source file: mills3.txt Date: Tue, 23 Dec 1997 14:43:21 +0100 Subject: RE: Gibson digest 1274 From: "Paul H. Erlich" > >}Gregg Gibson said: >}> >>The ideal value for the octave itself is of course 1200/19 =3D = 63.16 ; >}> >>here are the values for the other six consonances. >}> >>3:2 701.96/11 =3D 63.82 >}> >>4:3 498.04/8 =3D 62.26 >}> >>5:4 386.31/6 =3D 64.39 >}> >>5:3 884.36/14 =3D 63.17 >}> >>6:5 315.64/5 =3D 63.13 >}> >>8:5 813.69/13 =3D 62.59 >}> > >}> >>If one considers the consonances to be all of equal importance, = one can >}> >>simply take the average of the above seven figures, which is = 63.22, to >}> >>arrive at a figure for the octave of 1201.2. > >}Paul Ehrlich said: >}> >Gregg, this is not correct. You are not giving equal importance to = the >}> >intervals above. Since whatever compromise is made in taking the = average >will >}> >be multiplied 14-fold for the 8:5 but only 5-fold for the 6:5, you = are >being >}> >more permissive for tuning errors in the larger intervals. I am = certain >}> >Fokker would not have made this mistake, being a brilliant = physicist. I >know >}> >you will probably gloss over this post the way you have dismissed, = with >total >}> >lack of understanding, my other comments. But if you are = interested, I >will >}> >tell you the other things wrong with your calculations here, and = how I >}> >arrived at an optimal octave of 1202.7 cents for 19-equal. > >}you presume to assert that I have "glossed over" > >You have -- for instance, what is your response to my question as to = why >26-equal shouldn't be considered as a diatonically usable temperament? = What >is your response to my question as to why just intonation is far = superior to >53-equal? > >}and dismissed >}your posts with "total lack of understanding" > >You have and continue to do so -- here's evidence: > >}I advise you to study the cycle of 22-tone equal consonant fifths. See >}if you can find a consonant third, four fifths above the tonic. If you >}cannot (and you cannot) your arguments fall to the ground. I repeat = that >}when I was a beginner, I made the same mistake you are making.=20 > >which proves that you don't have an inkling of what my "arguments" are, >although I have evidently presented them with enough clarity so that = Graham >Breed, Paul Hahn, and Steven Rezsutek have understood them. > >}1. Fokker's method is as I have stated. Read the references for >}yourself. If you do not have access to Fokker's original texts, use >}Mandelbaum. > >I have not been able to gain access to Mandelbaum's writings, or to the >particular ones of Fokker in question. Believe me I have tried to read = every >shred of tuning-related literature, including the Yale, Harvard, NYU, = and NY >Public libraries, partly in an attempt to see if anyone had discovered = the >decatonic scales before me. Surprisingly, no one has, as far as I know. >However, since the issue is mathematical, not historical, we need not = appeal >to the literature. If Fokker was using a convenient approximation to >illustrate a point, so be it, but we can still do better. I have other >disagreements with Fokker, such as his use of square rather than = triangular >lattices, as well as his use of 31-equal to represent ratios of odd = numbers >(such as 13) that are not consistently expressed by the tuning. >=20 >}2. Fokker could have used _no_ method to achieve equal weighting of >}deviations of the consonances. > >Excuse me, but did you not say in the above that you were treating the >consonances as equally important, and therefore taking an average of = the >figures? > >}The most _basic_ principle of >}temperament, which is taught to all _beginners_ in this discipline, is >}that there can be _no_ temperament which compromises the consonances >}equally. > >Of course, one cannot make all the errors equal, but one can make them >equally important in determining a temperament. > >}This arises from the mathematics of the case. _Learn_ >}mathematics before you make such statements, which again, only show = your >}ignorance. > >}Now that I have administered the bitter pill, let me add a large >}tablespoon of sugar. You are quite bright. You were led into this = faulty >}conclusion by assuming that the consonances are independent, and so, >}vary independently when tempered. I use non-mathematical language here >}deliberately. But in fact the consonances consist of three pairs, 5:4 = & >}8:5, 6:5 & 5:3, and 3:2 & 4:3, each of which pair varies as if it were = a >}single interval, so far as temperaments are concerned. Therefore, to >}seek some system which shall weight deviations from the consonances >}equally, even in the case in which the octave is itself tempered, is = the >}classic pons asinorum of temperament, for no such system can exist. > >Again, _you_ were the one who attempted to weight the deviations = equally, >which is a perfectly fine thing to attempt to do. I simply pointed out = that >you did not do so correctly. If you wish, I will go through the math = and show >that you did not find the optimal octave for your initial example. Your = error >propagates through your other refinements and the fact that we arrive = at a >similar value for the optimal 19-tone octave (1202.7 cents) is either = sheer >coincidence or evidence that both of us were listening. > >}Finally, let me observe that the value of 1202.7 cents for the octave, >}while apparently very precise, and possibly indicative that one person >}may have copied the other without acknowledgement, > >That is so absurd as to be laughable. I calculated that number for the = first >time just before I posted it, and I clearly indicated its derivation, = which >again is quite different than yours.=20 > >}> > Gregg Gibson is posting such a lot of highly opinionated material, >}> > it is difficult not to join in the argument. >bil said: >}> Or wish my mailer had a filter function -- such a flood of junk. > >}For those who have been eager to attack me as uncivil, I give this as = an >}example of true incivility. It is not uncivil to say a tuning is = without >}value. > >Imagine telling an oboist that the oboe is a worthless instrument. If = that >were a civil thing to say, you wouldn't need to protect yourself from = an >incoming double reed. To describe yourslef as "meekly humble" only = shows that >you are hiding behind the impersonal screen of the Internet while = firing off >your insulting material. Evidently you find my tone objectionable as = well -- >I'd be happy to just discuss tuning and not make personal remarks, if = only >you'd attempt to understand, rather than dismiss as beneath your >consideration, the counterarguments that I and others offer. > >}As I have abundantly shown in previous posts, this temperament (and >}analogous ones such as 29- 41- & 53-tone equal,) is impossible for the >}diatonic, as well as for the chromatic modes (see the four chromatic >}modal genera or scales I have contributed; these include several of = the >}forms of the 'minor' of the West) which obviously require thirds and >}sixths vis-=E0-vis the tonic, it being granted that the fifth and/or >}fourth must also be present.=20 > >On the contrary, you have not shown anything, while I have shown that = three >of your four "chromatic" genera are indeed adequately represented in >non-meantone temperaments. > >}The case of the enharmonic is less clear-cut, and requires more = detailed >}consideration. For after all, one might ask, surely the divers >}enharmonic modes have fewer thirds or sixths vis-=E0-vis the tonic, = and so >}the inconsistency between the fifth cycle and those of the thirds = would >}appear to be less troublesome; in certain enharmonic modes it might >}never arise at all.=20 > >That's right -- just as it never arises in three of your four = "chromatic" >genera. > >}For example, take one form of the enharmonic: >}C C# Db F G G# Ab C (not the 22-tone equal notation) >}or in 22-tone equal: >}0 55 109 491 709 764 818 1200 > >}There is a consonant major third between Db & F, and another between = Ab >}& C, and a consonant minor third between F & Ab.=20 > >}In reality, of course, the notation of the 22-tone differs from the >}above notation: >}C Db B# F G Ab Fx C >}which BTW is not a mere artefact of notation, but on the contrary >}explains very succinctly the utterly non-diatonic and non-chromatic >}nature of this system, > >Which is why I would use a decatonic rather than heptatonic notation = for >22-equal. > >}The effect of this particular Greek enharmonic mode - the most famous = of >}them as I recollect - is in 22-tone not wholly dissimilar to its = effect >}in 19-tone equal, except that the latter is much more singable because >}of the wider 63.3 cent interval. > >But which is closer to the ancient Greek tuning of this mode? > >}Other enharmonic modes in 22-tone equal such as: >}C D D# F G G# Ab C (not 22-tone equal notation) >}0 218 273 491 709 764 818 1200 >}have problems however. Here D-F is dissonant, and this would tend to >}disagregate the already weak tonality of the enharmonic. If D is = lowered >}to make D-F consonant, D-G becomes dissonant. This is similar to the >}break in the fifths that besets just intonation itself. This is not a >}trivial matter; it is very noticeable to the ear. > >Does this mode have any practical or theoretical importance? > >}The difference of 9 cents between >}54.5 and 63.3 cents is very important, because it occurs near the >}melodic threshold or limen of perception. > >So if you go down 54 cents, and then up 63 cents, you've gone down an >imperceptible interval, and then up a perceptible interval, so you're = up a >perceptible interval. Therefore, 9 cents is a perceptible melodic = interval by >your logic. You see why we don't think your limen makes much sense? = Context >and training are far more important for what can or cannot be = distinguished >in melody than any precise cents value. SMTPOriginator: tuning@eartha.mills.edu From: "Paul H. Erlich" Subject: Reply to Bob Lee PostedDate: 23-12-97 14:45:34 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 14:43:21-23-12-97 14:43:22,23-12-97 14:42:53-23-12-97 14:42:53 DeliveredDate: 23-12-97 14:42:53 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 C1256576.004B5F64; Tue, 23 Dec 1997 14:45:12 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA24159; Tue, 23 Dec 1997 14:45:34 +0100 Date: Tue, 23 Dec 1997 14:45:34 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA24312 Received: (qmail 9976 invoked from network); 23 Dec 1997 05:44:33 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 23 Dec 1997 05:44:33 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu