[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-566619-105":53,"doc-detail-566619-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","tuning-tonality-and-twenty-two-tone-temperament-paul-erlich","Tuning, Tonality, and Twenty-Two-Tone Temperament - Paul Erlich","","Explores Western tuning and tonality through prime-limit interval theory, tracing how 5-limit consonance replaced the earlier 3-limit framework in musical practice. Describes the rise of recognized consonant thirds, formation of major and minor triads, and the meantone temperament’s relation to equal temperaments. Extends discussion toward 7-limit consonant units, then evaluates equal-tempered tuning accuracy via probabilistic assumptions and practical limits on tone count.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/tuning-tonality-and-twenty-two-tone-temperament-paul-erlich/566619/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/tuning-tonality-and-twenty-two-tone-temperament-paul-erlich/566619.png","ImageObject",442,249,{"name":88,"@type":89},"WPS_1790064749","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":79},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What prime-limit framework does the paper use to explain Western tuning and consonance?","Question",{"text":108,"@type":109},"It starts from the common 5-limit explanation, then clarifies that intervals involving primes no higher than 5 (with 2-handling from inversion/extension) are treated as consonant and therefore require accurate tuning.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How did the 5-limit change recognized consonant intervals and harmony?",{"text":113,"@type":109},"By introducing the major third (5:4) and minor third (6:5), it enabled consonant triads made only of non-equivalent consonant intervals, leading to major and minor triads as primary harmony structures.",{"name":115,"@type":106,"acceptedAnswer":116},"What is the role of equal temperaments in the paper’s analysis?",{"text":117,"@type":109},"It examines how well simple equal-tempered tunings approximate justly tuned intervals across multiple limits, using an accuracy score based on the likelihood that an equal-tempered approximation will be heard as the intended just interval.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},566619,1792033779,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":79,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":135,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":8},3985747859154,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Tuning, Tonality, and Twenty-Two-Tone Temperament  \nPaul Erlich ([perlich@aya.yale.edu](perlich@aya.yale.edu))  \n(originally published in Xenharmonikôn 17, Spring 1998; revised 4/2/02)  \nIntroduction  \nWestern modal and tonal music generally conforms to what is known as the 5-limit; this is often explained (e. g., by Hindemith) by stating that the entire tuning system is an expression of integer frequency ratios using prime factors no higher than 5. A more correct description is that all 5-limit intervals, or ratios involving numbers no higher than 5, aside from factors of 2 arising from inversion and extension, are treated as consonances in this music, and thus need to be tuned with some degree of accuracy. The diatonic scale of seven notes per octave has been in use since ancient Greek times, when harmonic simultaneities (other than unisons and octaves) were not used and the only important property of the scale was its melodic structure. As melodic considerations continued to play a dominant role in Western musical history, scales other than the diatonic had very limited application, and harmonic usage was intimately entangled with diatonic scale structure. It is fortuitous that the diatonic scale allowed for many harmonies that displayed the psychoacoustic phenomenon of consonance, defined by smallinteger frequency ratios, as well as many that did not.  \nMedieval music conformed to the 3-limit, the only recognized consonances being the octave (2:1) and the perfect fifth (3:2) (plus, of course, their octave inversions and extensions, obtained by dividing or multiplying the ratios by powers of 2; this “octave equivalence” will be implicit from here on. 1) The diatonic scales were therefore tuned as a chain of consecutive 3:2s (Pythagorean tuning). By the end of the 15th century,2 the 5-limit had eclipsed the 3-limit as the standard of consonance and two new consonant intervals were recognized: the major third (5:4) and the minor third (6:5). It was now possible to create consonant triads, consisting of all three non-equivalent consonant intervals and no dissonant intervals. There are two such triads: major (6:5:4), and minor (1/4:1/5:1/6).3 The advent of the 5-limit brought with it a new temperament of the diatonic scale, the meantone. Typical varieties of meantone are well approximated in 19-and 31-tone equal temperament; 12-tone equal temperament is approximately 1/11-comma meantone,4 lying about halfway between typical meantone temperamentsand Pythagorean tuning.  \nSoon anything less than a triad was deemed an incomplete harmony. Use was also made of linked 5-limit (or stacked-thirds) chords (“major and minor seventh chords”) and linked 3-limit chords (“suspended chords”), which contained only one dissonant interval but had to resolve. More dissonant triads (“diminished”) and tetrads (“dominant, diminished, and half-diminished seventh chords”), formed from the same pattern of scale steps as the 5-limit sounds, were normally restricted to cadential or  \n1The fact that the phenomenon of octave similarity does not lead to exact equivalence of harmonic function is assumed to have a minor impact on the general conclusions reached by this paper.  \n2Fokker, Adriaan D. 1975. New Music with 31 Notes. Verlag für systematische Musikwissenschaft GmbH, Bonn, p. 35. 3Note that the 6:4 interval in these chords is equal to 3:2, the perfect fifth. Recall also that we are implicitly defining all octave inversions and extensions of these intervals and chords as consonant. The fact that certain styles treat 4:3 and its extensions as a dissonance when it includes the bass part is a detail that need not concern us here.  \n4Barbour, J. Murray. 1951. Tuning and Temperament: A Historical Survey. Michigan State College Press, East Lansing, p. 35.  \nmodulatory progressions. These chords resulted from the existence of a diminished fifth within the diatonic scale. Unlike most fifths in the scale, the diminished fifth was considered dissonant, since it","cbCaieHLZycg7L3n","https://ap.wps.com/l/cbCaieHLZycg7L3n","pdf",296885,31,"English","# Introduction\n## 5-limit and 3-limit consonance\n## Triads, temperament, and meantone\n## Expanding harmony to 7-limit units\n## Equal temperaments and accuracy model","[{\"question\":\"What prime-limit framework does the paper use to explain Western tuning and consonance?\",\"answer\":\"It starts from the common 5-limit explanation, then clarifies that intervals involving primes no higher than 5 (with 2-handling from inversion/extension) are treated as consonant and therefore require accurate tuning.\"},{\"question\":\"How did the 5-limit change recognized consonant intervals and harmony?\",\"answer\":\"By introducing the major third (5:4) and minor third (6:5), it enabled consonant triads made only of non-equivalent consonant intervals, leading to major and minor triads as primary harmony structures.\"},{\"question\":\"What is the role of equal temperaments in the paper’s analysis?\",\"answer\":\"It examines how well simple equal-tempered tunings approximate justly tuned intervals across multiple limits, using an accuracy score based on the likelihood that an equal-tempered approximation will be heard as the intended just interval.\"}]","Tuning, Tonality, and Twenty-Two-Tone Temperament - Paul Erlich | PDF",1791863034]