Twelve TheoryMath + Music Playground

Music is of two kinds, theoretical and practical. Theoretical music is concerned with contemplating musical matters. According to Boethius in Book I, chapter 2, it has three divisions: cosmic music, which considers the harmony of the whole universe and of its parts; human music, which treats the proportions of body and soul and of the parts of each in relation to one another; and a third kind said to reside in instruments, which that author discusses in five books.

Practical music consists in the performance of song. It is found in rhythms, meters, and sounds. Sounds occur partly in instruments, which are themselves various, and partly in the human voice. It is chiefly the latter that we shall now attempt to discuss: regulated music as it concerns song.

Song is of two kinds. One is simple and uniform, the kind commonly used now in churches; plainsong, called Gregorian music, treats of it. The other is varied and manifold; the music treating it is now called by some figured music and by others mensural music. Since, so far as I know, nothing certain about this latter song is found among the ancients, we shall follow the teaching of modern authors in the later books.

Definition of theoretical music. Music is the faculty that weighs, by sense and reason, the differences between high and low sounds. Boethius, Book V, chapter 2.

Definition of practical music. Music is the knowledge of modulating correctly. Saint Augustine.

Since every mathematical discipline rests on demonstration, while the things themselves cannot always be brought into a discussion nor sounds written down, musicians invented signs for sounds: partly figures, which we now call notes, and partly syllable names. Six have come into use: ut, re, mi, fa, sol, la. They now commonly call these syllables voices, using the sign for the thing signified. The places occupied by these voices are called clefs.

These clefs are distinguished on lines and spaces in songs, rather like measurements made visible by parallel lines, although the pitches are not all equally distant from one another, as a later discussion of intervals will show more fully. Ancient musicians called the sounds phthongoi and the clefs strings.

Guido of Arezzo, a man of exceptional learning whom our age follows, arranged the clefs in order like a ladder, after the ancient Greek disposition of strings. On the lowest degree, upon a line, he placed the voice ut and prefixed the third Greek letter, gamma. We should remember that this discipline, like all the others, came to us from the Greeks. In the space above the first line he placed re with the letter A; on the second line, mi with B. Some believe B should be drawn square, in the form used by Franchinus of Lodi, the foremost musician of our age, to distinguish the two voices mi and fa at the octave clef B-fa-mi.

In the space above the second line Guido then placed two voices, fa and ut, preceded by C, so that a new six-note order would begin and take up the ascent where the preceding order became insufficient. It does not begin at the end of the earlier order but in its middle, so that voices similar in nature are located in the same clef: ut with fa, re with sol, and mi with la.

I believe the reason for this arrangement was that the semitone should always occur in the third position, between mi and fa, according to the diatonic genus, which we shall discuss later. Our musicians teach that ut-fa are soft voices, re-sol neutral, and mi-la hard; whether usage rather than nature is responsible will be considered in its place.

From A onward come the seven clefs musicians call essential: A, B, C, D, E, F, G. The remaining clefs repeat these, though with additional solmization syllables because of mi and fa at B. Lowercase letters are used next— a, b, c, d, e, f, g—followed by five doubled letters: aa, bb, cc, dd, ee.

Mi and fa at the B clef are not equally distant from the lower a. Mi stands a whole tone above it, while fa stands a larger semitone above it. Thus the two sounds placed, as it seems, upon the same clef are farther apart from one another than mi and fa in the small semitone. We shall demonstrate this more fully when discussing the division of the tone.

If B is counted as a single clef, Guido’s scale therefore contains twenty clefs in this order: Gamma-ut, A-re, B-mi, C-fa-ut, D-sol-re, E-la-mi, F-fa-ut, G-sol-re-ut, a-la-mi-re, b-fa-mi, c-sol-fa-ut, d-la-sol-re, e-la-mi, f-fa-ut, g-sol-re-ut, aa-la-mi-re, bb-fa-mi, cc-sol-fa, dd-la-sol, ee-la. In writing music, Gamma is put on the first line, A in the space above it, and thereafter line and space take the voices by degrees until the end. The result is the image of a ladder, whether one prefers to ascend or descend it, as the accompanying general diagram makes clear.

I should warn the reader that it was impossible to avoid altogether the language accepted by everyone who treats these matters. Since no such ancient description exists, I had either to invent new terms and incur the charge of arrogance or concede something to long-established usage. When a subject is in some respect new, I see no reason to avoid words that are likewise new but accepted by usage. A teacher must speak plainly and strive above all to be understood.

The general diagram arranges music in its genera and spaces—otherwise according to Pythagorean measurement—and in the diatonic genus, with Guido of Arezzo as authority. It assigns a letter to every string and marks the Greek and Latin clefs by number; we shall discuss that notation more fully in chapter 18 of this book.

First, one should know that letters were not placed at the beginnings of the clefs without reason; they are extremely useful to learners. There are seven clefs distinct by nature and marked by the seven letters A, B, C, D, E, F, G. By repetition they become twenty. Musicians distinguish the first seven with capital letters, the next seven with lowercase letters, and the last five with doubled letters. Thus we say great or lowest A, small or acute a, and doubled aa, and likewise with the other clefs.

Second, the distances between the clefs must be considered. Each adjacent clef stands a second from the next; the next-but-one a third; the next a fourth; and so on. In general, any letter of the same kind stands an octave from its nearest repetition: Gamma-ut from G-sol-re-ut, A-re from a-la-mi-re, and similarly for the others. Here we should note the rule that the same judgment applies to octaves. Whatever can be sung at G-sol-re-ut can rightly be sung at Gamma-ut; what is sung at a-la-mi-re can be sung at A-re; what occurs at b-fa-mi can occur at B-mi, and so with the upper clefs.

Third, this arrangement could have been extended without limit if the stated law were preserved, though some beginning and end were necessary. Suppose a note is placed below Gamma-ut. To determine what should be sung there, look to the clef an octave away: because F-fa-ut is below G-sol-re-ut, F-fa-ut will likewise stand below Gamma-ut. The same octave reasoning applies if a note occurs above ee-la. The human voice does not exceed these limits, though music for four voices and musical instruments often exceed the arrangement.

One should not call every unfamiliar syllable improper, but only those neither used in a clef nor contained in its octave—for example mi at F-fa-ut, sol at a-mi, or mi at a-la-mi-re. Finally, the clefs must be learned thoroughly and committed to memory, so that without deliberation we know which voices belong to each and do not sing mi where fa is required, or the reverse. The rest is learned through practice.

The word clef is plainly a metaphor from an iron key: just as the latter opens a lock, the former opens a song. Since it would be absurd to place every letter at the beginning of a piece, musicians mark one or two in the initial margin so that the others may be recognized from them; these are called signed clefs. Some, including Sebald Heyden, an outstanding musician of our age, call them characteristic signs—a term I would not dislike if it were established in usage.

The F-fa-ut and C-sol-fa-ut clefs are drawn most frequently, although G-sol-re-ut sometimes appears in the highest voice of polyphonic compositions. To sign Gamma-ut or dd-la-sol is superfluous. Once the signed clef is known, one must see on which clef the song begins and follow the preceding rules: in ascending, take the lower solmization syllables; in descending, take the higher.

One must also learn to recognize distances or intervals, lest, when the indicated voices are fa-re, one instead sing fa-mi, or when they are sol-ut, sing sol-re. Simple hexachord exercises are therefore taught to children and are exceptionally useful in forming their voices. They become accustomed to lines and spaces from bottom to top as ut-re-mi-fa-sol-la is repeated seven times through the scale. Whoever supplied these formulas truly served beginners. By attaching simple things to simple things and varying everything in small steps, a true teacher and guide gradually leads the novice to attempt greater matters.

Our Ludwig Senfl has set these exercises successfully for four voices to the tenor Fortuna. Beginners should by no means neglect exercises of this kind. Glarean gives seven deductions beginning from F-ut, C-ut, F-fa-ut, G-sol-re-ut, c-sol-fa-ut, f-fa-ut, and g-sol-re-ut.

Great care must also be taken to impress firmly upon the fresh memories of the young the distinction between tones and semitones—that is, as people commonly say, that they know the difference between mi and fa. Glarean gives a varied example made by placing mi and fa against one another in the notation: ‘Go into all the world; preach, saying Alleluia.’

It may also be useful to transpose one or two songs into different clefs and show pupils how far their harmony departs when they are moved from their proper clefs, as a door from its hinge—for example, if someone begins the introit Requiem at a-la-mi-re. It is enough to have pointed this out to a diligent teacher.

Franchinus treats note shapes carefully and learnedly, giving three descriptions: simple, compound, and intermediate. A simple note is not joined to another. It is drawn with a square body, or with a downward stem on its right side in the manner of the mensural longa. A compound note is joined to another and consequently receives a different form. The intermediate note is rhomboid, like the mensural semibreve. He believes it is never placed alone but in groups of two or more, especially in descent.

For compound notes he distinguishes beginning, middle, and end. At the beginning there are two forms. If the second note ascends, the first is written without a stem; if the second descends, the first has a stem descending on the left. In the middle, all notes lack stems but are drawn either square or oblique. At the end a note is written in three ways: if the final note descends it has no stem; if it ascends, it is drawn either directly above the preceding note or on its right side, in which case it always has a stem.

The usage of certain people, especially in Germany, has corrupted these old forms, although monasteries there preserve antiquity to some extent. These signs are now left to the judgment of singers, scarcely any class of people being more capricious.

According to Nicomachus as reported by Boethius, music was once so simple that it consisted entirely of four strings; this endured until Orpheus. Such a body of four strings was called a tetrachord. Later it grew gradually into the pentachord, hexachord, heptachord, and beyond, finally reaching fourteen strings. A fifteenth string was then added, completing the double octave; for this reason it was called the added string. Nevertheless, the ancient division into tetrachords remained.

Five Greek tetrachords are therefore placed in the complete order of this scale. Each has the ratio 4:3 and sounds a diatessaron, that is, a fourth, mi-la. The first begins after proslambanomenos and is called hypaton, the tetrachord of the first or principal strings; in Guido’s scale its notes are B, C, D, E. The second is meson, the tetrachord of the middle strings. It is conjunct with the first because they share a string; in Guido it is E, F, G, a.

The third, synemmenon, or conjunct, has a, B-flat, c, d. The fourth, diezeugmenon, or disjunct, is not joined to those before it but is separated from meson by a whole tone; its strings correspond to b-natural, c, d, e. The fifth, hyperboleon, or highest, shares a string with the preceding tetrachord and corresponds to e, f, g, aa.

The individual Greek string names indicate their positions: nete is the extreme or highest string; paranete is next to nete; trite is third from nete; mese is the middle; paramese is beside mese; lichanos is the forefinger string, a name Suidas derives from the finger.

Because two conjunct tetrachords do not fill an octave, a tone is added below the two lowest tetrachords, from hypate hypaton to proslambanomenos. Another tone is placed between the two highest, from paramese to mese, so that an octave system is completed on either side. Together, the strings extending through the double octave from proslambanomenos to nete hyperboleon make the Greater Perfect System.

Mese has a double-octave relation shown by the numbers 9,216 and 4,608; the same relation from mese to nete hyperboleon is shown by 4,608 and 2,304. Thus the ratio from proslambanomenos to nete hyperboleon is fourfold. Boethius uses such large numbers because of the parts involved in the schisma and comma; these will be discussed more fully when we treat the division of the tone.

I know that many think Guido’s order of strings inverted and contrary to the natural motion of the heavens, since the higher celestial bodies give the deeper sound because they are larger. Others thought the contrary: that higher heavenly bodies give the higher sound because they move faster. This is the account Cicero gives in Book VI of On the Republic. Out of respect for so many distinguished authorities, I do not wish to seem either to affirm or refute what antiquity imagined or taught about the music of the celestial spheres.

Whether one places the deeper sounds above as Boethius sometimes does, or below as Guido’s scale, the strings of the cithara, and modern organs require, it makes no difference to musical reasoning. In my judgment the human voice more often begins from below than from above, as rhetoricians advise for firmness of voice and as the openings of great orators show.

One point remains: the three melodic genera. Today we use only one, the diatonic, and perhaps not in its ancient purity. It proceeds by a smaller semitone, tone, and tone. The chromatic consists of a smaller semitone, larger semitone, and three semitones—or a semiditone. The enharmonic consists of a diesis, another diesis, and a ditone. A diaschisma is half of a smaller semitone.

The latter two genera have perished, although Boethius and other musicians took care to describe them. In our age I have seen no one who could—or rather, who even tried—to direct a song according to them. Yet the task is not especially difficult if one can find the semitones, as those who divide organs today readily know how to do.

By mutation we mean a consonant change of a voice into another voice. By voices, says Franchinus, I understand the syllables of the hexachords. A pitch is therefore not changed into another pitch by raising or lowering; rather, one syllable is changed into another on the same pitch.

Musicians invented mutation because a single deduction—that is, a progression through the six syllables—does not suffice for the ascent and descent found in songs. Seven deductions were therefore arranged so that they assist one another and come to the aid of an incomplete scale. Voices of the same nature are made to meet. Thus the first deduction ends at E-la-mi, where mi appears; this has the same nature as la, since the two stand a fourth apart.

I do not wish a matter that is not especially difficult to be confused by countless precepts, as one sees in certain modern musicians who teach almost monstrous rules. All mutations are made, in ascent, from the higher syllable into the lower and, in descent, from the lower into the higher. Thus at C-fa-ut, ascending, fa changes to ut; descending, ut changes to fa. At D-sol-re, sol changes to re and re to sol. At E-la-mi, la changes to mi and mi to la. At F-fa-ut the procedure is exactly as at C-fa-ut.

In clefs containing three voices there is one chief concern: inspect B-fa-B-mi carefully, for everything must be directed according to it. It sometimes has fa, whenever the soft or round b is written in the margin, and sometimes mi, when no b is written. The remaining voices must be arranged accordingly.

If one ascends while mi is sung at B, ut, not re, must be taken at G; descending, that ut changes again into sol. If fa is sung at B, re, not ut, is taken at G; descending, re changes into sol. Similar reasoning applies at a. In upper clefs, d-la-sol-re has la if fa occurs at B, but sol if mi occurs there. The same must be considered at c.

Musicians of our age place the round b at the beginning of a song whenever fa is to be sung at the B clef, but customarily mark nothing when mi is intended. Someone skilled in the modes will easily detect errors in manuscripts; we shall teach this later.

Sometimes the qualities of songs are altered: a soft quality is mixed into a hard song. Franchinus testifies that Ambrosian singers often did this, and we see it today in chants called graduals. There, for a time, the third species of the diapente is sung between c and F; toward the end it becomes the fourth species, ut-sol, by changing mi to fa at the B clef.

To avoid the tritone—harsh, unpleasant, and unheard in the diatonic genus as a single leap—the same change from mi to fa must often occur. Yet unless there is a definite reason, like the one mentioned in the graduals, I consider such harsh constructions fit to be banished to the furthest Garamantes.

It is altogether foolish to make a mutation at B-fa-B-mi when the sounds are not at the same pitch. A song so constructed that there is nowhere to escape by mutation except at B is itself inept, for the diatonic genus, the only genus we now use, does not permit it.

One final point: no mutation is made in long leaps such as octaves, sevenths, and sixths, nor in fifths of mi-mi or fa-fa. These exceed the order of the hexachords; the syllables must simply be taken as they are found in the clefs.

Musicians of this age have generally maintained the practice of not writing a song on more than five lines, and often only four. A song sometimes exceeds these, so transposition by clefs was invented. The clefs are changed upon the lines, and as they move, the notes vary with them.

One thing alone must be observed: consider the clef on which the earlier note stood and likewise the one on which the later note stands. From this comparison their distance is easily recognized. If the earlier note was at Gamma-ut and the later at c-sol-fa-ut, it is certain that they stand a fifth apart. Whenever possible, those who sign the clefs in the margins should act carefully, lest the song itself be disrupted by a singer’s error.

Musicians give this useful rule: as much as the clef ascends, so much does the note descend. Conversely, as much as the clef descends, so much does the note ascend. They plainly mean this of harmony or sound, not of the written figures: the note must be sung correspondingly higher or lower.

This practice has now almost disappeared among us; people prefer to add a line below or above rather than transpose clefs. I have nevertheless wished to say a little about it so that, if it is encountered somewhere, the student will not lack an answer.

I therefore add one example in two forms: one simple, with lines added above and below, and the other with transposed clefs, in the manner practiced some years ago and not yet wholly abolished. The song belongs to the Ionic or lascivious mode, now commonly called the fifth. In its middle it contains the fourth species of the diapente, ut-sol; above this is joined the third species of the diatessaron, ut-fa, and below it the semiditone.

The piece was most elegantly composed in four voices by Jean Mouton, the excellent French composer of our age. For the progress and practice of beginners I have reduced it to a simple melody. It was taken from its old place in the Ionic mode and moved by a fourth; its proper final was C, but because of the synemmenon tetrachord it can also occupy a place on F, as most songs of this kind do.

The sung text is Alma Redemptoris Mater: ‘Loving Mother of the Redeemer, who remains the open gate of heaven and star of the sea, come to the aid of a falling people that strives to rise. You who, while nature marveled, bore your holy Creator; Virgin before and after, receiving that Ave from Gabriel’s mouth, have mercy on sinners.’

This is a great subject, and one in which the greater part of musical thought is engaged, since it concerns the elements and principles of the discipline. We shall nevertheless review it briefly: first what our own age teaches, and then at least a taste of the ancient tradition.

According to Boethius, an interval is the distance between a higher and a lower sound, as between ut and re, ut and mi, ut and fa, and so forth. It will be useful to state the name of each interval and explain what each name signifies. Musicians of our age recognize fifteen: unison; tone; smaller semitone; ditone; semiditone; tritone; diatessaron; diapente; semidiapente; tone with diapente; semitone with diapente; ditone with diapente; semiditone with diapente; diapason; and semidiapason.

They call the unison an interval in the same way arithmeticians call unity a number. All the remaining intervals occur in perfect and imperfect pairs; in these names the prefix semi- does not signify an exact half, but imperfection. Within the seven essential clefs and the limits of the octave, all intervals are composed of tones and smaller semitones. This applies to the diatonic genus, into which the apotome, or larger semitone, does not enter.

I. Unison occurs when the same sound is repeated—for example, ut-ut-ut, re-re-re, or mi-mi-mi. It is used especially in present-day psalmody, where an entire verse may remain on one pitch except for the note before the medial pause and the final cadence.

II. The tone, or perfect second, proceeds from one note to the next, as from ut to re or re to mi. The linked syllables mi-fa are excepted: together they make less than half a tone. When separated and joined to other syllables, however, they make a tone no less than the others—for example mi-re or fa-sol.

III. The smaller semitone, or imperfect second, is the interval from mi to fa. From the tone and the smaller semitone all the remaining intervals are formed; these two must therefore be committed first to memory, especially the locations of the smaller semitones, for the species of intervals are distinguished chiefly by their position.

IV. The ditone, or perfect third, consists of two tones. It has two species, ut-mi and fa-la. In the enharmonic genus this same interval separates nete from paranete. V. The semiditone, the softer or imperfect third, consists of a tone and a smaller semitone. It likewise has two species, re-fa and mi-sol; in the chromatic genus this consonance separates nete from paranete.

VI. The tritone, a harsh fourth entirely unsuited to the diatonic genus, consists of three tones and must not be taken in one leap. It occurs within the second and third species of the diapente, bounded on one side or the other by a smaller semitone, as from F-fa-ut to mi at the B clef.

VII. The diatessaron is the softer fourth and is suitable for every kind of melody. It is the distance between the outer strings of every tetrachord and consists of two tones and a smaller semitone. Its three species follow the three possible positions of that semitone: re-sol, mi-la, and ut-fa. VIII. The diapente, or perfect fifth, consists of three tones and a smaller semitone. It has four species, according to the four positions of the semitone: re-la, mi-mi, fa-fa, and ut-sol. These species either add a tone to a species of the fourth or add a smaller semitone to the tritone. Hardly any other consonance sounds sweeter to the human ear when placed properly.

IX. The semidiapente, or imperfect fifth, consists of two tones and two smaller semitones, as often from E to fa at the B-flat clef. It is not admitted in a single leap, but may occur in a continued ascent or descent. Although it spans a fifth, it falls short of the tritone by a comma and exceeds the diatessaron by only a smaller semitone.

X. The tone with diapente—one interval described by two words—is the perfect sixth, composed of four tones and a smaller semitone. It is scarcely admitted in one leap because it is exceedingly harsh. Guido established his deductions upon this interval. XI. The semitone with diapente is the imperfect sixth, composed of three tones and two smaller semitones, as from mi at E-la-mi to fa at c-sol-fa-ut and in similar places. It is especially familiar to the Phrygian mode and has a wonderful grace when used in its proper place.

XII. The ditone with diapente is the major seventh, composed of five tones and a smaller semitone. It should be avoided as a single leap; it falls short of the octave by a smaller semitone and exceeds the semidiapason by a comma. XIII. The semiditone with diapente is the minor seventh, composed of four tones and two smaller semitones, as from D to c or E to d. It too is rarely useful as a single leap and falls short of the octave by a tone.

XIV. The diapason, queen of all consonances, is the perfect and complete octave, composed of five tones and two smaller semitones. Its seven species extend from each of the seven capital letters to its lowercase counterpart. It is formed by combining a diatessaron and a diapente and is divided into those two parts: either the fifth lies below the fourth, the division called harmonic, or the fourth lies below the fifth, the division called arithmetic. The word itself means ‘through all,’ because it joins all the essential strings; beyond the seven clefs the same judgment applies by octave equivalence.

XV. The semidiapason is the imperfect octave, composed of four tones and three smaller semitones—for example from E-la-mi to fa at the upper B-flat clef. Although called an octave, it falls short of the true octave by a comma, just as the semidiapente and tritone were compared above.

Other intervals are named by adding an octave to those already listed, so the octave should be fixed most firmly in the memory. The theorist Johannes Tinctoris lists only the intervals commonly used in singing, omitting the tritone, semidiapente, both sevenths, and semidiapason because they are almost never useful as single leaps or should be avoided. He counts unison but does not regard it as an interval. He also calls intervals ‘modes’ improperly; modes are what our age calls tones, though that usage too is imprecise, since tone properly names the second described above.

The concluding musical example sets the practical list in sound: unison, semitone, tone, semiditone, ditone, diatessaron, diapente, semitone with diapente, and tone with diapente, followed by the octave. Its text tells the singer that whoever delights in psalmody should learn these intervals.

The species of intervals, Leonides teaches in his harmonic introduction, are determined chiefly by the varying position of the semitone. Thus the first species of the fourth, re-sol, has the semitone in the middle; the second, mi-la, at the beginning; and the third, ut-fa, at the end. In the four species of the fifth the semitone occupies, respectively, the second, first, last, and third places. The seven octave species, each joining a fourth and fifth, likewise vary the positions of their two semitones.

The general rule is that every interval has one fewer species than it has sounds. Boethius states it this way: a fourth has four sounds and three species; a fifth has five sounds and four species; and an octave has eight sounds but seven species. Glarean’s diagram places these three, four, and seven species before the eye.

Boethius says that a phthong is the melodically suitable fall of the voice upon a single pitch. Others define it as a certain smallest part of melody, the definite vibration of one string, or—as Ptolemy does—a sound maintaining one and the same pitch. As unity is the beginning of number, a point of a line, and an instant of time, so the phthong is the beginning of harmony: an indivisible sound serving as an element from which every melody is made and into which it is resolved. Phthongs that answer one another with sounds agreeable to the ear are emmeles; the contrary are emmeles’ opposite, ecmeles.

Boethius defines consonance as a mixture of high and low sounds that reaches the ear sweetly and uniformly; elsewhere, as the concord of dissimilar voices reduced into one. Some think he followed Plato in the first definition and Nicomachus in the second.

Plato explains consonance in the ear by the different speeds of sounds: the higher and faster sound enters first, then, having slackened and returned as though struck by a repeated motion, meets the lower sound in likeness, and the two mingle into one consonance. Nicomachus does not accept this as exact. Consonance is not made from like sounds but from unlike ones arriving at one and the same concord; low joined to low produces no consonance.

Nicomachus therefore investigates the matter through strings. When two unlike strings are struck, their sounds meet. If they are commensurable, their shared measure mingles and makes one consonance of the voices. If they are incommensurable and each hastens on its own when struck together, dissonance necessarily results. Hence Boethius defines dissonance as the harsh and unpleasant impact upon the ear of two sounds mixed together: each resists the other, strives to remain whole, and obstructs its companion, so both reach the sense disagreeably.

The teaching of our age differs remarkably from antiquity in the number of consonances. Among the most approved ancient authors one scarcely finds more than six named. Boethius treats five: the octave in the double ratio, such as 12:6 or 24:12; the octave plus fifth in the triple ratio, such as 18:6 or 24:8; the double octave in the quadruple ratio, such as 24:6; the fifth in the sesquialteral ratio, 3:2; and the fourth in the sesquitertian ratio, 4:3. To these is added the tone in the sesquioctave ratio, 9:8, though Boethius says it is not thereby a consonance. Glarean’s diagram displays the ratios beside the corresponding clefs of Guido’s scale.

In our age the tone has been expelled from the consonances. It is admitted only in what musicians call syncopations—a new name for no new thing—where the tone itself is not heard as the controlling simultaneity. The fourth too is rejected unless it has beneath it a fifth, ditone, or semiditone; Glarean supplies examples from Franchinus.

Modern musicians divide consonances into perfect and imperfect, calling all the remaining distances dissonances. The five perfect consonances are unison, fifth, octave, twelfth, and fifteenth. In measured music, pairs of the same species often follow one another and bring a passage—like voices weary from labor and received into peaceful quiet—to a close.

The four imperfect consonances are the third, sixth, tenth, and thirteenth, which I do not know to occur anywhere among the ancients. They are extremely suitable for imitation between two voices moving together and possess much grace if they finally resolve into perfect consonances and come home, as it were, from wandering. They were also adopted after writing for four simultaneous voices arose, to relieve the weariness produced by continual repetition of perfect consonances.

The six dissonances that strongly trouble and offend the hearing are the second, fourth, seventh, ninth, eleventh, and fourteenth. This classification concerns intervals within the double octave. Wider distances do not make a true blending of phthongs even when some notes agree. Of the seventeenth, nineteenth, and twentieth, musicians count the middle one among the perfect consonances and the outer two among the imperfect. Learned composers of our age use the seventeenth often and the nineteenth and twentieth more rarely; in my judgment they sometimes do so more to let the highest voices sport aloft than to produce a true mixture of sounds.

Because commas and semitones have already been mentioned repeatedly, it is worthwhile to explain them more clearly, as promised. We shall give Boethius’s division of the tone briefly and in bare diagrams.

Musicians demonstrate by fixed ratios that a tone cannot be divided into two equal parts, because no superparticular ratio—and the tone belongs to this class—can be divided equally. A tone, established in the sesquioctave ratio 9:8, is therefore divided into a larger and a smaller semitone. The Greeks call the larger semitone apotome and the smaller diesis or lemma.

The smaller semitone is divided into two diaschismata. The excess by which the larger semitone surpasses the smaller is called the comma; Philolaus divides the comma into two parts called schismata. He defines the diesis as the space by which the sesquitertian ratio, 4:3, exceeds two tones. The comma is the space by which the sesquioctave ratio, 9:8, exceeds two dieses—that is, two smaller semitones. A schisma is half a comma, and a diaschisma half a diesis or smaller semitone.

From these definitions and Glarean’s following diagram one can determine into how many diaschismata and other minute spaces a tone may be divided. Boethius shows several possible divisions in Book III, chapter 5. Our purpose is not to pursue the furthest reaches of the art, but only what seems most necessary. The diagram shows that these relations are so; explaining why they are so belongs to a deeper inquiry. Here diesis is used properly for the smaller semitone; the ancient enharmonic use of the word for a diaschisma is improper.

In the diagram the tone is divided into its smaller semitone or diesis and its larger semitone or apotome; the diesis is divided into two diaschismata, and the comma—the difference between the two semitones—into two schismata. Applied to the Guidonian clefs, the interval from re at a-la-mi-re to fa at B-fa-B-mi is a smaller semitone, while the interval from mi at a-la-mi-re to mi at B-fa-B-mi is a whole tone.

It follows that the two sounds written at B-fa-B-mi, although they seem to occupy the same clef, stand farther from one another than either stands from its neighboring clef: B-flat and B-natural are separated by the larger semitone, whereas each outer neighboring step is only a smaller semitone. This theory should by no means be disregarded.

Boethius further teaches that the smaller semitone does not contain a full four commas, though it exceeds three; the larger semitone does not contain a full five, though it exceeds four. The tone consequently exceeds eight commas but does not complete a ninth.

I consider no other part of music equally worth relating, equally necessary, and at the same time equally delightful as the discussion of the musical modes that now follows. It accords so closely with human nature that it seems almost innate in many people. It is useful not only for judging every song, but also for setting poets’ verses to melody and understanding many passages in excellent authors. I therefore urge every student to attend closely; once these matters are thoroughly understood, their fruit will never cause regret.

The musical modes are nothing other than the species of the octave itself, and these species are composed of the various species of the fifth and fourth discussed above. Anyone who has understood that teaching will grasp the nature of the modes without difficulty, though other means of recognizing them will be given later.

We shall first set out the account taught by musicians of our own time, then in Book II add the ancient tradition, so that anyone may see how the modern teaching agrees with the old. Fourteen modes arise from the seven octave species. Our age knows only eight of them, although it uses thirteen—some constantly, others rarely—as will be shown later. Nor does it distinguish those eight by true reasoning and settled laws, but confines them with certain rules that wander about irregularly.

Modern musicians call the modes ‘tones’ with such stubborn consistency that, unless we speak likewise, some may think us ignorant of the principles of music. I shall quarrel with no one over the word, but prefer ‘modes,’ as all the ancients did. The name tone may have arisen by Boethius’s time, for he says that the octave species produce what are called modes, also named tropes or tones; his wording does not appear strongly to approve the innovation.

The modern precepts begin by dividing the eight modes into odd-numbered and even-numbered. The first, third, fifth, and seventh are called authentic; the second, fourth, sixth, and eighth are called plagal or subordinate. Every mode consists of an octave species. An authentic mode has its whole octave above the final string; a plagal shares the same fifth above the final but places its fourth below it.

D, E, F, and G are the finals of every regular, untransposed song, and two modes are assigned to each: first and second to D; third and fourth to E; fifth and sixth to F; seventh and eighth to G. The authentic ranges are D–d for the first, E–e for the third, F–f for the fifth, and G–g for the seventh. The plagal ranges are A–a for the second, B–b for the fourth, C–c for the sixth, and D–d for the eighth. The eighth therefore occupies the same outer span as the first, but its system is inverted: its fifth lies above its fourth.

Some distinguish the fourth and fifth of each mode by their solmization syllables, a useful aid to memory. Naming downward from the upper note, they give the odd modes’ fourth first and fifth second, but reverse the order for the even modes: first, sol-re and la-re; second, la-re and sol-re; third, la-mi and mi-mi; fourth, mi-mi and la-mi; fifth, fa-ut and sol-ut; sixth, sol-ut and fa-ut; seventh, sol-re and sol-ut; eighth, sol-ut and sol-re.

Modern teachers prescribe the finals of melodies in all the modes as follows. Every song ends on re, mi, or ut; ut may be either conjunct or disjunct. They call it conjunct when fa is used at B-fa-B-mi and disjunct when mi is used there. Songs of the first and second modes end on re; those of the third and fourth on mi; those of the fifth and sixth on conjunct ut, as now practiced; and those of the seventh and eighth on disjunct ut.

Anyone who knows the octave species well will easily judge the mode of any song. I repeat this willingly because songs vary through transposition. Although D-sol-re is the proper seat of the first and second modes, their songs are often placed at G-sol-re-ut, especially in four-voice writing, but not without fa at the B clef. This keeps the lowest voice, which sounds an octave with the middle one, within Guido’s scale. Though not strictly necessary, why let voices wander outside the scale when they can be placed comfortably within it?

The first and second modes are sometimes placed at a-la-mi-re, but only in melodies that do not exceed a fifth. Otherwise the first mode would lose its fourth above and the second its fourth below, and la-mi would occur in place of sol-re, contrary to the nature of those modes. Church singers use excessive freedom in transposing melodies, though they could readily do without it. Why transpose a whole song because of one or two altered notes, especially when such alterations are introduced more by custom than by reason?

The third and fourth modes have their finals at E, but may also end at a-la-mi-re if fa is sung at B-fa-B-mi. They cannot end at the B clef while the nature of the system is preserved, whatever certain writers have claimed. The fifth and sixth modes are now placed at F-fa-ut, though their proper ancient seat was at C. The seventh and eighth are nowhere more comfortably situated than at G-sol-re-ut, although they may also be placed at small c when fa is used at the B clef.

Some have therefore offered a false rule: that every mode may have another ending a fifth above its final, which they call the cofinal. This is not true in any mode wherever the fourth contradicts it. The true rule is instead that the song of any mode may end a fourth above its final through the diatessaron, provided fa is used at the B clef.

This has already been shown for the first two modes, and the preceding diagram makes it clear for the third and fourth: the same two-part system extends from mi at great E to la at a-la-mi-re as from mi at great B to la at small e. The fifth and sixth are likewise commonly regarded as transposed by a fourth, for their true seat was C. The same reasoning applies to the seventh and eighth. Glarean closes the chapter by pointing to a familiar and very sweet chant whose transposition makes the rule evident: it is moved to small c with fa at the B clef.

Songs may also be recognized from their effects, as philosophers say, by certain easier and thoroughly common rules. Although these rules were handed down by teachers of this art who were not especially learned, they should not be omitted, because they help the memory.

Songs of the first mode frequently leap from re to la. Examples are Gaudeamus omnes, Salve regina, and Ave maris stella. Those of the second move from re to fa: Salve sancta parens, Terribilis est, and Emendemus in melius. The third moves from mi to fa at the sixth above, as in Pange lingua, Discubuit Jesus, and Omnia quae fecisti nobis Domine. The fourth moves from mi to la: Tota pulchra es, Resurrexi, and Spiritus ubi vult spirat.

The fifth mode moves from mi to sol: O sacrum, Regnum mundi, and Illuminare Jerusalem. The sixth moves from fa to la: O quam admirabile and Homo quidam fecit. The seventh moves from ut to sol: Puer natus est nobis, Viri Galilaei, and Summae Trinitatis. The eighth moves from ut to fa: Veni sancte Spiritus, Spiritus Domini, and Vespere autem sabbati.

For memory, teachers add these rough verses, which should not be rejected: ‘First, re-la; second, re-fa; third, mi-fa; fourth likewise mi-la; fifth, mi-sol; sixth, fa-la; seventh, ut-sol; eighth holds ut-fa.’

To make this clearer than daylight, Glarean adds Franchinus of Lodi’s formulas for all the modes. He never names Franchinus without honor, for that musician seems to have approached Ambrosian restraint more closely than others who treated the modes. These formulas clearly display a beginning, middle, and ending and contribute greatly to judging songs.

The eight notated formulas carry these texts: ‘Kind Father Ambrose, hear our prayers; Christ, hear us’; ‘In honor of the apostles, Badianus built the Lord a new church’; ‘Ambrose, priest of Augustus, baptizes, while the anabole sounds Te Deum laudamus’; ‘Marcellinus the priest and Peter the exorcist, martyrs of Christ, intercede for us’; ‘By the prayers and merits of the blessed martyr Blaise, defend us, O God, from every disease of the throat’; ‘Saint Erasmus, glorious martyr, pour out prayers to our Lord for our salvation’; ‘Protasius and Gervasius—the same faith and suffering truly made them brothers’; and ‘O Mary, Virgin of virgins and star of the sea, help us in our misery.’

Among the earliest churchmen, melodies were at first so simple that they scarcely filled a diapente—or, as others prefer to say, a hexachord—in ascent and descent. The Ambrosians are said to have remained closest to this custom. Gradually music reached the octave, the true system of every mode. Then, as commonly happens, melodies did not remain within even those limits but exceeded the octave both above and below, as may be seen everywhere.

In my judgment, modes moving within the octave resemble a river flowing between its banks. Heat or another cause may diminish a river so that it does not always fill its channel; rain or storms may swell it until it overflows. In the same way, whenever a composer chooses, a mode may fail to fill its octave or may exceed it according to the character of the song.

Church songs frequently add a tone below the octave of odd-numbered modes, as in the first and seventh; sometimes a ditone below the third; and a smaller semitone below the fifth. Even-numbered modes instead add a tone above, as in the sixth and eighth. The second adds a semitone above, though rarely; the fourth does so very frequently, as is clear in many songs of that mode and in Jeremiah’s Lamentations, which were composed in it.

Sometimes the systems of two modes are joined. In the sequence Victimae paschali laudes, the first verse has the diapente re-la common to the first and second modes. The second and third verses begin with the first mode’s diatessaron, re-sol; the next two begin with the second mode’s corresponding fourth; and the final two return to the first. All retain the common fifth re-la.

The systems of the third and fourth modes are joined in the antiphon Pulchra es amica mea. The old fifth and sixth modes, which used mi at the B clef, are joined in the melody in which much of Germany and the Rhine country sings the Passion. The Evangelist occupies the third species of the fifth, fa-fa, common to both modes and fitting for narration; Judas and the others speaking to Christ take the upper fourth, ut-fa; Christ himself takes the lower fourth.

The modern fifth and sixth modes, which use fa at the B clef, are joined in the sequence Ave praeclara maris stella. The seventh and eighth are joined in Lauda Sion Salvatorem and Benedicta sit sancta Trinitas. Many antiphons drawn from the Song of Solomon, and many pieces Germans sing after the Alleluia, likewise have mixed modes.

Whenever the systems of two modes are joined, the modes are said to be mixed. Some assign every such song to a plagal mode. I think instead that the nature of both modes must be examined, for no one can deny that the modes are truly mixed. We shall treat this subject more fully in the following book.

For practical use, especially in choral singing, the first consideration is that beginners should learn the intonations or modal formulas and hold firmly in memory the endings of the psalm verses, called Euouae. Once these are known, they are compared with the beginnings of antiphons; the interval is judged, and the song is begun accordingly. Nature itself assists this effort when the mind is not altogether dull.

Different writers give different formulas. I present them plainly and shall quarrel with no one about them, since the matter is nearly arbitrary. What has been taught thus far about the modes is the precept of others, not my own. Let us therefore continue with what belongs to this practice: formulas for every mode, both in the forms called complete and altered; intonations for the lesser and greater psalms; and then verses for responsories and introits, as Franchinus also supplied, though in another way.

I have taken care that the middle of each verse clearly distinguishes every mode, avoiding the confusion that often occurs among singers. For the lesser psalms, each of the eight notated intonations sets the texts ‘The Lord said to my Lord: sit at my right hand’ and ‘I believed, and therefore I spoke.’

For the greater psalms, the eight formulas set ‘My soul magnifies the Lord’ and ‘Blessed be the Lord God of Israel.’ The responsory formulas all set the doxology ‘Glory be to the Father, and to the Son, and to the Holy Spirit.’ Glarean prints every version in notation so their different beginnings, reciting tones, mediations, and endings can be compared directly.

The introit verses use: ‘My heart has uttered a good word; I speak my works to the king’; ‘May everyone who celebrates your commemoration feel your relief’; ‘Bless the Lord, O my soul, and let everything within me bless his holy name’; ‘Attend, my people, to my law; incline your ears to the words of my mouth’; ‘Blessed are the undefiled in the way, who walk in the law of the Lord’; ‘Do not envy evildoers or be jealous of those who work iniquity’; ‘Sing to the Lord a new song, for he has done marvelous things’; and ‘Confirm, O God, what you have accomplished in us, from your holy temple in Jerusalem.’

Perhaps I have treated these matters at greater length than necessary, but some allowance must be made for ordinary readers, to whom nothing ever seems sufficiently explained. The question of modal ‘differences’ is more troublesome and, in my judgment, superfluous. The idea arose either because learned authorities did not agree about the principal formulas, so that differences were used as formulas and formulas as differences, or—more probably—through the excessive subtlety of people who sought an easy way to intone antiphons and made a perfectly clear matter exceedingly obscure.

It would take too long to explain how these differences were invented and which difference belongs to each antiphon—a task for the curious, not to say the idle. Different nations and churches use different ones; a single town may disagree sharply, and even one church varies between its books and customs. I therefore present them only as variant formulas devised according to learned judgment. Franchinus, a learned man of sound judgment, did not disdain them, so I have not entirely neglected them either. Let whoever wishes use them. The common body of singers should not be irritated, for I would prefer it well disposed toward me.

To show how far our age has degenerated from ancient simplicity, I place before the reader the Ambrosian modulation as Franchinus gives it. It has two formulas—what he calls a difference—for the seventh and eighth modes, and one for each of the others. Our formulas agree with these in the first and eighth modes; in the others they agree scarcely at all.

Musicians also add the peregrine tone, sung in these regions on Easter Day at the baptistery to Psalm 113. Franchinus says much about it, not altogether accurately, as the next book will show. I reproduce it as it is commonly sung so that students are not deprived of it here: ‘When Israel went out of Egypt, the house of Jacob from a people of strange language.’

Many other things could be added, but in trying too hard to satisfy the tastes of others I might rightly appear foolish and ridiculous. I therefore think it best to stop here. I shall append a little about dividing the monochord, partly from Boethius and partly from other authors—a subject as delightful as it is necessary.

I had reached the desired end of this part of the work, in which I treat the teachings of others more than my own. Yet I decided that the commentary should proceed somewhat further and add what is necessary about the manifold division of the monochord. First I must explain why I sometimes use words differently from the ordinary practice of musicians in our age, lest the fair-minded reader blame me or attribute it to arrogance.

No one can deny that for more than six hundred years every discipline suffered the calamity of corruption and mutilation. In music, much has been taught wrongly for many years and much published falsely for the general public. Some writers could not even inflect the names belonging to this discipline, yet attempted to teach the art in books, wielding a severe Aristarchan rod and a philosopher’s brow. Their foolish pursuit of glory has often moved me to laughter and more often to anger.

Even if we interpret them kindly and say they wrote from good zeal and a readiness to serve the public, the person who rashly undertakes to teach a subject he neither knows nor can treat worthily is not free from fault. Horace did not address professional poets alone when he wrote: ‘You who write, choose a subject equal to your powers, and consider long what your shoulders can bear and what they refuse.’

No one asks for splendid eloquence or ornamented speech here, since the subject refuses decoration and is content to be taught. Every good reader may nevertheless require an author to use the discipline’s own words—its technical terms—correctly. Some treated Greek names such as neuma, antiphona, and magada as first-declension Latin nouns, deceived by Greek accusative forms. Poetic license may excuse certain ancient Latin adaptations, but the terms of a discipline must remain stable and proper; otherwise error becomes inextricable and endless.

Men such as Guido, Otto, Berno, Bishop Theogerus, William, and John—later made pope—readily obtain pardon because they lived when honorable learning lay almost wholly prostrate together with humane letters. Our own age, which everywhere seems to speak Greek and Latin more than its native language, has no such excuse. I spare names and identify the fault only for students’ benefit. I regret that this stammering ruins many promising minds, especially when young learners cannot distinguish a good teacher from a bad one and value a single rhetorical flower above whole flourishing fields of learning.

No one taught music more learnedly or diligently than Boethius; I have seen no one plainly his equal. Franchinus alone followed him successfully after a long interval and greatly helped the discipline through his labor, yet even he did not always speak correctly. He declined diapente as though its genitive were diapetes, and treated diatessaron and diapason similarly. I marvel that none of the many excellently learned men then flourishing in Italy warned him. The neglected Greek language thus avenged itself by making even a most learned man ridiculous.

I therefore urge every young person who wishes to be initiated into this science and become worthy of its mysteries to bring three chief things. First, have at hand both the theoretical and practical rules of arithmetic. Second, do not be wholly ignorant of Greek, for most of the art’s vocabulary is Greek. Third, keep an instrument nearby with which every sound can also be measured by ear. I presuppose arithmetic and Greek here; I can, however, teach the use of an instrument without leaving the bounds of this discipline.

In Book IV, final chapter, Boethius—the true and singular master of this subject—shows how the ratios of consonances can be gathered with certainty by the briefest and simplest instrument. A string is stretched between two fixed bridges, with a third movable bridge between them. Moving the middle bridge according to numerical divisions permits the two resulting portions of the string to be plucked and compared.

Divide the span into three parts and set the movable bridge so one part lies on one side and two on the other: the 2:1 ratio sounds the octave. Divide it into four and separate one part from three: the 3:1 ratio sounds the octave plus fifth. Divide it into five and separate one from four: the 4:1 ratio sounds the double octave. These are the consonances in the multiple class.

Keeping a five-part division but separating two parts from three gives the fifth in the first superparticular ratio, 3:2. Divide the span into seven parts and separate three from four: 4:3 sounds the fourth. Finally, divide the span into seventeen parts and separate eight from nine: 9:8 displays the tone. Glarean labels the rule A–D, the two fixed bridges E and F, and the movable bridge K, then supplies a diagram so every division may be seen.

The name monochord undoubtedly comes from its single string, just as trichord comes from three strings, tetrachord from four, pentachord from five, and so on. Yet monochord means not only the one string itself but the entire instrument upon which it is stretched. The two points from which a string’s sound begins and ends are necessary because no string can be infinite; these supports are called magades.

According to Suidas, however, the whole instrument that carries the magades is also called magas. Guido seems to render Suidas when he says: ‘A monochord is a long, square piece of wood, hollow within, with a string drawn over it; by its sound we apprehend the differences among pitches.’ He probably learned this from someone who knew Greek, though he did not translate every word exactly and sometimes explained the art more intelligently than he reproduced Greek usage.

Greek writers preserve an immense vocabulary of musical instruments and an extraordinary confusion not only of things but of names. Athenaeus alone mentions a vast number of instruments, names, and authorities, yet—with one or two exceptions—does not explain clearly enough for a reasonably intelligent reader what form or use each had. Perhaps some were too familiar in his day to require description and others had already disappeared. Writers often omit what is obvious as unworthy of description, while what has vanished and become unknown gives them even less reason to write.

Magadis, for example, may be called a kind of pipes in one place, a kind of cithara in another, and elsewhere a psaltery or another instrument. Athenaeus reports conflicting authorities: Aristoxenus and Menaechmus identify it with the pectis, while Diogenes the tragedian and Phyllis of Delos distinguish them. Sopater describes the pectis as a two-stringed instrument, whereas Telestes calls the magadis a five-stringed one. Anacreon is even said to have given it twenty strings. The same word may refer to an instrument producing high and low sounds together, and some identify it with the sambuca.

Pindar calls it an instrument of paired voices because men and unbroken boys sing together, an idea Erasmus likewise explains. Suidas calls the magadis a double-voiced instrument; a string stretched between two fixed bridges and divided by a movable bridge produces two sounds. Because the sources disagree so strongly, I ask the reader’s pardon and state my judgment frankly.

First, magas appears to me the proper name for the hemispherical or bridge-like supports, as Boethius teaches. Suidas’s wording agrees: others call the same object a support, a restraint of sound, or—as modern players say—a little bridge. From this usage comes a verb meaning to play upon the magas. Second, by Suidas’s authority, magas may also name the instrument that holds a string upon two such supports. I do not strongly oppose those who prefer magadis as the nominative name of the whole instrument.

Some distinguish five similar words entirely: one for the support; one for the instrument; one for playing it; one for a kind of pipe; and one for the performer who accompanies the magadis. I leave fuller investigation to more diligent scholars. I do not claim to have completely untangled the matter and will gladly correct what I have said if a more reliable explanation is found in approved authors.

The instrument itself can be made in various ways. Georgius Valla describes a small table called a chordotonon because strings are stretched and explained upon it: a rectangular parallelogram ten palms long and five wide, twice as long as it is broad. Others make it a foot long; others two cubits long and two fingers wide and thick.

When I first decided to test the matter and had nothing suitable at hand, I ordered a carpenter to make, in a rough practical fashion, an instrument three feet long, a quarter-foot thick, and three inches wide, with two fixed bridges at the ends and one movable bridge. I fitted four strings so that one might serve as a reference to which the others—shortened according to the divisions of the three melodic genera—could be compared. I later found that two strings suffice. The instrument served, and still serves, our purpose very conveniently.

Germans and people along the Rhine use a related eight-foot instrument made from three boards joined into a triangular or gently tapering pyramidal body. They call it a Schiza tympanum. A string is stretched across one surface between bridges and sounded with a bow like that used on a lyre, its horsehair rubbed with resin. Some add a second string half as long so the octave may sound more strongly in dance music.

Street performers carry the pointed end against the chest. The left hand holds the instrument and touches the string with a finger—especially the thumb—at marked divisions, usually fourths and fifths and sometimes thirds, while the right hand draws the bow. The sounding portion is always the shorter segment within the touched boundary. The instrument’s sound is far more pleasing from a distance than nearby. It plays the Ionic and Hypoionic modes extremely well, but not the others, much as trumpeters are limited; this will be discussed in the next book.

The players find fourths, fifths, tones, and semitones only roughly, since they do not know musical theory. At first they tried to persuade me that tones and semitones could not be found on the instrument. Experiment showed that this belief arose partly from their inability to divide spans more precisely than with a thick finger and partly because the longer string produces a buzzing sound. The buzz is not present at every division but especially at fifths and thirds, not at seconds—that is, at the tone and semitone—so in that respect their observation was true.

They create the buzzing with an arched bridge whose broader, thicker foot supports the string at the triangular base and whose extended foot rests upon a hard, polished piece of ivory or similar material. Sometimes a very slender nail is set in the heel of the extended foot so the vibration rings more strongly against the solid surface, as a cithara string buzzes against its lower fastening. I laughed at the players’ contrivance, though I still seek the true cause why not every division permits the buzz.

These triangular monochords are now about five feet long. Each of the three boards is roughly a quarter-foot plus half an inch wide at the base and about two and a half inches wide at the tip. I have discussed the instrument at unusual length because I believe it closely resembles the ancient monochord—if that instrument ever had common rather than merely scholarly use—and because the chromatic, enharmonic, and diatonic genera can all be found upon it and demonstrated to the eye, something scarcely possible on the instruments used in our age.

Athenaeus contains much more on musical instruments, but since it does not assist the present inquiry I leave it for learned readers to examine and proceed to the remaining matters.