Positive Systems

John Pusey: I apologize for my own errors and would like to stipulate
that we use Bosanquet's definitions. Positive fifths are thus those
that are sharper than 700 cents, negative, less than 700 and positive
systems are those with positive fifths, etc.

I think that J. M. Barbour was so committed to 12-tet that it is
not surprising that he sometimes thought of the perfect fifth as
having 700 cents. Thus, I view his definition of positive as sharper
than 3:2 rather than 7 degrees of 12-tet as a lapse, not a fundamental misunderstanding. BTW, he did grudgingly admit that a piece by Bach
sounded magnificent in the 1/6th comma "meantone" tuning (notes to the
Musurgia series of didactic recordings he and Kuttner put put in the
60's).

As for "doubly positive," see pages 60-63 of Bosanquet. _"A regular cyclic
system is of the rth order, negative or positive, if 12 fifths fall short
or exceed 7 octaves by r degrees of the system"_. Hence for 22-tet,
12 x 13 = 156 (12 x the 22-tet fifth) is two degrees of 22-tet sharper
than 7 x 22 =154. Therefore, 22-tet is a doubly positive system.

R is thus the difference between the diatonic (5 Fifths) and chromatic
(7 Fifths) semitones in ET degrees or alternatively, the difference between
B# and C if the cycle is started on C. In other words, r is the number of
degrees in the Pythagorean ditonic comma in each ET (see Paul Rapoport's
article "The Notation of Equal Temperaments," in XH16, 1995).

Another parameter useful for characterizing ET's is the number of degrees
in the apotome or Pythagorean chromatic semitone (the #) (Rapoport, XH16,
1995), i.e., 7 Fifths minus 4 Octaves. This quantity is the number of
degrees between C# and C, 1 in the case of 12 and 19-tets, 2 in 17, 24
and 31-tets, and 3 in 22, 29, 36 and 50-tets. To my knowledge there is
no accepted name for this number as a parameter (Erv Wilson simply calls
the systems singulary, binary, ternary, etc. as the number is 1, 2, or 3,
etc., but it is essential to fitting various ET's to the Bosanquet or
multiple Bosanquet keyboards or rationally assigning accidentals (Wilson,
XH2, 1974; Rapoport, XH16, 1995).

Of course, this discussion presumes 12-tet is the standard system. If one
were to base one's classification on 19-tet, for example, then "r" would
be calculated relative to the 19-tet fifth (19 Fifths -11 Octaves) and
the systems would be arranged according to their ditonic commas (12 Fifths
- 7 Octaves). Similarly, one might chose 17, 22, 31 or even 5 and 7
(Wilson, XH3, 1975; XH1, 1974) as bases for notation and keyboards.
With less historical and musical justification, one might choose cycles
of other intervals than the 4th or 5th such as 5/4, 7/4 and a low number ET
which approximates them well. One could even use some tempered interval like
8 or 10 degrees of 13-tet (and higher homologs) to define r and the
(pseudo) chromatic semitone.

--John

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