF":1.1,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85590868","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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