[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-566614-105":53,"doc-detail-566614-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","12-music-scales","12 Music Scales","","This text examines musical scales through the lens of measurement theory and music theory, treating scales as interval scales and asking how those intervals are measured. It explains pitch as the basis of music, linking human pitch perception and production to communication and language. It introduces the major scale as a widely known Western example, emphasizing that scale-values like do and re are not fixed pitches or specific frequencies, and contrasting scales with conventional ratio measurement.",{"@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/12-music-scales/566614/",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/12-music-scales/566614.png","ImageObject",442,249,{"name":88,"@type":89},"A glass of water","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-21","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":76},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How does the text connect musical scales to measurement theory?","Question",{"text":108,"@type":109},"It frames a musical scale as essentially an interval scale, and focuses on how musical intervals are measured.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What role does pitch play in musical scales?",{"text":113,"@type":109},"Music is based on pitch produced by vibrating sources (e.g., a plucked guitar string). The text also emphasizes human pitch recognition and how pitch distinctions support communication.",{"name":115,"@type":106,"acceptedAnswer":116},"Why do scale-values like do and re not correspond to fixed frequencies?",{"text":117,"@type":109},"The text explains that scale-values do and re are not tied to particular pitches or specific frequency values such as 400 Hz; the scale is not a conventional ratio-scale.","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},566614,1792316605,{"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":76,"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":141},962090760832,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Scales in Music Gary Hardegree  \nDepartment of Philosophy University of Massachusetts Amherst, MA 01003  \n1. Introduction...................................................................................................................................... 1  \n2. The Fundamental Unit – The Octave ...............................................................................................2  \n1. An Aside on Logarithmic Scales .........................................................................................5  \n2. Cents and Sensibility............................................................................................................6  \n3. The Pythagorean Construction of the Major Scale ..........................................................................7  \n4. Ptolemaic Tuning ........................................................................................................................... 10  \n5. Mean-Tone Temperament (Tuning by Major Thirds) ...................................................................11  \n6. Problems with Perfect Tuning........................................................................................................ 12  \n1. An Aside on The Circle of Fifths and the Circle of Thirds................................................14  \n2. Back to the Problem of Tuning by Fifths ...........................................................................14  \n3. Ptolemaic Tuning Makes Matters Even Worse! ................................................................15  \n4. Mean-Tone Temperament’s Wolves..................................................................................17  \n7. What to do? .................................................................................................................................... 18  \n1. Give up Universality .......................................................................................................... 18  \n2. Give up Practicality – The Purist Approach ......................................................................18  \n3. Give up Perfect Tuning – Equal Temperament .................................................................19  \n4. Alternative Instruments ...................................................................................................... 19  \n5. Reconsidering Universality – Well-Tempered Scales .......................................................20  \n8. Summary of Temperament Systems ..............................................................................................21  \n9. Other Seven-Tone Scales ...............................................................................................................22  \n10. Pentatonic Scales............................................................................................................................23  \n11. Equal-Interval Scales .....................................................................................................................24  \n1. 31ET...................................................................................................................................24  \n2. 53ET...................................................................................................................................26  \n1. Introduction  \nAs noted earlier, the use of the word ‘scale’ in measurement theory is closely related to its use in music; in particular, this use of ‘scale’ derives from the Latin word sc³lae [ladder] . In this section we examine musical scales from the viewpoint of measurement theory (and of course music theory) . As we will see, from the viewpoint of measurement theory, a musical scale is basically an interval scale; indeed, in music theory, the distances between notes are even called ‘intervals’! The question then is – how are these intervals measured?  \nMusic is based on pitch. For example, when a guitar string is plucked, it vibrates, thereby producing sound with a characterist","cbCaiaBlgT4Q9PMQ","https://ap.wps.com/l/cbCaiaBlgT4Q9PMQ","pdf",227027,26,"English","# Introduction\n## The Fundamental Unit – The Octave\n## An Aside on Logarithmic Scales\n## Cents and Sensibility\n## The Pythagorean Construction of the Major Scale\n## Ptolemaic Tuning\n## Mean-Tone Temperament (Tuning by Major Thirds)\n## Problems with Perfect Tuning\n## Other Temperament and Tuning Topics\n## Summary of Temperament Systems\n## Other Seven-Tone Scales\n## Pentatonic Scales\n## Equal-Interval Scales","[{\"question\":\"How does the text connect musical scales to measurement theory?\",\"answer\":\"It frames a musical scale as essentially an interval scale, and focuses on how musical intervals are measured.\"},{\"question\":\"What role does pitch play in musical scales?\",\"answer\":\"Music is based on pitch produced by vibrating sources (e.g., a plucked guitar string). The text also emphasizes human pitch recognition and how pitch distinctions support communication.\"},{\"question\":\"Why do scale-values like do and re not correspond to fixed frequencies?\",\"answer\":\"The text explains that scale-values do and re are not tied to particular pitches or specific frequency values such as 400 Hz; the scale is not a conventional ratio-scale.\"}]","12 Music Scales | PDF",1791863025,9]