{"id":4290,"date":"2022-05-22T09:42:57","date_gmt":"2022-05-22T17:42:57","guid":{"rendered":"https:\/\/bendwavy.org\/wp\/?p=4290"},"modified":"2022-10-02T10:14:33","modified_gmt":"2022-10-02T18:14:33","slug":"31-frets","status":"publish","type":"post","link":"https:\/\/bendwavy.org\/wp\/?p=4290","title":{"rendered":"31 frets"},"content":{"rendered":"

Here is the neck of a guitar that I’d like to have made someday, if I should ever develop the dexterity to make it worthwhile. The blue stripes show where standard frets would be, for comparison.<\/p>\n

The tuning is my tweaked version of meantone: compared to just intonation, each factor of 2 is sharp by 1\/16 comma, each factor of 3 is flat by 1\/8 comma, and each factor of 5 is sharp by 1\/4 comma. (A comma<\/a> is the difference between 64:81, the Pythagorean major third derived from compounding fifths, and the more harmonious 4:5.) This makes the thirds and sixths much truer than in equal temperament, and the fifths slightly truer than in traditional meantone, which puts all the error in the 3s.<\/p>\n

This design has 31 frets in the first octave: 12 flats, 7 naturals, 12 sharps. The bent frets span the difference (~151:152) between 18 of my sharp octaves and 31 of my flat fifths.<\/p>\n

To reduce crowding, the second octave has only three flats and three sharps. The bent frets span the difference (~50:51) between G\u266f and A\u266d.<\/p>\n

The charts below should look familiar to players, if you squint a little.
\n <\/p>\n

The dots are placed according to a slightly different scale, which divides a factor of 12 into 111 equal steps; this is a local optimum among cyclic scales by the same criterion I used to choose the non-cyclic intervals for the frets.<\/p>\n","protected":false},"excerpt":{"rendered":"

Here is the neck of a guitar that I’d like to have made someday, if I should ever develop the dexterity to make it worthwhile. The blue stripes show where standard frets would be, for comparison. The tuning is my … Continue reading →<\/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-4290","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\/4290","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=4290"}],"version-history":[{"count":45,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/4290\/revisions"}],"predecessor-version":[{"id":4403,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/4290\/revisions\/4403"}],"wp:attachment":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4290"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4290"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4290"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}