source file: mills3.txt Date: Tue, 13 Jan 1998 19:49:01 +0100 Subject: Re: Tritone ratios From: gbreed@cix.compulink.co.uk (Graham Breed) Bob Lee wrote: > I had always assumed that 7/5 was the natural ratio for an augmented fourth, > and that 10/7 was the natural ratio for the diminished fifth. What do you > mean when you say they "do not arise in a diatonic context"? Traditional diatonic music uses 7-note scales and 5-limit harmony. In this context tritones are dissonant. The simplest 5-prime limit ratios are 25/18 and 36/25, or (-1 -2 2)H and (2 2 -2)H. The latter comprises two minor thirds 6/5 or (1 1 -1)H. That means it must be a kind of fifth. The difference between these quintal tritones is just over a 19th of an octave. That means they're well represented in 19-equal and the difference is melodically important. Staff notation is a consistent way of notating quintal intervals, so the distinction is preserved. When a tritone is used in isolation, or is treated as consonant, the best tuning is septimal. The difference between 10/7 and 7/5 is only 35 cents. 7/5 is chosen as the augmented fourth because it's the smaller. However, as this isn't a diatonic interval, there's no standard way of notating it. The obvious example is a 4:5:6:7 chord. For 7/5 to be an augmented 4th, 7/4 must be an augmented sixth. If you're writing septimal music in meantone temperament or notation, you have to be aware of this. Unfortunately, most composers do treat 10/7 and 7/5 as enharmonies, so the usual notation of 7/4 is as a minor seventh. You have to work out from the context whether a 7th chord should be quintal or septimal. SMTPOriginator: tuning@eartha.mills.edu From: "Paul H. Erlich" Subject: b5th ratios PostedDate: 13-01-98 21:27:51 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $MessageStorage: 0 $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: 13-01-98 21:27:14-13-01-98 21:27:15,13-01-98 21:27:03-13-01-98 21:27:03 DeliveredDate: 13-01-98 21:27:03 Categories: $Revisions: Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2 9-3-1997)) with SMTP id C125658B.00705A5C; Tue, 13 Jan 1998 21:27:40 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA07598; Tue, 13 Jan 1998 21:27:51 +0100 Date: Tue, 13 Jan 1998 21:27:51 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA04626 Received: (qmail 10470 invoked from network); 13 Jan 1998 12:27:44 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 13 Jan 1998 12:27:44 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu