source file: mills3.txt Date: Sat, 17 Jan 1998 10:32:54 +0100 Subject: Tempering issues From: gbreed@cix.compulink.co.uk (Graham Breed) Paul Erlich wrote: > This "sameness" and "in-tune-ness" are surprisingly flexible, especially > when it comes to multiple octaves. Playing the same melody on opposite > ends of a piano simultaneously, you can get away with a semitone > transposition without much disturbing the "sameness" or "in-tune-ness" > (Yasser gives a melody to use for this, and I didn't believe it until I > tried it!) Let's assume that the suitability of a temperament depends upon the most poorly tuned interval in the relevant consonant limit. If you take the Partch odd limit as the definition of consonance, then all octave transpositions of the main intervals have to be counted. If the octave is not perfect, some equivalent intervals will be tuned worse as a result. So, the suitability of the temperament diminishes. If you've got the same pitch set in all registers, it will be difficult to avoid the bad intervals. A small octave stretching may work to favour the most frequently used intervals but, for this to be acceptable, it must have a very small harmonic effect. As an example, if you play two 4:5:6:8 chords an octave apart, you get the intervals 3/1, 5/2, 8/3 and 16/5 coming out. A mistuning of the octave will become threefold apparent in two of these intervals. Harmony isn't really octave invariant, as I've argued before, but all these intervals will occur between melody and harmony. How big is a piano keyboard? One semitone at 8 octaves implies a formal octave 100/8=12.5 cents sharp. That's better than the thirds in 12-equal. Paul's example of the 8:10:12:15 chord is a good one. Taking 15/8 as a dissonance, this chord will still work best the better the fifths and thirds are tuned, so that their consonance is more evident. I think 15/8 has some degree of consonance, though, so if this chord is to be used frequently this interval should be considered, so that the difference tones are in tune. While experimenting with this, I found that 8:10:15 sounds better than 4:5:7, probably because of the fifth. 4:6:7 is preferable to 8:12:15, so 7-limit still rules! If the presence of consonances can make up for a dissonance in a chord, it follows that a strong consonance is important even in the presence of other consonances. So, octaves and fifths should be considered first in choosing a temperament. I think 12-equal works partly by 3-limit harmony plus poorly tuned thirds functioning as weak dissonances. Also, because memory corrects the tuning of familiar chords. You're still going to notice if one interval is way out, though. The worst tuned interval in a chord is still the most important. SMTPOriginator: tuning@eartha.mills.edu From: mr88cet@texas.net (Gary Morrison) Subject: CDRs and DVDs PostedDate: 17-01-98 15:18:37 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: 17-01-98 15:18:37-17-01-98 15:18:38,17-01-98 15:17:40-17-01-98 15:17:41 DeliveredDate: 17-01-98 15:17:41 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 C125658F.004E9A42; Sat, 17 Jan 1998 15:18:32 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA28007; Sat, 17 Jan 1998 15:18:37 +0100 Date: Sat, 17 Jan 1998 15:18:37 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA28050 Received: (qmail 4376 invoked from network); 17 Jan 1998 06:18:34 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 17 Jan 1998 06:18:34 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu