Pub Date : 2022-11-10DOI: 10.1080/17459737.2022.2131918
{"title":"Teponazcuauhtla, or “Forest of Resonances” Mesoamerican Plot of Harmony","authors":"","doi":"10.1080/17459737.2022.2131918","DOIUrl":"https://doi.org/10.1080/17459737.2022.2131918","url":null,"abstract":"","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87224167","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 : 2022-11-08DOI: 10.1080/17459737.2022.2128450
Emmet Crowley, Francisco Gómez-Martín
Whilst not widely extended, non-octave-repeating scales are present in a variety of musical settings, yet have received scarce attention in the existing literature. This paper provides a brief general historical contextualization before focusing on a specific group of two-octave scales based on properties in common with the most widely used scales in Western music. After characterizing them in mathematical terms, an exhaustive list of such scales is provided, being the first exhaustive list of non-octave-repeating scales of any given characteristics. A scale endowed with structural properties attributed to the diatonic collection in the field of diatonic theory – such as well-formed, Myhill property, maximally even or diatonic – is singled out for the first time in this paper.
{"title":"Structural properties of multi-octave scales","authors":"Emmet Crowley, Francisco Gómez-Martín","doi":"10.1080/17459737.2022.2128450","DOIUrl":"https://doi.org/10.1080/17459737.2022.2128450","url":null,"abstract":"Whilst not widely extended, non-octave-repeating scales are present in a variety of musical settings, yet have received scarce attention in the existing literature. This paper provides a brief general historical contextualization before focusing on a specific group of two-octave scales based on properties in common with the most widely used scales in Western music. After characterizing them in mathematical terms, an exhaustive list of such scales is provided, being the first exhaustive list of non-octave-repeating scales of any given characteristics. A scale endowed with structural properties attributed to the diatonic collection in the field of diatonic theory – such as well-formed, Myhill property, maximally even or diatonic – is singled out for the first time in this paper.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85961755","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 : 2022-11-07DOI: 10.1080/17459737.2022.2136776
Saba Goodarzi, W. Sethares
This paper studies the resonant frequencies of three-string networks by examining the roots of the relevant spectral equation. A collection of scaling laws are established which relate the frequencies to structured changes in the lengths, densities, and tensions of the strings. Asymptotic properties of the system are derived, and several situations where transcritical bifurcations occur are detailed. Numerical optimization is used to solve the inverse problem (where a desired set of frequencies is specified and the parameters of the system are adjusted to best realize the specification). The intrinsic dissonance of the overtones provides an approximate way to measure the inherent inharmonicity of the sound.
{"title":"Three-string inharmonic networks","authors":"Saba Goodarzi, W. Sethares","doi":"10.1080/17459737.2022.2136776","DOIUrl":"https://doi.org/10.1080/17459737.2022.2136776","url":null,"abstract":"This paper studies the resonant frequencies of three-string networks by examining the roots of the relevant spectral equation. A collection of scaling laws are established which relate the frequencies to structured changes in the lengths, densities, and tensions of the strings. Asymptotic properties of the system are derived, and several situations where transcritical bifurcations occur are detailed. Numerical optimization is used to solve the inverse problem (where a desired set of frequencies is specified and the parameters of the system are adjusted to best realize the specification). The intrinsic dissonance of the overtones provides an approximate way to measure the inherent inharmonicity of the sound.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88457514","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 : 2022-10-18DOI: 10.1080/17459737.2022.2120214
Luis Nuño
In post-tonal theory, set classes are normally elements of and are characterized by their interval-class vector. Those being non-inversionally-symmetrical can be split into two set types related by inversion, which can be characterized by their trichord-type vector. In this paper, I consider the general case of set classes and types in and their -class and -type vectors, ranging from to , which are properly grouped into matrices. As well, three relevant cases are considered: (hexachords), (heptatonic scales), and (chromatic scale), where all those type and class matrices are computed and provided in supplementary files; and, in the first two cases, also in the form of tables. This completes the corresponding information given in previous publications on this subject and can directly be used by researchers and composers. Moreover, two computer programs, written in MATLAB, are provided for obtaining the above-mentioned and other related matrices in the general case of . Additionally, several theorems on type and class matrices are provided, including a complete version of the hexachord theorem. These theorems allow us to obtain the type and class matrices by different procedures, thus providing a broader perspective and better understanding of the theory.
{"title":"Type and class vectors and matrices in ℤ n . Application to ℤ6, ℤ7, and ℤ12","authors":"Luis Nuño","doi":"10.1080/17459737.2022.2120214","DOIUrl":"https://doi.org/10.1080/17459737.2022.2120214","url":null,"abstract":"In post-tonal theory, set classes are normally elements of and are characterized by their interval-class vector. Those being non-inversionally-symmetrical can be split into two set types related by inversion, which can be characterized by their trichord-type vector. In this paper, I consider the general case of set classes and types in and their -class and -type vectors, ranging from to , which are properly grouped into matrices. As well, three relevant cases are considered: (hexachords), (heptatonic scales), and (chromatic scale), where all those type and class matrices are computed and provided in supplementary files; and, in the first two cases, also in the form of tables. This completes the corresponding information given in previous publications on this subject and can directly be used by researchers and composers. Moreover, two computer programs, written in MATLAB, are provided for obtaining the above-mentioned and other related matrices in the general case of . Additionally, several theorems on type and class matrices are provided, including a complete version of the hexachord theorem. These theorems allow us to obtain the type and class matrices by different procedures, thus providing a broader perspective and better understanding of the theory.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86197849","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 : 2022-09-02DOI: 10.1080/17459737.2022.2157060
J. Douthett, Richard Cohn, Dani Zanuttini-Frank
Jack Douthett wrote a number of letters to John Clough and Richard Cohn concerning Cohn's “P-Relations,” single-semitone voice-leading relationships. The ideas in these letters led to graph-theoretic and geometric models. The following selection has been edited and prepared for publication by Richard Cohn and Dani Zanuttini-Frank.
{"title":"Letters on P-Relations, 1992–1997","authors":"J. Douthett, Richard Cohn, Dani Zanuttini-Frank","doi":"10.1080/17459737.2022.2157060","DOIUrl":"https://doi.org/10.1080/17459737.2022.2157060","url":null,"abstract":"Jack Douthett wrote a number of letters to John Clough and Richard Cohn concerning Cohn's “P-Relations,” single-semitone voice-leading relationships. The ideas in these letters led to graph-theoretic and geometric models. The following selection has been edited and prepared for publication by Richard Cohn and Dani Zanuttini-Frank.","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":"81714883","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 : 2022-09-02DOI: 10.1080/17459737.2022.2157061
Richard Cohn
This publication transcribes seven previously unpublished letters from Jack Douthett. The earliest four letters, from 1992–94, were written to John Clough, who was the Slee Professor of Music Theory at SUNY Buffalo from the early 1980s to 2001. They are hand-written, mailed by postal service, and are among the founding documents of the sub-field that emerged in the 1990s that has come to be known under the catch-all label of “neo-Riemannian theory.” The three remaining letters, word processed and circulated via email to a working group of researchers known informally as the “Buffalo group,” are from 1997 and 2000, and responded to emerging work of a remarkable cohort of PhD students then working on neo-Riemannian topics. The first three letters were the first of a flurry of eleven that were precipitated by some ideas that I sketched over lunch with John and Jack in late October 1992 at the Society for Music Theory annual meeting in Kansas City, and subsequently detailed in a written document that I mailed to them shortly thereafter.1 In that document, I defined (1) a “P relation” when two pitchclass sets are connected by single-semitonal displacement, for example {C, E, G} P {C, E, G }, (2) a “P property” for any Tn/TnI set class2 that contains P-related pairs, and (3) a “PP property” for any set class that partitions into cycles of three or more P-related pairs. These definitions, which are presented more systematically at the end of this introduction, supported my central finding: that the structures that co-anchor the European tonal system, major/minor triads and major/minor scales, together with their complements, uniquely possess the PP property (nontrivially).3 In that document, I also identified set classes with the PP property in universes with fewer than twelve elements, and advanced some theorems about the relation of the PP property to other properties and relations central to theories of atonality and diatonicism. Douthett’s first response, from 11/24/92, is primarily concerned with connections between my central finding and his research with Clough on maximally even sets (Clough and Douthett 1991). That letter is primarily algebraic, but ends with a graph-theoretic turn that is developed a few days later in the brief letter of 11/30/92, and emerges in mature form in the longer letter of 12/12/92, where the two graphs that became respectively known as “Cube Dance” and “Power Towers” are first described. These three letters, together with eight subsequent letters from the
本出版物转录了杰克·杜特以前未发表的七封信。最早的四封信写于1992年至1994年，写给约翰·克拉夫(John Clough)，他在20世纪80年代初至2001年期间担任纽约州立大学布法罗分校的音乐理论Slee教授。它们是手写的，通过邮政服务邮寄，是20世纪90年代出现的子领域的创始文件之一，后来以“新黎曼理论”的笼统标签而闻名。剩下的三封信，经过文字处理后，通过电子邮件发送给了一个非正式的“布法罗小组”的研究小组。这封信写于1997年和2000年，是对一群研究新黎曼主题的杰出博士生的回应。1992年10月下旬，在堪萨斯城举行的音乐理论学会年会上，我在与约翰和杰克共进午餐时勾勒出了一些想法，并在随后不久寄给他们的一份书面文件中详细说明了这些想法在该文档中，我定义了(1)当两个音调类集通过单半位移连接时的“P关系”，例如{C, E, G} P {C, E, G}，(2)对于任何包含P相关对的Tn/TnI集class2的“P性质”，以及(3)对于任何划分为三个或更多P相关对的循环的集类的“PP性质”。这些定义，在本导言的最后更系统地呈现，支持了我的中心发现:共同锚定欧洲音调系统的结构，大调/小调三和弦和大调/小调音阶，以及它们的补语，独特地拥有PP属性(非平凡的)在那篇论文中，我还在少于12个元素的宇宙中确定了具有PP性质的集合类，并提出了一些关于PP性质与其他性质和关系的定理，这些性质和关系是无调性和全音阶理论的核心。Douthett的第一个回应是在1992年11月24日，主要关注我的中心发现和他与Clough关于最大偶数集的研究之间的联系(Clough and Douthett 1991)。这封信主要是代数的，但在几天后的1992年11月30日的简短信中以图论的转变结束，并在1992年12月12日的较长信中以成熟的形式出现，其中两个图分别被称为“立方体舞蹈”和“动力塔”。这三个字母，加上后面的八个字母
{"title":"Jack Douthett’s letters","authors":"Richard Cohn","doi":"10.1080/17459737.2022.2157061","DOIUrl":"https://doi.org/10.1080/17459737.2022.2157061","url":null,"abstract":"This publication transcribes seven previously unpublished letters from Jack Douthett. The earliest four letters, from 1992–94, were written to John Clough, who was the Slee Professor of Music Theory at SUNY Buffalo from the early 1980s to 2001. They are hand-written, mailed by postal service, and are among the founding documents of the sub-field that emerged in the 1990s that has come to be known under the catch-all label of “neo-Riemannian theory.” The three remaining letters, word processed and circulated via email to a working group of researchers known informally as the “Buffalo group,” are from 1997 and 2000, and responded to emerging work of a remarkable cohort of PhD students then working on neo-Riemannian topics. The first three letters were the first of a flurry of eleven that were precipitated by some ideas that I sketched over lunch with John and Jack in late October 1992 at the Society for Music Theory annual meeting in Kansas City, and subsequently detailed in a written document that I mailed to them shortly thereafter.1 In that document, I defined (1) a “P relation” when two pitchclass sets are connected by single-semitonal displacement, for example {C, E, G} P {C, E, G }, (2) a “P property” for any Tn/TnI set class2 that contains P-related pairs, and (3) a “PP property” for any set class that partitions into cycles of three or more P-related pairs. These definitions, which are presented more systematically at the end of this introduction, supported my central finding: that the structures that co-anchor the European tonal system, major/minor triads and major/minor scales, together with their complements, uniquely possess the PP property (nontrivially).3 In that document, I also identified set classes with the PP property in universes with fewer than twelve elements, and advanced some theorems about the relation of the PP property to other properties and relations central to theories of atonality and diatonicism. Douthett’s first response, from 11/24/92, is primarily concerned with connections between my central finding and his research with Clough on maximally even sets (Clough and Douthett 1991). That letter is primarily algebraic, but ends with a graph-theoretic turn that is developed a few days later in the brief letter of 11/30/92, and emerges in mature form in the longer letter of 12/12/92, where the two graphs that became respectively known as “Cube Dance” and “Power Towers” are first described. These three letters, together with eight subsequent letters from the","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":"80642189","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 : 2022-09-02DOI: 10.1080/17459737.2022.2143590
J. Douthett, Richard Cohn, Dani Zanuttini-Frank
After encountering Julian Hook's work on uniform triadic transformations, Jack Douthett wrote letters to John Clough suggesting further group-theoretic generalizations of Hook's idea. These letters have been edited and prepared for publication by Richard Cohn and Dani Zanuttini-Frank.
{"title":"Letters on Hook's group, 2000","authors":"J. Douthett, Richard Cohn, Dani Zanuttini-Frank","doi":"10.1080/17459737.2022.2143590","DOIUrl":"https://doi.org/10.1080/17459737.2022.2143590","url":null,"abstract":"After encountering Julian Hook's work on uniform triadic transformations, Jack Douthett wrote letters to John Clough suggesting further group-theoretic generalizations of Hook's idea. These letters have been edited and prepared for publication by Richard Cohn and Dani Zanuttini-Frank.","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":"77349626","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 : 2022-09-02DOI: 10.1080/17459737.2022.2124461
J. Douthett, P. Steinbach, R. Peck, R. Krantz
We extend the theory of maximally even sets to determine the evenness of partitions of the chromatic universe . Interactions measure the average evenness of colour sets (partitioning sets) of . For 2-colour partitions the Clough-Douthett maximal-evenness algorithm determines maximally even partitions. But to measure the evenness of non-maximally even partitions, it is necessary to use computational methods. Moreover, for more than two colour sets there is no simple algorithm that determines maximally even partitions. Again, we rely on computational methods. We also explore collections of partitions and partition-classes (orbits under a dihedral group) and construct tables that order partition-classes according to the evenness of their partitions. We use Bell numbers, Stirling numbers of the second kind, and integer partitions to enumerate relevant combinatorial objects related to our investigation.
{"title":"Partitions, their classes, and multicolour evenness","authors":"J. Douthett, P. Steinbach, R. Peck, R. Krantz","doi":"10.1080/17459737.2022.2124461","DOIUrl":"https://doi.org/10.1080/17459737.2022.2124461","url":null,"abstract":"We extend the theory of maximally even sets to determine the evenness of partitions of the chromatic universe . Interactions measure the average evenness of colour sets (partitioning sets) of . For 2-colour partitions the Clough-Douthett maximal-evenness algorithm determines maximally even partitions. But to measure the evenness of non-maximally even partitions, it is necessary to use computational methods. Moreover, for more than two colour sets there is no simple algorithm that determines maximally even partitions. Again, we rely on computational methods. We also explore collections of partitions and partition-classes (orbits under a dihedral group) and construct tables that order partition-classes according to the evenness of their partitions. We use Bell numbers, Stirling numbers of the second kind, and integer partitions to enumerate relevant combinatorial objects related to our investigation.","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":"76285326","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 : 2022-09-02DOI: 10.1080/17459737.2022.2157059
Roger Asensi Arranz, T. Noll
This paper explores Jack Douthett's model of dynamical voice leading on the level of harmonic states. It investigates global contiguous and stroboscopic dynamical systems on the entire state space and introduces a measure of effectiveness for the trajectories under consideration. Special attention is paid to the harmonic state spaces behind second-order Clough-Myerson scales, such as diatonic triads and seventh chords. Finally, a set of trajectories as a tiling of exemplifies the connection with the Spectral Conjecture.
{"title":"Turning the volvelle: Exploring Jack Douthett's voice leading dynamics","authors":"Roger Asensi Arranz, T. Noll","doi":"10.1080/17459737.2022.2157059","DOIUrl":"https://doi.org/10.1080/17459737.2022.2157059","url":null,"abstract":"This paper explores Jack Douthett's model of dynamical voice leading on the level of harmonic states. It investigates global contiguous and stroboscopic dynamical systems on the entire state space and introduces a measure of effectiveness for the trajectories under consideration. Special attention is paid to the harmonic state spaces behind second-order Clough-Myerson scales, such as diatonic triads and seventh chords. Finally, a set of trajectories as a tiling of exemplifies the connection with the Spectral Conjecture.","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":"83769444","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 : 2022-09-02DOI: 10.1080/17459737.2022.2143589
Jason Yust
Jack Douthett's work over three decades was central to defining an era in mathematical theory. The present special issue attests to his abiding influence over the field, as well as the energy he brought to research in all areas of mathematical music theory through his collaborations, correspondence, and relationships.
Jack Douthett三十多年的工作对于定义数学理论的一个时代至关重要。本期特刊证明了他对该领域的持久影响，以及他通过合作、通信和关系为数学音乐理论的各个领域的研究带来的活力。
{"title":"Jack Douthett and mathematical music theory","authors":"Jason Yust","doi":"10.1080/17459737.2022.2143589","DOIUrl":"https://doi.org/10.1080/17459737.2022.2143589","url":null,"abstract":"Jack Douthett's work over three decades was central to defining an era in mathematical theory. The present special issue attests to his abiding influence over the field, as well as the energy he brought to research in all areas of mathematical music theory through his collaborations, correspondence, and relationships.","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":"79600429","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：