Positively confused

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



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