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represented by tones. The sounds of which these Hexachords are composed are sung, by the rules of this system, to the syllables ut, re, mi, fa, sol, la, the semitone falling always between the syllables mi and fa. But, in addition to this syllabic distinction, the notes of each entire octave are provided with alphabetical names, exactly similar to those now in use—A, B, C, D, E, F, G; and, these names being immutable, it follows, that, as the Hexachords begin on different notes, and constantly overlap each other, the same syllable is not always found in conjunction with the same letter. At this point arises the only complication with which the system is burthened—a complication so slight that it is well worth the student's while to master it, seeing that its bearing upon the treatment of the Ecclesiastical Modes, and the management of Real Fugue, is very important indeed. [See Real Fugue.]

The first, or Hard Hexachord (Hexachordon durum), begins on G, the first line in the bass: a note which is said to have been added, below the Greek scale, by Guido, who called it Γ (gamma), whence the word gamma-ut, or gamut:—

{ \override Score.TimeSignature #'stencil = ##f \clef bass \cadenzaOn g,1_"Γ" _\markup { \smaller \italic ut } a,_"A" _\markup { \smaller \italic re } b,_"B" _\markup { \smaller \italic mi } c_"C" _\markup { \smaller \italic fa } d_"D" _\markup { \smaller \italic sol } e_"E" _\markup { \smaller \italic la } }

The second, or Natural Hexachord (Hexachordon naturale), begins on C, the second space:—

{ \override Score.TimeSignature #'stencil = ##f \clef bass \cadenzaOn c1_"C" _\markup { \smaller \italic ut } d_"D" _\markup { \smaller \italic re } e_"E" _\markup { \smaller \italic mi } f_"F" _\markup { \smaller \italic fa } g_"G" _\markup { \smaller \italic sol } a_"A" _\markup { \smaller \italic la } }

On comparing these two examples it will be seen that the note which, in the first Hexachord, was sung to the syllable fa, is here sung to ut. Hence, this note, in the collective gamut, is called C fa ut. And the same system is followed with regard to all notes that occur in more than one Hexachord.

The third, or Soft Hexachord (Hexachordon molle), begins on F, the fourth line: and, in order to place the semitone between its third and fourth sounds, the note, B, must be made flat.

{ \override Score.TimeSignature #'stencil = ##f \clef bass \cadenzaOn f1_"F" _\markup { \smaller \italic ut } g_"G" _\markup { \smaller \italic re } a_"A" _\markup { \smaller \italic mi } bes_\markup { B\flat } _\markup { \smaller \italic fa } c'_"C" _\markup { \smaller \italic sol } d'_"D" _\markup { \smaller \italic la } }

The note, sung, in the second Hexachord, to the syllable fa, is here sung to ut, and is therefore called F fa ut. The next note, G, is sung to sol, in the second Hexachord, re, in the third, and ut, in the next Hard Hexachord, beginning on the octave G; hence, this note is called G sol re ut. And the same rule is followed with regard to all notes that appear in three different Hexachords. The note B♭, occurring only in the Soft Hexachord, is always called B fa. B♮ is called B mi, from its place in the Hard Hexachord, where alone it is found.

The four remaining Hexachords—for there are seven in all are mere recapitulations of the first three, in the higher octaves. The entire scheme, therefore, may be represented, thus—

Hex. 7 The Gamut
Hex 6. E la E la.
D la D sol D la sol.
C sol C fa C sol fa.
Hex. 5 B♭ fa B♮ mi B fa. B mi.
A la A mi A re A la mi re.
G sol G re G ut G sol re ut.
Hex. 4 F fa F ut F fa ut.
Hex. 3 E la E mi E la mi.
D la D sol D re D la sol re.
C sol C fa C ut C sol fa ut.
Hex. 2 B♭ fa B♮ mi B fa. B mi.
A la A mi A re A la mi re.
G sol G re G ut G sol re ut.
Hex. 1 F fa F ut F fa ut.
E la E mi E la mi.
D sol D re D sol re.
C fa C ut C fa ut.
B mi B mi.
A re A re.
Γ ut Γ ut.

The art of correctly adapting the syllables to the sounds is called Solmisation. So long as the compass of a single Hexachord is not exceeded, its Solmisation remains immutable. But, when a melody extends from one Hexachord into the next, or next but one, the syllables proper to the new series are substituted—by a change called a Mutation—for those of the old one. In the following example, the bar shows the place at which the syllables of the Hexachord of C are to be sung in place of those belonging to that of G; the syllables to be omitted being placed in brackets.

{ \override Score.TimeSignature #'stencil = ##f \clef bass \cadenzaOn g,1_\markup { \smaller \italic ut } a,_\markup { \smaller \italic re } b,_\markup { \smaller \italic mi } c_\markup { \smaller \italic fa } ^\markup { \smaller \italic (ut) } \bar "|" d_\markup { \smaller \italic (sol) } ^\markup { \smaller \italic re } e_\markup { \smaller \italic (la) } ^\markup { \smaller \italic mi } f^\markup { \smaller \italic fa } g^\markup { \smaller \italic sol } a^\markup { \smaller \italic la } \bar "||" }

The Hexachord of C passes, freely, either into that of G, or F: but no direct communication between the two latter is possible, on account of the confusion which would arise between the B♭ and B♮. The mutation usually takes place at re, in ascending; and sol [App. p.672 "la"], in descending.

We have said that this subject exercises an important bearing upon the treatment of Real Fugue, in the Ecclesiastical Modes. Without the aid of Solmisation, it would sometimes be impossible to demonstrate, in these Modes, the fitting answer to a given subject; for, in order that the answer may be a strict one, it is necessary that its Solmisation shall correspond, exactly, in one Hexachord, with that of the subject, in another. Failing this characteristic, the passage degenerates into one of mere imitation. The