{"id":2543,"date":"2014-09-10T22:05:45","date_gmt":"2014-09-10T14:05:45","guid":{"rendered":"http:\/\/www.tongtuning.com\/wordpress\/?p=2543"},"modified":"2025-10-18T00:25:43","modified_gmt":"2025-10-17T16:25:43","slug":"entropy-based-tuning-of-musical-instruments","status":"publish","type":"post","link":"https:\/\/www.tongtuning.com\/wordpress\/2543","title":{"rendered":"Entropy-based Tuning of Musical Instruments"},"content":{"rendered":"<h5><!--StartFragment --><\/h5>\n<h5>This is a fascinating, research-driven approach to tuning that blends <strong>physics, information theory, and psychoacoustics<\/strong>. Instead of relying purely on human ears or traditional electronic tuners, it uses <strong>entropy minimization<\/strong> as the guiding principle.<!--EndFragment --><\/h5>\n<h5><!--more--><\/h5>\n<h5><!--StartFragment --><\/h5>\n<h5>Core Idea<\/h5>\n<ul>\n<li>\n<h5>The human ear perceives sounds as \u201cin tune\u201d when their <strong>harmonic spectra are correlated<\/strong>\u2014that is, when the overtones line up in a way that creates order in the auditory system.<\/h5>\n<\/li>\n<li>\n<h5><strong>Entropy<\/strong> (in the Shannon sense) measures disorder or unpredictability.<\/h5>\n<\/li>\n<li>\n<h5>By minimizing the entropy of the combined Fourier spectra of notes, one can mathematically approximate the same \u201corder\u201d that the human ear perceives as consonance.<\/h5>\n<\/li>\n<\/ul>\n<h5><\/h5>\n<h5>How It Works<\/h5>\n<ol>\n<li>\n<h5><strong>Record the sound<\/strong> of the instrument (e.g., piano notes).<\/h5>\n<\/li>\n<li>\n<h5><strong>Fourier transform<\/strong> the signal to analyze its harmonic content.<\/h5>\n<\/li>\n<li>\n<h5><strong>Preprocess<\/strong> the spectra to highlight relevant harmonics.<\/h5>\n<\/li>\n<li>\n<h5><strong>Calculate Shannon entropy<\/strong> of the combined spectra.<\/h5>\n<\/li>\n<li>\n<h5><strong>Adjust tuning<\/strong> to minimize entropy across intervals.<\/h5>\n<\/li>\n<\/ol>\n<h5>This process naturally produces:<\/h5>\n<ul>\n<li>\n<h5>The <strong>stretch curve<\/strong> (slight widening of octaves) that piano tuners use aurally.<\/h5>\n<\/li>\n<li>\n<h5>Small <strong>pitch fluctuations<\/strong> similar to those found in high-quality human tuning.<\/h5>\n<\/li>\n<\/ul>\n<h5><\/h5>\n<h5>Why It Matters<\/h5>\n<ul>\n<li>\n<h5>Traditional electronic tuners often fail to capture the subtle stretch and compromises that human tuners introduce.<\/h5>\n<\/li>\n<li>\n<h5>Entropy-based tuning provides a <strong>scientifically grounded, automated method<\/strong> that mimics human aural tuning more closely than simple frequency-matching devices.<\/h5>\n<\/li>\n<li>\n<h5>It bridges <strong>objective measurement<\/strong> with <strong>subjective perception<\/strong> of harmony.<\/h5>\n<\/li>\n<\/ul>\n<h5><\/h5>\n<h5>Reference:<\/h5>\n<h5><a href=\"https:\/\/arxiv.org\/abs\/1203.5101\">Original paper: Entropy-based Tuning of Musical Instruments by Haye Hinrichsen (2012) &#8211; https:\/\/arxiv.org\/abs\/1203.5101<\/a><\/h5>\n<hr \/>\n<h5><!--EndFragment --><\/h5>\n<h5 style=\"text-align: center;\"><a href=\"https:\/\/www.facebook.com\/media\/set\/?vanity=Piano.Technician.HK&amp;set=a.4713921331959485\"><strong>Details<\/strong><\/a><\/h5>\n<div class=\"embedsocial-album\" data-ref=\"bc20735181a02f81d3803d308315ff18acbd9b5a\"><a class=\"feed-powered-by-es\" title=\"Powered by EmbedSocial\" href=\"https:\/\/embedsocial.com\/facebook-albums\/\" target=\"_blank\" rel=\"noopener\">Powered by EmbedSocial\u2192<\/a><\/div>\n<p><script>(function(d, s, id){var js; if (d.getElementById(id)) {return;} js = d.createElement(s); js.id = id; js.src = \"https:\/\/embedsocial.com\/embedscript\/ei.js\"; d.getElementsByTagName(\"head\")[0].appendChild(js);}(document, \"script\", \"EmbedSocialScript\"));<\/script><\/p>\n<p>Haye Hinrichsen<br \/>\nUniversitat Wurzburg<br \/>\nFakultat fur Physik und Astronomie<br \/>\nD-97074 Wurzburg, Germany<\/p>\n<hr \/>\n<h5>Further Reading:<\/h5>\n<h5><a title=\"\u56de\u7b54\u554f\u984c \u2013 \u8abf\u97f3\u6642\u662f\u5426\u7528\u96fb\u5b50\u5100\u5668\u6703\u8abf\u5f97\u66f4\u6e96\u78ba?\" href=\"https:\/\/www.tongtuning.com\/wordpress\/421\">\u8abf\u97f3\u6642\u662f\u5426\u7528\u96fb\u5b50\u5100\u5668\u6703\u8abf\u5f97\u66f4\u6e96\u78ba?<\/a><\/h5>\n<h5><a title=\"\u8abf\u97f3\u6642\u7528\u7684\u96fb\u5b50\u5100\u5668\u6709\u591a\u6e96\u78ba?\" href=\"https:\/\/www.tongtuning.com\/wordpress\/417\">\u8abf\u97f3\u6642\u7528\u7684\u96fb\u5b50\u5100\u5668\u6709\u591a\u6e96\u78ba?<\/a><\/h5>\n<h5><a title=\"\u96fb\u5b50\u62cd\u5b50\u6a5f \u7684 Check \u97f3\u6e96\u529f\u80fd \u6709\u591a\u6e96\u78ba?\" href=\"https:\/\/www.tongtuning.com\/wordpress\/11\">\u96fb\u5b50\u62cd\u5b50\u6a5f \u7684 Check \u97f3\u6e96\u529f\u80fd \u6709\u591a\u6e96\u78ba?<\/a><\/h5>\n<h5><a title=\"\u570b\u969b\u8a8d\u53ef\u7684\u92fc\u7434\u6280\u8853\u4eba\u54e1\u8a8d\u53ef\u7684\u8abf\u97f3\u65b9\u6cd5\" href=\"https:\/\/www.tongtuning.com\/wordpress\/419\">\u570b\u969b\u8a8d\u53ef\u7684\u92fc\u7434\u6280\u8853\u4eba\u54e1\u8a8d\u53ef\u7684\u8abf\u97f3\u65b9\u6cd5\u662f\u4ec0\u9ebc?<\/a><\/h5>\n<hr \/>\n<h5>\u6e6f \u5148\u751f\/ Mr. Tong<\/h5>\n<h5>\u82f1\u570b\u8a3b\u518a\u92fc\u7434\u6280\u5e2b \/ Registered Piano Technician (MPTA)<\/h5>\n<h5>Mobile \/ Whatsapp: 6304 4081<\/h5>\n<h5>Website:\u00a0<a href=\"https:\/\/www.tongtuning.com\/\">www.tongtuning.com<\/a><\/h5>\n<h5>Blog:\u00a0<a href=\"https:\/\/www.tongtuning.com\/wordpress\">www.tongtuning.com\/wordpress<\/a><\/h5>\n<h5>Facebook:\u00a0<a href=\"https:\/\/www.facebook.com\/Piano.Technician.HK\">https:\/\/www.facebook.com\/Piano.Technician.HK<\/a><\/h5>\n<h5>YouTube Channel:\u00a0<a href=\"http:\/\/www.youtube.com\/user\/tongtuning\">http:\/\/www.youtube.com\/user\/tongtuning<\/a><\/h5>\n<h5>Instagram:\u00a0<a href=\"http:\/\/instagram.com\/Piano_Technician\">http:\/\/instagram.com\/Piano_Technician<\/a><\/h5>\n<div class=\"sharedaddy sd-sharing-enabled\"><div class=\"robots-nocontent sd-block sd-social sd-social-icon-text sd-sharing\"><h3 class=\"sd-title\">Share this:<\/h3><div class=\"sd-content\"><ul><li class=\"share-facebook\"><a rel=\"nofollow noopener noreferrer\" data-shared=\"sharing-facebook-2543\" class=\"share-facebook sd-button share-icon\" href=\"https:\/\/www.tongtuning.com\/wordpress\/2543?share=facebook\" target=\"_blank\" title=\"Click to share on Facebook\"><span>Facebook<\/span><\/a><\/li><li class=\"share-twitter\"><a rel=\"nofollow noopener noreferrer\" data-shared=\"sharing-twitter-2543\" 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Instead of relying purely on &hellip; <a class=\"kt-excerpt-readmore\" href=\"https:\/\/www.tongtuning.com\/wordpress\/2543\" aria-label=\"Entropy-based Tuning of Musical Instruments\">Read More<\/a><\/p>\n<div class=\"sharedaddy sd-sharing-enabled\"><div class=\"robots-nocontent sd-block sd-social sd-social-icon-text sd-sharing\"><h3 class=\"sd-title\">Share this:<\/h3><div class=\"sd-content\"><ul><li class=\"share-facebook\"><a rel=\"nofollow noopener noreferrer\" data-shared=\"sharing-facebook-2543\" class=\"share-facebook sd-button share-icon\" href=\"https:\/\/www.tongtuning.com\/wordpress\/2543?share=facebook\" target=\"_blank\" title=\"Click to share on Facebook\"><span>Facebook<\/span><\/a><\/li><li class=\"share-twitter\"><a rel=\"nofollow noopener noreferrer\" data-shared=\"sharing-twitter-2543\" class=\"share-twitter sd-button share-icon\" href=\"https:\/\/www.tongtuning.com\/wordpress\/2543?share=twitter\" target=\"_blank\" title=\"Click to share on Twitter\"><span>Twitter<\/span><\/a><\/li><li class=\"share-jetpack-whatsapp\"><a rel=\"nofollow noopener noreferrer\" data-shared=\"\" class=\"share-jetpack-whatsapp sd-button share-icon\" href=\"https:\/\/www.tongtuning.com\/wordpress\/2543?share=jetpack-whatsapp\" target=\"_blank\" title=\"Click to share on WhatsApp\"><span>WhatsApp<\/span><\/a><\/li><li class=\"share-linkedin\"><a rel=\"nofollow noopener noreferrer\" data-shared=\"sharing-linkedin-2543\" class=\"share-linkedin sd-button share-icon\" href=\"https:\/\/www.tongtuning.com\/wordpress\/2543?share=linkedin\" target=\"_blank\" title=\"Click to share on LinkedIn\"><span>LinkedIn<\/span><\/a><\/li><li class=\"share-email\"><a rel=\"nofollow noopener noreferrer\" data-shared=\"\" class=\"share-email sd-button share-icon\" href=\"https:\/\/www.tongtuning.com\/wordpress\/2543?share=email\" target=\"_blank\" title=\"Click to email this to a friend\"><span>Email<\/span><\/a><\/li><li class=\"share-print\"><a rel=\"nofollow noopener noreferrer\" data-shared=\"\" class=\"share-print sd-button share-icon\" href=\"https:\/\/www.tongtuning.com\/wordpress\/2543\" target=\"_blank\" title=\"Click to print\"><span>Print<\/span><\/a><\/li><li class=\"share-end\"><\/li><\/ul><\/div><\/div><\/div>","protected":false},"author":1,"featured_media":7779,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false,"spay_email":""},"categories":[88,3],"tags":[2903,2904,10,2730,2905,13],"aioseo_notices":[],"jetpack_featured_media_url":"https:\/\/www.tongtuning.com\/wordpress\/wp-content\/uploads\/2021\/09\/Piano-Slider-2.jpg","jetpack_shortlink":"https:\/\/wp.me\/p4ik8h-F1","jetpack_sharing_enabled":true,"jetpack_likes_enabled":false,"jetpack-related-posts":[{"id":8274,"url":"https:\/\/www.tongtuning.com\/wordpress\/8274","url_meta":{"origin":2543,"position":0},"title":"\u92fc\u7434\u8abf\u97f3\u5e2b \u8a71\u4f60\u77e5 &#8211; \u4e00\u53f0\u8d70\u97f3\u7684\u92fc\u7434\u70ba\u4ec0\u9ebc\u4ee4\u4f60\u5fc3\u7169\u610f\u4e82","date":"April 1, 2025","format":false,"excerpt":"\u92fc\u7434\u8abf\u97f3\u5e2b \u8a71\u4f60\u77e5 - \u4e00\u53f0\u8d70\u97f3\u7684\u92fc\u7434\u70ba\u4ec0\u9ebc\u4ee4\u4f60\u5fc3\u7169\u610f\u4e82 Video: https:\/\/youtu.be\/QhN6gZRb_ps \u4ee5\u76e1\u91cf\u7c21\u55ae\u7684\u65b9\u6cd5\u8aaa\u660e\uff1a 1. \u92fc\u7434\u8abf\u97f3\u7684\u6b65\u9a5f 2. \u5c0d\u6bd4 unison & 8\u5ea6 \u8d70\u97f3\u6642 \u5c0d\u92fc\u7434\u7528\u5bb6\u7684\u97f3\u611f\u9ad4\u9a57\u5f71\u97ff \u66f4\u591a\u984d\u5916\u8cc7\u6599: \u00a0 Beat Frequency \u793a\u8303: https:\/\/youtu.be\/V8W4Djz6jnY?si=pchNDSob0xuGcFSR \u00a0 Beat Frequency \u7406\u8ad6: https:\/\/youtu.be\/4M72kQulGKk?si=qiJPv5-HNSlMQFoD \u00a0 False Beat in Piano: https:\/\/youtu.be\/e_UvfoUevy8?si=1s91fP-PviZNrEPp \u00a0 \u6709\u95dc stretching (Entropy-based Tuning of Musical Instruments): http:\/\/www.tongtuning.com\/wordpress\/2543 \u6709\u95dc\u670d\u52d9: \u92fc\u7434\u8abf\u97f3\/\u6aa2\u67e5: https:\/\/www.tongtuning.com\/wordpress\/487 Special Thanks for Marketing Consultation\u2026","rel":"","context":"In &quot;\u6e6f\u751f\u7684 YouTube Channel\/ My YouTube Channel&quot;","img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.tongtuning.com\/wordpress\/wp-content\/uploads\/2021\/09\/Piano-Slider-2.jpg?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]},{"id":341,"url":"https:\/\/www.tongtuning.com\/wordpress\/341","url_meta":{"origin":2543,"position":1},"title":"Why Your Piano requires tuning?","date":"January 28, 2014","format":false,"excerpt":"A piano is a complex instrument with over 200 tightly stretched strings, each under high tension. That tension, combined with environmental factors, means the instrument naturally drifts out of tune over time. Here\u2019s why regular tuning is essential: Environmental Changes Humidity & Temperature: Wood and felt inside the piano expand\u2026","rel":"","context":"In &quot;English Articles&quot;","img":{"alt_text":"","src":"https:\/\/i1.wp.com\/www.tongtuning.com\/wordpress\/wp-content\/uploads\/2014\/02\/A440_Tuning_Fork.jpg?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]},{"id":494,"url":"https:\/\/www.tongtuning.com\/wordpress\/494","url_meta":{"origin":2543,"position":2},"title":"Piano Tuning Service (Mr. Tong &#8211; Mobile: 6304 4081)","date":"February 4, 2014","format":false,"excerpt":"\ufeff\ufeff\ufeff\ufeff A professional piano technician doesn\u2019t just \u201cfix notes\u201d\u2014they preserve your piano\u2019s health, protect your investment, and ensure it produces the rich, balanced sound it was built for. 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Tuning Piano tuning is the process of adjusting the tension of the strings inside an acoustic piano so that the notes sound correct and\u2026","rel":"","context":"In &quot;English Articles&quot;","img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.tongtuning.com\/wordpress\/wp-content\/uploads\/2014\/01\/Heater.jpg?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]},{"id":1700,"url":"https:\/\/www.tongtuning.com\/wordpress\/1700","url_meta":{"origin":2543,"position":4},"title":"History of Piano Tuning \u2013 A NEED FOR TUNERS","date":"May 2, 2014","format":false,"excerpt":"Prior to the advent of the piano, most musicians tuned their own instruments. \u00a0 This was a necessary part of owning one; to call someone in to tune a harpsichord would have been as preposterous an idea as calling someone in to tune a violin for a professional violinist. \u00a0\u2026","rel":"","context":"In &quot;English Articles&quot;","img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":417,"url":"https:\/\/www.tongtuning.com\/wordpress\/417","url_meta":{"origin":2543,"position":5},"title":"\u8abf\u97f3\u6642\u7528\u7684\u96fb\u5b50\u5100\u5668\u6709\u591a\u6e96\u78ba?","date":"February 1, 2014","format":false,"excerpt":"\u92fc\u7434\u8abf\u97f3\u9700\u8981\u6975\u9ad8\u7684\u7cbe\u6e96\u5ea6\uff0c\u56e0\u70ba\u92fc\u7434\u6709 88 \u500b\u7434\u9375\uff0c\u9084\u6d89\u53ca\u300c\u5ef6\u4f38\u8abf\u97f3\uff08stretch tuning\uff09\u300d\u8207\u300c\u6cdb\u97f3\u4e0d\u8ae7\u548c\u6027\uff08inharmonicity\uff09\u300d\u7b49\u8907\u96dc\u56e0\u7d20\u3002\u8a31\u591a\u4eba\u6703\u597d\u5947\uff1a\u5165\u9580\u7d1a\u96fb\u5b50\u8abf\u97f3\u5668\uff08ETD\uff09\u662f\u5426\u8db3\u5920\u6e96\u78ba\u4f86\u8abf\u92fc\u7434\uff1f \u4ee5\u4e0b\u70ba\u8a73\u7d30\u8aaa\u660e\u3002 \u5165\u9580\u7d1a ETD \u7684\u6e96\u78ba\u5ea6 \u89e3\u6790\u5ea6\uff1a\u5927\u591a\u6578\u5165\u9580\u7d1a ETD\uff08\u5982\u624b\u6a5f\u61c9\u7528\u7a0b\u5f0f\u6216\u4fbf\u651c\u5f0f\u8abf\u97f3\u5668\uff09\u7cbe\u6e96\u5ea6\u7d04\u70ba \u00b11 \u97f3\u5206\uff08cent\uff09\u3002 \u92fc\u7434\u9700\u6c42\uff1a\u5c08\u696d\u92fc\u7434\u8abf\u97f3\u901a\u5e38\u9700\u8981 \u5c0f\u65bc 1 \u97f3\u5206\u7684\u7cbe\u5ea6\uff0c\u751a\u81f3\u9054\u5230 0.1 \u97f3\u5206\uff0c\u624d\u80fd\u78ba\u4fdd\u97f3\u7a0b\u8207\u548c\u8072\u7684\u5e73\u6ed1\u5ea6\u3002 \u5be6\u969b\u6548\u679c\uff1a\u5165\u9580\u7d1a ETD \u53ef\u4ee5\u8b93\u92fc\u7434\u300c\u5927\u81f4\u4e0a\u300d\u5408\u97f3\uff0c\u9069\u5408 \u7c97\u7565\u8abf\u97f3\u3002 \u6280\u5e2b\u7684\u89c0\u9ede \u591a\u6578\u5c08\u696d\u92fc\u7434\u8abf\u97f3\u5e2b, \u5373\u4f7f\u4f7f\u7528, \u4e5f\u53ea\u6703\u5c07 ETD \u7576\u4f5c\u53c3\u8003\u5de5\u5177\uff0c\u4f46\u4ecd\u4f9d\u9760**\u807d\u89ba\u8abf\u97f3\uff08aural tuning\uff09**\u4f86\u5fae\u8abf\u516b\u5ea6\u8207\u540c\u97f3\u3002 \u7e3d\u7d50 \u5165\u9580\u7d1a ETD\uff1a\u7cbe\u5ea6\u7d04 \u00b11 \u97f3\u5206\uff0c\u9069\u5408\u5b78\u7fd2\u3001\u7df4\u7fd2\u6216\u5bb6\u5ead\u7c97\u7565\u8abf\u97f3\u3002 \u6700\u4f73\u505a\u6cd5\uff1a\u5373\u4f7f\u4f7f\u7528\u5c08\u696d ETD\uff0c\u4ecd\u9700\u7d50\u5408\u807d\u89ba\u8abf\u97f3\uff0c\u624d\u80fd\u7372\u5f97\u771f\u6b63\u81ea\u7136\u3001\u6085\u8033\u7684\u97f3\u8272\u3002 Reference: ETD Piano Tuning with PianoMeter - How to Tune Pianos. https:\/\/howtotunepianos.com\/etd-piano-tuning-with-pianometer\/ What ETD\u2026","rel":"","context":"In &quot;\u92fc\u7434\u8b77\u7406\u77e5\u8b58\/ Technical Advice&quot;","img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]}],"_links":{"self":[{"href":"https:\/\/www.tongtuning.com\/wordpress\/wp-json\/wp\/v2\/posts\/2543"}],"collection":[{"href":"https:\/\/www.tongtuning.com\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.tongtuning.com\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.tongtuning.com\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.tongtuning.com\/wordpress\/wp-json\/wp\/v2\/comments?post=2543"}],"version-history":[{"count":0,"href":"https:\/\/www.tongtuning.com\/wordpress\/wp-json\/wp\/v2\/posts\/2543\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.tongtuning.com\/wordpress\/wp-json\/wp\/v2\/media\/7779"}],"wp:attachment":[{"href":"https:\/\/www.tongtuning.com\/wordpress\/wp-json\/wp\/v2\/media?parent=2543"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.tongtuning.com\/wordpress\/wp-json\/wp\/v2\/categories?post=2543"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.tongtuning.com\/wordpress\/wp-json\/wp\/v2\/tags?post=2543"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}