1 A LETTER TO A NOBLE WOMAN

Even though the study of intervals and their relations within the harmonic elements is intricate and difficult to embrace in a single account, and though I am at present unsettled and distracted by travel and its attendant haste, I cannot ignore your request. You have asked me to set forth, in brief, the main points of the science of music. And so, despite my present circumstances, I shall exert myself with all zeal, since it is you who have bidden me to do so, Your Ladyship.

What I send now is only a sketch, a manual to guide your memory — a light outline of matters explained more fully elsewhere — so that at a glance you may recall the principles that lie behind each chapter. It will lack the fullness and proof you deserve, yet it will serve as a chart until I can provide a more abundant and careful teaching.

When leisure and repose return to me, I shall compose for you a longer and more detailed introduction to these same subjects — one more closely reasoned and divided into several volumes. At the first opportunity I shall send it to you, wherever you and your household are residing. In order that it be easier for you to follow, I shall begin at the same point where your instruction first began, so that all may be consistent with what you have already heard.

2 THE TWO SPECIES OF THE VOICE

Now, Your Ladyship, know first that the voice, though one in genus, is divided by the ancients into two species: the continuous and the intervallic. These names are drawn from the character of each.

The intervallic voice is the voice of melody. It pauses at each note and makes the change between them distinct, free from confusion, and measurable by the magnitudes of the intervals that lie between. In this way it proceeds step by step, like a series of well-marked stations, each note separate and never dissolving into its neighbor. To a trained ear this species makes clear every tone and the size of the interval each tone contains. If anyone departs from this way of producing voice, he ceases to sing and begins merely to speak.

The continuous voice, by contrast, is that of ordinary speech and reading. In this kind we do not separate the pitches into explicit steps but run them together into a single flow, from our first utterance until silence. If someone in speaking or reading makes evident the sizes of the intervals by distinct steps, such a one is no longer speaking but singing.

Since the human voice has these two species, it also has two regions through which they pass. The region of the continuous voice is indeterminate, for it takes its limit from whatever point the speaker begins until he stops. It is, for the most part, regulated by us.

But the region of the intervallic voice is determined by nature and not by our will. It begins at the first sound perceptible to the ear and ends at the highest sound that can be produced by the voice. Our awareness of notes and of the changes between them begins at the point where our hearing first receives its stimulus, although fainter sounds may exist in nature beyond our perception. As with weight: a balance does not move under a few grains of chaff, but when enough is added the beam tilts and we recognize the first measurable quantity. So too with the singing voice: as faint sound is strengthened little by little, we calculate the first degree audible to our ears as the beginning of the region of song. But it is the human voice that fixes the end of this region, for we define the final limit where the melodic ascent reaches its utmost point.

Let it make no difference to us at present whether we speak of the human voice or of instruments—stringed, wind, or percussion—for these imitate the voice. We will pass over their differences for now, lest we disperse our inquiry at its very beginning.

Therefore, having now distinguished the two kinds of voice and the two regions they inhabit, you are prepared to learn of the intervals, ratios, and harmonies that arise from these foundations.

3 THE MUSIC OF THE PLANETS

Now, Your Ladyship, having grasped the two species of the voice and the regions through which they move, you must next learn the model from which our music takes its pattern. Among all things perceptible to sense, the motions of the seven wandering stars — the planets — are held to be the archetype of our harmonies. For our earthly music imitates theirs as image imitates model.

It is said that all swiftly moving bodies produce sound when what they pass through yields to them. These sounds differ from one another according to three factors: the weight of the moving body, the swiftness of its motion, and the position in which the motion takes place. These differences are seen most clearly in the heavenly bodies, for the planets differ from one another in size, speed, and orbital place as they whirl unceasingly through the aether.

Because of this, the ancient sages named the notes of our scale after the seven moving stars. From the furthest and slowest, Kronos, came the name Hypate (“highest up”), which is yet the lowest tone of the system. From the next, Zeus, came Parhypate (“next to the highest”). From Ares, who stands between Zeus and the Sun, came Hyper­mese or Lichanos (“above the middle” or “finger note”). The Mese (“middle”) came from the Sun, the fourth in order and the central place of the system. From Hermes, between the Sun and Aphrodite, came Paramese (“next to the middle”), also called Trite. From Aphrodite, who stands above the Moon, came Paranete (“next to the lowest”). And from the Moon, the swiftest and nearest to Earth, came Nete (“lowest”), which is yet the highest pitch.

Thus our seven notes mirror the seven planets:

These seven together form the heptachord, the ancient pattern of two diatonic tetrachords joined at Mese. In this way the heavens supply not only the names but the very structure of our scale. We shall give the detailed numbers and demonstrations of these intervals in a later chapter, but for now you must hold fast the principle: the order of the planets corresponds to the order of the notes.

And if you wonder why this majestic music of the spheres does not reach your ears, know that the tradition of the ancients says either that the sound is so constant and immense that we no longer perceive it, or that only a soul purified like Pythagoras could ever hear it. In due course we will address this question more fully.

Thus you see, student, that our music imitates the harmony of the heavens. For motion is the cause of sound, and the purest motion is that of the celestial bodies. As the slowest bodies produce the lowest notes and the swiftest the highest, so the Moon — nearest and fastest — sings the highest pitch, while Kronos — farthest and slowest — utters the lowest. Earth itself, as the unmoving center, produces no note but receives all.

In this way, the mathematical laws that govern the heavens also govern our science of music. The very spacing of the planets reflects the intervals of the consonances. Our instruments, voices, and scales are therefore shadows of this greater music, echoing the cosmic order that number and ratio establish among the stars.

Remember this well, for from this chapter onward we shall show how the exact numerical ratios underlying pitch and interval arise, and how they bind the whole universe in a single harmony.

4 The Discovery of Consonances

Now, Your Ladyship, having learned how our music reflects the harmony of the heavens, you must understand by what principle our notes are formed. For nothing in music is without order, and the principle of its order is number. We say that sound in general is the percussion of air moving unbroken to the ear, while a note is a pitch without breadth, a point on which the melodic voice comes to rest. An interval is the passage from low to high, or from high to low, and a system is a weaving together of more than one interval into a whole. Thus every note, interval, and system has its basis in motion and is measurable.

When a strong or ample blow strikes the air, the sound is loud; when a small or weak blow strikes it, the sound is soft. When the motion is even and steady, the sound is smooth; when uneven, harsh. When the movement is slow, the sound is low; when swift, it becomes high. By this you may see that the speed and strength of the motion determine both pitch and loudness.

These principles hold true not only in the voice but in all instruments. Among the winds—the aulos, the trumpet, the panpipes, and the hydraulic organ—the longer and wider the air-column, the slower the movement of the breath and the lower the note produced. Among the strings—the cithara, the lyre, the spadix, and others—the greater the tension and the finer the string, the quicker its return when plucked and the higher the note; the lesser the tension, the slower its return and the lower the note. In each case, greater size or length produces slower vibration and lower pitch; lesser size or length produces quicker vibration and higher pitch.

Some instruments share the properties of both kinds. The monochord, called by the Pythagoreans the canon, allows one to shorten or lengthen a single string and thereby measure the intervals exactly. The triangular harp likewise exhibits the principle of length and tension. In these instruments you can see plainly how number governs sound, by doubling or halving a string, by lengthening or shortening an air-column, by tightening or slackening tension.

From these examples you should recognize the universal principle that the properties of musical notes are regulated by quantity, and quantity is the property of number. All differences of pitch, loudness, and tone arise from measurable differences of length, mass, and speed. As Pythagoras showed, to each pitch corresponds a number, to each interval a ratio, to each system a pattern of ratios.

Hold this firmly in mind, for in the following chapters we shall show how these numbers, when set in order, produce the consonances and scales. In this way you will see not only that music imitates the heavens, but that both alike obey the same laws of number and harmony.

5 The Architecture of The Octave

Now, Your Ladyship—Pythagoras, perceiving that the concord of the fourth alone did not exhaust the order of harmony, sought to give the lyre its perfect measure by setting forth the consonance of the octave as a principle. For as long as the ancient heptachord endured, mese produced with hypate at one end and with nete at the other only the consonance of the fourth. Yet he judged that the extremes themselves should also sound together with the fullest concord, namely the octave, whose ratio of two to one could not be derived from the two tetrachords alone.

Therefore, with reverence for number and the proportions of nature, he intercalated an eighth string between mese and the note formerly called paramese, separating it from mese by a whole tone and from that other by a semitone. In this way he set the two tetrachords apart by a tone without disturbing their internal order, and thus gave to the whole system its perfect attunement, diapason. The string once called paramese now being counted the third from nete, trite by name and place, while the interposed string assumed the fourth place and restored to the system the full consonance of the fourth that mese had once formed with hypate. And as this new tone stood between the two tetrachords, whether reckoned above or below, it completed in each case the consonance of the fifth, which consists of the fourth and a whole tone together. Thus was displayed in audible form the ratios which reason had discerned: the epitritic ratio of the fourth [4:3], the hemiolic ratio of the fifth [3:2], and between them the whole tone in its sesquioctave ratio [9:8].

This act of Pythagoras formalized in teaching what had long been sensed in practice. For tradition had so consecrated the seven-stringed lyre that any addition seemed an impiety. The number seven, manifest in the heavens and in human affairs alike—the seven wandering stars, the seven sages, the seven gates of Thebes—was thought to carry the harmony of the cosmos within itself. Yet Pythagoras, appealing not to mere sense-pleasure but to the intellect, showed that the laws of number demand an attunement extending through all the notes, diapason, and that such attunement reflects the fitting-together of concordant intervals ordained by nature.

Others before him had striven for the octave. Terpander, first among them, is said to have contrived an octave on his seven-stringed instrument by altering the places of his notes rather than adding the forbidden string, and others after him, Simonides and Lycaon, openly attached an eighth string but incurred censure for their audacity. Only Pythagoras, in adding his string, escaped reproach, for he acted not to gratify the senses but to teach the immutable order of harmonia.

For he maintained that music’s virtue could be apprehended only by the mind and not judged by the ears alone, and that whatever is true of one octave holds true of all octaves in succession. Thus he limited his teaching to the octave as the perfect pattern, the archetype of all musical extension. By taking the two tetrachords of the heptachord and separating them by a whole tone without disturbing their internal form, he set forever the Dorian foundation from which all later Greek scales were to develop.

And thus we learn that the perfection of music lies not in multiplying its sounds at random but in discerning within them the ratios that bind them. The octave contains within itself the fourth and the fifth, and their difference, the whole tone. By the law of number, the whole tone thus stands as the measure between the lesser and the greater, and every consonance may be known as the fitting-together of these intervals. Therefore let the student of harmony, guided by Pythagoras, begin from the octave as from a first principle, for in it is reflected the true order of the cosmos and the harmony that rules all things.

6 Species and Inversions of The Tetrachord

Thus Pythagoras, guided by natural necessity and by reason, discerned how the octave is divided. You have already learned how the eighth string was added to the lyre. Now I shall show you how these eight notes are arranged in the diatonic genus.

First, know this: every tetrachord proceeds by a fixed order of intervals—first a semitone, then a whole tone, then another whole tone. This produces the consonance of the fourth, the epitritic ratio of 4:3. When you add yet another whole tone beyond this, you arrive at the fifth, the hemiolic ratio of 3:2. Thus the fourth plus the whole tone yields the fifth, and the fourth plus the fifth yields the octave in the double ratio of 2:1.

In the ancient heptachord, each note stood at the distance of a fourth from the one four degrees above it. This required the semitone to occupy, by transference, now the first, now the middle, now the third place within the tetrachord, according to the species of the scale. Thus, when you play upon the monochord or lyre, you may find these species of the fourth:

• Semitone + Whole tone + Whole tone
• Whole tone + Semitone + Whole tone
• Whole tone + Whole tone + Semitone

Each of these species preserves the total of two whole tones and one semitone but shifts their order. This is what we call the species of the fourth.

Now when we extend the heptachord to an octachord, whether you imagine it as a tetrachord and a pentachord joined in conjunction or as two tetrachords separated by a whole tone in disjunction, the principle remains the same. Each note will stand at the interval of a fifth from the one five degrees above it. This requires the semitone, as you ascend, to shift through four places. In other words, just as there are three species of the fourth, so in the octave there are multiple species of the fifth, depending on where the semitone lies.

To see this clearly, mark out the notes on your monochord. Begin with the lower hypate and ascend:

E — F — G — A — C — D — E

Here, from E to A is a fourth (E–F semitone, F–G whole, G–A whole). Add the whole tone to reach the fifth: E to B. Add the remaining fourth to reach the octave: E to E’.

Now shift the semitone forward by one step and you obtain another species:

E — F♯ — G — A — C — D — E

And again, shift it:

E — G♭ — G — A — C — D — E

Each time, though the semitone moves, the total structure of two tetrachords joined by a whole tone remains intact, and the octachord is preserved. This is what I mean when I say the semitone “wanders” or “transfers” its place but does not change its nature.

In this way, the octave embraces all the consonances of the fourth and the fifth, joined by the whole tone, and from these arise the multiple species and modes. For every mode or harmonia is nothing other than a particular ordering of these intervals within the octave. You may think of them as different paths over the same terrain, each with its own ethos and color, yet all arising from one and the same natural order.

Thus Pythagoras, by adding the eighth string, did not merely extend the lyre but revealed the natural divisions of the octave itself. He showed that the octave is a harmonia of fixed proportions, and that its interior notes, though many in appearance, are one in law and measure. This is why the diatonic genus, oldest and most natural of all, remains the foundation upon which every other genus is built.

7 Plato and the Harmonics of the Soul

It is useful, now that we have reached this point, to open at this opportune moment the passage in the Psychogony where Plato expressed himself as follows:

“So that within each interval there are two means, the one superior and inferior to the extremes by the same fraction, the other by the same number. He [the Demiurge] filled up the distance between the hemiolic interval [3:2] and the epitritic [4:3] with the remaining interval of the sesquioctave [9:8].”

For the double interval is as 12 is to 6, but there are two means, 9 and 8. The number 8, however, in the harmonic proportion is midway between 6 and 12, being greater than 6 by one-third of 6 (that is, 2), and being less than 12 by one-third of 12 (that is, 4). That is why Plato said that the mean, inasmuch as it is of the harmonic proportion, is greater and lesser than the extremes by the same fraction. For the greatest term compared with the smallest is thus in a double proportion; and so it follows that the difference between the greatest term and the middle is 4, compared with the difference between the middle and the smallest, which is 2; for these differences are in a double proportion, 4 to 2.

The peculiar property of such a mean is that, when the extremes are added to one another and multiplied by the middle term, a product is yielded which is the double of the product of the extremes; for 8 multiplied by the sum of the extremes, that is, 18, gives 144, which is double the product of the extremes, that is, 72.

The other mean, 9, which is fixed at the paramese degree, is observed to be at the arithmetic mean between the extremes, being less than 12 and greater than 6 by the same number (3). And the peculiar property of this mean is that the sum of the extremes is the double of the middle term itself, and the square of the middle term (which is 81) is greater than the product of the extremes (that is, 72) by the whole square of the differences, that is, by 3 times 3, or 9, for this is the difference.

One can also point out the third mean, more properly called “proportion,” in both the middle terms, 9 and 8. For 12 is in the same proportion to 8 as 9 is to 6; for both are in a hemiolic proportion. And the product of the extremes is equal to the product of the middle terms, 12 times 6 being equal to 9 times 8.

Thus Plato, through these means, exhibited the ordering of the concordant intervals and showed how the octave is bound together by fourths and fifths, as we ourselves have demonstrated through the ratios of the monochord. Yet the careful examiner knows it is otherwise: for in filling up the intervals Plato left a remainder, the so-called leimma, by which it is evident that the semi-tone is not exactly the half of a whole-tone, nor is there a single point which divides the octave into equal and opposite parts. But since our present discourse concerns the ordering of the larger consonances, we shall for now attend to the harmonious arrangement itself, and at another time return to what remains.

8 TUNING BY NUMBERS

Let us now return to the matter we began earlier and consider how one may verify with the senses what reason has already taught us. For the numbers which order the heavens also govern the trembling of strings and the breath within pipes. When you stretch a string or blow into a pipe, you are not merely making a sound—you are making number audible.

Take first the stretched string, for here the truth is easiest to see. A string fixed at both ends vibrates at right angles to its length, and its pitch is determined principally by three things: its length, its tension, and its thickness. If you keep the tension constant but shorten the length, the pitch rises; if you lengthen it, the pitch falls. Thus the length and the pitch stand in inverse proportion to one another. The thicker the string under equal tension, the slower its vibration and the lower its pitch; the thinner, the higher.

If, therefore, you pluck the whole string and compare it to the sound of only half that string—stopped at its exact center by a bridge—you will hear an octave. The sound of the half-string stands to the whole as 2:1.

Mark off one-third of the string and prevent the vibration from extending beyond that: the remaining two-thirds will yield the consonance of a fifth with the whole, or 3:2. Do the same with three-fourths, and you obtain the fourth, or 4:3. In this way the hand and the ear confirm what the mind has already deduced.

The same law holds with the aulos. Think of it as a pipe whose air-column vibrates lengthwise rather than crosswise. Its pitch too depends upon the length and thickness of its bore. When all the finger holes are closed, you hear the fundamental note. Lift your finger from the middle hole, and the air-column shortens: you will hear the octave above. Lift the finger from the hole next to the end, and you will find the proportion of a fifth; compare this to the whole, and you find the fourth. The pipes of the syrinx work likewise. Longer tubes, or those with more air in them, sound lower; shorter or narrower tubes sound higher.

Consider also the weight of a string, for that corresponds to the volume of air in a pipe. A string of two strands emits a sound in duple proportion to one made of four strands, just as a narrow pipe produces a pitch different from a wide one even when both are of the same length. The tension of the string or the pressure of the air may alter these results slightly, but the underlying ratios remain.

Thus every well-made instrument—string or pipe—reveals the same order. The octave, the fifth, and the fourth are not opinions but necessities, grounded in the very physics of vibration. When you stop the string at its half or third or fourth, or when you uncover a hole at the appropriate place on a pipe, you are making visible and audible the invisible numbers. And the careful examiner will know it is otherwise than chance: for in all cases the numbers 2, 3, and 4 stand behind the sweetness of sound.

In this way you will see that our art of tuning is nothing other than the art of measuring. What we call music is but the shaping of air and string by number, and the same proportions that form the soul of the cosmos also guide your hands when you tune your lyre or blow into your aulos.

9 THE DOUBLE OCTAVE

Now that you understand how one octave is formed by ratios and how the notes of the tetrachord take their places, let us widen our gaze to embrace the double octave. For just as the heavens extend beyond the bounds of a single sphere, so the voice extends beyond a single octave. The ancients called this wide arrangement the “Greater Perfect System,” and in the diatonic genus it is built from five linked tetrachords and a single extra note at the bottom.

Think first of the heptachord you already know: two conjunct tetrachords sharing the middle note mese. Above this familiar span the older theorists attached another tetrachord, the hyperbolaion, whose notes “overshoot” the original high point. Below the heptachord they added the hypaton, a mirror image on the low side. In this way the range of the voice was extended both upward and downward until it spanned two full octaves.

Each of these tetrachords kept its own internal structure—semi-tone followed by two whole-tones—and each acquired its own set of names so the musician could tell at once which register he was in. Above the old nete came trite hyperbolaion, paranete hyperbolaion, and nete hyperbolaion. Below the old hypate came hypate hypaton, parhypate hypaton, and lichanos hypaton. Between the familiar lower and upper tetrachords a fifth tetrachord called synemmenon could be inserted to link ranges smoothly and give the singer or kithara player alternate pathways between pitches.

The ancients also found it necessary to add one last, lowest note, called proslambanomenos, “the one added,” a whole tone below hypate hypaton. This ensured that the system on either side of mese was truly an octachord and that mese itself was the genuine middle, the eighth from either end. When arranged in this way, the whole system becomes perfectly symmetrical: four linked tetrachords, plus the proslambanomenos, yielding fifteen notes and a double octave in the proportion 4:1.

It is worth remembering that the Greek singer or lyre-player did not think of this as a “scale” in our modern sense but as a living terrain of fixed points and movable steps. The guiding degree was always mese; from it the tetrachords could be reckoned upward or downward. The specific modes or harmoniai (what later writers called the species of the octave) were carved out of this larger framework like constellations out of the night sky.

This double octave, with its carefully named notes and its chain of tetrachords, is the culmination of our study of the diatonic genus. In my more extensive treatment I will show you how the same system can be mapped in the chromatic and enharmonic genera, and how the so-called Pythagorean canon divides at its twenty-seventh multiple. But for now you should simply fix in your mind the structure: five tetrachords, one added note, and the symmetry around mese. For only then will the larger consonances reveal their full order, and you will see how the art of the ancients joined the extremes of pitch into a single harmonic cosmos.

10 THE THREE GENERA

Let us now bring together all we have learned and examine how the notes progress and divide themselves across the three genera. Only by setting them side by side can you understand how our earlier discussions of the tetrachord, the octave, and the double octave all resolve into a single view of melody.

A single note, as you have already learned, is like a point: indivisible in its pitch, a monad of sound. An interval is the space between two such points. A relationship is the ratio measuring that space, while the difference is the excess or deficiency between the two pitches. Do not confuse relationship with difference: the ratio 2:1 has the same “difference” as 1:2 but a contrary relationship — one is twice as large, the other only half as large. This is why the ratios, and not just the raw distances, govern harmonic order.

A system is a weaving-together of intervals. Within it, no two neighboring notes are perfectly consonant; they are always set off from each other. Consonance arises only when several notes together make a stable proportion — the fourth, the fifth, or the octave — which the ear blends as if into a single voice. This is why the ancient theorists always returned to the tetrachord, the first and simplest consonant span, for it contains the seed of all three genera.

In the diatonic genus — which we have used as our standard so far — the four notes of the tetrachord are arranged as:

• a semi-tone, then
• a whole-tone, then
• another whole-tone.

In the chromatic genus, the span contracts inward:

• a semi-tone, then
• another semi-tone, then
• a larger incomposite third (a trihemitone).

And in the enharmonic genus, the contraction is at its greatest and subtlest:

• a quarter-tone, then
• another quarter-tone (together equaling a semi-tone), then
• a remaining large ditone.

Thus all three genera still cover roughly “two whole-tones and a semi-tone” but distribute the steps differently. The diatonic moves by tones, the chromatic by “colors” of semi-tones, and the enharmonic by micro-steps beyond ordinary hearing. The extremes of the tetrachord never change — they are the fixed notes — but the middle notes slide inward or outward depending on genus, earning them the name movables. In this way the chromatic stands between the diatonic and the enharmonic, deviating only by a single semi-tone from the first but much more from the second.

You may now also see why the octave is not six whole-tones, as some of the moderns believe, but five whole-tones and two semi-tones. If the semi-tones were truly halves of a whole-tone, then indeed two of them would equal one tone and the octave would be six tones; but the ratios do not permit this—the two semi-tones that do not fuse neatly into a tone.

Please forgive the haste and brevity of these words — for you know well that you summoned me while I was still unsettled on my journey. Yet, trusting in your kindness and your most intelligent and gentle nature, I have sent you these first fruits of my labor as a token of friendship. When at last I am granted the calm and leisure that proper study demands, I shall compose for you a more ample and complete treatise, setting forth in full the matters I have here only outlined. At the earliest opportunity I will send it to you, wherever you and your household may be.

In this, Your Ladyship, as in all things, I remain your guide in the path first illuminated by our mutual and eternal friend through the ages.