Hexachords, solmization, and musica ficta

1. The basic hexachord system and its origins

In simplest terms, a hexachord is a set of six notes arranged to form intervals of two whole-tones, a central semitone, and two more whole-tones. We may represent this arrangement as T-T-S-T-T, with "T" standing for a whole-tone (Latin tonus), S for a semitone (semitonium).

In Guido's system, as we shall see, these six notes of a given hexachord are assigned the syllables ut-re-mi-fa-sol-la, with the semitone always occurring between the syllables mi-fa. While this system was codified beginning with Guido and his followers in the earlier 11th century, there is an interesting precedent going back to about 150 years earlier.

1.1. Hucbald and the six-stringed lyre

The theorist Hucbald of St. Amand, in his treatise De Harmonica Institutione (c. 880), describes a six-stringed cithara or lyre tuned so as to form the intervals T-T-S-T-T, and uses this instrument as the basis for a notation showing exact pitches. In this notation, lines represent the strings of the cithara, and spaces the intervening intervals, labeled T or S.

Here is an ASCII approximation of Hucbald's cithara diagram for the antiphon Ecce vere Israhelita. While the notation shows the tones and semitones of the melody (a characteristic of the later hexachord system also) rather than specific notes, Hucbald states that instruments of this kind have C as their lowest note: Thus the six lines could stand for the notes C-A, to be known in the Guidonian system as the "natural hexachord." As in the original - but much less elegantly in ASCII - the syllables are connected with diagonal lines, providing a kind of visual graph of the melodic contour. Melismas, passages in which a single syllable is set to many notes, are shown in Hucbald's notation by repeating the syllable:

                    ta                        e    e         e
|T|                /  \                       | \/  |       /  \
| |              li  in q[u]o    lus          e  e  e       e   e
|T|             |          \    /  \          |       \   /       \
| | Ec     Isra |           \  o    no        e        e  e        e
|S|   ce   |  he             do       on      |          e           e
|T|      \/                              \    |                       \
| |     vere                              e   |                       est
|T|                                         \/
| |                                          e
| |----------------------------------------------------------------------

Christopher Page describes this notation as a kind of "tablature" [1], and a similar system is used in treatises of this same epoch providing the first known descriptions of Western European polyphony. Hucbald's treatise, reflecting the Carolingian revival of classical studies, shows a lively interest in music not only as an academic or philosophical disciple, but as an art practiced by instrumentalists as well as singers. It remains an interesting question whether a six-stringed instrument of the kind described by Hucbald, and his use of it to notate a melody, may have helped to inspire the hexachord system of Guido and his followers.

1.2. Guido's solmization syllables: Ut queant laxis

While Hucbald's example notated a melody in terms of the six strings of an instrument arranged T-T-S-T-T, it was the innovation of Guido to associate this series of six notes with a set of easily remembered syllables. For the purpose he used the hymn Ut queant laxis [2], which like Hucbald's chant has a range of C-A, and exhibits a protus or Dorian modality, with D as the final or point of repose. This hymn has the additional feature, crucial for Guido's purposes, of having each of the first six phrases start on the ascending steps of the hexachord from C to A:

 C    D   F  D  E  D
Ut que-ant la - xis

 D   D  C  D  E   E
re-so-na-re fi-bris

 E F G E  D   E  C  D
mi - ra ge-sto - rum

 F   G  A  G  F D  D
fa-mu-li tu-o - rum

 G A G F E F   G  D
sol- ve pol-lu-ti

 A   G  A F  G A  A
la-bi-i re - a-tum,

 G  F  E D C  E   D
San - cte Io-an-nes.

Thus the six steps of the hexachord are associated with these initial syllables of the first six phrases of the hymn:

   T    T   S   T     T
Ut   Re   Mi Fa   Sol   La

This hexachord structure places a special emphasis on the central semitone mi-fa, the step E-F in the hexachord of C-A, known as the "natural" hexachord. This step serves as a pivotal landmark in navigating a melody; mi is always located immediately below a semitone, and fa immediately above it.

1.3. The three standard hexachords: Musica recta

Either Hucbald's six-stringed lyre or a single hexachord of Guido has limited musical possibilities: Many medieval chants, as well as secular melodies, have a larger range. In the case of a lyre, as Hucbald himself notes, more strings can be and often are added to accommodate the ranges of the various modes. In the case of Guido's hexachord system, the solution is to have more than one hexachord available. Then a singer negotiating a melody can change or "mutate" as required from one hexachord to another.

In the standard Guidonian system, designed for chant melodies using the seven diatonic notes plus Bb, musical space is in effect "tiled" between three hexachords. The lowest note of the system, if we notate middle C as C4, is "gamma ut" at G2; the highest note is typically E5, giving an overall range of two octaves plus a sixth. Here it bears emphasis that medieval pitch is flexible, and that singers were free to sing a melody at any comfortable pitch level.

From "gamma ut" comes the term gamut, referring to the full range of the system and to its complete set or network of hexachords; more generally, for example, we speak of "running the gamut" in the sense of covering a full range of possibilities.

As the following diagram shows, the standard medieval gamut or system of musica recta ("proper music") spans its musical space with three kinds of hexachords. Hexachords on G (with B-natural) are known as durum or "hard"; hexachords on C as "natural"; and hexachords on F (with Bb) as molle or "soft." Note that both "flavors" of the fluid or mutable step B/Bb - the "hard" flavor B-natural and the "soft" flavor Bb - are integral elements of the system.

    G-E   C-A     F-D
    hard  natural soft         note name

E5  E la     --    --          Ela
D5  D sol    --    la          Dlasol
C5  C fa     --    sol         Csolfa
B4  B mi     --    --          Bmi
Bb4 B --     --    fa          Bfa
A4  A re     la    mi          Alamire
G4  G ut     sol   re          Gsolreut
F4  F --     fa    ut          Ffaut
E4  E la     mi    --          Elami
D4  D sol    re    la          Dlasolre
C4  C fa     ut    sol         Csolfaut
B3  B mi     --    --          Bmi
Bb3 B --     --    fa          Bfa
A3  A re     la    mi          Alamire
G3  G ut     sol   re          Gsolreut
F3  F --     fa    ut          Ffaut
E3  E la     mi                Elami
D3  D sol    re                Dsolre
C3  C fa     ut                Cfaut
B2  B mi                       Bmi
A2  A re                       Are
G2  Gammaut                    Gammaut

In medieval and Renaissance usage, it is common to name a particular note on the gamut by specifying the set of possible syllables it can represent in any available hexachords. Thus D3, for example is known as dsolre, since it might be sung as either sol in the hard hexachord on gammaut (G2), or re in the natural hexachord on C3.

Note that D4 an octave higher, however, is known as dlasolre, since it may serve not only as sol or re, but additionally as la in the soft hexachord on F3; D5 is dlasol, since it can be sung within the standard system only as la of the soft hexachord on F4 or sol of the hard hexachord on G4.

Similarly, C3 is cfaut, being either fa of the hard hexachord on gammaut or ut of the first natural hexachord in the gamut; C4 is csolfaut, serving not only as fa or ut but also as sol of the soft hexachord on F3; and C5 is csolfa, serving only as sol in the soft hexachord on F4 or fa in the hard hexachord on G4.

When a note belongs to more than one hexachord, it is conventional to name its alternative syllables in descending order from la to ut: Thus E3 is elami; G3 is gsolreut; and A3 (or A4) is alamire. The euphonious "alamire," incidentally, was taken up as the pen name of one music scribe of the early Renaissance now honored, for example, by the name of the early music ensemble Alamire.

The fluidity or mutability of B/Bb may be clearer if we consider its name "bfabmi" - that is, this note may have either a "hard" aspect (shown by a "square B," the origin of the natural and sharp signs, and also of the German "h" for B-natural), or a "soft" aspect (shown by a "rounded B," the origin of the flat-sign, here "b").

In fact, the square-B sign is virtually identical to the modern natural sign, and the round-B sign to the modern flat sign. However, following the helpful usage of Peter Urquhart, we might best refer to these two hexachord-associated symbols as "mi-signs" and "fa-signs" respectively. They indicate that a given note - specifically B/Bb, in the standard untransposed gamut we are now considering - should be sung either as the hexachord step mi with a semitone above, or as fa with a semitone below. As the hexachord system expands, these mi-signs and fa-signs may apply to other degrees than B/Bb, applications involving transposition of the gamut (Section 1.6) or the "invention" of new accidental steps not part of the basic gamut (Section 2).

1.4. Negotiating the gamut: Hexachord mutation

To sing a melody with a range wider than that of a single hexachord, we must mutate or switch between two or more hexachords at opportune points. Normally, this involves choosing a step of the melody common to both the old hexachord and the new as a "bridge" between the two hexachords.

For example, suppose we wish to sing the ascending octave of the Dorian mode, D3-D4. One possible solution would be to sing as follows:

                  D3    E3  F3   G3   A3   B3  C4   D4

Natural : C3-A3   re    mi  fa   sol (la)
Hard    : G3-E4                       re   mi  fa   sol

Starting in the natural hexachord, we would sing the first notes of the ascending Dorian octave D3-E3-F3-G3 as re-mi-fa-sol, then making a mutation or shift on the fifth note A3 to the hard hexachord on G3, singing this and the remaining notes (A3-B3-C4-D4) as re-mi-fa-sol. The "bridge" note, A3, is common to both hexachords, serving as la in the first, and re in the second.

Some treatises suggest that singers, at least as a kind of training exercise, might sing the "bridge" note of a mutation with the syllables of both the old hexachord and the new, for example in our last example (N showing "natural hexachord" and H showing "hard hexachord"):

D3     E3   F3    G3    A3     B3   C4    D4
re     mi   fa    sol   la-re  mi   fa    sol
N      N    N     N     N - H   H    H     H

In usual practice, one might guess it more likely that only the syllable of the new hexachord would actually be sung; but this pedagogical technique emphasizes the point that in order for a usual or "proper" mutation to occur, the bridge note must belong to both the old hexachord and the new.

While mutations between natural and hard hexachords, or natural and soft hexachords, may be most common, it is also possible at times to mutate between hard and soft hexachords. This might occur, for example, in chants in the Lydian or F mode, also known as tritus, where both bmi (B-natural) and bfa (Bb) may appear in the same melody. Here S stands for the soft hexachord on F3, and H for the hard hexachord on G3:

F3   A3   C4   D4  C4     B3  C4  A3    Bb3 A3  G3  F3
ut   mi   sol  la  sol-fa mi  fa  re-mi fa  mi  re  ut
S    S    S    S   S - H  H   H   H -S  S   S   S   S

Certain treatises offer guidelines on where and how to make a mutation: For example, mutate between hexachords no sooner than is necessary. However, this technique was likely more of a fine art than an exact science, being somewhat analogous to decisions regarding phrasing - or possibly to choices of when to reposition the hands on a keyboard.

While normal mutation can solve routine problems of singing chant, and also of negotiating medieval and Renaissance polyphony, there remain certain unusual intervals and passages which require an "improper" or "false" mutation - a "quantum jump" between notes without the benefit of a common "bridge" between hexachords.

A direct tritone leap such as F3-B3, although rather rare, presents this kind of problem: The first note Ffaut is found only in the natural hexachord on C3 or soft hexachord on F3, while the second note Bmi is found only in the hard hexachord on G3.

Another problematic interval in many positions is the major sixth, which occurs at the opening of Guillaume de Machaut's monophonic virelai Ay mi!: A3 B3 D3. Here B3 or Bmi is found only in the hard hexachord on G3, and D3 or Dsolre only in the hard hexachord on G2 or the natural hexachord on C3.

Additionally, especially in the Renaissance, singers may use certain "shortcuts" to sing a single note a step above or below the range of the current hexachord, where normal mutation would be quite possible (see Section 3.4).

1.5. Mapping the gamut: The Guidonian hand

An interesting tool for visualizing and memorizing the regular gamut is the Guidonian hand: The drawing of a hand with the notes of the gamut and their possible solmization syllables placed near the joints of the thumb and fingers. The following ASCII diagram, while quite schematic and also compressed in the vertical dimension, may give an idea of a typical basic arrangement:

                   ________     /        \     ________
                  /        \   |          |   /        \     ________
    ____         |  elami   |  | dlasolre |  | csolfaut |   / bfabmi \
   /    |        |          |  |          |  |          |  |          |
  /      |       |          |  |          |  |          |  |          |
 |        \      |  ffaut   |  | ddlasol  |  | ccsolfa  |  | alamire  |
 |         \     |          |  |          |  |          |  |          |
 \ Gammaut  \    |          |  |          |  |          |  |          |
  \          \   | gsolreut |  | aalamire |  | bbfabbmi |  | Gsolreut |
   \   Are    \  |          |  |          |  |          |  |          |
    \          \ |          |  |          |  |          |  |          |
     \   Bmi    \|  Cfaut   |__| Dsolre   |__|  Elami   |__|  Ffaut   |
      \                                                               \
       |                                                               |
       |                                                               |
       |                                                               |

In navigating the 20 steps of the regular gamut shown on the hand, we may find it helpful to note that medieval octaves are often counted from A to A. Thus above gammaut (represented by the capital Gamma of the Greek alphabet), or G2, we have A-G (A2-G3), a-g (A3-G4), and aa-ee (A4-E4). Note that while B2 occurs only as Bmi in this regular gamut, B3 and B4 are counted as single fluid steps realizable as either mi (B-natural) or fa (Bb) - thus bfabmi and bbfabbmi.

Touring the complete gamut, we start at Gammaut near the top of the thumb and move in a spiral-like pattern, first proceeding down the thumb (Gammaut-Bmi, G2-B2), and then from left to right along the base joints of the fingers (Cfaut-Ffaut, C3-F3), continuing up the smallest finger (Gsolreut-bfabmi, G3-Bb3/B3); then across the tops of the other three fingers (csolfaut-elami, C4-E4), then down the index finger to the second lowest joint (ffaut-gsolreut, F4-G4); then across the same joints of the third and fourth fingers (aalaremi-bbfabbmi, A4-Bb4/B4), then up the fourth finger a joint to csolfa (C5), then to the same joint on the third finger, ddlasol (D5); and finally a leap to the highest note eela (E4), located above the tip of the middle finger.

It is sometimes suggested that a choir director may actually have pointed to the joints of this hand - or of an actual human hand - in order to indicate the desired notes. In any case, the hand could serve as a convenient mnemonic device.

It symbolized the "regular" notes of music, or musica recta, including the seven diatonic tones plus Bb. The additional "invented" accidentals were described as extra manum or "outside the hand" - belonged to new hexachords not part of Guido's basic system (see Section 2).

1.6. Signatures as gamut transpositions

For plainsong and polyphony, an interesting feature of the system is the use of a signature - most commonly one of Bb, and next most commonly one of Bb and Eb - effectively to transpose the system of the gamut and its standard hexachords to a different level.

Such transpositions have the effect of generating new notes within the transposed version of the regular gamut (musica recta), for example Eb in melodies with a Bb signature, while placing other notes such as B-natural outside the set of musica recta notes in such a gamut.

For example, if Dorian is transposed from its natural position of D3-D4 up a fourth to G3-G4, then a signature of Bb serves in effect as a "gamut signature" moving the entire system of hexachords by a fourth.

Like the untransposed gamut, our transposed system consists of three hexachords, but each shifted by a fourth: Instead of the hexachords G-E (hard), C-A (natural), and F-D (soft), we have corresponding hexachords C-A, F-D, and Bb-G. Let us look at this relocated system:

       C-A       F-D      Bb-G

       .....   etcetera  .....
A4     la        mi       --
G4     sol       re       la
F4     fa        ut       sol
E4     mi        --       --
Eb4 E  --        --       fa
D4  D  re        la       mi
C4  C  ut        sol      re
Bb3 B  --        fa       ut
A3  A  la        mi
G3  G  sol       re
F3  F  fa        ut
E3  E  mi
D3  D  re
C3  C  ut

One consequence of the transposition is to make Bb analogous to F in the untransposed system: It may serve either as fa in our F-D hexachord, or as ut in our Bb-G hexachord. This result not only enriches but complicates the system.

In the untransposed version of the gamut, the "rounded-B" or fa-sign invariably indicates that the affected note is to be sung as fa (i.e. Bb as fa of the F-D hexachord); but in the transposed version, our "rounded-B" (Bb) may additionally be sung as ut in the new Bb-G hexachord.

While Bb in its dual fa/ut role thus becomes analogous to F in the untransposed system, we find that transposition has also resulted in a fluid or flexible degree E/Eb analogous to B/Bb in the original system. Thus in our C-A hexachord (analogous to the usual hard hexachord of G-E) we have Emi; in our new Bb-G hexachord (analogous to the usual soft hexachord of F-D) we have Efa (Eb); and in our F-D hexachord (analogous to the usual natural hexachord of C-A) we have neither version of E/Eb.

While our transposition "naturalizes" Eb as an integral note of the gamut analogous to the usual Bb, it additionally "denaturalizes" B-natural; our relocated system no longer includes the hexachord of G-E where B-natural serves as mi. In this transposed system, a G-E hexachord would constitute an extra fourth hexachord "outside the hand" - like a D-B hexachord, with F# as mi, in the untransposed system. Thus B-natural here, like F# there, is not a "usual" note, but must be "invented" as an addition to the standard three-hexachord gamut (see Section 2).

Similarly, a gamut signature of two flats (Bb and Eb) establishes recta hexachords on F-D, Bb-G, and Eb-C, the last providing a flexible degree A/Ab (A-mi for the F hexachord, A-fa for the Eb hexachord). Such signatures, while going beyond the normal system of plainsong, do occur from time to time in medieval and Renaissance polyphony.

1.7. Inflections: Written and unwritten

While accidental signatures serve to transpose the whole gamut system, local accidentals may simply indicate that a given note should be sung as fa (e.g. Bb or "rounded B" in the usual untransposed system) or mi (e.g. B-natural in the same system). Since, in the untransposed gamut, both B and Bb are integral or musica recta notes, the choice between these two "flavors" may sometimes be a matter of performer discretion where no inflection is indicated.

One complication in considering such discretionary inflections in plainsong is that practices and tastes may have varied in different times and places.

For example, one later medieval viewpoint suggests that B-fa (Bb) is rather frequently used in the protus or Dorian modes on D; rather infrequent in the deuterus or Phrygian modes on E; most frequent in the tritus or Lydian modes on F, where it provides a perfect fourth above the final as an alternative to the tritone or augmented fourth formed by B-mi (B-natural); and least frequent in the tetrardus or Mixolydian modes on G, where if used often it would make this mode resemble protus (in effect approaching the same result as G Dorian with a Bb signature).

One author compares an explicit fa-sign for Bb with a bell rung to announce a meal; one does not always need such a sign to eat a meal, or to sing Bfa when it seems appropriate.

Generalizations about inflections in the different modes are indeed generalizations: For example, one beautiful version of the chant Veni Creator Spiritus in tetrardus or G Mixolydian features a variation between Bmi and an appearance of Bfa in the final phrase. Such guidelines do suggest, for example, that one should not promiscuously use Bb in this mode simply in order to avoid the characteristic melodic outline of the tritone between Bmi as the third degree of the mode and F as the step below the final G. [3]

While tritus or Lydian is indeed a mode which often favors Bb, there can be great variation between chants, which may lean toward Bfa or Bmi. There may be a tendency in chants favoring both flavors of BfaBmi to use Bmi (B) in ascending and Bfa (Bb) in descending, but the exceptions may be as significant as such a generalization.

It has been suggested that the use of Bb tends to increase in later medieval interpretations of plainsong, with this accidental (part of the musica recta system) often introduced in places where B-natural was favored in earlier practice, for example in some chants in the Lydian mode on F.

How freely performers may have introduced Bb inflections not indicated in the chant manuscripts remains an open question; while modern editions may typically specify such inflections explicitly, with the understanding that they should occur only when specified, the remarks of some medieval theorists suggest an element of performer discretion, at least in some times and places.

To Section 2 - Expanding the gamut: Musica ficta and "invented" hexachords.

To Table of Contents.

Margo Schulter