source file: mills3.txt Date: Tue, 23 Dec 1997 22:44:10 +0100 Subject: Reply to R. C. Valentine From: "Paul H. Erlich" > >}> On the list, I believe someone =17 by no means devoid of understanding = >=17 >}> said that "Gibson's preference for mean-tone temperaments is >}> well-established" as if there are any others worth discussing (for >}> long). Well, there is just intonation and Pythagoreanism, but they are >}> not temperaments. There are 22- 29- 41- & 53-tone equal, but I do hope >}> that by now I have explained why these 'temperaments' are valueless, for >}> they cast out the consonant thirds from their tonal fabric as soon as >}> one tries to preserve the consonant fifths. =20 > >}I really don't understand this statement "cast out the consonant thirds". > >} 12TET 7 M4 =3D=3D 4 P5 >} 19TET 11 M4 =3D=3D 6 P5 >} 53TET 31 M4 =3D=3D 17 P5=20 > >}Is the "problem" that the 53TET (and others) has a choice of thirds to >}use depending on context (in this run of 31 M4, we probably came close >}to a variety of M4 that were diatonically correct to the original root, >}if we built the equivalent of I, IV, V)? > >I can't understand your notation. What is M4? Anyway, Gregg is referring to >the fact that 22-, 29-, 41-, 53-equal and JI are not meantone temperaments. >You're going to tie yourself (and the music) in knots if you try to use these >tunings for traditional diatonic music. Modulations to nearby keys become >extremely awkward, as do single-key chord progressions such as I-IV-ii-V-I >and I-vi-ii-V-I. > >}Of course, this choice gives you lots of VERY consonant thirds, and,=20 >}using them when it is important harmonically (to fix audible beating) >}does not mean we really care whether we hit them exactly melodically >}(since we already have determined that melody can be a lot sloppier >}than harmony). > >JI is perfect harmonically but contains melodic errors of a syntonic comma. >(The other non-meantone temperaments mentioned are worse than JI in both >respects). If one treats sustained tones as single pitches (not altering them >by a comma mid-stream) these melodic errors can accumulate over the course of >a piece, often landing the piece as much as ten commas below where it >started. 1/4-comma meantone temperament is perfect melodically but contain >harmonic errors of no more than 1/4 syntonic comma. These errors do not >accumulate since consonant intervals of the same type do not combine to form >other consonant intervals. So the relevant question is not whether melody can >be a lot sloppier than harmony but whether melody can be four (locally) to >forty (globally) times sloppier than harmony. I think the answer depends on >the exact sizes of the errors and, of course, the musical context, but I >think meantone wins in most cases. Of course, one can write piece >specifically for JI which avoids most of its disturbing characteristics, but >the vast majority of classical composers did not do this. > >In a performance context one would have to deal with far fewer pitches in a >meantone rendition. These practical benefits of meantone may have been the >decisive factor in making meantone the most important tuning in the West from >1500-1800. > >31-equal is close to 1/4-comma meantone; 19-equal is virtually 1/3-comma >meantone, having harmonic errors no larger than 1/3 comma. > >}Ezra Simms is currently advocating 72TET (he approached it from a few >}other directions) and has articles in Perspectives of New=20 >}Music (vol 29, their microtonal series) and Computer Music Journal >}in recent years. As a REAL tuning system to advocate vs 12TET, this=20 >}one meets all the tests above. > >I do like your argument (deleted). Ezra Sims, though, attributes some ratios >to 72TET that it cannot express consistently. 72TET is only consistent >through the 17-limit. > >}Regarding 19TET in particular, perhaps it is the last temperment which >}preserves "sameness" of the "big" and "small" major seconds (9:8, 10:9). > >} major scale... > >} 12TET : 2 2 1 2 2 2 1 >} 19TET : 3 3 2 3 3 3 2 >} 22TET : 4 3 2 4 3 4 2 >} 29TET : 5 4 3 5 4 5 3 >} 41TET : 7 6 4 7 6 7 4 >} 53TET : 9 8 5 9 8 9 5 > >}But... is that a good thing? Thanks a lot. > >This sameness is what characterizes a meantone temperament, and as I have >stated above, I think it is a good thing for diatonic music. More: > >15TET: 3 2 1 3 2 3 1 >26TET: 4 4 3 4 4 4 3 >31TET: 5 5 3 5 5 5 3 >34TET: 6 5 3 6 5 6 3 >72TET: 12 11 7 12 11 12 7 > >For one thing, the ii triad is lousy in those of the above "major scales" >where there are two sizes of major second. It is unfortunate that the best >meantone scales that 72TET has to offer are the same as 12TET scales. This >often prevents one from taking advantage of the superb 5:4 of 72TET in a >diatonic context. > >The "major scales" in 15- and 22-equal sound really out-of-tune melodically >because of the two different sizes of major second. So even if the ii triad >and modulation are avoided, these tunings will sound weird for diatonic >music. > >I have non-diatonic reasons for liking 22-equal. Blackwood has written about >good reasons for liking 15- and 16-equal in Perspectives of New Music. > SMTPOriginator: tuning@eartha.mills.edu From: Gregg Gibson Subject: More on Just Intonation 2 PostedDate: 23-12-97 23:44:53 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: 23-12-97 23:42:40-23-12-97 23:42:40,23-12-97 23:42:11-23-12-97 23:42:11 DeliveredDate: 23-12-97 23:42:12 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 C1256576.007CBFE8; Tue, 23 Dec 1997 23:44:32 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA24700; Tue, 23 Dec 1997 23:44:53 +0100 Date: Tue, 23 Dec 1997 23:44:53 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA24718 Received: (qmail 12496 invoked from network); 23 Dec 1997 14:44:50 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 23 Dec 1997 14:44:50 -0800 Message-Id: <34A0A105.3CED@ww-interlink.net> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu