{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,8]],"date-time":"2025-02-08T05:28:25Z","timestamp":1738992505387,"version":"3.37.0"},"reference-count":20,"publisher":"MIT Press","issue":"3","content-domain":{"domain":["direct.mit.edu"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2023,9,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Quadratic difference tones belong to a family of perceptual phenomena that arise from the neuromechanics of the auditory system in response to particular physical properties of sound. Long deployed as \u201cghost\u201d or \u201cphantom\u201d tones by sound artists, improvisers, and computer musicians, in this article we address an entirely new topic: How to create a quadratic difference tone spectrum (QDTS) in which a target fundamental and harmonic overtone series are specified and in which the complex tone necessary to evoke it is synthesized. We propose a numerical algorithm that solves the problem of how to synthesize a QDTS for a target distribution of amplitudes. The algorithm aims to find a solution that matches the desired spectrum as closely as possible for an arbitrary number of target harmonics. Results from experiments using different parameter settings and target distributions show that the algorithm is effective in the majority of cases, with at least 99% of the cases being solvable in real time. An external object for the visual programming language Max is described. We discuss musical and perceptual considerations for using the external, and we describe a range of audio examples that demonstrate the synthesis of QDTSs across different cases. As we show, the method makes possible the matching of QDTSs to particular instrumental timbres with surprising efficiency. Also included is a discussion of a musical work by composer Marcin Pietruszewski that makes use of QDTS synthesis.<\/jats:p>","DOI":"10.1162\/comj_a_00687","type":"journal-article","created":{"date-parts":[[2024,9,3]],"date-time":"2024-09-03T18:37:19Z","timestamp":1725388639000},"page":"19-34","update-policy":"https:\/\/doi.org\/10.1162\/mitpressjournals.corrections.policy","source":"Crossref","is-referenced-by-count":0,"title":["Generating Sonic Phantoms with Quadratic Difference Tone Spectrum Synthesis"],"prefix":"10.1162","volume":"47","author":[{"given":"Esteban","family":"Guti\u00e9rrez","sequence":"first","affiliation":[{"name":"Department of Information and Communications Technologies Universitat Pompeu Fabra Carrer de Roc Boronat 138, Sant Mart\u00ed, Barcelona 08018, Spain egutierreo@mat.uc.cl"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christopher","family":"Haworth","sequence":"additional","affiliation":[{"name":"Department of Music University of Birmingham Birmingham B15 2TT, United Kingdom c.p.haworth@bham.ac.uk"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rodrigo F.","family":"C\u00e1diz","sequence":"additional","affiliation":[{"name":"Music Institute and Department of Electrical Engineering Pontificia Universidad Cat\u00f3lica de Chile Av. Vicu\u00f1a Mackenna 4860, Macul, Santiago, Chile rcadiz@uc.cl"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"281","published-online":{"date-parts":[[2023,9,1]]},"reference":[{"volume-title":"Sound Characters: Making the Third Ear","year":"1999","author":"Amacher","key":"2025020718451279400_B1"},{"key":"2025020718451279400_B2","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/j.jsc.2014.09.025","article-title":"On the Complexity of the F5 Gr\u00f6bner Basis Algorithm","volume":"70","author":"Bardet","year":"2015","journal-title":"Journal of Symbolic Computation"},{"volume-title":"Algorithms in Real Algebraic Geometry","year":"2005","author":"Basu","key":"2025020718451279400_B3"},{"key":"2025020718451279400_B4","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198165057.001.0001","volume-title":"The Musician's Guide to Acoustics","author":"Campbell","year":"1994"},{"issue":"11","key":"2025020718451279400_B5","doi-asserted-by":"publisher","DOI":"10.1121\/10.0015244","article-title":"Cubic and Quadratic Distortion Products in Vibrations of the Mouse Cochlear Apex","volume":"2","author":"Dewey","year":"2022","journal-title":"JASA Express Letters"},{"issue":"4","key":"2025020718451279400_B6","doi-asserted-by":"crossref","first-page":"750","DOI":"10.1137\/0219053","article-title":"The Structure of Polynomial Ideals and Gr\u00f6bner Bases","volume":"19","author":"Dub\u00e9","year":"1990","journal-title":"SIAM Journal on Computing"},{"key":"2025020718451279400_B7","first-page":"237","article-title":"Generating Quadratic Difference Tone Spectra for Auditory Distortion Synthesis","author":"Guti\u00e9rrez","year":"2023","journal-title":"Proceedings of the International Computer Music Conference"},{"volume-title":"Signals, Sound, and Sensation","year":"2004","author":"Hartmann","key":"2025020718451279400_B8"},{"key":"2025020718451279400_B9","first-page":"342","article-title":"Composing with Absent Sound","author":"Haworth","year":"2011","journal-title":"Proceedings of the International Computer Music Conference"},{"key":"2025020718451279400_B10","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1162\/LMJ_a_00099","article-title":"Ear as Instrument","volume":"22","author":"Haworth","year":"2012","journal-title":"Leonardo Music Journal"},{"issue":"1","key":"2025020718451279400_B12","doi-asserted-by":"publisher","first-page":"17","DOI":"10.3109\/01050398309076220","article-title":"Evoked Acoustic Emissions from the Human Ear: III. 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