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Encyclopedia of Microtonal Music Theory

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vapro

[Joe Monzo]

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.

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 coefficient of -1 produces [ 4 -4, 1 > = 80/81 = ~ -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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