A bipromo represents an infinite 2-dimensional lattice of monzos.
A "vanishing bipromo" represents an infinite 2-dimensional lattice of unison-vectors which vanish in a certain temperament or family of temperaments.
The unison-vectors may be modeled geometrically as points lying on parallel lines in a plane. These lines represent the two independent promos which comprise the bipromo.
For example:
When the familiar 12-tone equal-temperament (12-et) is intended to represent the 5-limit lattice of JI pitches and intervals, two of the many unison-vectors which vanish are:
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.
and
2,3,5-monzo ratio ~cents
etc.
[ 7 0, -3 > * 3 = [ 21 0, -9 > 2097152/1953125 123.1765752
[ 7 0, -3 > * 2 = [ 14 0, -6 > 16384/15625 82.11771681
[ 7 0, -3 > * 1 = [ 7 0, -3 > 128/125 41.05885841
[ 7 0, -3 > * 0 = [ 0 0, 0 > 1/1 0
[ 7 0, -3 > * -1 = [ -7 0, 3 > 125/128 -41.05885841
[ 7 0, -3 > * -2 = [-14 0, 6 > 15625/16384 -82.11771681
[ 7 0, -3 > * -3 = [-21 0, 9 > 1953125/2097152 -123.1765752
etc.
Since the [ -4 4, -1 > and [ 7 0, -3 > promos are independent, and their wedge product has a GCD of 1, adding together all of their integer multiples (negative and positive) generates all of the unison-vectors which vanish in 12-et. Some examples:
2,3,5-monzo ratio name
[ -4 4, -1 > * 2 + [ 7 0, -3 > * -1 = [-15 8, 1 > = 32805:32768 skhisma
[ -4 4, -1 > * -1 + [ 7 0, -3 > * 1 = [ 11 -4, -2 > = 2048:2025 diaschisma
[ -4 4, -1 > * 3 + [ 7 0, -3 > * -1 = [-19 12, 0 > = 531441:524288 pythagorean-comma
[ -4 4, -1 > * 1 + [ 7 0, -3 > * 1 = [ 3 4, -4 > = 648:625 major diesis
etc.
(A large selection of these unison-vectors can be seen on the 12-et bingo-card lattice.)
These unison-vectors and all their integer multiples form a 2-dimensional lattice of unison-vectors which may be modeled geometrically as points lying on a plane.
The two points in that lattice which lie closest in prime-space to 1/1 and have a positive pitch-height, are those two listed above which have coefficients of +1. Thus, the bipromo for 5-limit 12-et is designated by the bimonzo [[ 28 -19 12 >> which is calculated from the two promos.
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