In musical tuning theory, a "vanishing promo" (or vanishing unison-vector), which represents an entire infinite set of vanishing monzos, which are all integer multiples of the promo, and which may be characterized geometrically as a line in prime-space. The vapro's GCD, like that of all promos, is 1.
If the interval which is used to represent the vapro is not tempered out, then it is simply a unison-vector and not a representation of a vapro: the essential fact about a vapro is that it physically disappears.
The pair of unison-vectors closest to the origin-point (monzo [ 0 0, 0 > and ratio 1:1) lie at equal distances from it, on either side of it, along the line. It is always the case that one of this pair of intervals will have a positive pitch-height and the other a negative pitch-height (measured, for example, in cents). By convention, tuning theorists generally use the positive one as the promo and vapro.
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For example: The most common and familiar vapro is [ -4 4, -1 >, which is the monzo of the syntonic-comma of ratio 81/80. This vapro represents all of the following unison-vectors:
2,3,5-monzo ratio ~cents
etc.
[ -4 4, -1 > * 3 = [-12 12, -3 > 531441/512000 64.51886879
[ -4 4, -1 > * 2 = [ -8 8, -2 > 6561/6400 43.01257919
[ -4 4, -1 > * 1 = [ -4 4, -1 > 81/80 21.5062896
[ -4 4, -1 > * 0 = [ 0 0, 0 > 1/1 0
[ -4 4, -1 > * -1 = [ 4 -4, 1 > 80/81 -21.5062896
[ -4 4, -1 > * -2 = [ 8 -8, 2 > 6400/6561 -43.01257919
[ -4 4, -1 > * -3 = [ 12 -12, -3 > 512000/531441 -64.51886879
etc.
On a prime-space lattice, all of these coordinates would be points which lie on a single line, both ends of which extend into infinity. The two unison-vectors which lie closest to the origin are those with coefficients of +1 and -1.
The coeffecient of 1 produces the monzo [ -4 4, -1 > = 81/80 = ~ 21.5062896 cents. The latter interval's pitch-height is negative, so [ -4 4, -1 > = 81/80, whose pitch-height is positive, is used to designate the vapro. All temperaments in which any of these intervals vanishes belong to the meantone family; thus, since the [ -4 4, -1 > vapro represents all of them, it is part of the definition of all meantone tunings. |
For temperaments which temper out two or more vapros, a prefix is added before "vapro" to denote the dimensionality of the set of vapros.
Thus, a bivapro represents a 2-dimensional lattice (plane) of vanishing unison-vectors; a trivapro represents a 3-dimensional lattice (cube) of vanishing unison-vectors; etc.