Pub Date : 2022-09-02DOI: 10.1080/17459737.2022.2157062
Julian L. Hook
Two letters that Jack wrote soon after I introduced uniform triadic transformations in 1999 exemplify the energy with which he threw himself into work on subjects that stirred his interest. As I note in a tribute elsewhere in this issue (Krantz 2022), I first met Jack at a meeting of the Society for Music Theory in the fall of 1999 – a conference at which I also presented my work on UTTs (later published in Hook 2002). By April of 2000, Jack had distributed two substantial letters with his evolving thoughts on the subject. The wreath-product structure of the UTT group, the simply transitive subgroups of that group, and “skew groups” (which combine mode-preserving transformations of one kind with mode-reversing transformations of another kind) were subjects of special fascination to him from the start, as they remained a few years later when we produced our joint paper on applications of UTTs to serialism (Hook and Douthett 2008). The second letter is particularly notable for a few ideas that have not been pursued elsewhere. Here Jack relates UTTs to the seventh-chord transformations that he had studied with Peter Steinbach (Douthett and Steinbach 1998), and also to Jonathan Kochavi’s work (Kochavi 1998) on contextual inversions (presenting what Jack calls “Kochavi diagrams” for some groups of UTTs). He also refines my embryonic suggestion for generalizing UTTs to situations where there are more than two classes of objects (such as inversionally related chord qualities).
1999年，在我介绍均匀三元变换后不久，杰克写了两封信，这两封信体现了他对那些激起他兴趣的课题投入的精力。正如我在本期其他地方(Krantz 2022)的致敬中所指出的那样，我第一次见到杰克是在1999年秋天的音乐理论学会的一次会议上——在那次会议上，我还展示了我关于utt的工作(后来发表在Hook 2002)。到2000年4月，杰克已经发出了两封内容丰富的信，表达了他在这个问题上不断发展的想法。UTT群的环积结构、该群的简单传递子群和“偏群”(将一种保模变换与另一种反模变换结合在一起)从一开始就是他特别着迷的主题，因为几年后我们发表了关于UTT在序列论中的应用的联合论文时，它们仍然存在(Hook and Douthett 2008)。第二封信特别值得注意的是，它提出了一些其他地方没有提出过的观点。在这里，杰克将utt与他与彼得·斯坦巴赫(Douthett and Steinbach, 1998)一起研究的七和弦变换以及乔纳森·科查维(Jonathan Kochavi, 1998)关于上下文倒位的工作(为某些utt组呈现杰克所谓的“科查维图”)联系起来。他还改进了我的初步建议，将utt推广到存在两类以上对象的情况(例如反向相关的和弦质量)。
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Pub Date : 2022-09-02DOI: 10.1080/17459737.2022.2140214
R. Krantz
Remembrances of Jack Douthett from his students and colleagues.
学生和同事对杰克·杜赛特的缅怀。
{"title":"Colleague, collaborator, friend Jack Douthett (1942–2021)","authors":"R. Krantz","doi":"10.1080/17459737.2022.2140214","DOIUrl":"https://doi.org/10.1080/17459737.2022.2140214","url":null,"abstract":"Remembrances of Jack Douthett from his students and colleagues.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86022956","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 : 2022-08-31DOI: 10.1080/17459737.2022.2111612
J. Arias-Valero, E. Lluis-Puebla
Thanks to Mazzola's notion of gestures on topological categories we can appreciate how the notion of gesture transcends its manifestation as the movement of the body's limbs and takes a more abstract form that blends diagrammatic (discrete gesturality) and bodily aspects (continuous gesturality). These two aspects are strongly related to two main branches of mathematical music theory, namely, a discrete branch and a continuous branch. The discrete branch corresponds to the diagrams of transformational theory, that is, to networks in musical analysis. The continuous branch corresponds to the movement of the musical performer's body. Informally, a gesture on a topological category is a diagram of continuous paths of morphisms in the category. This definition amounts to that of topological category of gestures, whose structure we study in this article. Specifically, we study the presentation of topological categories of gestures as suitable categories of topological functors and as suitable categories of sequences, and the explicit presentation of morphisms of a typical topological category of gestures. In particular, we present an exhaustive study, not included in previous publications, of the topological category of continuous paths of an arbitrary digraph. This article can be regarded as a continuation of a previous publication, in a previous issue of this journal, on the presentation of spaces of gestures as function spaces. We include an application of the theory to the variations in Mozart's Piano Sonata K. 331. We provide an Online Supplement, in which we include some technical passages.
{"title":"Explicit presentations of topological categories of gestures","authors":"J. Arias-Valero, E. Lluis-Puebla","doi":"10.1080/17459737.2022.2111612","DOIUrl":"https://doi.org/10.1080/17459737.2022.2111612","url":null,"abstract":"Thanks to Mazzola's notion of gestures on topological categories we can appreciate how the notion of gesture transcends its manifestation as the movement of the body's limbs and takes a more abstract form that blends diagrammatic (discrete gesturality) and bodily aspects (continuous gesturality). These two aspects are strongly related to two main branches of mathematical music theory, namely, a discrete branch and a continuous branch. The discrete branch corresponds to the diagrams of transformational theory, that is, to networks in musical analysis. The continuous branch corresponds to the movement of the musical performer's body. Informally, a gesture on a topological category is a diagram of continuous paths of morphisms in the category. This definition amounts to that of topological category of gestures, whose structure we study in this article. Specifically, we study the presentation of topological categories of gestures as suitable categories of topological functors and as suitable categories of sequences, and the explicit presentation of morphisms of a typical topological category of gestures. In particular, we present an exhaustive study, not included in previous publications, of the topological category of continuous paths of an arbitrary digraph. This article can be regarded as a continuation of a previous publication, in a previous issue of this journal, on the presentation of spaces of gestures as function spaces. We include an application of the theory to the variations in Mozart's Piano Sonata K. 331. We provide an Online Supplement, in which we include some technical passages.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80091538","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 : 2022-08-11DOI: 10.1080/17459737.2022.2105411
Elsa De Luca
Lisboa, Biblioteca Nacional, Alcobaça 446, fols. 32v-33r (reproduced with permission) Etymologiae is one of the most popular works of the Middle Ages. It represents Isidore of Seville’s encyclopaedic effort to collect information from all branches of knowledge by analysing the principal terms employed. Etymologiae was copied all over Europe and more than 1000 copies of the manuscripts survive. In a small group of manuscripts from the Iberian Peninsula, diagrams have been interpolated, possibly in the eight century. The diagrams, which have only been added to these Iberian copies, are in Book III, between De Geometria and De
{"title":"The scale of the Old Hispanic chant","authors":"Elsa De Luca","doi":"10.1080/17459737.2022.2105411","DOIUrl":"https://doi.org/10.1080/17459737.2022.2105411","url":null,"abstract":"Lisboa, Biblioteca Nacional, Alcobaça 446, fols. 32v-33r (reproduced with permission) Etymologiae is one of the most popular works of the Middle Ages. It represents Isidore of Seville’s encyclopaedic effort to collect information from all branches of knowledge by analysing the principal terms employed. Etymologiae was copied all over Europe and more than 1000 copies of the manuscripts survive. In a small group of manuscripts from the Iberian Peninsula, diagrams have been interpolated, possibly in the eight century. The diagrams, which have only been added to these Iberian copies, are in Book III, between De Geometria and De","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77925929","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 : 2022-07-15DOI: 10.1080/17459737.2022.2084569
Darrell Conklin
Published in Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance (Vol. 17, No. 1, 2023)
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Pub Date : 2022-05-16DOI: 10.1080/17459737.2022.2068687
Florence Levé, G. Micchi, J. Allouche
A melody is often described as a line of music that evolves through time and, therefore, it is possible to draw its 2D pitch-time representation as a series of points implicitly defining a curve. We introduce to computational musicology a descriptor of this music curve: the inconstancy, a function that gives information on the curve's smoothness as well as some of its topological properties. A mathematical analysis of the inconstancy of music is provided, followed by a lengthy application of inconstancy to musicological tasks. We compare the inconstancy of melodic lines with that of typical accompaniment patterns such as the Alberti bass; this analysis, together with the case study of W.A. Mozart's Variations on Ah! vous dirai-je, maman, suggests a significant difference in the value of the inconstancy of a music line depending on its function. The inconstancy seems to be correlated also with the compositional style: the analysis on almost 10,000 musical themes of the common practice repertoire shows that Baroque music has higher inconstancy. Finally, we also define a windowed version of the inconstancy for studying longer scores and show the insights one can gain into, for example, structural analysis and cadence detection.
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Pub Date : 2022-05-04DOI: 10.1080/17459737.2022.2088875
Jason Yust, C. White, Leigh VanHandel
Godfried Toussaint occupied a unique place in music theory. The contributions in this special issue honour his legacy and continue the work that he started in his uniquely creative approach to introducing mathematical and computational tools for the analysis of cyclic rhythms.
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Pub Date : 2022-05-04DOI: 10.1080/17459737.2022.2071491
J. Rahn
Ostinatos of sub-Saharan Africa, South America, and the Caribbean are continually repeated rhythms also known as “timelines.” Taking as a starting point the ostinato termed the “Standard Pattern,” generic and specific features of Black-Atlantic ostinatos are analyzed: in European-derived theory, these features correspond to the quantity and quality of musical intervals. Like the standard pattern, other ostinatos that have been employed in several Black-Atlantic traditions and comprise a rhythmic structure that corresponds to the structure of hyperdiatonic scales in contemporary European-derived pitch theory; other widespread ostinatos considered have a complementary, hyperpentatonic structure. Interpreted in terms of Gestalt grouping principles, both of these kinds of ostinatos maximize similarity and proximity. Other ostinatos of these traditions can be construed as variants of hyperdiatonic or hyperpentatonic ostinatos. Differences of similarity and proximity that these variants manifest are analyzed in terms of adjacent swaps, fusions and fissions, and moduli that encompass two or more ostinatos’ least common multiples. In turn, similarity and proximity measures for rhythmic ostinatos are shown to parsimoniously clarify aspects of pitch relations, including the general notion of evenness, within 1- dimensional frameworks for pitch and time.
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Pub Date : 2022-05-04DOI: 10.1080/17459737.2022.2075946
C. White, Joe Pater, Mara Breen
This paper compares two corpora of melodies drawn from premillennial and postmillennial American popular music, and identifies several notable differences in their use of rhythm. The premillennial corpus contains melodies written between 1957 and 1997 [deClercq and Temperley (2011. “A Corpus Analysis of Rock Harmony.” Popular Music 30 (1): 47–70)], while the postmillennial corpus (compiled for this study) consists of songs popular between 2015 and 2019. For both corpora, we analysed (1) the distribution of note onsets within measures; (2) the distribution of four-note rhythmic cells, (3) the speed of melodic delivery, and (4) the tempo of the tactus. Our analyses indicated that the postmillennial melodies are delivered more quickly, are distributed more evenly throughout their measures, repeat rhythmic cells more frequently, and are annotated at slower tempos. Even when the tactus tempos were standardized into an allowable window of 70–140 BPM, this effect, though smaller, remained. We then use our techniques to observe the properties of three representative postmillennial tracks, finding that salient information can be located in both standardized and non-standardized tactus data, and that tempo-variant differences between corpora are closely connected to musical genre, with music designated as “pop” being more similar over both genres, and postmillennial rap and hip-hop introducing the most uniqueness.
本文比较了千禧年前和千禧年后美国流行音乐的两个旋律语料库，并确定了它们在节奏使用上的几个显著差异。千禧年语料库包含1957年至1997年间创作的旋律[deClercq and Temperley(2011)]。《摇滚和声的语料库分析》流行音乐30(1):47-70)]，而后千禧年语料库(为本研究编制)由2015年至2019年之间的流行歌曲组成。对于这两个语料库，我们分析了(1)音符开始在度量内的分布;(2)四音符节奏细胞的分布，(3)旋律传递的速度，(4)动作的速度。我们的分析表明，千禧年后的旋律传递得更快，在整个小节中分布得更均匀，重复节奏细胞的频率更高，并且以较慢的速度注释。即使将节拍标准化为70-140 BPM的允许范围，这种影响虽然较小，但仍然存在。然后，我们使用我们的技术观察了三首代表性的后千年曲目的属性，发现标准化和非标准化的tactus数据中都可以找到显著的信息，语料库之间的节奏变化差异与音乐类型密切相关，被指定为“流行”的音乐在这两种类型中都更加相似，而后千年的说唱和嘻哈音乐引入了最独特的特征。
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Pub Date : 2022-04-20DOI: 10.1080/17459737.2023.2219994
Alberto Alcal'a-Alvarez, P. Padilla-Longoria
In the present article we describe and discuss a framework for applying different topological data analysis (TDA) techniques to a music fragment given as a score in traditional Western notation. We first consider different sets of points in Euclidean spaces of different dimensions that correspond to musical events in the score, and obtain their persistent homology features. Then we introduce two families of simplicial complexes that can be associated with chord sequences, and leverage homology to compute their salient features. Finally, we show the results of applying the described methods to the analysis and stylistic comparison of fragments from three Brandenburg Concertos by J.S. Bach and two Graffiti by Mexican composer Armando Luna.
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