Hexachords - Basic System

Hexachords, solmization, and musica ficta1. The basic hexachord system and its originsIn 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 syllablesut-re-mi-fa-sol-la, with the semitone always occurring between the syllables mi-fa. While thissystem was codified beginning with Guido and his followers in the earlier 11th century, there is aninteresting 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 asix-stringed cithara or lyre tuned so as to form the intervals T-T-S-T-T, and uses this instrument asthe basis for a notation showing exact pitches. In this notation, lines represent the strings of thecithara, and spaces the intervening intervals, labeled T or S.

Here is an ASCII approximation of Hucbald's cithara diagram for the antiphon Ecce vereIsrahelita. While the notation shows the tones and semitones of the melody (a characteristic of thelater hexachord system also) rather than specific notes, Hucbald states that instruments of this kindhave C as their lowest note: Thus the six lines could stand for the notes C-A, to be known in theGuidonian system as the "natural hexachord." As in the original - but much less elegantly inASCII - the syllables are connected with diagonal lines, providing a kind of visual graph of themelodic contour. Melismas, passages in which a single syllable is set to many notes, are shown inHucbald'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 usedin treatises of this same epoch providing the first known descriptions of Western Europeanpolyphony. Hucbald's treatise, reflecting the Carolingian revival of classical studies, shows a livelyinterest in music not only as an academic or philosophical disciple, but as an art practiced byinstrumentalists as well as singers. It remains an interesting question whether a six-stringed

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instrument of the kind described by Hucbald, and his use of it to notate a melody, may have helpedto 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 arrangedT-T-S-T-T, it was the innovation of Guido to associate this series of six notes with a set of easilyremembered syllables. For the purpose he used the hymn Ut queant laxis [2], which like Hucbald'schant has a range of C-A, and exhibits a protus or Dorian modality, with D as the final or point ofrepose. This hymn has the additional feature, crucial for Guido's purposes, of having each of thefirst six phrases start on the ascending steps of the hexachord from C to A:

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

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

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

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

G A G F E F G Dsol- ve pol-lu-ti

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

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

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

T T S T TUt Re Mi Fa Sol La

This hexachord structure places a special emphasis on the central semitone mi-fa, the step E-F inthe hexachord of C-A, known as the "natural" hexachord. This step serves as a pivotal landmark innavigating a melody; mi is always located immediately below a semitone, and fa immediatelyabove 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, asHucbald himself notes, more strings can be and often are added to accommodate the ranges of thevarious modes. In the case of Guido's hexachord system, the solution is to have more than onehexachord available. Then a singer negotiating a melody can change or "mutate" as required fromone hexachord to another.

In the standard Guidonian system, designed for chant melodies using the seven diatonic notes plusBb, musical space is in effect "tiled" between three hexachords. The lowest note of the system, if


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we notate middle C as C4, is "gamma ut" at G2; the highest note is typically E5, giving an overallrange of two octaves plus a sixth. Here it bears emphasis that medieval pitch is flexible, and thatsingers 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 itscomplete set or network of hexachords; more generally, for example, we speak of "running thegamut" in the sense of covering a full range of possibilities.

As the following diagram shows, the standard medieval gamut or system of musica recta ("propermusic") 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) asmolle or "soft." Note that both "flavors" of the fluid or mutable step B/Bb - the "hard" flavorB-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 -- -- ElaD5 D sol -- la DlasolC5 C fa -- sol CsolfaB4 B mi -- -- BmiBb4 B -- -- fa BfaA4 A re la mi AlamireG4 G ut sol re GsolreutF4 F -- fa ut FfautE4 E la mi -- ElamiD4 D sol re la DlasolreC4 C fa ut sol CsolfautB3 B mi -- -- BmiBb3 B -- -- fa BfaA3 A re la mi AlamireG3 G ut sol re GsolreutF3 F -- fa ut FfautE3 E la mi ElamiD3 D sol re DsolreC3 C fa ut CfautB2 B mi BmiA2 A re AreG2 Gammaut Gammaut

In medieval and Renaissance usage, it is common to name a particular note on the gamut byspecifying the set of possible syllables it can represent in any available hexachords. Thus D3, forexample 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 orre, but additionally as la in the soft hexachord on F3; D5 is dlasol, since it can be sung within thestandard 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 naturalhexachord in the gamut; C4 is csolfaut, serving not only as fa or ut but also as sol of the softhexachord on F3; and C5 is csolfa, serving only as sol in the soft hexachord on F4 or fa in the hardhexachord on G4.

When a note belongs to more than one hexachord, it is conventional to name its alternativesyllables in descending order from la to ut: Thus E3 is elami; G3 is gsolreut; and A3 (or A4) is


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alamire. The euphonious "alamire," incidentally, was taken up as the pen name of one music scribeof the early Renaissance now honored, for example, by the name of the early music ensembleAlamire.

The fluidity or mutability of B/Bb may be clearer if we consider its name "bfabmi" - that is, thisnote may have either a "hard" aspect (shown by a "square B," the origin of the natural and sharpsigns, and also of the German "h" for B-natural), or a "soft" aspect (shown by a "rounded B," theorigin 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 tothe modern flat sign. However, following the helpful usage of Peter Urquhart, we might best referto these two hexachord-associated symbols as "mi-signs" and "fa-signs" respectively. Theyindicate that a given note - specifically B/Bb, in the standard untransposed gamut we are nowconsidering - should be sung either as the hexachord step mi with a semitone above, or as fa with asemitone below. As the hexachord system expands, these mi-signs and fa-signs may apply to otherdegrees 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 switchbetween two or more hexachords at opportune points. Normally, this involves choosing a step ofthe melody common to both the old hexachord and the new as a "bridge" between the twohexachords.

For example, suppose we wish to sing the ascending octave of the Dorian mode, D3-D4. Onepossible 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 octaveD3-E3-F3-G3 as re-mi-fa-sol, then making a mutation or shift on the fifth note A3 to the hardhexachord 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 ourlast example (N showing "natural hexachord" and H showing "hard hexachord"):

D3 E3 F3 G3 A3 B3 C4 D4re mi fa sol la-re mi fa solN 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 wouldactually 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


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most common, it is also possible at times to mutate between hard and soft hexachords. This mightoccur, 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 onF3, and H for the hard hexachord on G3:

F3 A3 C4 D4 C4 B3 C4 A3 Bb3 A3 G3 F3ut mi sol la sol-fa mi fa re-mi fa mi re utS 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, mutatebetween hexachords no sooner than is necessary. However, this technique was likely more of afine art than an exact science, being somewhat analogous to decisions regarding phrasing - orpossibly to choices of when to reposition the hands on a keyboard.

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

A direct tritone leap such as F3-B3, although rather rare, presents this kind of problem: The firstnote Ffaut is found only in the natural hexachord on C3 or soft hexachord on F3, while the secondnote 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 ofGuillaume de Machaut's monophonic virelai Ay mi!: A3 B3 D3. Here B3 or Bmi is found only inthe hard hexachord on G3, and D3 or Dsolre only in the hard hexachord on G2 or the naturalhexachord on C3.

Additionally, especially in the Renaissance, singers may use certain "shortcuts" to sing a singlenote a step above or below the range of the current hexachord, where normal mutation would bequite 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: Thedrawing of a hand with the notes of the gamut and their possible solmization syllables placed nearthe joints of the thumb and fingers. The following ASCII diagram, while quite schematic and alsocompressed in the vertical dimension, may give an idea of a typical basic arrangement:

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


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| | | |

In navigating the 20 steps of the regular gamut shown on the hand, we may find it helpful to notethat medieval octaves are often counted from A to A. Thus above gammaut (represented by thecapital 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 assingle 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 aspiral-like pattern, first proceeding down the thumb (Gammaut-Bmi, G2-B2), and then from left toright 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 acrossthe same joints of the third and fourth fingers (aalaremi-bbfabbmi, A4-Bb4/B4), then up the fourthfinger a joint to csolfa (C5), then to the same joint on the third finger, ddlasol (D5); and finally aleap 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 couldserve as a convenient mnemonic device.

It symbolized the "regular" notes of music, or musica recta, including the seven diatonic tonesplus Bb. The additional "invented" accidentals were described as extra manum or "outside thehand" - 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 - mostcommonly one of Bb, and next most commonly one of Bb and Eb - effectively to transpose thesystem 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 theregular gamut (musica recta), for example Eb in melodies with a Bb signature, while placing othernotes 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, thena signature of Bb serves in effect as a "gamut signature" moving the entire system of hexachordsby a fourth.

Like the untransposed gamut, our transposed system consists of three hexachords, but each shiftedby a fourth: Instead of the hexachords G-E (hard), C-A (natural), and F-D (soft), we havecorresponding 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 laF4 fa ut solE4 mi -- --Eb4 E -- -- fa


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D4 D re la miC4 C ut sol reBb3 B -- fa utA3 A la miG3 G sol reF3 F fa utE3 E miD3 D reC3 C ut

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

In the untransposed version of the gamut, the "rounded-B" or fa-sign invariably indicates that theaffected 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 findthat transposition has also resulted in a fluid or flexible degree E/Eb analogous to B/Bb in theoriginal system. Thus in our C-A hexachord (analogous to the usual hard hexachord of G-E) wehave Emi; in our new Bb-G hexachord (analogous to the usual soft hexachord of F-D) we haveEfa (Eb); and in our F-D hexachord (analogous to the usual natural hexachord of C-A) we haveneither version of E/Eb.

While our transposition "naturalizes" Eb as an integral note of the gamut analogous to the usualBb, it additionally "denaturalizes" B-natural; our relocated system no longer includes thehexachord of G-E where B-natural serves as mi. In this transposed system, a G-E hexachordwould constitute an extra fourth hexachord "outside the hand" - like a D-B hexachord, with F# asmi, in the untransposed system. Thus B-natural here, like F# there, is not a "usual" note, but mustbe "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 Ebhexachord). Such signatures, while going beyond the normal system of plainsong, do occur fromtime 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 maysimply indicate that a given note should be sung as fa (e.g. Bb or "rounded B" in the usualuntransposed 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" maysometimes be a matter of performer discretion where no inflection is indicated.

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

For example, one later medieval viewpoint suggests that B-fa (Bb) is rather frequently used in theprotus or Dorian modes on D; rather infrequent in the deuterus or Phrygian modes on E; mostfrequent in the tritus or Lydian modes on F, where it provides a perfect fourth above the final as analternative to the tritone or augmented fourth formed by B-mi (B-natural); and least frequent in the


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tetrardus or Mixolydian modes on G, where if used often it would make this mode resembleprotus (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 notalways 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 avariation 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 thecharacteristic melodic outline of the tritone between Bmi as the third degree of the mode and F asthe step below the final G. [3]

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

It has been suggested that the use of Bb tends to increase in later medieval interpretations ofplainsong, with this accidental (part of the musica recta system) often introduced in places whereB-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 manuscriptsremains 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 medievaltheorists 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


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Hexachords - Basic System

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