At the turn of the ninth century, the Carolingians recovered ancient writings about music that they had not known previously. These works introduced them to the pitch-based systems of Greek theory. Over the course of that century, they developed a theory of their own that adapted these concepts to sacred chant.Footnote ¹ Much has been written about the obstacles they faced in reconciling ancient systems of scales and modes with the music of their time.Footnote ² This article will make the case that their difficulties began at a more basic level: in the early ninth century, musicians did not yet have a functional concept of musical intervals. Most critically, they did not understand the difference between whole tones and semitones, which are essential building blocks of musical structures. In stark contrast, theory of the later ninth century and beyond discussed everything to do with melody, harmony, scales and modes in terms of intervals. We will explore the transition between these modes of thought, first in writings about music, then through the development of notation.

The first evidence of Carolingian engagement with intervals manifests in commentaries on Martianus Capella's De nuptiis Philologiae et Mercurii and Boethius's De institutione musica that began to appear by the second quarter of the ninth century.Footnote ³ These texts were the most comprehensive treatments of ancient theory available at the time. Martianus's first two books introduce the harmony of the spheres and the Pythagorean ratios that generate the octave, fifth, fourth and whole tone, but do not explain how these proportions translate into sound. The semitone appears only as an unspecified unit of distance between planets. Book 9, on harmony, is devoted largely to musical terminology and includes this definition: ‘a whole tone is an interval with a fixed size that is composed out of two mutually different sounds. That which is called a semitone contains half of a whole tone.’Footnote ⁴ We are not told how to recognise these intervals in audible music. Martianus names the individual pitches and octave species of Greek harmonic systems, but never specifies the intervals among them. His work alone was not an adequate summary of ancient pitch theory. Boethius, on the other hand, offered a thorough but highly technical treatment of pitch. The mathematical complexity of his work was a challenge to medieval readers, but his division of the monochord made the intervals of the musical scale explicit and audible, while the letters he used to designate those intervals were the first notation known to the Middle Ages that distinguished individual pitches.

Many early manuscripts of Martianus and Boethius contain explanatory glosses in the margins. Three generations of Martianus commentaries have been preserved. The first, dating from the 820s or 830s, appears to have been a collective effort.Footnote ⁵ The second, written in part by John Scottus Eriugena, is from the 840s,Footnote ⁶ while the third was compiled by Remigius of Auxerre around 900.Footnote ⁷ Glosses on music tend to concentrate on the mathematics of pitch relationships.Footnote ⁸ It is natural to suppose that once musicians were exposed to the idea of Pythagorean proportions, they must have understood how those proportions applied to sounding music. However, the first two sets of Martianus glosses do not confirm this. For instance, Martianus's book 7 (on arithmetic) remarks that ‘there are six whole tones of the whole harmony [i.e., the octave]; that is, five whole tones and two semitones’,Footnote ⁹ to which the oldest gloss tradition adds: ‘Two [whole tones] from the fourth and three from the fifth, [plus] a semitone from the fifth and a semitone from the fourth make six [whole tones].’Footnote ¹⁰ This writer obviously grasped how the proportions of consonances could be broken down into combinations of smaller ratios, but these ratios are not applied to musical practice anywhere in the commentary. The glosses associated with Eriugena are more sophisticated with respect to mathematics, but still do not go beyond what is found in Martianus's text. Of course, the primary function of these commentaries was to clarify Martianus's text, not to apply it to contemporary music, but it is important to recognise that they do not tell us what early Carolingian musicians understood about pitch. It is only at the end of the century, in Remigius, that we find examples from contemporary music being used to illustrate Martianus's intervals. When Martianus defines consonances in book 9,Footnote ¹¹ Remigius cites chants that begin with those intervals, at least according to transcribable manuscripts of the twelfth century and later. Here is his commentary concerning the fourth (Martianus's text is in small caps):

Here the practical knowledge of intervals goes beyond Martianus and the earlier sets of glosses.

Glosses on Boethius accumulated from the ninth to twelfth centuries and never coalesced into a standard text. In keeping with the character of Boethius's work, they focus even more narrowly on mathematical aspects of music. Glosses with practical content are rare and do not occur in manuscripts earlier than the tenth century.Footnote ¹³ The only apparent exception is a unique gloss preserved on a fragmentary leaf in a late ninth-century hand, bound with a slightly earlier copy of De institutione musica (D-Mbs, clm 14523). This comment has been associated with Boethius's wing diagram of the modes in chapter 16,Footnote ¹⁴ although the chapter itself is missing in this manuscript.Footnote ¹⁵ The gloss uses medieval names for the first two chant modes and attempts to define each with a melodic phrase. The first mode is described thus:

This text seems to be incomplete, garbled, or both. The meaning of ‘diapente proportion’ is obscure in context and the sequence of pitches does not clearly outline the first mode. Nonetheless, it does attempt to define church modes in terms of whole tones and semitones.

So we see that commentaries on Martianus and Boethius provide some evidence for the understanding of pitch at the end of the ninth century, but nothing definite for earlier periods. For that, we must turn to the oldest surviving medieval treatise on music, Aurelian of Réôme's Musica disciplina, which was written in 849 or early 850.Footnote ¹⁷ It is difficult to say how much of this work is original to Aurelian. The first nine chapters frequently quote or paraphrase material from ancient writers on music, and other sections may have been adapted from medieval sources that are no longer extant. However, we can at least credit Aurelian with the selection of content and some added text. For instance, I assume it was Aurelian who inserted biblical parallels with theoretical concepts whenever possible. These early chapters are haphazard in organisation and do not form a coherent theory of music, but there is a recurring theme: to explain ‘certain patterns of melodies, which they call tonos or tenores’.Footnote ¹⁸ Throughout chapters 1–9, Aurelian uses these terms to mean ‘modes’. This focus continues in chapters 10–19, which describe the cadential formulas that link psalm verses with their associated antiphons and responsories in each mode.

Aurelian's remarks about pitch all occur in discussions of mode. Chapter 2 features Pythagoras's legendary discovery of music's underlying principles from the sounds made by hammers of differing size at a smithy. Aurelian closely paraphrases Boethius's exposition of the proportional hammer weights that supposedly generated the octave, fourth, fifth and whole tone,Footnote ¹⁹ but he also links each ratio with an antiphon in one of the four authentic modes:

These chants seem chosen to illustrate the intervals named, but a closer reading casts doubt on that assumption. None begins with the associated intervals (if we can rely on later diastematic sources) and Aurelian does not say where in the melodies those intervals occur. Several historians have tried to show how the relevant intervals may be picked out of transcriptions of the chants in modern notation,Footnote ²¹ but it is questionable whether they would have been obvious to singers working from memory or from unheightened neumes. The first phrase of Inclina domine does span a rising octave, although there is a downturn along the way. Joseph Ponte pointed to a rising fourth between the first two syllables of Confessio et pulchritudo,Footnote ²² but the lower pitch of this leap occurs on the passing note of a liquescent neume. Clear evidence for how liquescence was performed in the early Middle Ages is lacking, but later theory suggests that passing notes often did not settle on a specific pitch.Footnote ²³ If that were the case here, this would not be an effective example of the interval. Circumdederunt me outlines a fifth with one intervening pitch at me ge(mitus), but the outer pitches are split across words and phrases. Every chant is full of whole tones, and Aurelian could have found many seventh-mode chants that started with a whole tone up or down, but he chose Puer natus, which starts with the leap of a fifth. If Aurelian was trying to illustrate intervals, these chants were not well chosen. Even more confusingly, he says that each of the four intervals is somehow characteristic of a particular mode, which makes no sense, since they occur in every mode.

The same chants occur in glosses to the relevant passage of De institutione musica in three tenth- and eleventh-century copies.Footnote ²⁴ Even though these manuscripts post-date Aurelian, it is conceivable that he found these chant examples in Boethius glosses.Footnote ²⁵ On the other hand, they make no better sense as glosses than they do in Musica disciplina. The deciding factor may be that Aurelian offers an explanation of sorts for how the chants relate to the intervals, which the glosses do not. Immediately following the passage just quoted, he asserts that there are really only four modes, not eight, because the authentic and plagal of each mode are too closely related to be independent. An abbreviated form of this argument also appears in the Boethius glosses, but Aurelian goes on to say that ‘these same four principles mentioned previously, discovered by Pythagoras, generate the kindred [modes] out of themselves’.Footnote ²⁶ This implies that the four Pythagorean ratios and the four mode-pairs of chant are manifestations of the same fundamental principle of music, although its nature is not explained. Hence the chants Aurelian cites are better understood as examples of modes, not intervals. This is consistent with his repeated statement that each ratio is associated with the chants of a particular mode, but it raises serious doubt as to whether Aurelian understood what Boethius was saying about intervals.

Aurelian returns to Pythagorean ratios in chapter 6, where he tries to demonstrate that numbers are essential to music. Specifically, he sets out to show that ‘all music consists of six consonances, fifteen sounds, and eight modes’.Footnote ²⁷ The consonances are the fourth, fifth, octave, octave plus fourth, octave plus fifth and double octave. To explain why there are six, he quotes Boethius again, directly and at length, this time from De institutione arithmetica, where Pythagorean ratios that generate the six consonances serve as examples of harmonic medial proportion.Footnote ²⁸ Boethius was not concerned here with the sounds these ratios produce, only their mathematical properties. Nonetheless, Aurelian takes Boethius's calculations as proof that there are exactly six consonances. For the balance of the chapter, Aurelian sets himself the task of demonstrating that fifteen sounds generate six consonances and eight modes. In this endeavour, he is confused by Cassiodorus, whose Institutiones is quoted at this point. Cassiodorus also identified six consonances and explained how many sounds (soni) are contained in each, leading up to the double octave, which spanned fifteen sounds.Footnote ²⁹ These soni represented individual pitches of the Greek Greater Perfect System, but Cassiodorus said nothing about the intervals between them. Immediately thereafter, he turned the discussion to modes (toni), starting with this definition: ‘tonus is a species of the whole harmonic system and a magnitude that resides in the accent or inflection of the voice’.Footnote ³⁰ That is, tonus can refer to a mode or an interval. Cassiodorus named fifteen toni, meaning the fifteen scalar modes (tonoi) of later Greek theory, each of which starts a semitone higher than the last. He laid out in relentless detail how many semitones, whole tones (also called toni), or combinations thereof every mode's starting pitch lies distant from each of the lower ones. Along the way, he highlighted those intervals that produce consonances. For example: ‘the Dorian mode surpasses the Hypolydian by a semitone, the Hypoaeolian by a whole tone, the Hypophrygian by a whole tone and a half, the Hypoiastian by two whole tones, [and] the Hypodorian by two whole tones and a half, that is, by the consonance diatessaron’.Footnote ³¹ At the end of this litany, he observed correctly that the starting pitches of the lowest and highest modes are seven whole tones apart.

Aurelian reproduces Cassiodorus's exposition of octave species, but adds a few sentences that fail to distinguish Cassiodorus's two meanings of toni:

Aurelian thought that Cassiodorus had confirmed his premise that the fifteen sounds generate specific numbers of consonances and modes. For him, tonus signified mode; this is his usage throughout the first half of Musica disciplina, except in quotations from ancient authors. He takes his cue from Cassiodorus when he says that each sound rises a semitone above the previous one. This is the only place where Aurelian speaks of sounds as pitches and the only place where semitones are mentioned outside of quotations from Cassiodorus. However, he asserts that the first and fifteenth sounds are eight toni apart, even though Cassiodorus had made clear that the highest and lowest modes (toni) are separated by seven whole tones (also toni). It is unlikely that Aurelian miscounted the number of whole tones, since Cassiodorus had done the math for him. When he refers to eight toni, he still seems to be talking about church modes, not intervals. What he made of semitones is impossible to say.

The same confusion afflicts the first part of chapter 8. A variant of this section is preserved independently in several manuscripts of the eleventh century and later under the title De octo tonis. One of these carries an attribution to Alcuin, but there is no compelling evidence that this material predates Musica disciplina.Footnote ³³ Aurelian's version begins with multiple definitions of tonus drawn from ancient sources:

The eight toni that bind melodies together are modes, but Aurelian also claims that a tonus is the smallest component of music, comparable to a letter in grammar or a number in arithmetic. The grammar–music analogy can be traced to Calcidius, who called the smallest musical elements ‘pthongi, which we call sounds [soni]’,Footnote ³⁵ not toni. Aurelian's tonus does not mean ‘sound’, since soni and toni are distinguished from one another in the next sentence. It is also unlikely to mean ‘whole tone’, since Aurelian's quotation from Cassiodorus in chapter 6 made it clear that the semitone, not the whole tone, is the smallest interval. Aurelian's remark that toni are ‘patterns’ suggests that he is still thinking of modes, not pitches or intervals. The definition in the last sentence comes from Cassiodorus (quoted previously), but Aurelian still does not acknowledge the difference between its two meanings for tonus.

Through the first nine chapters of Musica disciplina, Aurelian repeatedly fails to make sense of what his ancient sources had to say about intervals and evinces no understanding of how whole tones and semitones affect musical systems or even what they are. This conceptual lapse is even more conspicuous in the second half of the treatise. Here the undigested morsels of ancient learning that make up earlier chapters give way to systematic discussions of contemporary chant practice. These take the form of not one, but two catalogues of psalm-tone cadences (varietates or differentiae) that effect transitions between psalm endings and chants of the Mass and Office. The catalogues are sufficiently unlike in their presentation and vocabulary that they appear to be the work of different authors. It is not certain whether Aurelian wrote either of them. Both differ from the first nine chapters in that they do not mention ancient authors or musical concepts. The only nod to theory occurs early in chapter 10:

These lines are incongruous in context – they interrupt a practical discussion of first-mode varietates – and may have been inserted by Aurelian to provide a link, however tenuous, between the two halves of the work.

The first catalogue (chapters 10–18) seeks to differentiate melodic endings for the varietates in each of the eight modes. Its treatment of first-mode introits will serve as an example:

The author is trying to explain how these cadential phrases differ from one another, but he offers only vague remarks about melodic outlines. His difficulty is not just a lack of technical vocabulary. Nothing in his remarks here or elsewhere suggests that he conceived melody in terms of individual notes or the intervals between them. This is not to say that he could not hear patterns of intervals. He obviously had a clear idea of what chants belonged to what modes, as we see in his further comments about first-mode introits: ‘The first [introit] receives the end of the verses into itself because it begins on [the same] level, the voice does not flow through winding curves, and the tonus is found on the third syllable, namely on -te, as in Gaudete.’Footnote ³⁸ This tonus is located on a specific syllable. Later transcribable versions of Gaudete in Domino semper show that the first pitch sung on -te is the final of the mode, and this seems to be what the author means by tonus. Presumably the configuration of whole tones and semitones around the final produced a characteristic sound pattern that he recognised, even though he could not articulate what made that pattern distinctive.

The second catalogue (chapter 19) is much shorter. For the most part, it is limited to highlighting select melodic gestures of varietates, as in the first catalogue, but occasionally we find syllable-by-syllable descriptions of entire melodies, which often employ grammatical terms to suggest inflections. Here, for example, is the beginning of a passage devoted to the Gloria patri in the first authentic:

This tells us a little about melodic direction within some syllables, but not between them. Elsewhere, the author employs Greek pitch terms a few times, but it is unclear what he meant by them. Diesis figures in his syllable-by-syllable discussion of the Gloria patri melody of the fourth authentic: ‘In the verses of the introits, the first syllable, that is Glo-, will undergo an elevation of the voice [arsis], then the second, that is -ri-, will undergo a diesis’.Footnote ⁴⁰ The verb in this sentence, patior, suggests that the music sung to both syllables will experience some sort of change. Arsis, which Latin grammarians used to mean a rising vocal inflection, fits well. Diesis, however, normally designated one of several intervals smaller than a whole tone in ancient theory, and did not signify upward or downward motion. In chapter 19, it has probably been substituted in error for thesis, which means a downward inflection of the voice and was often paired with arsis.Footnote ⁴¹Hypodorius and hyperlydius appear twice each in this catalogue.Footnote ⁴² In Aurelian's chapter 6, these were the names of the lowest and highest Greek tonoi,Footnote ⁴³ but now they seem tied to individual notes or syllables. Both occurrences of hypodorius use it as an adjective modifying vox, for instance: ‘the verses of the antiphons of this mode [i.e., authentic deuterus] are yoked to the sonority of the first authentic's verses almost to the end, with the exception that this [mode] rejoices in the Hypodorian note more than the first [mode]’.Footnote ⁴⁴Hyperlydius modifies either sonus or melodia. The latter instance reads as follows: ‘for on the eighth syllable of the same [i.e., the response Dum staret Abraham ad ilicem mambre], namely on i-, a Hyperlydian melody of the voice should be adopted, so that we may sing ilicem with a short vowel’.Footnote ⁴⁵ There is no explanation of what was Hypodorian or Hyperlydian about a voice, sound or melody. Michael Bernhard suggested that they might refer to lowest and highest notes,Footnote ⁴⁶ but passages where they occur are not tied to specific melodies and no other classical or medieval writers use the words in this way. We have no means of determining what the author meant by them.

Chapters 10–19 of Musica disciplina provide the best evidence we have for how music was perceived before the Carolingians assimilated ancient theory. They reveal that musicians did not think of music in terms of individual pitches, and while they could hear differences among modal patterns, they did not understand that those differences were generated by variations in interval size. Chapters 1–9, on the other hand, show that Aurelian had been exposed to ancient writings on harmonics, but scarcely understood them on their own terms and could not relate them effectively to contemporary music.

A functional understanding of whole tones and semitones emerges for the first time in treatises from the last third of the ninth century, namely Musica enchiriadis, Scolica enchiriadis and Hucbald of St Amand's Musica.Footnote ⁴⁷ Unlike Aurelian's Musica disciplina, these works describe music as patterns of individual pitches.

Musica enchiriadis (hereafter ME) develops a scale system from a basic tetrachord, called finales because its component pitches are finals of the four authentic modes. Each of the four pitches is said to possess a distinctive quality, which depends on its distance from neighbouring pitches, as well as a potency that produces the authority of the eight modes. To distinguish the finales from one another, the author assigns distinctive symbols to each, all derived from the ancient Greek sign for rough breathing before initial vowels, the prosodia daseia, and specifies the intervals between them.Footnote ⁴⁸ The dasian symbols are then reoriented (inverted, reversed, inverted and reversed, or laid on their sides) to create three more identically structured tetrachords, one below the finales and two above, plus two further whole tones at the top, making a scale of eighteen pitches. Subsequent discussions of melody, mode and organum employ these symbols to articulate pitch relationships. Some music examples place t or s between adjacent dasian symbols to indicate whether they are a whole tone or semitone apart. The distinction between whole tones and semitones is not, however, addressed directly until chapter 9. This chapter is positioned between the two principal sections of ME, which are devoted respectively to monophony and polyphony, and is only loosely connected to either. It consists of definitions for Greek and Latin technical terms. The church modes are mentioned in passing, but there is no further discussion of contemporary music. Here we are told that ‘a whole tone is a fixed size of interval from sound to sound,Footnote ⁴⁹ and since this interval of musical sounds is in a sesquioctave [9/8] proportion, it is called epogdous by the Greek nation. … A semitone is not the full interval of a whole tone.’Footnote ⁵⁰ We are not told how a 9/8 proportion manifests in sound and no more precise measure of a semitone is forthcoming.

Throughout ME, pitch concepts are introduced at a rudimentary level as if they were new and unfamiliar, even though the treatise was clearly not intended for elementary instruction. It assumes substantial prior knowledge of chant, suggesting that it was intended for singers who knew the repertory well, not beginners. Just how unfamiliar these ideas were becomes clearer when we consider a related treatise, the Inchiriadon Uchubaldi Francigenae, which seems in large part to be a preliminary version of chapters 1–14 from ME.Footnote ⁵¹ It survives only in late tenth- and eleventh-century copies, making it difficult to know which deviations from the standard ME recension represent older writing and which might have been added later or distorted in transmission. Nonetheless, we shall see that Inchiriadon presents a less-developed understanding of intervals than ME. It opens with the collection of definitions that forms chapter 9 of ME, including those for the whole tone and semitone. This is the only reference to the semitone in Inchiriadon. The exposition of tetrachords and dasian symbols that follows (corresponding to ME, chapter 1) is similar in both works, but Inchiriadon says nothing about interval sizes or their role in pitch systems. The passage quoted above, detailing the alternation of whole tones and semitones in the finales tetrachord, has no counterpart in Inchiriadon, even though the surrounding discussion is essentially the same. Equally significant, music examples that accompany Inchiriadon's discussion of monophony, corresponding to ME chapters 1–8, do not use t and s between dasian symbols to distinguish whole tones and semitones. Without some indication of where whole tones and semitones fall, dasian symbols do not constitute a pitched notation; it would not be possible to sing an unfamiliar melody from them. T and s do appear in some examples of consonances and organum later in the treatise (as in ME, chapters 10–14), but their meaning is never explained in the text, raising the possibility that they may be later additions.

The difference between the two versions is clearest when we compare other passages that treat semitones in ME with their equivalents in Inchiriadon. In chapter 7, ME invites the reader to sing the versus Rex caeli domine from a graph that places dasian symbols over syllables of the text. The symbols are also labelled archoos, deuteros, tritos or tetrardos to indicate their positions in a tetrachord. The author advises what to do if the performer becomes unsure what interval to sing: ‘But where perchance there is doubt respecting any sound, [as to] which in sequence it should be, then the modes should be scrutinised successively with respect to the semitones, by which it is established the deuterus and tritus are always separated, and then soon it will be recognised which that [sound] should be.’Footnote ⁵² That is to say, the singer can determine the mode of a melody by locating the position of the semitones. Inchiriadon offers a similar treatment of a different example, the hymn Aeterna Christi munera et martyrum victorias, but when it suggests how the singer should proceed if he loses track of the mode, intervals do not figure in the discussion: ‘Where perchance there is doubt respecting any sound, it helps if three or four sounds should be scrutinised in order upwards or downwards to or from the same sound, and then soon it will be recognised which that [sound] should be.’Footnote ⁵³ Here it is assumed that the singer can find his place simply by moving up and down the scale, relying on his familiarity with how a given mode should sound rather than his recognition of intervals.

Chapter 8 of ME uses a graph resembling staff notation to show how each note of the finales tetrachord is also the final of a mode. The syllables of an alleluia melody are distributed across a series of horizontal lines labelled with dasian symbols. The same musical phrase is shown starting from each of the finales, and the accompanying text explains that each iteration of the phrase is in a different mode: ‘Since these four little graphs stand apart from one another by a single semitone or whole tone, that is, a harmonic interval, the individual [graphs] may be transposed from [one] mode to every [other] mode.’Footnote ⁵⁴Inchiriadon makes a similar comment, but states that the finales are all separated by whole tones: ‘These four little graphs stand apart from one another by a single epogdous or whole tone, that is, by the harmonic interval between strings. They may be transposed from one mode alone to every mode.’Footnote ⁵⁵ This author appears unaware that the intervals between dasian symbols or between strings on instruments (presumably those that were diatonically tuned) were not all the same.

The second half of ME (chapters 10–18) focuses on consonant intervals and their role in organum. In chapter 12, the author explains that when a chant is replicated a fifth higher or lower by an organal voice, both will share the same quality, meaning that whole tones and semitones will occur in the same sequence. However, that is not the case for melodies separated by any smaller interval:

In other words, parallel melodic lines separated from one another by a fourth or less do not share the same configuration of whole tones and semitones. The corresponding passage in Inchiriadon agrees that such parallel melodies do not share the same quality, but fails to explain why:

Once again, Inchiriadon does not differentiate scale intervals. The discussions of semitones in ME that we have reviewed all seem to be rewrites of material from Inchiriadon that do not mention semitones at all. The revisions reflect a dawning awareness of the functional difference between whole tones and semitones. Only a few passages in ME discuss semitones,Footnote ⁵⁸ but those few transform the meaning of the entire work.

In Scolica enchiriadis (hereafter SE), the distinction between whole tones and semitones is a fundamental concept throughout, although it is still presented as something unfamiliar to readers. The first third of SE parallels ME in dialogue format, but places much greater emphasis on knowing where the semitone is located in order to avoid singing wrong intervals. Multiple music examples demonstrate the consequences of misplacing a semitone. The letters t and s, employed so commonly in ME to distinguish whole tones from semitones between dasian symbols, appear only once in this part of SE; interval sizes are detailed in the text itself. The remainder of SE has an entirely different character. It marks the first serious medieval attempt to synthesise musical concepts and terminology from ancient number theory, and does so without reference to contemporary music. Even here, the whole tone and semitone are not defined with precision until the end, where we find the first medieval monochord tuning formula, giving the Pythagorean ratios for a complete diatonic octave.Footnote ⁵⁹

In Musica, Hucbald begins his treatment of musical intervals at the most fundamental level, explaining the difference between equal and unequal sounds. He then introduces nine intervals of increasing size, identifying each one by citing chants and explaining which syllables of their texts contain the relevant interval. The smallest intervals are defined in this fashion: ‘The first size is present when two pitches are close to each other with the separation of the smallest interval, to such a degree that the difference between them may scarcely be discerned. … The second [size] is more perceptible.’Footnote ⁶⁰ Only after examples of all nine have been presented does Hucbald tell us that the first two are called semitone and whole tone, and that each of the nine intervals is a semitone larger than the last, so the whole sequence spans what we would call a major sixth. Throughout the treatise, whole tones and semitones are the elements from which scales and tetrachords are built, but we never get more precise definitions of them. Hucbald also employs the letters t and s to differentiate these intervals in multiple graphical contexts: between syllables of chant texts, between parallel lines representing the strings of a cithara, between pitch letters designating the steps of a two-octave scale, between neumes, and between the Greek note names of the Greater Perfect System. It is clear that Hucbald, like the authors of SE and the standard recension of ME, understood the function of intervals, while it is equally plain that Aurelian and the author of Inchiriadon did not. On this evidence, interval awareness among Carolingian musicians was a development of the later ninth century.

The same transition may be traced in the development of non-pitched and pitched notations. Non-pitched notation consisted of unheightened, staffless neumes, which were placed above chant texts to help singers remember melodies. Neumes indicated syllable groupings, general melodic direction and some details of performance, but did not distinguish intervals. Unfamiliar chants could not be sung from neumes alone; neumes could only remind singers of melodies they already knew. Fully neumed chant books survive from the beginning of the tenth century,Footnote ⁶¹ but there are individual chants or small groups recorded in neumes across some thirty other early sources that have been dated to the ninth century.Footnote ⁶² Most of the latter were added to manuscripts after their principal content had been written, making it difficult to pinpoint their time of origin. However, a few examples have been more precisely dated from their palaeography. The oldest is a prosula, Psalle modulamina, which was inserted at the end of a manuscript containing Ambrose's Expositio Evangelii secundum Lucam (D-Mbs, clm 9543, 199v). Bernhard Bischoff determined that both words and neumes were written by the scribe who copied the main text, a cleric named Engyldeo, who is documented at St Emmeram in Regensburg between 817 and 834.Footnote ⁶³ In addition, there are surviving fragments of a few early antiphoners that were planned from the outset to leave space for neumes.Footnote ⁶⁴ One has been dated, again by Bischoff, to the second quarter of the ninth century (GB-Ob, Auct. F. 4. 26, flyleaf).Footnote ⁶⁵ This not only confirms that neumes were known in the first half of the century, but that they were being systematically applied to the central chant repertory generations before the oldest surviving complete service books were made. Pitched notation, on the other hand, is confined to musical examples in theory treatises of the later ninth century, such as we have seen in the Enchiriadis treatises and Hucbald.

The knowledge that neumes existed decades before pitched notations were developed impels us to revisit theories of why neumes took the form they did and what they were meant to convey. The prevailing view of Western chant development, first proposed in the early 1950s, is that the repertory we call Gregorian was the product of a concentrated effort by Pippin III and his son Charlemagne in the later eighth century to establish a consistent liturgy and its attendant music, modelled on Roman practice, and impose it on churches throughout the Frankish realm.Footnote ⁶⁶ Also in the early 1950s, Solange Corbin examined early manuscripts containing neumes and concluded that none could be securely dated before the end of the ninth century.Footnote ⁶⁷ As more examples of chant became known, she reluctantly accepted that neumes were known by the middle of the ninth century, although she remained sceptical of dating by palaeography alone.Footnote ⁶⁸ She pointed out that many early neumes occur in non-liturgical manuscripts, while others are found in service books that did not leave room for notation above the text. To her, this meant that neuming was at first a sporadic activity, used primarily to record melodies that were new or outside the liturgical mainstream, while singers continued to rely on memory for the central chant repertory until comprehensive neuming began at the end of the century. During the period between the establishment of the Frankish repertory and its writing down, chant was transmitted orally. By the time the first fully neumed antiphoners were compiled, according to Corbin, systems for notating pitch had already been developed for music theory, so presumably musicians could have devised a practical pitched notation for chant had they wished. The fact that they did not prompted speculation that the melodies had become so familiar over generations of oral transmission that cantors no longer found it necessary to record their precise pitch contours. However, it is now clear that neumes were in use by the second quarter of the ninth century and there is no reason to assume that our earliest examples represent the beginning of neuming. We have also seen that music theory of this early period displays no practical knowledge of interval sizes. This suggests powerfully that the absence of pitch in neumes was not a matter of choice; neumes are vague about pitch because their creators’ understanding of pitch was vague.

In the 1970s several writers, among whom Leo Treitler was the most prolific, looked to the oral tradition that antedated notation in order to interpret the character and function of early neumes.Footnote ⁶⁹ They posited that pre-Carolingian cantors did not try to compose and memorise individual melodies for every liturgical text. Instead, they developed conventions for reconstructing melodies as they performed them from a fund of melodic formulas and their knowledge of what was appropriate for particular categories of texts. Treitler argued that this fluid conception of melody persisted even after the introduction of notation. He assumed that the inventors of neumes were capable of recording exact pitches,Footnote ⁷⁰ but chose not to do so because they did not regard melodies as fixed entities. On the other hand, we have seen that neumes were developed when Carolingian understanding of intervals was still unformed, so it was not possible for them to create a pitched notation. Hence the character of neumes does not prove that musicians deliberately avoided fixing chant melodies. It is likely, however, that some reconstructive techniques remained necessary after the invention of neumes, owing to their lack of pitch content.

Kenneth Levy advanced a theory diametrically opposed to Treitler's. He argued that the Gregorian repertory was fixed to a high level of melodic detail when it was established in the later eighth century, with no allowance for reconstructive variation.Footnote ⁷¹ In his view, this achievement would not have been possible without notation to support cantors’ memories. Neumed antiphoners were distributed to churches across Francia by around 800 with the object of imposing a consistent chant practice and suppressing local variants. Levy's theory has not won universal acceptance, largely due to the absence of complete, fully neumed antiphoners from the early part of the century. His point is well taken, however, that the initial push to spread the new chant across the empire may have prompted the invention of notation to support that effort. There is no more likely historical circumstance for the creation and dissemination of neumes. Levy, like Treitler, assumed that Carolingian musicians were capable of devising a pitched notation, but ‘the church musicians who opted for the inexact aides-mémoire of staffless neumes … were content with incomplete representations of melodic substance because the full substance seemed safely lodged in memory’.Footnote ⁷² This exposes an anomaly in Levy's thinking. If the purpose of notation was to stabilise chant melodies, including some, at least, that would have been unfamiliar to many cantors, why invent a system that intentionally omitted the one factor – pitch – that contributed most towards that goal? This difficulty disappears when we consider that early ninth-century musicians lacked a functional concept of interval size, so that creating a pitched notation was not an option. Neumes may have been intended to help standardise the chant, but they were not adequate for the task.

Although early neumes are unspecific about pitch, they include numerous signs that indicate how certain melodic gestures should be sung. For some historians, this sort of performance detail calls to mind contemporary accounts by John the Deacon and Notker Balbulus, as well as the eleventh-century writer Adémar de Chabannes, concerning the difficulty Carolingian singers had assimilating the Roman style of singing.Footnote ⁷³ They suggest that it was the unfamiliar singing style that required the support of neumes, while the melodies had already been committed to memory. Neumes, according to this argument, were invented primarily to record nuances of vocal delivery rather than pitch.Footnote ⁷⁴ The evidence presented here offers another way of thinking about such nuances. Faced with their inability to notate pitch, musicians may have compensated by writing down other details that might help singers differentiate one chant melody from another. Once again, the lack of pitch information in neumes does not mean that musicians had no interest in recording precise melodic contours; they simply did not know how.

The evolution of Carolingian theory and notation point to the same conclusion: until the later the ninth century, musicians had no practical understanding of interval size. Aurelian of Réôme and the author of Inchiriadon had no concept of the functional difference between whole tones and semitones. Interval awareness emerged in the Enchiriadis treatises and Hucbald's Musica, making possible the first medieval systems of mode and scale. This conceptual divide explains why early neumes lack pitch content and reshapes our perception of why neumes assumed the forms they did.