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Music from the Earliest Notations to the Sixteenth Century


CHAPTER 3 Retheorizing Music
Richard Taruskin

Before exploring the implications of these statements, though, or taking a closer look at music set to vernacular poetry, or discussing the reasons why the word “secular” is being set off in this context by quotation marks, let us return briefly to the original subject of this chapter, the formulation of new theoretical concepts and their influence on musical practice. There is one more tale to tell.

For a long time, two of the Marian antiphons, Alma Redemptoris mater and Salve Regina, were attributed to Hermannus Contractus (Hermann the Lame, 1013–1054), a monk at the Swiss abbey of Reichenau. That attribution is no longer credited, but Hermann was a notable poet-composer (of sequences and Offices for local saints) and a major theorist. In his treatise, Musica, Hermann proposed surrounding the tetrachord of four finals (D, E, F, G) with a tone on either end, thus producing a six-note diatonic segment or hexachord from C to A, and with symmetrical intervallic content TTSTT.3 This module, Hermann implied, sums up with the greatest possible economy the tonal range of Gregorian chant. The tetrachord beginning with the first note, C, gives the beginning of the Mixolydian scale as well as that of the adjusted Lydian with B-flat: TTS. (In view of what we have observed about the F mode with B-flat, we could call this the major tetrachord.) If one begins on the second note of the hexachord, one gets the beginning of the Dorian scale, TST (we can call it the minor tetrachord). And by beginning on the third note one derives the essence of the Phrygian, STT. For all practical purposes, this model implies, there are only three finals—not four—and their scales are best thought of as beginning on C, D, and E. It was a step in the direction of what we call major-minor tonality.

Hermann appears to have been unaware of the fact, but his conceptual module had already been abstracted from the chant itself as part of a great pedagogical breakthrough—perhaps the greatest in the history of the literate tradition of music in the West. For it was precisely this breakthrough that at last made “sight-singing” possible and put Western music on a literate footing in truly practical terms. Its importance would be hard to overestimate.

Theory and the Art of Teaching

fig. 3-5 Guido of Arezzo instructing his pupil Theodal at the monochord, from a twelfth-century manuscript in the Austrian National Library, Vienna.

The man responsible for this signal achievement was the same Italian monk, Guido of Arezzo, who around 1030 (in the prologue to an antiphoner) first proposed placing neumes on the lines and spaces of a ruled staff to define their precise pitch content. Guido used special colors, later replaced by alphabet signs, to denote the C and F, “key” lines—claves in Latin—that have semitones below them; these letters survive as our modern “clefs.” We, who still rely on his inventions nearly a thousand years later, owe him a lot, as did all the generations of Western musicians preceding us. No wonder he was a legend in his own time, and by now is something of a myth, a musical Prometheus.

The actual Guido lived from about 990 to about 1033 and specialized for most of his fairly brief life in the training of choirboys. Like many teachers of ear training, he was ever on the lookout for melodies (in his case, chiefly chant antiphons) with which to exemplify the various intervals. Imagine his excitement, then, when (as he tells us) he chanced upon a tune that could exemplify all of them. This was the hymn Ut queant laxis (“So that tongues might loosen”), composed in the late eighth century by Paul the Deacon, a monk at the Benedictine abbey of Monte Cassino, in honor of the abbey’s patron saint, John the Baptist. This hymn tune is so constructed that the first syllable in each half-line is one scale degree higher than the one that precedes it, the whole series exactly tracing out the basic hexachord from C to A (Ex. 3-14). So well does it fit the pedagogical bill that scholars now suspect that Guido actually wrote the melody himself on the familiar words of the hymn.

Theory and the Art of TeachingTheory and the Art of Teaching

ex. 3-14 Hymn, Ut queant laxis; words by Paul the Deacon, music possibly by Guido d’Arezzo

This module gave a syllable-name (or vox, “voice”) to each degree (or locus, “place”) in the hexachord. Once internalized, the set of “musical voices” (voces musicales) served a double purpose for ear training. In the first place any interval, ascending or descending, could be demonstrated in terms of a vox combination (thus: ut–re, the tone; ut–mi, the major third; ut–fa the perfect fourth; re–fa the minor third; etc.). And, second, the difference between the tone and the semitone, the all-important definer of mode quality, could be mastered by drilling the interval mi–fa.

Around the beginning of the seventeenth century, the syllable si, derived from the initials of “Sancte Ioannes,” was added by some singing teachers to the Guidonian module so that a full major scale could be sung with model (“solmization”) syllables. (In modern practice, as every music student knows, si has been replaced by ti, and the closed syllable ut has been replaced by the open syllable do, sometimes spelled “doh” in English speaking countries to avoid confusion with the verb “to do.”) Guido, however, who did not as yet have or need the concept of the major scale, managed to complete the octave by transposing the basic module so that it began on G, the hexachord G–E being intervallically identical (or “affined,” to use Guido’s vocabulary) with C–A. In this new placement, the progression mi–fa corresponds with the semitone B–C. To solmize the full scale from C to c, one “mutates” at some convenient point (either on sol–ut or la–re) from one location of the module to the other, thus (dashes denoting semitones):

C D E — F G A B — C ……

ut re mi — fa sol la

    ut re mi — fa sol la

To take care of the F-with-B-flat situation, later theorists recognized another transposition of the module, beginning on F, that would place the mi–fa pair on A and B-flat. The whole range of hexachord transpositions thus achieved, mapping out the whole musical space within which Gregorian chant was habitually sung, finally looked like Ex. 3-15.

Theory and the Art of Teaching

ex. 3-15 The gamut, or full range of pitches represented on the Guidonian hand, together with the seven hexachords that are required for its solmization. The recurrent pitch names across the bottom of the diagram are called claves in medieval music theory; the recurrent solmization syllables are the voces. An individual pitch, or locus (“place” within the gamut), is specified by a combination of clavis and vox, from Gamma ut (whence “gamut”) to E la. What we now call “middle C” was C sol-fa-ut to medieval singers

In order to gain an ut at the bottom on which to begin the first set of voces, Guido placed a G below the A that normally marked the lower end of the modal system. This extra G was represented by its Greek equivalent, gamma. Its full name within the array of voces was “Gamma ut,” which (shortened to gamut) became the name of the array itself. (The word “gamut,” of course, has entered the common English vocabulary to denote the full range of anything.) The two versions of B (the one sung as mi over G, corresponding to our B natural, and the one sung as fa over F, corresponding to B-flat), were assigned to a single mutable space, whose actual pitch realization would depend on the context. The higher B was known as the hard one (durus), and was represented by a square-shaped letter that eventually evolved into the modern natural sign. The hexachord containing it was also known as the “hard” hexachord (hexachordum durum). The lower one, which softened augmented fourths into perfect ones, was known accordingly as soft (mollis) and was represented by a rounded letter that eventually evolved into the modern flat sign. The hexachord containing B-flat (B-mollis) was known as the “soft” hexachord (hexachordum molle; the original module, derived from the hymn, was called the “natural” hexachord.)

Eventually, the use to which Guido put the C–A hexachord module, and the concepts that arose from it, began to influence the more theoretical notion of the hexachord as expounded by Hermann. One now could distinguish pieces ending “on ut” (Regina caeli, for example) from pieces ending “on re” (like Salve Regina). A whole interval-species could be summoned up by a single syllable. This, too, reinforced the tendency to simplify the concept of mode and reduce it all the more to our familiar major-minor dualism. Eventually the “ut” modes (like G with a B natural) were called durus, and “re” modes (like G with a B flat, a “transposed” Dorian) were called mollis. This terminology survives to this day in some languages, like German and Russian, as equivalents for major and minor (thus in German G-dur means “G major” and g-moll means “G minor.”) In French and Russian, the word bémol (from “B-mollis”) denotes the flat sign.

As an aid toward internalizing the whole set of voces and applying them to the actual notes written on Guido’s other invention, the staff, Guido—or, more likely, later theorists acting in his name—adopted a mnemonic device long used by calendar makers and public speakers, whereby items to be memorized were mapped spiralwise onto the joints of the left palm (Fig. 3-6). (The once widespread use of such devices is still reflected in our daily language by expressions like “rule of thumb” and “at one’s fingertips.”) In its fully developed musical form (not actually reached until the thirteenth century, two hundred years after Guido), each location on the “Guidonian hand” (and one in space, above the middle finger) represented a musical locus, defined by the conjunction of two overlapping cycles: the octave-cycle naming the notes as written (the claves, or letter names), and the series of hexachord placements that assigned voces to each of the claves. A specific locus, then, represents the product of a clavis and a vox. C fa ut (lowest joint of index finger), for example, is the C below middle C (C), and only that C: it can be solmized only in the hard hexachord (in which it is fa) and the natural (in which it is ut); there is no F below it in the gamut, so it cannot be solmized as sol. Middle C (c, top joint of ring finger) is C sol fa ut: it can be solmized in all three hexachords. The C above middle C (cc, second joint of ring finger), and only that C, is C sol fa, for it can only be solmized in the soft and hard hexachords. To sing it as ut would imply that the gamut (or “hand,” as it was fondly called) continued past its upper limit.

Armed with the memorized and internalized gamut, a singer could parse a written melody into its constituent intervals without hearing it or hunting for it on a monochord. The first phrase of Salve Regina (Ex. 3-12b), for example, could be seen at a glance to lie exactly within the compass of the natural hexachord, in which it would be solmized with these voces: /la sol la re/ (Salve); /la sol fa mi fa sol fa mi re/ (Regina); /ut re re ut re mi fa sol re mi ut re / (mater misericordiae). All of Regina caeli (Ex. 3-12a) lies within a single soft hexachord. The beginning of the second phrase (“Quia quem meruisti”), the first that encompasses the entire range of the chant, would take these voces: /ut sol sol la la sol fa mi re ut re mi mi/. Finally, here are the voces for the first “Kyrie” acclamation and the first “Christe” in Kyrie IX (Ex. 3-5):

Kyrie: /re fa sol la sol fa mi re fa re ut re ut re fa sol fa mi re/(natural hexachord).

Christe: /mi mi re fa mi re re ut re fa re mi/ (soft hexachord).

Theory and the Art of Teaching

fig. 3-6 The “Guidonian hand” as represented in a thirteenth-century Bavarian manuscript.

Except for the beginning of the second “Kyrie” invocation, which extends down into the first hard (or “gamma”) hexachord (syllables: re fa sol sol), the whole of Kyrie IX can be solmized using one natural and one soft hexachord. It would be a good exercise for the reader. Another good exercise would be to seek out phrases in the chants used as examples in this book so far that exceed the interval of a sixth, and that therefore require a mutation for their proper solmization. Salve Regina contains a number of interesting examples of this type. The phrase “Ad te suspiramus, gementes et flentes” requires a mutation from natural to soft and back again, thus: /re fa la (think mi) sol re re ut re mi (think la), re fa sol sol re fa mi re ut/. The phrase “Eia ergo, Advocata nostra, illos tuos misericordes oculos” is tricky: it begins in the soft hexachord, and descends into the natural; but when the upper range is regained, mutation must be not to the soft hexachord but to the hard, since the melody (as the alert singer will have scanned ahead to notice) has a B-natural, not a B-flat, thus: /ut ut re ut re mi mi, sol re mi re ut (think fa) re sol la (think re), sol sol fa mi fa sol re sol fa re ut (think fa) la sol fa mi fa mi re ut/. At “nobis,” however, where the B-flat is called for, so is the soft hexachord: /re la (think mi) fa mi/.

Armed with these techniques, and with Guido’s hand stored in memory for ready reference, a singer could truly sing at sight, or (as Guido put it in the title of his famous epistle of 1032) “sing an unknown melody.” Reinforced over centuries of practice, this pedagogical aid wrought enormous changes in the way music was disseminated and thought about. When transmission from composer to performer could take place impersonally, without direct oral/aural contact, music became that much less a process or a social act, and that much more a tangible, autonomous thing. The notion of a “piece” of music could only arise when music began to be thought of in terms of actual pieces of paper or parchment. For these far-reaching conceptual changes, we have the legendary Guido, the greatest ear trainer of them all, to thank. He turned out to be even more a trainer of eyes and minds than of ears.


(3) See Richard Crocker, “Hermann’s Major Sixth,” Journal of the American Musicological Society XXV (1972): 19–37.

Citation (MLA):
Richard Taruskin. "Chapter 3 Retheorizing Music." The Oxford History of Western Music. Oxford University Press. New York, USA. n.d. Web. 7 Oct. 2022. <https://www.oxfordwesternmusic.com/view/Volume1/actrade-9780195384819-div1-003009.xml>.
Citation (APA):
Taruskin, R. (n.d.). Chapter 3 Retheorizing Music. In Oxford University Press, Music from the Earliest Notations to the Sixteenth Century. New York, USA. Retrieved 7 Oct. 2022, from https://www.oxfordwesternmusic.com/view/Volume1/actrade-9780195384819-div1-003009.xml
Citation (Chicago):
Richard Taruskin. "Chapter 3 Retheorizing Music." In Music from the Earliest Notations to the Sixteenth Century, Oxford University Press. (New York, USA, n.d.). Retrieved 7 Oct. 2022, from https://www.oxfordwesternmusic.com/view/Volume1/actrade-9780195384819-div1-003009.xml