source file: mills3.txt Date: Sat, 20 Dec 1997 05:42:22 +0100 Subject: Reply to Paul Ehrlich From: Gregg Gibson Gregg Gibson said: > >>The ideal value for the octave itself is of course 1200/19 = 63.16 ; > >>here are the values for the other six consonances. > >>3:2 701.96/11 = 63.82 > >>4:3 498.04/8 = 62.26 > >>5:4 386.31/6 = 64.39 > >>5:3 884.36/14 = 63.17 > >>6:5 315.64/5 = 63.13 > >>8:5 813.69/13 = 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. Since you presume to assert that I have "glossed over" and dismissed your posts with "total lack of understanding" and since you compound your folly by seeking to assert that the great Fokker would have fallen into the gross error into which you have fallen, sir, I am compelled to explain to you that: 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. 2. Fokker could have used _no_ method to achieve equal weighting of deviations of the consonances. 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. 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. Were however, you to confine yourself to the narrower (but unfortunately, perfectly irrelevant) objective of finding a tempered octave which should deteriorate both of the members of each _pair_ of the consonances equally, either from just or from their 19-tone equal values, independent of the other two pairs, then, and then only, would your objection be valid. 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. Your assertion that the chromatic modes of 19-tone equal are just as well expressible in 12-tone equal, could never be made by anyone who has heard them, and extensively used them in music with any harmonic element. Even in melody, and even insofar as the diatonic modes are concerned, the 12-tone equal modes so deviate from the tonal fabric of just intonation that they are very evidently deteriorated. But in the chromatic modes, where the augmented tone is so much used, the difference is enormous. 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, actually follows from the decision to round the 19-tone equal tuning degree, 63.16 cents, upward by the nearest tenth of a cent. 63.2 cents gives an octave of 63.2 x 19 = 1200.8 cents. 63.3 cents gives an octave of 63.3 x 19 = 1202.7 cents. 63.4 cents gives an octave of 1204.6 cents. Back in the early 90's I settled on the value of 63.3 cents as the value which seems best when taken to the nearest tenth of a cent. Almost anyone who thought of tempering the 19-tone octave would be led to 1202.7 cents as the most eligible of the three values. I repeat that I have no interest in disputing priority of this discovery. The true credit, insofar as it belongs to anyone, belongs partly to Fokker, who to my knowledge at least, first suggested tempering the octave with the purpose of improving the other consonances. He therefore put the germ of this idea in my own mind, and it is my habit to make occasional reference to this as a kind of tribute. This is a very important discovery, for by it the 19-tone equal becomes virtually as smooth harmonically as the 31-tone equal. I bear you _no_ personal ill will. I myself _do_ sometimes make bad, terribly embarassing mistakes. We all do. The best defense against despising ourselves is to adopt an elaborately civil tone in all public discourse. SMTPOriginator: tuning@eartha.mills.edu From: Gregg Gibson Subject: Mandelbaum's Use of 31-tone Equal PostedDate: 20-12-97 05:51:47 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: 20-12-97 05:49:45-20-12-97 05:49:45,20-12-97 05:49:20-20-12-97 05:49:21 DeliveredDate: 20-12-97 05:49:21 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 C1256573.001A82DE; Sat, 20 Dec 1997 05:51:30 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA22468; Sat, 20 Dec 1997 05:51:47 +0100 Date: Sat, 20 Dec 1997 05:51:47 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA22436 Received: (qmail 12002 invoked from network); 19 Dec 1997 20:51:44 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 19 Dec 1997 20:51:44 -0800 Message-Id: <349BB0D2.4BDC@ww-interlink.net> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu