source file: mills2.txt Date: Fri, 1 Mar 1996 12:20:44 -0800 Subject: Positively confused From: xen@tiac.net (J. Pusey) There seems to be some confusion about the definition of the term "positive system" (and, by extension, "negative system"). I've been doing quite a lot of tuning-related reading recently, yet have been encountering conflicting definitions for these concepts. According to R. H. M. Bosanquet in "An Elementary Treatise on Musical Intervals and Temperament," (1876, [1987 reprint]): "Fifths are called positive if they have positive departures, i.e. if they are greater than E. T. fifths; they are called negative if they have negative departures, i.e. if they are less than E. T. fifths. Perfect fifths are more than seven semitones; they are therefore positive. Systems are said to be positive or negative according as their fifths are positive or negative." (p. 60) By "E. T. fifths" he means 700 cent 12-tET fifths, of course. Similarly, "seven semitones" means seven 100 cent 12-tET semitones. This would appear to be rather clear. However, J. Murray Barbour can't seem to make up his mind in "Tuning and Temperament: A Historical Survey" (1951, [1972 reprint]). Right up front in the Glossary (p. ix), he defines a positive system as: "A regular system whose fifth has a ratio larger than 3:2." Similarly, he defines a negative system as: "A regular system whose fifth has a ratio smaller than 3:2." This, of course, is at odds with Bosanquet's definition. Barbour then goes on in the main text, when describing multiple divisions of the octave, to say: "The 34-division is a positive system, like the 22-division. That is, its fifth of 706 cents is larger than the perfect fifth ..." (p. 121). This is consistent with his definition of positive system in the glossary, although it is not at all clear whether he is using the term "perfect fifth" to refer to the pure ratio of 3/2 (701.955 cents) or to its 12-tET approximation of 700 cents. However, a few pages later, he describes 53-tET as "a positive system, with fifths sharper than those of equal temperament ..." (p. 124). The 53-tET fifth is 701.887 cents, which is slightly *smaller* than 3/2 (701.955 cents), so we must conclude that his definition in the glossary is incorrect. Unfortunately, the confusion has not been limited to Barbour. In his article, "Cents and Non-Cents: Logarithmic Measures of Musical Interval Magnitude" in Xenharmonikon 15, our own John Chalmers writes: "A negative system is one whose fifths are less than the 3/2 perfect fifth of 701.9550009 cents." (p. 82) Fortunately, John C. has since corrected his error in the "Notes & Comments" section of XH 16 (p. 2). Ironically, several pages later in this very same issue of XH, Paul Rapoport, in his otherwise excellent article "The Notation of Equal Temperaments," makes the same mistake, I believe, when he writes: "In many positive temperaments, i.e. where v is greater than the just perfect fifth (701.955 cents), k is the obvious choice of komma to use for notation, ..." (p. 66). He uses the variable "v" to represent the equal-tempered interval most closely approximating the just perfect fifth. I believe Bosanquet to be the final authority in this matter -- is he indeed the originator of the terms "positive system" and "negative system"? However, the question remains: Why is there so much confusion over these terms? Now, what in blazes is a "doubly positive system"??? John --- John G. Pusey xen@tiac.net Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 1 Mar 1996 22:08 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id NAA11535; Fri, 1 Mar 1996 13:08:07 -0800 Date: Fri, 1 Mar 1996 13:08:07 -0800 Message-Id: <313766DD.267F@dial.pipex.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu