  # negative system

[Joe Monzo]

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

Manuel Op de Coul suggested that the tuning's p-rating be included here, where p represents the Pythagorean comma in terms of number of that temperament's scale degrees:

Yahoo tuning-math, message 2580 (Fri Jan 11, 2002 5:08 am):

If v is the size of the fifth, and a is 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.

Some negative systems, listing successively closer equivalents:

```tuning  p       tempering of "5th"

19-edo  -1  ==  -1/3-comma
-((25+x)/(74+3x))-comma, x = (0...24)

31-edo  -1  ==  -1/4-comma
-((4+x)/(17+4x))-comma, x = (0...4); result: -4/17-, -5/21-, -6/25-, -7/29-comma
-13/54-comma

43-edo  -1  ==  -1/5-comma
-((21+x)/(106+5x))-comma, x = (0...20)

50-edo  -2  ==  -2/7-comma
-3/11-comma
-5/18-comma

53-edo  +1  ==  -(1/(158+x))-comma, x = (0...157); best result: -1/315-comma
(emulates both pythagorean and 5-limit just intonation tunings)

55-edo  -1  ==  -1/6-comma
-2/11-comma
-3/17-comma
-7/40-comma
-10/57-comma

65-edo  +1  ==  -1/52-comma
-2/103-comma
(emulates both pythagorean and 5-limit just intonation tunings)

67-edo  -1  ==  -1/6-comma
-((3+x)/(19+6x))-comma, x = (0,1); result: -3/19-, -4/25-comma
```

19-, 31-, 43-, 55-, and 67-edo and their equivalents, all with p = -1, and 50-edo with p = -2, are meantone systems.

53- and 65-edo, with p = +1, are approximations to just intonation.

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