{"id":4772,"date":"2026-04-30T16:31:14","date_gmt":"2026-05-01T00:31:14","guid":{"rendered":"https:\/\/bendwavy.org\/wp\/?p=4772"},"modified":"2026-05-03T00:42:06","modified_gmt":"2026-05-03T08:42:06","slug":"increasing-harmony-2","status":"publish","type":"post","link":"https:\/\/bendwavy.org\/wp\/?p=4772","title":{"rendered":"increasing harmony"},"content":{"rendered":"<p>In 1987 I conceived a theoretical improvement on the quarter-comma meantone scale: the generators are 2<sup>&minus;5\/4<\/sup> 3<sup>1\/4<\/sup> 5<sup>7\/16<\/sup> (a &lsquo;tone&rsquo;, 194.501 cents) and 2<sup>7\/2<\/sup> 3<sup>&minus;1\/2<\/sup> 5<sup>&minus;9\/8<\/sup> (a &lsquo;semitone&rsquo;, 114.420 cents).  I previously showed <a href=\"?p=4290\">a guitar concept<\/a> fretted accordingly; see that post for my motives.<\/p>\n<p>Next, can I extend the concept to higher primes?  One difficulty is that, while the lower primes have well-defined notation (2 is five tones and two semitones, 3 is eight tones and three semitones, 5 is twelve tones and four semitones), it is not so clear where to put higher primes: should a factor of 7 be approximated by 12 tones and 9 semitones, or 15 tones and 4 semitones, or some other combination?  This choice, which must be done again for every prime to be added to the scheme, affects the optimization.<\/p>\n<p>So I attacked a simpler problem: local optima &ndash; <a href=\"\/tuning\/euler.htm\">minimax temperaments<\/a> &ndash; with a <em>single<\/em> generator.<br \/>\nThe world mostly yawned.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In 1987 I conceived a theoretical improvement on the quarter-comma meantone scale: the generators are 2&minus;5\/4 31\/4 57\/16 (a &lsquo;tone&rsquo;, 194.501 cents) and 27\/2 3&minus;1\/2 5&minus;9\/8 (a &lsquo;semitone&rsquo;, 114.420 cents). I previously showed a guitar concept fretted accordingly; see that &hellip; <a href=\"https:\/\/bendwavy.org\/wp\/?p=4772\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12,21],"tags":[],"class_list":["post-4772","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-musicverse"],"_links":{"self":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/4772","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4772"}],"version-history":[{"count":4,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/4772\/revisions"}],"predecessor-version":[{"id":4784,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/4772\/revisions\/4784"}],"wp:attachment":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4772"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4772"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4772"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}