{"id":342,"date":"2012-11-15T10:55:31","date_gmt":"2012-11-15T18:55:31","guid":{"rendered":"http:\/\/www.garygarrett.me\/?p=342"},"modified":"2017-08-18T09:00:43","modified_gmt":"2017-08-18T16:00:43","slug":"the-lattice","status":"publish","type":"post","link":"https:\/\/www.garygarrett.me\/?p=342","title":{"rendered":"The Lattice"},"content":{"rendered":"

In 1739, the great mathematician Leonhard Euler<\/a>\u00a0published something he called a Tonnetz<\/a>, German for “tone network.” It looked like this:<\/p>\n

\"\"<\/a><\/p>\n

Euler’s Tonnetz organizes the notes into a matrix, instead of a scale. Moving down and to the left represents motion by an interval of a fifth (V) in musical space. Down and to the right shows movement by a major third (III).<\/p>\n

The lattice has been rediscovered and redrawn many times over the years. One of my favorites is the Duodenarium of Alexander Ellis<\/a>, which showed up in his appendix to Helmholtz’s<\/a> pioneering book, On the Sensations of Tone<\/em>, in the late 1800’s.<\/p>\n

\"\"<\/a><\/p>\n

Now we’re talkin’! C is at the center. The fifths go up and down, and thirds from left to right, leading to a square grid.<\/p>\n

One of W. A. Mathieu’s innovations in Harmonic Experience<\/a>\u00a0is to slant the axes and make them line up with the musical staff:<\/p>\n

\"\"<\/a><\/p>\n

Seriously, if this blog interests you, please get a copy of this book<\/a>. I have no stake in you doing this, except that I believe the more broadly understood this man’s work is, the more great music will be made.<\/p>\n

I’ve been messing around with the lattice for a year and a half now, and I’ve morphed it into a form that suits my own musical work.<\/p>\n

\"\"<\/a><\/p>\n

Further slanting the thirds axis to 60 degrees makes it a hexagonal lattice, and for me the relationships between the notes become more intuitive. The major chord is now, appropriately, a stable-looking triangle. And a new axis appears, northwest to southeast: movement by minor thirds<\/a>. I follow Mathieu’s example and show this one with a dotted line, because it isn’t a direct move: the minor third is a third down and a fifth up, a compound move on the lattice — a major (sorry) insight into the nature of the minor third. Much more on that one later.<\/p>\n

Japanese mathematician Shoh\u00e9 Tanaka<\/a>\u00a0drew a hexagonal tone lattice\u00a0in the 1800’s. I haven’t been able to find a picture.<\/p>\n

Movement to the right represents multiplication by 3, that is, up a fifth. Up and to the right means you’ve multiplied by 5, up a major third. Left means division by 3, down a fifth. Down left is division by 5, down a major third. The tonic<\/a>, 1, is at the center (below left of center in this portion). The grid goes out to infinity. This is the region encompassed by Flying Dream<\/a>, which in fact covers most of the territory I’ve found useful so far, a major reason I chose that song for the video.<\/p>\n

Next:\u00a0The Tonic Major Chord<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

In 1739, the great mathematician Leonhard Euler\u00a0published something he called a Tonnetz, German for “tone network.” It looked like this: Euler’s Tonnetz organizes the notes into a matrix, instead of a scale. Moving down and to the left represents motion by an interval of a fifth (V) in musical space. Down and to the right…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_kadence_starter_templates_imported_post":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"footnotes":""},"categories":[112],"tags":[298,75,74,72,36,28,77,62,76,27,31,63,37,73],"class_list":["post-342","post","type-post","status-publish","format-standard","hentry","category-thelattice","tag-book","tag-duodenarium","tag-ellis","tag-euler","tag-fifth","tag-flying-dream","tag-harmonic-experience","tag-harmonic-space","tag-hexagonal","tag-lattice","tag-mathieu","tag-staff","tag-third","tag-tonnetz"],"_links":{"self":[{"href":"https:\/\/www.garygarrett.me\/index.php?rest_route=\/wp\/v2\/posts\/342","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.garygarrett.me\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.garygarrett.me\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.garygarrett.me\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.garygarrett.me\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=342"}],"version-history":[{"count":10,"href":"https:\/\/www.garygarrett.me\/index.php?rest_route=\/wp\/v2\/posts\/342\/revisions"}],"predecessor-version":[{"id":1993,"href":"https:\/\/www.garygarrett.me\/index.php?rest_route=\/wp\/v2\/posts\/342\/revisions\/1993"}],"wp:attachment":[{"href":"https:\/\/www.garygarrett.me\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=342"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.garygarrett.me\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=342"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.garygarrett.me\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=342"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}