source file: mills3.txt Date: Thu, 25 Sep 1997 21:52:35 +0200 Subject: Re: Barbershop and Equal-Step Tuning From: Paul Hahn On Thu, 25 Sep 1997, Carl Lumma wrote: > >Peter, if you are interested I could dredge up some of the messages from > >the last time we discussed this. (Hey, is the list archived anywhere?) > > I for one would be interested in anything you've got on this. I think I > remember seeing this term "consistency" in an article in an old issue of the > Xenharmonikon, but I didn't understand it until now. Thanks, Paul! I'll start culling through my files to find the messages that are worth reposting. > There was also another term that Ivor used, "symmetry". He never defined it > too well, but he complained that 12-tone was too symmetric, and that when > one first began Xenharmonic exploration, it was crucial to "knock the ear > out" with a non-symmetric tuning like 19-tone. Could this be the same > thing, Ivor's term for consistency? I don't think so . . . > Wouldn't it be nice to have a chart that measured both the "approximation" > and "consistency" values for each limit for each temperament? Look at these URLs: http://library.wustl.edu/~manynote/consist.txt http://library.wustl.edu/~manynote/consist2.txt http://library.wustl.edu/~manynote/consist3.txt The first shows what Paul E. and I were calling "consistency level" for all ETs up to 1200TET. A tuning is level N consistent at a certain harmonic limit M if not only all the M-limit intervals are consistently represented, but also up to N combinations of them. Example: because 12TET is level 3 consistent at the 5-limit, we know that three 6/5s (3 steps), or 216/125, will be most closely approximated by 9 steps. The second shows the same information as the first, but only displays an ET if there is some limit at which it has a higher level of consistency than all lower-numbered ETs, going up to 10000. The third shows the greatest error in cents for each harmonic limit an ET is consistent within, up to 99TET. --pH http://library.wustl.edu/~manynote <*> O /\ "Well, so far, every time I break he runs out. -\-\-- o But he's gotta slip up sometime . . . " SMTPOriginator: tuning@eartha.mills.edu From: Paul-Hahn@library.wustl.edu Subject: Symmetry (was Re: Barbershop and Equal-Step Tuning) PostedDate: 25-09-97 22:13:30 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: 25-09-97 22:13:12-25-09-97 22:13:12,25-09-97 22:12:16-25-09-97 22:12:16 DeliveredDate: 25-09-97 22:12:16 Categories: $Revisions: Received: from ns.ezh.nl by notesrv2.ezh.nl (Lotus SMTP MTA v1.1 (385.6 5-6-1997)) with SMTP id C125651D.006F0FEF; Thu, 25 Sep 1997 22:13:05 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA31120; Thu, 25 Sep 1997 22:13:30 +0200 Date: Thu, 25 Sep 1997 22:13:30 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA31109 Received: (qmail 29272 invoked from network); 25 Sep 1997 13:13:27 -0700 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 25 Sep 1997 13:13:27 -0700 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu