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positive system

[Joe Monzo]

A system of temperament in which the "5th" is wider (larger) than the 12-edo "5th" of 700 cents.

(I had originally provided some examples here measured in terms of fraction-of-a-comma temperament. These have been removed in favor of the ones given below.)

. . . . . . . . .
[Paul Erlich, Yahoo tuning-math, message 2542 (Thu Jan 10, 2002 4:56 pm)]

Today, on these [internet tuning] lists, we tend to call negative systems "meantone" and positive systems "schismic". The reason 700 cents was chosen as the dividing line between "negative" and "positive" is that when the fifth is below 700 cents, the "meantone" (+4 fifths) approximation to the 5/4 is better than the "schismic" (-8 fifths) approximation to the 5/4. When the fifth is above 700 cents, the "schismic" approximation to the 5/4 is better than the "meantone" approximation to the 5/4. I might differ, saying that there is a "gray area", and also factoring the 6/5 into consideration . . . but the definitions are well-established and there is no reason to favor ones which could breed potential contradictions.

... Positive systems should be characterized by the fraction of a schisma that the fifths differ from just -- this is the relevant measure of them. Knowing what fraction of a syntonic comma a positive system's fifth might have been increased by is irrelevant for understanding the functioning of the system, and is potentially misleading.

. . . . . . . . .

Manuel Op de Coul also suggested that I add the tuning's "p"-rating, where "p" represents the Pythagorean comma in terms of number of that temperament's scale degrees:

[Manuel Op de Coul, Yahoo tuning-math message 2580 (Fri Jan 11, 2002 5:08 am)]
If v is the size of the fifth, and a the size of the octave, then p = 12v - 7a. For example in 31-tET, v = 18 and a = 31, so p = -1.

A negative value indicates a negative temperament, and a positive value indicates a positive temperament. Values greater than one require an additional qualifier, as, for example, 2 designates "doubly positive", -3 is "triply negative", etc.

. . . . . . . . .
[Joe Monzo]

So taking Paul's advice and Manuel's suggestion, here is the listing of the same EDO "5ths", but giving this time their deviation from 3:2 in fractions of a skhisma, and also their "p" value:

tuning  p      tempering of "5th"

17-EDO  +1  ==  + 2 skhismas

22-EDO  +2  ==  + 3 & 1/3 skhismas
                + 3 & 15/23 skhismas

29-EDO  +1  ==  + 4/5 skhisma
                + 7/9 skhisma
                + 13/17 skhisma

39-EDO  +3  ==  + 2 & 15/16 skhismas

41-EDO  +1  ==  + 1/4 skhisma
. . . . . . . . .

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