{"id":1793,"date":"2010-05-05T16:07:42","date_gmt":"2010-05-05T16:07:42","guid":{"rendered":"http:\/\/meeseeks:5080\/blog\/?p=1793"},"modified":"2010-05-05T16:07:42","modified_gmt":"2010-05-05T16:07:42","slug":"combinatorics-of-some-musical-objects","status":"publish","type":"post","link":"https:\/\/vukutu.com\/blog\/2010\/05\/combinatorics-of-some-musical-objects\/","title":{"rendered":"Combinatorics of some musical objects"},"content":{"rendered":"

Excerpts from Appendix C (page 164) from Keith [1991].\u00a0 All results assume a 12-tone equal-tempered scale.<\/p>\n

Number of diatonic scale classes: 3
\nNumber of note names (A-G); number of notes in a common scale; number of white keys per octave on a piano:\u00a0 7
\nNumber of scales one note different from the Major scale: 9
\nNumber of notes in the most common equal-tempered scale:\u00a0 12
\nNumber of common musical keys (C + 1-6 flats\/sharps):\u00a0 13
\nNumber of 7-note diatonic scales (=7 * 3):\u00a0 21
\nNumber of elementary 2-fold polychords: 23
\nA k-fold polychord is an n-note chord sub-divided into k non-empty subchords, for k=1,\u00a0 . . ., n.\u00a0 For example, the 6-note chord <C, D, E, F#, G, A> can be subdivided into the 3-note 2-fold polychords, <C, E, G> and <D, F#, A>.
\nNumber of 7-note chords:\u00a0 66
\nNumber of distinct interval sets (partitions of 12):\u00a0 77
\nNumber of 7-note triatonic scales (=7*35):\u00a0 245
\nNumber of notationally-distinct diatonic scales (=13 *21):\u00a0 273
\nNumber of distinct chord-types (= N(12) – 1):\u00a0 351
\nNumber of 7-note musical scales (=7*66):\u00a0 462
\nNumber of scales (=Number of n-note scales, summed over all n)\u00a0 (=2^(12-1) = 2^11): \u00a0 2048
\nNumber of chords without rotational isomorphism (= 2^12 – 1):\u00a0 4095
\nNumber of notationally-distinct scales (=13 * 462):\u00a0 6006
\nNumber of non-syncopated 8-bar 1\/4-note rhythmic patterns:\u00a0 458,330
\nNumber of non-syncopated 8-bar 1\/8-note rhythmic patterns:\u00a0 210,066,388,901<\/p><\/blockquote>\n

Reference:<\/em>
\nMichael Keith [1991]:\u00a0 From Polychords to Polya:\u00a0 Adventures in Musical Combinatorics.\u00a0 (<\/em>Princeton, NJ:\u00a0 Vinculum Press.)<\/p>\n","protected":false},"excerpt":{"rendered":"

Excerpts from Appendix C (page 164) from Keith [1991].\u00a0 All results assume a 12-tone equal-tempered scale. Number of diatonic scale classes: 3 Number of note names (A-G); number of notes in a common scale; number of white keys per octave on a piano:\u00a0 7 Number of scales one note different from the Major scale: 9 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[50,56],"tags":[],"class_list":["post-1793","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-music","p1","y2010","m05","d05","h16"],"_links":{"self":[{"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/posts\/1793","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/comments?post=1793"}],"version-history":[{"count":0,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/posts\/1793\/revisions"}],"wp:attachment":[{"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/media?parent=1793"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/categories?post=1793"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/tags?post=1793"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}