# bipromo

[Joe Monzo, with Paul Erlich]

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:

• the syntonic-comma (ratio 81/80, monzo [ -4 4, -1 > ) and all of its multiples and submultiples:
```                           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

• the enharmonic diesis (ratio 128/125, monzo [ 7 0, -3 > ) and all of its multiples and submultiples:
```                           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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