Bel(R) is a simplified form of the Bel notation described in:

• Bernard Bel. "Time and musical structures". Interface (Journal of New Music Research) Volume 19, Issue 2-3, 1990. (http://hal.archives-ouvertes.fr/hal-00134160)
• Bernard Bel. "Two algorithms for the instantiation of structures of musical objects". Centre National de la Recherche Scientifique, 1992. GRTC 458 (http://www.lpl.univ-aix.fr/~belbernard/music/2algorithms.pdf)

For patterns without tempo indications, the two notations should give equivalent phase diagrams, for instance (Bel 1990, §11, p.24):

> bel_ascii_pp "ab{ab,cde}cd"

Bel(R): "ab{ab,cde}cd", Dur: 7

a _ b _ a _ _ b _ _ c _ d _
        c _ d _ e _        

and:

> bel_ascii_pp "{a{bc,def},ghijk}"

Bel(R): "{a{bc,def},ghijk}", Dur: 5

a _ _ _ _ _ _ _ _ _ b _ _ _ _ _ _ _ _ _ _ _ _ _ _ c _ _ _ _ _ _ _ _ _ _ _ _ _ _
                    d _ _ _ _ _ _ _ _ _ e _ _ _ _ _ _ _ _ _ f _ _ _ _ _ _ _ _ _
g _ _ _ _ _ _ _ h _ _ _ _ _ _ _ i _ _ _ _ _ _ _ j _ _ _ _ _ _ _ k _ _ _ _ _ _ _

The Bel notation allows n-ary parallel structures, ie. {a_bcd_e,a_f_gh_,ji_a_i_} (Bel 1992, p.29), however Bel(R) allows only binary structures. The parallel interpretation rules are associative:

> bel_ascii_pp "{a_bcd_e,{a_f_gh_,ji_a_i_}}"

Bel(R): "{a_bcd_e,{a_f_gh_,ji_a_i_}}", Dur: 7

a _ b c d _ e
a _ f _ g h _
j i _ a _ i _

Bel(R) does allow unary parallel structures (see Iso), which can be used to isolate tempo changes:

> bel_ascii_pp "ab{*2cd}ef{*2/3gh}ij"

Bel(R): "ab{*2cd}ef{*2/3gh}ij", Dur: 10

a _ b _ c d e _ f _ g _ _ h _ _ i _ j _

Patterns with tempo indications have completely different meanings in Bel and Bel(R), though in both cases parallel nodes delimit the scope of tempo markings.

Bel(R) replaces the /n notation for explicit tempo marks with a *n notation to indicate a tempo multiplier, and a set of bracketing notations to specify interpretation rules for parallel (concurrent) temporal structures.

The tempo indication /1 in the expression ab{/1ab,cde}cd (Bel 1990, p.24) requires that the inner ab have the same tempo as the outer ab, which is implicitly /1. Setting the tempo of one part of a parallel structure requires assigning a tempo to the other part in order that the two parts have equal duration. Here the tempo assigned to cde is /1.5, but since fractional tempi are not allowed the expression is re-written as /2ab{/2ab,/3cde}/2cd.

Importantly the explicit tempo indications make it possible to write syntactically correct expressions in Bel that do not have a coherent interpretation, ie. {/1ab,/1cde}. Determining if a coherent set of tempos can be assigned, and assigning these tempos, is the object of the interpretation system.

In comparison, all syntactically valid Bel(R) strings have an interpretation. The expression {*1ab,*1cde} is trivially equal to {ab,cde}, and tempo marks in parallel parts do not interact:

> bel_ascii_pp "{a*2b,*3c/2d/3e}"

Bel(R): "{a*2b,*3c*1/2d*1/3e}", Dur: 3

a _ _ _ _ _ b _ _
c d _ e _ _ _ _ _

Here a is twice the duration of b, and e is three times the duration of d, which is twice the duration of c (in Bel(R) /n is equivalent to *1/n). The duration of any Bel(R) expression can be calculated directly, given an initial Tempo:

bel_dur 1 (bel_char_parse "a*2b") == 3/2
bel_dur 1 (bel_char_parse "*3c/2d/3e") == 3

Therefore in the composite expression the left part is slowed by a factor of two to align with the right part.

The Bel string ab{/1ab,cde}cd can be re-written in Bel(R) as either ab~{ab,cde}cd or ab(ab,cde)cd. The absolute tempo indication is replaced by notations giving alternate modes of interpretation for the parallel structure.

In the first case the ~ indicates the opposite of the normal rule for parallel nodes. The normal rule is the same as for Bel and is that the duration of the whole is equal to duration of the longer of the two parts. The ~ inverts this so that the whole has the duration of the shorter of the two parts, and the longer part is scaled to have equal duration.

In the second case the parentheses () replacing the braces {} indicates that the duration of the whole is equal to the duration of the left side, and that the right is to be scaled. Similarly, a ~ preceding parentheses indicates the duration of the whole should be the duration of the right side, and the left scaled.

> bel_ascii_pp "ab~{ab,cde}cd"

Bel(R): "ab~{ab,cde}cd", Dur: 6

a _ _ b _ _ a _ _ b _ _ c _ _ d _ _
            c _ d _ e _            

There is one other parallel mode that has no equivalent in Bel notation. It is a mode that does not scale either part, leaving a hole at the end of the shorter part, and is indicated by square brackets:

> bel_ascii_pp "ab[ab,cde]cd"

Bel(R): "ab[ab,cde]cd", Dur: 7

a b a b   c d
    c d e    

The Bel string /2abc/3de (Bel 1992, p.53) can be written as *2abc*1/2*3de, or equivalently as *2abc*3/2de:

> bel_ascii_pp "*2abc*3/2de"

Bel(R): "*2abc*3/2de", Dur: 13/6

a _ _ b _ _ c _ _ d _ e _

It can also be written using the shorthand notation for rest sequences, where an integer n indicates a sequence of n rests, as:

> bel_ascii_pp "(9,abc)(4,de)"

Bel(R): "(---------,abc)(----,de)", Dur: 13

- - - - - - - - - - - - -
a _ _ b _ _ c _ _ d _ e _

In the Bel string {ab{/3abc,de},fghijk} (Bel 1992, p.20) the tempo indication does not change the inter-relation of the parts but rather scales the parallel node altogether, and can be re-written in Bel(R) notation as:

> bel_ascii_pp "{ab*3{abc,de},fghijk}"

Bel(R): "{ab*3{abc,de},fghijk}", Dur: 6

a _ _ _ _ _ b _ _ _ _ _ a _ b _ c _
                        d _ _ e _ _
f _ _ g _ _ h _ _ i _ _ j _ _ k _ _

Curiously the following example (Bel 1990, p. 24) does not correspond to the phase diagram given:

> bel_ascii_pp "{i{ab,cde},jk}"

Bel(R): "{i{ab,cde},jk}", Dur: 4

i _ a _ _ b _ _
    c _ d _ e _
j _ _ _ k _ _ _

The paper assigns tempi of /6 to both i and ab, which in Bel(R) could be written:

> bel_ascii_pp "{i~{ab,cde},jk}"

Bel(R): "{i~{ab,cde},jk}", Dur: 3

i _ _ _ _ _ a _ _ _ _ _ b _ _ _ _ _
            c _ _ _ d _ _ _ e _ _ _
j _ _ _ _ _ _ _ _ k _ _ _ _ _ _ _ _