source file: mills2.txt Date: Mon, 5 May 1997 07:16:11 -0700 Subject: Post: tuning thirds in triads From: PAULE The reason I differ with Paul Rapoport when he defines the m3 as simply "left over" from the P5 and the M3 is that in tuning a major triad, both thirds are important to my ears. Some of the experiments I discussed with John Chalmers provide evidence for this belief. For example, John (in Tuning Digest 1008) and I both heard that a chord tuned 0 400 720 was clearly "better" than a chord tuned 0 372 720, even though the M3 and P5 are equally false in both chords. (To refute an explanation based on a preference for stretched intervals, use the first inversion of these chords.) Therefore it matters whether you're using the best possible m3 in a particular tuning; just using the best M3 and the best P5 does not guarantee the best major triad. For example, try 64-tet. There the best perfect fifth is 37 steps, and the best major third is 21 steps. Changing either of these by 1 to give a better minor third of 17 steps improves the major triad. 64-tet is actually an interesting tuning; it is the simplest equal temperament that contains all "tonal" systems (defined according to my paper on 22-tET, the systems are pentatonic, diatonic, and a new system first found in 22-tET and described at length in that paper). The above assumes that the "best" tuning of a major triad is 4:5:6, a statement which causes little serious controversy. Unfortunately things are not so simple for the minor triad. The 10:12:15 (or 1/6:1/5:1/4) tuning is lowest in roughness, but for a typical root-position voicing, (4:8:12:)16:19:24 will sound more stable. Gary Morrison has pointed out that there can be two different types of consonance judgments operating at the same time; I believe that one is based on roughness while the other is based on various factors having to do with how well the tones fit into a single harmonic series. 16:19:24 is typically still within the "dip" in the roughness function whose local minimum is at 10:12:15, so 16:19:24 is still less rough than just about any triad besides the major. To me a root-position minor triad sounds out-of-tune as 10:12:15, although I think 12:15:20 (1/5:1/4:1/3) is a very sweet first-inversion minor triad. So it is possible that the best thirds for the major triad are different that the best thirds for the minor triad. It destroys a lot of the mathematical beauty of 5-limit theory by bringing in the 19th harmonic, but I have to let my ears be the final arbiter, and in this case they tell be that the 19th harmonic is important. A much-ignored study by Pierce and Roberts (I think; it's from the book _Harmony and Tonality_) found that untrained listeners generally fall into two categories, "pure" and "rich." The pure listeners liked just 4:5:6 and 3:5:7 triads better than versions of those chords where the middle tone was displaced by +/-15 or +/-30 cents, while the rich listeners like the 15-cents-off versions of the chords best, with no particular preference between a +15 and a -15 cent "error" However, for 10:12:15 (or 1/6:1/5:1/4) chords, both classes of listeners preferred the version where the middle tone was lowered by 15 cents. The authors of the article did not provide much of an explanation of this phenomenon. Note that although the 15-cent shift has about the same effect on roughness for both major and minor triads, the listeners may have been motivated not so much by the small differences in roughness but by a preference for harmonic series. The preferred version of the minor triad is very close to 16:19:24, which has a note octave-equivalent to the fundamental in the bottom voice. Thus in a sense it better resembles a harmonic series than the "just" or 10:12:15 minor triad. The rich listeners may have also been attracted to the beating, which is on the order of 7 times per second for the +/- 15 cent versions of all of these chords. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 5 May 1997 19:27 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA05050; Mon, 5 May 1997 19:27:42 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA05039 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id KAA05261; Mon, 5 May 1997 10:20:19 -0700 Date: Mon, 5 May 1997 10:20:19 -0700 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu