{"id":"https://openalex.org/W4414409276","doi":"https://doi.org/10.3390/axioms14090707","title":"Extended Expressive Intonation: An Application of the Convergents and Semiconvergents in Pythagorean Tuning","display_name":"Extended Expressive Intonation: An Application of the Convergents and Semiconvergents in Pythagorean Tuning","publication_year":2025,"publication_date":"2025-09-19","ids":{"openalex":"https://openalex.org/W4414409276","doi":"https://doi.org/10.3390/axioms14090707"},"language":"en","primary_location":{"id":"doi:10.3390/axioms14090707","doi":"https://doi.org/10.3390/axioms14090707","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14090707","pdf_url":"https://www.mdpi.com/2075-1680/14/9/707/pdf?version=1758521808","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/14/9/707/pdf?version=1758521808","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5013211808","display_name":"Rafael Cubars\u00ed","orcid":"https://orcid.org/0000-0001-7748-1322","observed_orcids":["https://orcid.org/0000-0001-7748-1322"]},"institutions":[{"id":"https://openalex.org/I9617848","display_name":"Universitat Polit\u00e8cnica de Catalunya","ror":"https://ror.org/03mb6wj31","country_code":"ES","type":"education","lineage":["https://openalex.org/I9617848"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Rafael Cubarsi","raw_affiliation_strings":["Departament de Matem\u00e0tiques, Universitat Polit\u00e8cnica de Catalunya, 08028 Barcelona, Spain"],"raw_orcid":"https://orcid.org/0000-0001-7748-1322","affiliations":[{"raw_affiliation_string":"Departament de Matem\u00e0tiques, Universitat Polit\u00e8cnica de Catalunya, 08028 Barcelona, Spain","institution_ids":["https://openalex.org/I9617848"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5013211808"],"corresponding_institution_ids":["https://openalex.org/I9617848"],"apc_list":{"value":2400,"currency":"CHF","value_usd":2896},"apc_paid":{"value":2400,"currency":"CHF","value_usd":2896},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.25281993,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"14","issue":"9","first_page":"707","last_page":"707"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10941","display_name":"Musicology and Musical Analysis","score":0.823199987411499,"subfield":{"id":"https://openalex.org/subfields/1210","display_name":"Music"},"field":{"id":"https://openalex.org/fields/12","display_name":"Arts and Humanities"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10941","display_name":"Musicology and Musical Analysis","score":0.823199987411499,"subfield":{"id":"https://openalex.org/subfields/1210","display_name":"Music"},"field":{"id":"https://openalex.org/fields/12","display_name":"Arts and Humanities"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11349","display_name":"Music Technology and Sound Studies","score":0.08590000122785568,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12170","display_name":"History and Theory of Mathematics","score":0.006899999920278788,"subfield":{"id":"https://openalex.org/subfields/2614","display_name":"Theoretical Computer Science"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/convergents","display_name":"convergents","score":0.8190000057220459},{"id":"https://openalex.org/keywords/continued-fractions","display_name":"continued fractions","score":0.8769999742507935},{"id":"https://openalex.org/keywords/pythagorean-tuning","display_name":"Pythagorean tuning","score":0.9829999804496765},{"id":"https://openalex.org/keywords/microtones","display_name":"microtones","score":0.5410000085830688},{"id":"https://openalex.org/keywords/automorphisms","display_name":"automorphisms","score":0.5950000286102295}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7294999957084656},{"id":"https://openalex.org/C2780992000","wikidata":"https://www.wikidata.org/wiki/Q17016113","display_name":"Generator (circuit theory)","level":3,"score":0.6432999968528748},{"id":"https://openalex.org/C149629883","wikidata":"https://www.wikidata.org/wiki/Q660926","display_name":"Fraction (chemistry)","level":2,"score":0.5551999807357788},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.546500027179718},{"id":"https://openalex.org/C118712358","wikidata":"https://www.wikidata.org/wiki/Q782566","display_name":"Automorphism","level":2,"score":0.5091000199317932},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.5060999989509583},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4277999997138977},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.4172999858856201},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.3833000063896179},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.36169999837875366},{"id":"https://openalex.org/C2988750069","wikidata":"https://www.wikidata.org/wiki/Q60790315","display_name":"Automorphism group","level":3,"score":0.33730000257492065},{"id":"https://openalex.org/C136197465","wikidata":"https://www.wikidata.org/wiki/Q1729295","display_name":"Variety (cybernetics)","level":2,"score":0.335999995470047},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.33379998803138733},{"id":"https://openalex.org/C2777382242","wikidata":"https://www.wikidata.org/wiki/Q6017816","display_name":"Index (typography)","level":2,"score":0.32580000162124634},{"id":"https://openalex.org/C2780595030","wikidata":"https://www.wikidata.org/wiki/Q3860309","display_name":"Multiplication (music)","level":2,"score":0.3075000047683716},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.2854999899864197},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.2827000021934509},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.27079999446868896},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.26350000500679016},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.25440001487731934}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/axioms14090707","doi":"https://doi.org/10.3390/axioms14090707","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14090707","pdf_url":"https://www.mdpi.com/2075-1680/14/9/707/pdf?version=1758521808","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:upcommons.upc.edu:2117/465962","doi":null,"is_oa":true,"landing_page_url":"https://www.researchgate.net/publication/395738496_Extended_Expressive_Intonation_An_Application_of_the_Convergents_and_Semiconvergents_in_Pythagorean_Tuning","pdf_url":"https://www.researchgate.net/journal/Axioms-2075-1680/publication/395738496_Extended_Expressive_Intonation_An_Application_of_the_Convergents_and_Semiconvergents_in_Pythagorean_Tuning/links/68d6c5d69383755fd7066aa1/Extended-Expressive-Intonation-An-Application-of-the-Convergents-and-Semiconvergents-in-Pythagorean-Tuning.pdf","source":{"id":"https://openalex.org/S4377196262","display_name":"UPCommons institutional repository (Universitat Polit\u00e8cnica de Catalunya)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I9617848","host_organization_name":"Universitat Polit\u00e8cnica de Catalunya","host_organization_lineage":["https://openalex.org/I9617848"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"pmh:oai:doaj.org/article:398f97ac81954b26a7c82b3d56288bd6","doi":null,"is_oa":true,"landing_page_url":"https://doaj.org/article/398f97ac81954b26a7c82b3d56288bd6","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 14, Iss 9, p 707 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms14090707","doi":"https://doi.org/10.3390/axioms14090707","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14090707","pdf_url":"https://www.mdpi.com/2075-1680/14/9/707/pdf?version=1758521808","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4414409276.pdf","grobid_xml":"https://content.openalex.org/works/W4414409276.grobid-xml"},"referenced_works_count":28,"referenced_works":["https://openalex.org/W129974563","https://openalex.org/W134348770","https://openalex.org/W762889747","https://openalex.org/W1163235289","https://openalex.org/W1971237591","https://openalex.org/W1982442868","https://openalex.org/W2004577463","https://openalex.org/W2029081457","https://openalex.org/W2032510966","https://openalex.org/W2060186227","https://openalex.org/W2071556498","https://openalex.org/W2095191196","https://openalex.org/W2122980236","https://openalex.org/W2149010017","https://openalex.org/W2171866795","https://openalex.org/W2269717150","https://openalex.org/W2320951299","https://openalex.org/W2321179210","https://openalex.org/W2597675794","https://openalex.org/W2799886001","https://openalex.org/W2891122909","https://openalex.org/W3010100139","https://openalex.org/W4243347036","https://openalex.org/W4291746372","https://openalex.org/W4367548623","https://openalex.org/W4399054336","https://openalex.org/W4407749898","https://openalex.org/W4411384195"],"related_works":[],"abstract_inverted_index":{"Cyclic":[0],"scales":[1,85],"are":[2,36],"associated":[3,117],"with":[4,86,118,132],"convergents":[5],"and":[6,24,70,89,114],"semiconvergents":[7],"of":[8,13,33,66,100,104,111],"the":[9,14,64,67,101,105,112,119,133],"continued":[10],"fraction":[11],"expansions":[12],"generator":[15],"tone.":[16],"After":[17],"each":[18,55],"convergent,":[19],"a":[20,28,30,79,124],"scale":[21],"lineage":[22,65],"ends":[23],"another":[25],"begins.":[26],"Along":[27],"lineage,":[29],"constant":[31],"number":[32],"generic":[34],"accidentals":[35],"successively":[37],"added":[38,87],"to":[39,52,58,63,75,83],"its":[40],"first":[41],"scale,":[42],"becoming":[43],"regularly":[44],"interspersed.":[45],"In":[46],"this":[47],"way,":[48],"it":[49,90],"is":[50,57,73,81,91],"easier":[51],"know":[53],"where":[54],"note":[56],"go.":[59],"This":[60],"process,":[61],"applied":[62],"7-,":[68],"12-,":[69],"17-tone":[71],"scales,":[72],"related":[74],"expressive":[76],"intonation.":[77],"Such":[78],"concept":[80],"extended":[82],"larger":[84],"microtones":[88],"described":[92],"how":[93],"they":[94],"can":[95],"be":[96],"chosen":[97],"in":[98,109],"terms":[99,110],"starting":[102],"index":[103],"scale.":[106],"An":[107],"automorphism":[108],"step":[113],"co-step":[115],"indices":[116],"two":[120],"elementary":[121],"intervals":[122],"provides":[123],"two-dimensional":[125],"representation":[126],"that":[127],"shares":[128],"some":[129],"common":[130],"features":[131],"musical":[134],"staff.":[135]},"counts_by_year":[],"updated_date":"2026-10-16T09:59:22.235173","created_date":"2025-10-10T00:00:00"}