source file: mills3.txt Date: Fri, 19 Dec 1997 19:07:22 +0100 Subject: RE: 22-tone equal & chromaticism From: "Paul H. Erlich" >}Gregg Gibson said: >}> >}but 22-tone >}> >}equal is not a temperament at all, but a mere tuning artefact that >}> >}reproduces the worst defects of just intonation. > >}Paul Ehrlich replied; >}> >But these "defects" (the non-vanishing of the syntonic comma) can only be >}> >seen as such in the context of diatonicism! Since we are speaking of >escaping >}> >diatonicism, the relevant properties of a tuning system will be quite >}> >different. > >}The failure of 22-tone equal to provide both consonant thirds and fifths >}has nothing to do with diatonicism; the presence of consonant melody and >}harmony is no less vital for the use of chromatic modes than for the use >}of diatonic ones; I assume you will agree if you consider the matter. If >}anything, because chromatic modes have but four (at most), not six >}degrees (as in the diatonic modes) on which consonant triads are >}erigible, chromatic modes in 22-tone equal would be even more >}disorganized than diatonic ones. For enharmonic modes the matter is less >}clear-cut, I concede. > >OK folks, switch to a constant-width font -- it's ASCII diagram time! > >We will represent the 3:2 ratio ("perfect fifths") by lines running left to >right. >We will represent the 5:4 ratio ("major thirds") by lines running from lower >left to upper right. >We will represent the 6:5 ratio ("minor thirds") by lines running from upper >left to lower right. > >Consonant triads will therefore be represented by isosceles triangles. Major >triads have the base of the triangle on the bottom; minor triads have the >base on top. > >First of all, let's look at the diatonic scale -- the white keys on a piano. >It has six consonant triads: > > >D---------A---------E---------B > \ / \ / \ / \ > \ / \ / \ / \ > \ / \ / \ / \ > \ / \ / \ / \ > F---------C---------G---------D > >Notice that "D" occurs twice. > >In 19-equal, the 3:2 is 11 steps, the 5:4 is 6 steps, and the 6:5 is 5 steps. >Setting the leftmost D=0, the diatonic scale is then > >0---------11--------3---------14 > \ / \ / \ / \ > \ / \ / \ / \ > \ / \ / \ / \ > \ / \ / \ / \ > 5---------16--------8---------0 > >Both instances of "D" are represented by "0." The diatonic scale "works" in >19-equal. > >In 22-equal, the 3:2 is 13 steps, the 5:4 is 7 steps, and the 6:5 is 6 steps. >Again setting the leftmost D=0, the diatonic scale becomes > >0---------13--------4---------17 > \ / \ / \ / \ > \ / \ / \ / \ > \ / \ / \ / \ > \ / \ / \ / \ > 6---------19--------10--------1 > >"D" is now represented by both "0" and "1". As most classical composers made >use of the fact that "D" or its equivalent in keys other that C major is one >and only one pitch, 22-equal is unusable for most classical music. Other >tunings unusable for most classical music are 15, 27, 34, 41, 53TET, and JI >-- these are the tunings where the syntonic comma (representing 81:80) does >not vanish, causing two different "D"s to arise at a very small interval from >one another. Tunings that are usable at least for the early stuff are 12, 19, >26, 31, 43, 50, and 55TET, as well as meantone tunings such as Kornerup's >Golden Tuning, LucyTuning, 1/4-comma meantone, 2/7-comma meantone, 3/14-comma >meantone, etc. > >Now let's look at Gregg Gibson's four "new" scales. They each contain four >consonant triads: > >D melodic minor ascending: > > C# > / \ > / \ > / \ > / \ >A---------E---------B > \ / \ > \ / \ > \ / \ > \ / \ > G---------D---------A > \ / > \ / > \ / > \ / > F > >This scale has the same step sizes as the diatonic scale. Gregg has made a >big deal about 12-equal not really representing the "chromatic" scales >adequately, since the minor third is conflated with the augmented second in >12-equal. This scale has no augmented second or any other non-diatonic steps, >so I presume the term "chromatic" applies to the other three: > >A harmonic minor: > > G# > / \ > / \ > / \ > / \ >D---------A---------E---------B > \ / \ / > \ / \ / > \ / \ / > \ / \ / > F---------C > > >C harmonic major: > > E---------B > / \ / \ > / \ / \ > / \ / \ > / \ / \ >F---------C---------G---------D > \ / > \ / > \ / > \ / > Ab > >E major Gypsy: > > G#--------D# > / \ / > / \ / > / \ / > / \ / > A---------E---------B > / \ / > / \ / > / \ / > / \ / >F---------C > >The reader is encouraged to work out these diagrams in 19- and 22-equal. Note >that in these last three scales, there are no notes that appear twice in the >diagram, and thus no potential for the scales not to "work" in any tuning >that has good 5-limit approximations, including 22-equal. The harmonic major >and harmonic minor scales have four step sizes in 22-equal but three in 12- >and 19-equal; if any tuning is avoiding "conflations" within these scales it >is 22-equal, though I prefer the greater uniformity of step sizes in 12- and >19-equal. For the last of these scales, the number of different step sizes is >the same in all three tunings (12, 19, 22) so it is really, really hard to >claim that it "works" in some but not all of these tunings. > >My interest in 22-equal has nothing to do with these scales. I use these >scales all the time in 12-equal (the last one is great for surf music) and >see no great advantage to justify the expense of moving to 19-equal (though >I'd be the first to pick up a 19-tone guitar, keyboard, and bass if they were >commercially available). In fact, two of my favorite scales do not work in >19-equal at all: > >C augmented (six consonant triads): > > G#--------D# > / \ / > / \ / > / \ / > / \ / > E---------B > / \ / > / \ / > / \ / > / \ / > C---------G > / \ / > / \ / > / \ / > / \ / >Ab--------Eb > >This is like the diatonic scale turned on its side. In 12-equal Ab and G# are >the same and Eb and D# are the same. In almost any other tuning, this scale >doesn't work; you get eight instead of six tones and some really small >intervals. 27-equal is a rare counterexample, but 27-equal does not support >the diatonic scale. > >C diminished (eight consonant triads): > >F#--------C# > \ / \ > \ / \ > \ / \ > \ / \ > A---------E > \ / \ > \ / \ > \ / \ > \ / \ > C---------G > \ / \ > \ / \ > \ / \ > \ / \ > Eb--------Bb > \ / \ > \ / \ > \ / \ > \ / \ > Gb--------Db > >This scale doesn't work unless F#=Gb and C#=Db, i.e., in almost any tuning >other than 12-equal. 28-equal is a rare counterexample which doesn't, >however, support the diatonic scale. > >So far all the scales discussed are well-represented in 12-equal. No other >tuning can represent them all, expect maybe one with a vast number of notes. >However, my interest was to see if one could expand consonant harmony to >include the 7-limit (Partch seemed confident that at least the 11-limit could >possess some measure of consonance) while maintaining a diatonic-like >structure. This entails throwing out the diatonic scale and all concepts >associated with it, such as classifying the consonances as "major thirds", >"perfect fifths", etc. -- the 3:2 is now a "perfect seventh" and the 5:4 and >6:5 are major and minor "fourths." > >To add the 7-limit to these diagrams, a note in the center of a major triad >will signify a 7 added to the 4:5:6 major triad to yield a 4:5:6:7 otonal >tetrad, and a note in the center of a minor triad will signify a 1/7 added to >the 1/6:1/5:1/4 minor triad to yield a 1/7:1/6:1/5:1/4 utonal tetrad. In >22-equal, the 7:4 is 18 steps, the 7:5 is 11 steps, and the 7:6 is 5 steps. >Here are the two decatonic scales in 22-equal: > >Symmetrical decatonic scale (8 consonant tetrads): > > 13--------4---------17 > / \ / \ / > / \ 8 / \ 21 / > / 2---------15--------6 > / / \ \ / / \ \ / / >6--/---\--19-/---\--10 / > / 13 \ / 4 \ / > / \ / \ / >17--------8---------21 > > >Pentachordal decatonic scale (6 consonant tetrads): > > 2 > / \ > / \ >0---------13--------4 > \ / / \ \ / \ > \ 17-/---\--8 / \ > \ / 2---------15 \ > \ / \ \ / / \ > 6------\--19-/------10 > \ / > \ / > 8 > >These scales have 10 tones and 2 step sizes, and many, many important >properties in common with the diatonic scale (see my paper). In most any >other tuning with good 7-limit approximations, such as 27-, 31-equal, and JI, >there will be more than 10 tones and more than 2 step sizes, including some >really small steps. These steps represent what are known in JI as the >septimal sixth-tone (49:48) and, in the case of the symmetrical decatonic >scale, the septimal comma (64:63). That these intervals vanish in 22-equal is >as essential for the functioning of the decatonic scales as the vanishing of >the syntonic comma is for the functioning of the diatonic scale. > >In 26-equal the septimal comma does not vanish but the septimal sixth-tone >does. So the pentachordal decatonic scale does have 10 tones: > > 2 > / \ > / \ >0---------15--------4 > \ / / \ \ / \ > \ 20-/---\--9 / \ > \ / 2---------17 \ > \ / \ \ / / \ > 7------\--22-/------11 > \ / > \ / > 9 > >However, there are three step sizes instead of two. Besides making it >melodically awkward, this fact makes modulation difficult. An interval which >is a "second" in one key can become a "third" in a neigboring key (3:2 away) >-- which is disturbing when one considers that the "thirds" are supposed to >represent consonances: a major "third" is 7:6, and a minor "third" is 8:7. >The scalar grammar falls apart. So even for the pentachordal decatonic scale, >both the septimal sixth-tone and the septimal comma must vanish. For neither >of the decatonic scales must the syntonic comma vanish, however. SMTPOriginator: tuning@eartha.mills.edu From: "Paul H. Erlich" Subject: Error in "Stretching the 19-tone Equal" PostedDate: 19-12-97 19:08:15 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: 19-12-97 19:06:06-19-12-97 19:06:06,19-12-97 19:05:42-19-12-97 19:05:42 DeliveredDate: 19-12-97 19:05:42 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 C1256572.00636E38; Fri, 19 Dec 1997 19:07:59 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA19711; Fri, 19 Dec 1997 19:08:15 +0100 Date: Fri, 19 Dec 1997 19:08:15 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA19737 Received: (qmail 12252 invoked from network); 19 Dec 1997 09:52:52 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 19 Dec 1997 09:52:52 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu