{"title":"Mathematical foundations of complex tonality","authors":"Jeffrey R. Boland, Lane P. Hughston","doi":"10.1080/17459737.2023.2228546","DOIUrl":null,"url":null,"abstract":"<p>Equal temperament, in which semitones are tuned in the irrational ratio of <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0001.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0001.gif\"}' src=\"//:0\"/><span></span></span><span><span style=\"color: inherit; display: none;\"></span><span data-mathml='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mn>2</mn><mrow><mn>1</mn><mrow><mo>/</mo></mrow><mn>12</mn></mrow></msup><mo>:</mo><mn>1</mn></math>' role=\"presentation\" style=\"position: relative;\" tabindex=\"0\"><nobr aria-hidden=\"true\"><span style=\"width: 3.951em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.322em; height: 0px; font-size: 118%;\"><span style=\"position: absolute; clip: rect(1.292em, 1003.25em, 2.504em, -1000em); top: -2.359em; left: 0em;\"><span><span><span style=\"display: inline-block; position: relative; width: 1.989em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.186em, 1000.45em, 4.141em, -1000em); top: -3.997em; left: 0em;\"><span style=\"font-family: MathJax_Main;\">2</span><span style=\"display: inline-block; width: 0px; height: 3.997em;\"></span></span><span style=\"position: absolute; top: -4.39em; left: 0.5em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">1</span><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">/</span></span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">12</span></span><span style=\"display: inline-block; width: 0px; height: 3.997em;\"></span></span></span></span><span style=\"font-family: MathJax_Main; padding-left: 0.278em;\">:</span><span style=\"font-family: MathJax_Main; padding-left: 0.278em;\">1</span></span><span style=\"display: inline-block; width: 0px; height: 2.359em;\"></span></span></span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.057em; border-left: 0px solid; width: 0px; height: 1.203em;\"></span></span></nobr><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mn>2</mn><mrow><mn>1</mn><mrow><mo>/</mo></mrow><mn>12</mn></mrow></msup><mo>:</mo><mn>1</mn></math></span></span><script type=\"math/mml\"><math><msup><mn>2</mn><mrow><mn>1</mn><mrow><mo>/</mo></mrow><mn>12</mn></mrow></msup><mo>:</mo><mn>1</mn></math></script></span>, is best seen as a serviceable compromise, sacrificing purity for flexibility. Just intonation, in which intervals are given by products of powers of 2, 3, and 5, is more natural, but of limited flexibility. We propose a new scheme in which ratios of Gaussian integers form the basis of an abstract tonal system. The tritone, so problematic in just temperament, given ambiguously by the ratios <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0002.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0002.gif\"}' src=\"//:0\"/><span></span></span><span><span style=\"color: inherit;\"><span><span><span style=\"vertical-align: 0.25em;\"><span><span>45</span></span><span style=\"margin-top: -0.9em;\"><span><span><span style=\"height: 1em; border-top: none; border-bottom: 1px solid; margin: 0.1em 0px;\"></span></span><span><span><span>32</span></span></span></span></span></span></span></span></span><span tabindex=\"0\"></span><script type=\"math/mml\"><math><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mfrac><mn>45</mn><mn>32</mn></mfrac></mstyle></math></script></span>, <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0003.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0003.gif\"}' src=\"//:0\"/><span></span></span><span><math><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mfrac><mn>64</mn><mn>45</mn></mfrac></mstyle></math></span>, <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0004.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0004.gif\"}' src=\"//:0\"/><span></span></span><span><math><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mfrac><mn>36</mn><mn>25</mn></mfrac></mstyle></math></span>, <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0005.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0005.gif\"}' src=\"//:0\"/><span></span></span><span><math><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mfrac><mn>25</mn><mn>18</mn></mfrac></mstyle></math></span>, none satisfactory, is in our scheme represented by the complex ratio <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0006.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0006.gif\"}' src=\"//:0\"/><span></span></span><span><math><mn>1</mn><mo>+</mo><mrow><mi mathvariant=\"normal\">i</mi></mrow><mo>:</mo><mn>1</mn></math></span>. The major and minor whole tones, given by intervals of <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0007.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0007.gif\"}' src=\"//:0\"/><span></span></span><span><math><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mfrac><mn>9</mn><mn>8</mn></mfrac></mstyle></math></span> and <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0008.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0008.gif\"}' src=\"//:0\"/><span></span></span><span><math><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mfrac><mn>10</mn><mn>9</mn></mfrac></mstyle></math></span>, can each be factorized into products of complex semitones, giving us a major complex semitone <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0009.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0009.gif\"}' src=\"//:0\"/><span></span></span><span><math><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mfrac><mn>3</mn><mn>4</mn></mfrac></mstyle><mo>(</mo><mn>1</mn><mo>+</mo><mrow><mi mathvariant=\"normal\">i</mi></mrow><mo>)</mo></math></span> and a minor complex semitone <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0010.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0010.gif\"}' src=\"//:0\"/><span></span></span><span><math><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo>(</mo><mn>3</mn><mo>+</mo><mrow><mi mathvariant=\"normal\">i</mi></mrow><mo>)</mo></math></span>. The perfect third, given by the interval <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0011.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0011.gif\"}' src=\"//:0\"/><span></span></span><span><math><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mfrac><mn>5</mn><mn>4</mn></mfrac></mstyle></math></span>, factorizes into the product of a complex whole tone <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0012.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/tmam20/0/tmam20.ahead-of-print/17459737.2023.2228546/20230717/images/tmam_a_2228546_ilm0012.gif\"}' src=\"//:0\"/><span></span></span><span><math><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>(</mo><mn>1</mn><mo>+</mo><mn>2</mn><mrow><mi mathvariant=\"normal\">i</mi></mrow><mo>)</mo></math></span> and its complex conjugate. Augmented with these supplementary tones, the resulting scheme of complex intervals based on products of low powers of Gaussian primes leads to the construction of a complete system of major and minor scales in all keys.</p>","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":"605 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematics and Music","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/17459737.2023.2228546","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
Equal temperament, in which semitones are tuned in the irrational ratio of 21/12:1, is best seen as a serviceable compromise, sacrificing purity for flexibility. Just intonation, in which intervals are given by products of powers of 2, 3, and 5, is more natural, but of limited flexibility. We propose a new scheme in which ratios of Gaussian integers form the basis of an abstract tonal system. The tritone, so problematic in just temperament, given ambiguously by the ratios 4532, , , , none satisfactory, is in our scheme represented by the complex ratio . The major and minor whole tones, given by intervals of and , can each be factorized into products of complex semitones, giving us a major complex semitone and a minor complex semitone . The perfect third, given by the interval , factorizes into the product of a complex whole tone and its complex conjugate. Augmented with these supplementary tones, the resulting scheme of complex intervals based on products of low powers of Gaussian primes leads to the construction of a complete system of major and minor scales in all keys.
期刊介绍:
Journal of Mathematics and Music aims to advance the use of mathematical modelling and computation in music theory. The Journal focuses on mathematical approaches to musical structures and processes, including mathematical investigations into music-theoretic or compositional issues as well as mathematically motivated analyses of musical works or performances. In consideration of the deep unsolved ontological and epistemological questions concerning knowledge about music, the Journal is open to a broad array of methodologies and topics, particularly those outside of established research fields such as acoustics, sound engineering, auditory perception, linguistics etc.