From jmb184@servtech.com Wed Jun 19 11:42:30 CDT 1996 Article: 75937 of sci.math Path: muir.math.niu.edu!fnnews.fnal.gov!uwm.edu!news-res.gsl.net!news.gsl.net!news.mathworks.com!newsfeed.internetmci.com!newsfeeder.servtech.com!murphy2.servtech.com!news From: jmb184@servtech.com (John Bailey) Newsgroups: sci.math Subject: Re: Fundamentals of music Date: 19 Jun 1996 02:08:57 GMT Organization: CAP Lines: 27 Message-ID: <4q7nfp$cif@murphy2.servtech.com> References: <4pkg6o$5gm@news.abs.net> <4pm64j$au4@eccles.dsbc.icl.co.uk> <4q4gen$hmf@muir.math.niu.edu> NNTP-Posting-Host: zenith.roc.servtech.com X-Newsreader: WinVN 0.92.6+ In article <4q4gen$hmf@muir.math.niu.edu>, rusin@vesuvius.math.niu.edu (Dave Rusin) says: > >In article <4pm64j$au4@eccles.dsbc.icl.co.uk>, >Roy Lakin wrote: > >>Helmholtz's "Sensations of Tone" showed a 53-keys-to-the-octave keyboard >>where you would have to move backwards and forwards to transpose keys, so >>as to position your hands above a slightly different subset. >> > >The 'best' number of tones per interval can be deduced from a >continued-fraction argument, as another poster noted, although in >practice there have been some reasonably successful experiments with, >say, a 19-tone scale. I find the key layout for a 19 tone scale utterly charming (mathematically) If you visualize the piano keys for the 12 tone, equi-tempered scale there are seven white keys and five black in one octave. From any white key you go up a half tone to the black key for a sharp, and down a half tone to a black key for a flat. If you now want a 19 tone equi-tempered scale, you split every black key. For each of the white keys, when you go up for a sharp, it can now be a different key than going down for a flat from the next higher white key. Except, now the white keys that did not have a black key in between before can now have a black key between them.(only one) Where before there were 7 white and 5 black there are now 7 white, 5x2 split black keys and 2 more new single black keys. Making 19 in all. This is probably very confusing. If anyone is interested, I will make a diagram so it is clearer.