source file: mills3.txt Date: Sat, 16 Aug 1997 17:55:50 +0200 Subject: syntonic and skhismatic notation From: gbreed@cix.compulink.co.uk (Graham Breed) I mentioned some of this before in a post about quartertones that should have been in TD1153, but wasn't. I class all just intervals and approximations thereof based on syntonic meantone temperament using the letters t, s, r, q and p for major seconds, minor seconds, augmented unisons, diminished seconds and "commas" respectively. The syntonic comma p5 is dimensionless in meantone temperament. I usually think of the dimensions of intervals in terms of q and r. I now use the term "demisemitone" for intervals with dimensions of q and r-q, although I would rather call them quartertones. A chromatic semitone is a kind of r, and an enharmonic diesis q. For 7/4 to be an augmented sixth, a septimal comma has dimensions of q. In terms of fifths, octave reduced, t=2 steps, s=-5, r=t-s=7, q=s-r=-12. The # symbol means "raise this note by r" and the / symbol can mean raising by q. These symbols are consistent when 0<=q<=r, where 0 is a unison. This is true for all meantone temperaments where the fifth is between that for 19tet and 12tet. When the fifth is sharper than 7/12 oct, a skhismatic approximation is better. 12tet is the scale that is both a syntonic and skhismatic meantone, and can be defined as such using Fokker's method. The symbols / and \ can then be added to a 12 note piano roll view to denote boosts of q from the following scale: r s s r s r s r s s r s C C# D Eb E F F# G G# A Bb B C' There is a one to one mapping between this and the standard staff notation. The symbols # and b can still be used if they are found to be convenient. Over the past couple of days, I have extended this idea to syntonic intervals using Greek letters. The major tone is a kind of tau, and a diatonic semitone is a kind of sigma. Then, rho=tau-sigma and pi=sigma-rho. In Pythagorean terms, sigma's are apotomes and rho's are limmas. A minor tone has dimensions of tau-pi or 2rho. A chromatic semitone has dimensions rho-pi, and an enharmonic diesis is 2pi. The syntonic and Pythagorean commas are both pi's, and I have also defined the septimal comma to be a pi. The piano roll view can also be adapted to skhismatic scales with / and \ denoting boosts of pi from a scale like this: rho rho sigma rho sigma rho sigma rho rho sigma rho sigma C D E F G A B C The capital letters here name keys rather than notes, as the standard note names do not work with skhismatic scales. This is compatible with the syntonic version in that the same scale will occupy the same positions in both notations, but with different modifiers. They are also compatible with 12 note keyboards. One day, I intend to implement these views in software, so this isn't all abstract. It is also easier to get dynamic tuning to work with 2-D than just scales. Now, does anyone know of a 2-D approximation (linear temperament) that covers 22tet? Approximating the enharmonic diesis (7 0 -3)H and syntonic comma (-4 4 -1)H to be equal doesn't work. If it did, it would cover 12, 22, 34 and 46tet. It is, incidentally, significant that these are all even numbers. SMTPOriginator: tuning@eartha.mills.edu From: kollos@cavehill.dnet.co.uk Subject: Re: 1/4 tones and 1/n "comma" meantones - naming PostedDate: 16-08-97 19:18:05 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH RouteTimes: 16-08-97 19:18:08-16-08-97 19:18:09,16-08-97 19:16:16-16-08-97 19:16:16 DeliveredDate: 16-08-97 19:16:16 Categories: $Revisions: Received: from ns.ezh.nl by notesrv2.ezh.nl (Lotus SMTP MTA v1.1 (385.6 5-6-1997)) with SMTP id C12564F5.005F0912; Sat, 16 Aug 1997 19:18:02 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA31900; Sat, 16 Aug 1997 19:18:05 +0200 Date: Sat, 16 Aug 1997 19:18:05 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA31891 Received: (qmail 12314 invoked from network); 16 Aug 1997 17:17:58 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 16 Aug 1997 17:17:58 -0000 Message-Id: <33F58C99.7063@cavehill.dnet.co.uk> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu