It can be easily proved that the power of numbers is prior to that of music.

Performers are mere slaves. A musician is he who commands a knowledge of the mathematical basis of the art.

Cited in Calvin M. Bower, Boethius' "The Principles of Music": an Introduction, Translation, and Commentary (Ph.D. dissertation, George Peabody College. 1966), p. 29

A mere singer seems to stand before a true musician as a prisoner before a judge.

Cited in Irving Godt, “New Voices and Old Theory,” Journal of Musicology 3: 312-19.

In our times, of all men, singers are the most foolish.

Cited in Irving Godt, “New Voices and Old Theory,” Journal of Musicology 3: 312-19.

The mere singer goes like a drunken man who does indeed get home but does not in the least know by what path he returns.

Cited in Irving Godt, “New Voices and Old Theory,” Journal of Musicology 3: 312-19.

Most of the scholars admitted to the pursuit of the Seven Liberal Arts in a Medieval university were, or expected to become clerics who would spend their lives in the church, and music was once an enormous part of the church’s business. A little simple arithmetic makes that truism especially impressive. If we multiply the number of musical items in the Mass (say, eleven) by the number of days in the year (ignoring duplications) we get a figure of 4,015 items. Taking the number of musical items sung each day in the Offices as 41 (again ignoring duplications), we get a total of 14,965. This massive total of 19,980 musical ’events’ per year does not include the 12,775 (or more) items sung to Psalm Tones, Lesson Tones, Gospel Tones, Canticle Tones, etc.; nor does it take account of special occasions such as baptisms, marriages, funerals, ordinations, etc. Even a non-musical cleric seeking preferment in the bureaucracy of the Medieval church might have had to supervise the organization, staffing, financing, or discipline of the vast and almost ceaseless musical activity of the churches within his administration. A theologian might have been inclined to grant more importance to the symbolic and speculative side of music, but he too might have had some responsibility for the day-to-day conduct of worship (which was mostly sung). It is difficult to imagine anything other than a fundamentally practical attitude toward music among churchmen—even among those scholar-clerics who wrote the most speculative treatises.

From Odo of Cluny (tenth century) down to John Dowland’s 1609 translation of Ornithoparcus, we encounter precise mechanical instructions for the calibration of a monochord so that it would sound true musical intervals. We may profitably digress here to remind ourselves that precise measurement is not a completely modern accomplishment. Galileo, for example, was able to make his observations of the acceleration of falling bodies without the benefit of modern chronometry thanks to the musical training he received from his father. If Ancient and Medieval musicians could not count vibration frequencies, their monochord allowed them to measure and to reproduce intervals as small as a syntonic comma, or about 1/9 of an equal tempered whole tone ( = 21.5 cents; a very sensitive ear can detect a difference as fine as 6 cents, less than a third of a comma).

We may feel some reluctance to recognize the monochord as a kind of “concerting” instrument, but the comment of Odo leaves no doubt that, as far as he was concerned, the monochord measured not only abstract theoretical intervals, but their performance by singers. It is needless to review here all the theorists who prescribe it, but Odo does not stand alone. More than half a millenium after him, writers continued to advocate it as a practical tool for teaching, not musical thinkers, but singers. Conrad von Zabern (fl. 1426-1474) appealed for a revival of the use of the monochord in the teaching of plainchant. Gafurius (1496) recommends that “it is exceedingly worthwhile for the human voice to imitate the harmonious strings that are sounded by the action of the lyre, cithara, or monochord.” For Ornithoparcus (1517) and for Dowland, his translator almost a century after him, the monochord still served as a basic teaching tool:

     OF THE PROFIT OF THE MONOCHORD

     The monochord was chiefly invented for this purpose,
     to be the judge of Musical voices and intervals
     as also to try whether the song be true or false;
     furthermore, to show haire-braind Musitians their errors,
     and the way of attaining the truth. Lastly that
     children which desire to learn Musicke, may have
     an easie meanes to it, that it may intice beginners,
     direct those that be forward, and so make
     of vnlearned learned.

Gafurius (and Dowland/Ornithoparcus) seem to imply by their choice that keyboard instruments (and the lute), because they usually required some kind of tempered tuning, could not satisfactorily answer their didactic purposes. All of the writers mentioned seem to have believed that the monochord alone produced those true sounds against which they expected singers to measure themselves.

Oliver Strunk, Source Readings in Music History (New York: W. W. Norton & Co., 1950), 106 n.: “A monochord made according to Odo’s directions will give the so-called Pythagorean intonation, the semitone (256/243) being obtained by subtracting the sum of two whole tones (9/8 plus 9/8 equals 81/64) from the fourth (4/3). Somewhat differently worded, Odo’s directions are given also by Guido in the Micrologus (GS, II, 4-5) and in the Epistola de ignoto cantu (GS II, 46).”

Book 3, Chapter Three. The Eight Mandates or Rules of Counterpoint.

    Nevertheless, this fifth, as organists maintain, permits the diminution of a small, concealed, and somewhat vague quantity which these organists call temperament.

Book 3, Chapter Fifteen. How a Singer Ought to Behave When He Performs.

    Finally we are led to propose to young singers for the purpose of instruction and admonition that in performing they should not project their voices with an unusual and unsightly opening of their mouths, or with an absurd loud bellowing when they strive after melodies, especially in the divine mysteries. They should also spurn excessive vibrato and voices which are too loud, for they are not compatible with other voices similarly pitched. In short, because of their own continual instability they cannot maintain harmonious proportions with the other voices.

—Translated by Irwin Young, The Practica musicae of Franchinus Gafurius, Translated and edited with musical transcriptions (Madison, Milwaukee, and London: The University of Wisconsin Press, 1969), 133, 160.

Chapter 1. How Any Consonance May Be Matched to Its Proportion.

    The ancients, and especially the Pythagoreans, matched only two genera of proportions to consonances: namely, the multiplex and the superparticular. From the multiplex they applied duple to the octave, triple to the octave and fifth, and quadruple to the double octave. In addition, they matched sesquialtera to the fifth and sesquitertia to the fourth. No other [intervals] than these did they consider to be consonances, as appears from their beliefs as we know them. But this position, notwithstanding that it arose from the greatest authority, nevertheless strikes me as false, for it contradicts the senses.

Who, unless he is deficient in the sense of hearing, can fail to find more consonances than the five just named? For within the octave are not beyond those [five] found the semiditone, the ditone, the minor hexachord and the major hexachord; and outside the octave [beyond those five] found the octave plus semiditone and the octave plus ditone and the octave plus fourth, just as Ptolemy proposed? And also the octave plus hexachord and the octave plus major hexachord: these, which I am adding, are also consonances, and are called by those who practice music the minor third, the major third, the minor sixth, the major sixth, the minor tenth, the major tenth, the eleventh, the minor thirteenth, and the major thirteenth. For it cannot be denied that all these intervals are true and very agreeable consonances except by a denial of the senses, which is incongruous.

Moreover, all composers of polyphony in their compositions, and all organists, and all string players, and all singers without other skills, make use of consonances of this kind, as anyone who knows anything at all about this art realizes. I say, therefore, that many more consonances are found than the ancients posited, and that more than two genera of proportions can be matched to musical consonances.

On the contrary, all genera of proportions are found to be valid in producing them [consonances]. Indeed, it is necessary for the minor hexachord and the major hexachord to be made from the superpartient genus; and for the octave plus ditone, the octave plus minor hexachord, the octave plus semiditone, and the octave plus fourth to be made from the multiplex superpartient genus; while the octave plus ditone, the octave plus minor hexachord, and the octave plus major hexachord to derive their origin from the multiplex superparticular genus. Thus, it is definitely not the case that only two species of the superparticular genus may be matched to consonances. For beyond these are two others of this same species of genus, namely, sesquiquarta [5:4] and sesquiquinta [6:5], which generate the ditone and semiditone, as will be seen below. How any given consonance may be matched to its proportion is now to be described. First we will speak concerning those derived from the multiplex, then those from the superparticular genus, and thus successively, observing the order of the genera.

Chapter 6. The Division of the Consonances into Perfect and Imperfect.

    ...The perfect are the unison, fourth, fifth, and octave, with their compounds.... The imperfect are the third, sixth, and their compounds.... Perhaps the perfect are so called because their intervals are determined by ratios contained in the number 4, and these proportions belong to the multiplex and superparticular classes. The number 4 (as I have said elsewhere) was held to be perfect by the Pythagoreans, because from its aliquot and nonaliquot parts, 4, 3, 2, 1, there resulted another number, which they called perfect: the number 10. Actually they called these perfect because, alone or accompanied by other consonances, they have the quality of satisfying the ear instantly and fully when it is affected by them. Whether placed high or low, provided it is in its true ratio, the perfect consonance so fortifies and pleases the ear that the sense desires nothing further to improve its sweet, welcome perfection....

    The other consonances they called imperfect, for their ratios are those found in numbers beyond 4, namely 6, 5, 4. Thus the ditone springs from the proportion sesquiquarta [5:4] and the semiditone from the sesquiquinta [6:5], both in the superparticular class. These two intervals added to the diatessaron generate the sixths; that is, one forms the major and the other the minor sixth.

Chapter 10. The Property or Nature of the Imperfect Consonances.

    The property or nature of the imperfect consonances is such that some of them are lively and cheerful, accompanied by great sonority; and others, although sweet and smooth, tend to be sad and languid. The former are the major third and sixth, with their compounds; the latter are the minor forms. All these have the capacity to alter every composition and to make it sad or cheerful, according to their respective natures. This may be seen from the fact that certain compositions are lively and full of cheer, whereas others on the contrary are somewhat sad and languid. In the first named the major imperfect consonances are often heard on finals or medians of certain modes or tones, namely the fifth, sixth, seventh, eighth, eleventh, and twelfth, as we shall see. These modes are very gay and lively, because in them the consonances are frequently arranged according to the nature of the sonorous number, that is, the fifth is harmonically divided into a major and minor third, which is very pleasing to the ear. [Note: The consonances are arranged according to the sonorous number when they occur in the order in which they are produced by the successive divisions of a string into 2, 3, 4, 5, and 6 parts. The fourth, fifth, and sixth divisions produce a series of notes in which a major third stands below a minor third. For example, if the string lengths that produce the three notes C, E, G are 30, 24, and 20, and the fifth is divided harmonically, the middle term is known as a harmonic mean. The three terms in a harmonic proportion are such that the proportion of the extremes is equal to the proportion of the differences. For example, in the series, 30, 24, 20, the differences 6 and 4 are in the same proportion as the extremes 30 and 20, namely 3:2.]

    I say that the consonances are arranged according to the nature of the sonorous number because they are placed in their natural positions. Then the mode is gayer and greatly pleases the sense, which greatly appreciates and enjoys proportionate things, and despises and abhors the disproportionate. In the other modes, then, which are the first, second, third, fourth, ninth, and tenth, the fifth is divided otherwise. It is arithmetically divided by a middle note, in such a way that often one hears the consonances arranged contrary to the nature of the sonorous number. [Note: In an arithmetic series the difference between successive terms are equal. So the notes A, C, E, in the proportions 6:5:4, form an arithmetic series, and the fifth is divided by an arithmetic mean.]

Chapter 61. Common Rules.

    ...and [the third] must be major if the harmony is to be happier and fuller. Should it be minor, as it so often is, the harmony will be sadder.

—Translated by Guy A. Marco and Claude V. Palisca, Gioseffo Zarlino. The Art of Counterpoint. Part Three of Le Istitutioni harmoniche, 1588 (New York: Da Capo Press, 1983), 15-16, 21-22, 196.

    It is established by experience, and those who have even the slightest ear for music cannot deny, that consonances in the above proportions [he has just given the just ratios] are very perfect and better than when one deviates from these true numerical proportions. And those who have dared to maintain the contrary and that the fifth does not consist in the ratio 3 to 2, either did not have an ear capable of judging it, or thought they had a reason for their opinion, but their conclusion was false.
Cited in D.P. Walker, “Seventeenth-Century Scientists’ Views on Intonation,” p. 113.