A "projective monzo" (ratio), which represents an entire infinite set of monzos, which are all integer multiples of the monzo used to define the promo, and which may be characterized geometrically as a line in prime-space. The promo's GCD = 1.
If the interval which is used to represent the promo is not tempered out, then it is simply a unison-vector and not a promo: the essential fact about a promo is that it physically vanishes.
The pair of monzos closest to the origin-point on the lattice (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. For example:
|
The most typical promo is probably [ -4 4, -1 >, which is the monzo of the syntonic-comma of ratio 81/80, which vanishes in all temperaments belonging to the meantone family. This "vanishing promo" represents all of the following "vanishing monzos":
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 [ -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 promo. All temperaments in which any of these intervals vanishes belong to the meantone family; thus, since the [ -4 4, -1 > promo represents all of them, it is part of the definition of all meantone tunings, and of the entire meantone family. (for further discussion of this particular promo, see "tempering out the syntonic-comma".) |
For sets of more than one promo (for example, temperaments which temper out two or more promos), a prefix is added before "promo" to denote the dimensionality of the set.
Thus, a bipromo represents a 2-dimensional lattice of monzos; a tripromo represents a 3-dimensional lattice of monzos; etc.
The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.