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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 prime-space must include prime-factor 2 in order for the vapro (or promo) to appear as a line; in the fairly common usage of lattices which do not show 2 as a dimension, all the vapros/promos would appear as a single point. The vapro's GCD, like that of all promos, is 1.
If the interval which may be 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 (which includes 2 as a dimension), 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.
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