THE SECRET LIFE OF THE BECUADRO SIGN
We are familiar with the becuadro, our "natural sign" erasing the functions of the flat and sharp signs in the notation of our modern music, and sending it on the white keys of the piano. In fact it is doing what it had always done before in pre-modal music of the Renaissance, to cue the reading of music by solmization and not necessarily affect the pîtch of a specific note. In the medieval gamut we find the becuadro sign attached to the letter B, for the note B-natural, and even in some illustrations in place of the letter itself; but apparently the sign had a function of triggering mental transpositions which make up the secret life of the sign, as we shall see in the chanson Adieu m'amour et ma maistresse by Gilles Binchois in Oxford Canonici Misc., 213, fol. 86.
Click Here for Binchois' Chanson Musical Example
With the B-flat sign in the signature of the two lower voices, the Tenor and the Contratenor, the octave f-ff of the flat scale bisected by the note c can be seen at many places and especially in the last cadence of the piece. Soon at letter [A] the music modulates on the relative sharp side of the system with the three voices descending by a fifth. A becuadro sign inserted before the note g cued a sol of G-ut for the hexachordal octave C-g-c in the first cadence in m. 3, for the Ionian mode in regular position.
The next cadence and the one after are in the Ionian mode once flat(f-g-a-bf-c-d-e-ff) in the hexachordal octave C-f-c-ff at letters [B] and [C]. But at letter [C] there is a modulation to octaves C-g-c-gg and g-c-gg for a cadential stop, re-la/mi-la in Hypodorian (a-bf-c-d-e-f-g-aa) at letter [D]. It is obvious, here in the Tenor of m. 9, that the octave C-g-c ascends by a fourth to the g-Hexachord negating a flat hexachord in the other voices at the cadence of letter [D]. From this point there is a modulation leading to a cadence in Hypodorian once flat that was cued to the singers by the becuadro signs inserted in the Tenor and the Cantus to indicate a mental transposition to an upper fifth. The becuadro sign on the line of the note c between the notes c and d cued the solmization fa-mi-la-sol-la-fa-la for the descent from the c-Hexachord to the f-Hexachord leading to letter [E] in the Cantus. Likewise the becuadro sign under the note B, in the staff in the Tenor, cued a mental transposition by an upper fifth giving the descending sol-fa-la-sol-la fourth in the f-Hexachord leading to the cadence at letter [E]. This is a logical transformation of the Hypodorian passage heard in mm. 9 to 12 imitated at a fifth lower in m. 12. The leap down by a fifth in the Tenor, on the first beat of m. 14, re-establishes the C-g-c-gg octaves followed by the cadence of the Ionian mode in regular position at letter [F], shortly followed by the final cadence of the chanson in the Ionian mode once flat.
The "secret becuadro signs" cueing a mental transposition above---the analysis of which seems to have avoided the eye of modern transcribers---would have been superfluous to seasoned professional singers able to "read" the F-Hexachord and use solmization by the three hexachord Guidonian hand. Although rarely seen in sources, the becuadro sign meant to cue a transposed solmization to avoid “reading” the F-Hexachord was not peculiar to Binchois’ chanson and we find it in two examples of virelais attributed to Guillaume de Machaut a century earlier, and illustrated in the Davidson & Apel Historical Anthology of Music, No. 46
In this chanson by Binchois we find an interesting mélange of Hypolydian and Hypoionian modes, with the Hypolydian octave fa-fa-ut in the Contratenor from m. 11 to m. 15, and the Hypoionian fifth sol-fa-ut in the Tenor from m. 1 to 4, and especially in the Contratenor from m. 16 to the end.
This chanson is also very clever in that it modulates on two levels from the natural scale to its transposition on the flat side of the system, and it contrasts melodically the ut-sol and fa-fa fifths of the Hypolydian and Ionian modes at cadential points [A][B][C][F] and last cadence, with the re-la fifth of the Dorian mode at letters [D] and [E].
There are four wholetones in a hexachord, the melodic unit for the notation of music in the Middle Ages and the Renaissance, but since there is only one semitone, it was logical to mark it by the becuadro sign. Since there are two semitones in an octave of the natural scale, the mi-fa of the
fundamental tetrachord G-A-B-C and the la-fa semitone of the mutation tetrachord C-D-E-f of the same scale, the same becuadro sign was used to cue them in the solmization by the early two hexachord Guidonian hand. Because the mutation tetrachord can become a fundamental tetrachord and vice versa, the natural scale can be transposed on the flat side of the system by starting it on C-ut, F-ut, and Bf-ut and also on the sharp side of the system starting from D-ut and A-ut. Thus the "invention" of a three hexachord Guidonian hand made it possible through transpositions by shifting mental clefs, to "read" just as many transposed natural scales plus the B-flat fa-supra-la of the C-ut scale as needed.
This is what happens to the natural scale transposed up by a fourth featuring a b-flat cued in by a becuadro in six places of L'antefana by Lorenzo Masini (GB. Lib. Add. 29987, fol. 55r). Similarly the becuadro sign facing the note d in the Motetus of Machaut's Kyrie I of Missa Notre Dame in the source Fonds français 1585, measure 26, cued the syllable of solmization fa for the mi-fa of A-ut, while this happens in many places of that source as well as Fonds français 9221 of that Mass. It is important to note that other sources, like 1584 and 22546 of the same work, have the becuadro placed for the mi of the tetrachord. Characteristic of the solmization by the two hexachord Guidonian hand are the superimposed becuadros at a fourth or a fifth from one another in superimposed voices of the polyphonic web. Those forming a tritone fa under mi certainly cued a harmonic division of the octave for the listening singers, just as a diminished fifth mi under fa cued an arithmetic division of the octave.
Click Here for Machaut's Contrasting Cadences