In the western world we have the predominance of the heptatonic scale<\/a>.<\/p>\n I always wondered why the Major and the Minor scale are the dominating scales, whether one can derive this from some simple principle. It is kinda strange that there are 5 whole tone steps and 2 halftone steps: One explanation is the use of the perfect 5th (Pythagorean tuning<\/a>). The ratio of 2:1 is one octave. 1. 3\/2 = 1.5 When we order these values we get In the western world we have the predominance of the heptatonic scale. I always wondered why the Major and the Minor scale are the dominating scales, whether one can derive this from some simple principle. It is kinda strange that there are 5 whole tone steps and 2 halftone steps: Whole:Whole:Half:Whole:Whole:Whole:Half One explanation is the […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-109","post","type-post","status-publish","format-standard","hentry","category-general"],"_links":{"self":[{"href":"http:\/\/math.eretrandre.org\/htwp\/index.php?rest_route=\/wp\/v2\/posts\/109","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/math.eretrandre.org\/htwp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/math.eretrandre.org\/htwp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/math.eretrandre.org\/htwp\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/math.eretrandre.org\/htwp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=109"}],"version-history":[{"count":18,"href":"http:\/\/math.eretrandre.org\/htwp\/index.php?rest_route=\/wp\/v2\/posts\/109\/revisions"}],"predecessor-version":[{"id":130,"href":"http:\/\/math.eretrandre.org\/htwp\/index.php?rest_route=\/wp\/v2\/posts\/109\/revisions\/130"}],"wp:attachment":[{"href":"http:\/\/math.eretrandre.org\/htwp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=109"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/math.eretrandre.org\/htwp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=109"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/math.eretrandre.org\/htwp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=109"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nWhole:Whole:Half:Whole:Whole:Whole:Half<\/p>\n
\nThe interval of a perfect fifth, corresponds to a ratio of 3:2 of the frequencies. This ratio is experienced as quite harmonic.<\/p>\n
\nSo if we repeat perfect fifths and transpose them back into the octave, i.e. into values between 1 and 2 (by dividing by 2), we get the following values:<\/p>\n
\n2. 3\/2 * 3\/2 \/ 2 = 9\/8 = 1.125
\n3. 3\/2 * 3\/2 * 3\/2 \/ 2 = 27\/16 = 1.6875
\n4. 3\/2 * 3\/2 * 3\/2 * 3\/2 \/ 4 = 81\/64 = 1.265625
\n5. 3\/2 * 3\/2 * 3\/2 * 3\/2 * 3\/2 \/ 4 = 243\/128 = 1.8984375
\n6. (3\/2)^6 \/ 8 = 729\/512 = 1.423828125<\/p>\n
\n0. < 2. < 4. < 6. < 1. < 3. < 5.\n1 < 9\/8 < 81\/64 < 729\/512 < 3\/2 < 27\/16 < 243\/128 < 2\n1 < 1.125 < 1.265625 < 1.423828125 < 1.5 < 1.6875 < 1.8984375 < 2\n\nthe multiplicative differences in each step are\n9\/8 \/ 1 = 9\/8\n81\/64 \/ (9\/8) = 9\/8\n729\/512 \/ (81\/64) = 9\/8\n3\/2 \/ (729\/512) = 256\/243\n27\/16 \/ (3\/2) = 9\/8\n243\/128 \/(27\/16) = 9\/8\n2 \/ (243\/128) = 256\/243\n\nNow 9\/8 corresponds to a whole tone and 256\/243=1.053497942386831... corresponds to a half tone.\nSo we have the characteristic pattern of 3 and 2 whole tones seperated by a half tone.\n\nf=1, g=9\/8, a=81\/64, b=729\/512, c=3\/2, d=27\/16, e=243\/128, f=2\nOr if we start with c=1, by multiplying with 2\/3:\nc=1, d=9\/8, e=81\/64, f=4\/3, g=3\/2, a=27\/16, b=243\/128, c=2\n\nThe following tones are \n7. (3\/2)^7 \/ 16 = 1.06787109375\n8. (3\/2)^8 \/ 16 = 1.601806640625\n9. (3\/2)^9 \/ 32 = 1.20135498046875\n10. (3\/2)^10 \/ 32 = 1.802032470703125\n11. (3\/2)^11 \/ 64 = 1.35152435302734375\n12. (3\/2)^12 \/ 128 = 1.0136432647705078125\n\nThe last value is called the Pythagorean comma<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"