[Margin: The Origin of Music, from Jubal.]

[I.] How great a kinship there is between the Pastoral [shepherd’s] art, the Hammering or Iron-smithing art, and the Musical art may be traced from the very first origin of them, since it is established that these three arts were born in one and the same house of Lamech (who was of the stock of Cain) at a near, if not at a single, birth. For holy Moses narrates (Genesis ch. 4), concerning Lamech: “Who took two wives, the name of the one Adah, and the name of the other Zillah. And Adah bore Jabal, who was the father of those dwelling in tents, and of shepherds; and the name of his brother [was] Jubal, who was the father of those who play on the cithara and the organ. Zillah also bore Tubal-cain, who was a hammerer and smith in every work of bronze and iron.” — As though even then Music exercised its powers both among the soft kind of songs and sounds, which suits shepherds, and among the hard, which suits the hammerers of metals; and as though it were given to understand that neither can [gentle] minds rightly [be moved], nor fierce [tempers] be ruled or tamed, without a certain harmony.

[Margin: Pythagoras the second discoverer of the consonances.]

And although Berosus the Chaldean and Josephus the Hebrew affirm that the Musical consonances were discovered by Jubal, while with attentive ear he observed the sound of Tubal-cain’s hammers; yet afterward — whether that art suffered shipwreck in the Flood, or lay hidden among a few Hebrews — it is said to have arisen again under Mercury, the inventor of the lyre [testudo] (as Diodorus Siculus and Lucian think), or under Apollo, the inventor of the Lyre (as Lactantius prefers), or from Amphion (as Pliny has it). But at last it was reborn to its pristine birth in the iron-smith’s workshop, and — as it were half-buried — being roused again by the sound of hammers, it revived. For, as Macrobius reports (bk. 2 on the Dream of Scipio, ch. 1), and from him Boethius: when Pythagoras, passing by a workshop in which smiths were softening glowing iron with their strokes, had heard from the sounds of the hammers (answering one another in a fixed order, and by the alternation of high and low) that something harmonious fell upon his ears, in order to explore whence that consonance arose — whether from the strength and vigor of the smiths’ arms, or from the weight of the hammers themselves — he ordered the smiths to exchange the hammers among themselves; and since he heard the same harmoniousness, he did not doubt that it must be ascribed to the diversity of the weights, especially since, when other weights were added, he obtained diverse and not so consonant sounds. Having therefore examined the weights of the hammers, he found that sound stood to sound just as weight to weight of the same material. Then, turning from hammers to strings made from the intestines of sheep and the sinews of oxen, he hung upon them weights in the proportion which he had detected in the hammers; and since those too rendered a similar consonance, he established that universal axiom:

[Margin: The Musical Axiom.] As magnitude to magnitude, so Sound to Sound.

[II.] Furthermore, among the several consonances [thus found], he received into the Canon [scale] only those which are simplest — because (as Hérigone says, in the Introduction to Music) he wished to remove from things all confusion and inconstancy, and, like a bee, to gather the purest [flowers] and to sip only the dew of the concords. Wherefore he judged that the musical consonances are not to be estimated from the unreliable and inconstant arbitration of the ears, but from certain causes and the proportion of numbers; for he reckoned simplicity to be the mother of consonance, but composition and mixture to be the source of dissonance, and so of inconstancy and uncertainty. Hence it came about that he received into the Canon only those consonances which are drawn from the Multiplex or Superparticular Proportion, within the quadruple inclusive, and no further: namely those between 2 and 1 [the Octave], between 3 and 2 [the Fifth], between 4 and 3 [the Fourth], between 3 and 1 [the Twelfth], and between 4 and 1 [the Double-octave] — which consonances almost alone Boethius too approved (bk. 2, from ch. 16), with the Tone added as a consonance (as he himself affirms, bk. 1), namely by the ratio 9 to 8, or 18 to 16, etc. Furthermore, from the quaternary of numbers within which Pythagoras kept himself, arose the Tetractys, or Pythagorean Quaternary, by which — as by a perennial fountain of the beauty of the intellectual soul — they were wont to swear; of which Zarlino treats more learnedly (Harmonics, ch. 2), and Kepler (bk. 3 of the Harmonics, part 2, ch. 4).

[Margin: The Canonical and Harmonic Musicians.]

Hence likewise arose the sect of the Canonical Musicians, who were called Canonici, because in determining the harmonic intervals and consonances they attributed more to the ratios of numbers than to the senses; who are also called Pythagoreans. To which sect was opposed the sect of the Harmonics — namely of the Aristoxenians — who attributed more to the sense of the ears than to the ratios of numbers abstracted from sensible matter. Among whom Ptolemy, walking a middle way, gave as much to the senses as Aristoxenus, and as much to the ratios as Pythagoras, and therefore approved more consonances than Pythagoras — though not all that Aristoxenus did. And so Ptolemy (bk. 1 of the Harmonics), having said that the harmonic faculty of the soul is to be recognized in discerning the difference of low and high in sounds, and that sound is an affection of struck air, [adds that] the judges of harmony are Hearing and Reason — under the same condition, but Hearing [directed] to the [material], and Reason to the form and cause, and in general: for to the senses it is peculiar that they find the approximate but receive the exact [from reason], whereas Reason receives the approximate [from sense] but finds the exact. From ch. 6 he corrects the Pythagoreans, from ch. 9 the Aristoxenians, from ch. 13 Archytas the Pythagorean, and from ch. 15 (and through the whole [second] book) [he treats] of Reason and Hearing. And the same [Ptolemy], in bk. 1, ch. 2, after he had compared the harmonist with the astronomer in this — that each must, as it were a priori, know [the causes] of the apparent motions, so that the things which appear to the senses be saved — adds: “But in these errors both the Pythagoreans and the Aristoxenians were deceived: for the Pythagoreans, not following the comparison [with sense] in all the cases in which they ought, fitted to the differences of the sounds ratios least congruent with certain experiments [when these were] employed; wherefore they brought a calumny upon this their judgment among the men of the other sect. But the Aristoxenians, attributing too much to those things which they had received through the senses, misused Reason,” etc.

Yet not even Ptolemy himself so [corrected the two sects] as to leave nothing for later Musicians to perfect. For whereas the Pythagoreans received only the five consonances called Diapason [the Octave], Dia-tessa-…

[…continues on p. 508 (PDF 543) with the catchword “tessa-” — “Dia-tessaron” (the Fourth) and the rest of the five Pythagorean consonances, still within Chapter IV.]

(printed p. 508 — Chapter IV continued. The nomenclature of the consonances is completed — the five Pythagorean consonances, the Aristoxenian additions (thirds and sixths), and Ptolemy’s further admissions — followed by Kepler’s critique that number alone cannot explain why some ratios are consonant, and his derivation of the consonances from divisions of the circle by regular figures. A sub-section then gives definitions and axioms pertaining to harmonics: consonance, dissonance, unison, the grades of consonances, and the praise of the Senary.)

[Header: BOOK IX. SECTION V. — 508]

…Dia-tessaron, Diapente, Diapason-diapente, and Disdiapason — which the modern Musicians call the Octave, the Fifth, the Fourth, the Twelfth, and the Fifteenth — the Aristoxenians, for their part, had adopted the Ditone, Semiditone, and the greater and lesser Hexachord (that is, the major Third, the minor Third, the major Sixth, and the minor Sixth), which Ptolemy rejected as not melodious, although all Musicians [now] receive them; and in turn he admitted among the consonances the Diapason-with-Semiditone and the Diapason-with-Ditone, and the Diapason-diatessaron — that is (to speak with the moderns) the minor Tenth, the major Tenth, and the Eleventh, which are harmonious. But besides these he added that which [he reckoned] harmonious between 6 and 7, and that between 7 and 8, and others like them, which are abhorrent to the ears and to the practice of singing; and he omitted the minor and major Third, which today all Musicians accept.

[Margin: The errors, and the cause of the errors. — Ptolemy’s consonances.]

[III.] Kepler (bk. 3 of the Harmonics, p. 8) judges the cause of Ptolemy’s error to have been that he sought the principle of the consonances from abstract numbers, such as the numbers are; and yet a reason cannot be rendered by numbers why the numbers 1, 2, 3, 4, 5, 6, etc. concur to [form] harmonic intervals, but 7 and 13 and the like do not. For indeed that which in Music is consonant arises from sound, and sound from the quantity of the sounding body — from continuous, not discrete, quantity. Wherefore, as Zarlino notes (part 1, ch. 41), and Valgulius (on Plutarch’s Music), and Kepler (bk. 3 of the Harmonics, p. 9), one must seek the causes of the consonances from continuous quantity, since number is not their proximate cause; rather [the cause is] the proportion between the high and the low sound. Granted that that proportion, being rational, is expressed by numbers, and that by numbers the differences of the sounds are found more exactly than by dividing a continuous quantity with compasses.

[Margin: Whence Kepler derives the Consonances.]

[IV.] For the aforesaid causes, therefore, Kepler — having investigated the cause of the consonances in continuous quantity — sought them from the division (or divisions) of a circle into aliquot parts, which he demonstrated can be done Geometrically and scientifically: that is, from the regular and demonstrable plane Figures, so that those sides of the figure, compared with the whole circumscribed circle, [yield] certain consonances which are knowable and have their own demonstration, as we shall show in the Scholia of this chapter. Meanwhile, let us set forth the more common doctrine.

Certain Definitions and Axioms pertaining to Harmonics

[Margin: What is Consonance?]

[V.] CONSONANCE (in Greek συμφωνία, symphonia) is a Ratio between a high and a low sound, pleasant to the hearing — as is gathered from section 19 of Aristotle’s Problems; or, as Severinus Boethius defines it (bk. 1, ch. 8): “It is a mixture of high and low, falling sweetly and uniformly upon the ears,” to which definition subscribe Glareanus (bk. 3 of the Dodecachordon, ch. 9), Zarlino (part 2 of the Harmonic Institutions), and Kircher (bk. 3 of the Musurgia, in the preface). Otherwise, however, Boethius said: “Consonance is a concord of dissimilar voices reduced into one” — doing this, as some think, so that by the first definition he might cleave to Plato (who referred consonance to likeness), and by the second to Nicomachus (who [referred it] to unlikeness). But Daniel Barbaro (on Vitruvius, bk. 5, ch. 4) said: “Consonance, or Concentus, is a tempered mixture of low and high sounds, pertaining with pleasantness to the ears, arising from the comparison [of a multiplex or superparticular ratio].”

[Margin: Dissonance. — What is a Phthongus? — Unison.]

DISSONANCE, by Kircher’s definition, is the unpleasant perception of two sounds mingled with one another. Phthongus (in Greek) is a musical sound; it is called emmeles [in-tune] if [it falls] by [a fitting] case [interval]. Unison is an altogether equal sound, or a twin equal [sound] — which is, in Music, as the point is in Geometry.

There are agreed among the experts in Music these Axioms: (1) A part [of a string] gives a higher sound than the whole; (2) The whole harmonic [string] gives a lower sound than the part — namely because the air, constrained by a part, is vibrated more quickly than by the whole, and more frequently; (3) the higher sound is the quicker, and a large, long, and thick body gives a lower sound. — Tension (in Greek τάσις, tasis) is the state of a voice set in a tone fit for singing. Intension of a string (Greek ἐπίτασις, epitasis) is motion toward a higher sound; but Remission (Greek ἄνεσις, anesis) [is motion] from high to low. Elevation (Greek ἄρσις, arsis) is the pronunciation of a syllable or word with the acute accent; θέσις (thesis) is pronunciation with the [grave] accent.

A Simple Consonance is one which does not contain another consonance; a Composite, one which contains other consonances. By which reckoning the Diapason, the Diatessaron, and likewise the Ditone and Semiditone, and the greater and lesser Hexachord, are called simple; but the Diapason-diapente, the Disdiapason, the Ditone-with-diapason, and the rest of this kind are called composite. A Perfect Consonance is one which so affects the hearing that the appetite rests in it and desires nothing further; an Imperfect, when it does not [bring the appetite to rest], but [the appetite] still desires something further (as Fogliano teaches, Musica Theorica, ch. 5).

Furthermore, among the consonances themselves there are certain grades. For those which arise from the Multiplex or Superparticular genus, and do not extend beyond the ternary, and are simple, are the sweetest and in the first grade — namely the Diapason, whose ratio is 2 to 1, and the Diapente, whose ratio is 3 to 2 (of which the Diapason is the more perfect). But those which arise from the Multiplex or Superparticular genus yet extend beyond the ternary, or [are] within the ternary but composite, are in the second grade — as the Diatessaron, whose ratio is 4 to 3, the Disdiapason (i.e. 4 to 1), etc. And all the rest are in the third grade — which are indeed simple, and arise from the genus of Superparticular proportion, but extend up to the Senary [6] inclusive, beyond the quaternary: of which kind are the Ditone, whose ratio is 5 to 4, and the Semiditone, whose ratio is 6 to 5. All the others are imperfect, either because they do not arise from the Multiplex or Superparticular genus, or because they are beyond the Senary.

[Margin: The Praises of the Senary.]

For although not all the proportions of numbers contained within the Senary beget perfect consonances, yet all the perfect consonances, considered in their least and radical terms, are contained within the Senary — which the Senary, therefore, Zarlino deservedly extols (part 1 of the Harmonic Institutions, chs. 15 and 16), and Athanasius Kircher (bk. 4 of the Musurgia, ch. 4, from p. 186). These perfect [consonances], which are simple, are born from the proportions among the numbers of the Senary [1–6], taken in order, as you see in the following little table — although Zarlino afterward excludes the Ditone and Semiditone from the perfect, that is, from the more perfect:

But Kepler (bk. 3, ch. 2, of the Harmonics) [recognizes] only six [consonances], or, with the Unison, seven.

[Margin: What is a dissonant interval?]

Dissonant Intervals — but Musical and Concinnous [in-tune ones] — are the differences of the Consonances, or parts of them, which, although accord-…

[…continues on p. 509 (PDF 544) with the catchword “secun-” (secundùm) — the definition of the musical dissonant intervals, and the Table of Dissonant Intervals, still within Chapter IV.]

(printed p. 509 — Chapter IV continued, then the first great summary table. The definition of dissonant intervals is finished — they are the degrees by which one passes between consonances — and the rule for finding the difference between two consonances is worked, yielding the sesquioctave 9:8 or greater Tone. The layout of the coming table is explained (names, proportions, species, and genus of each interval), with a philological caution on the Greek names Diapason, Diapente, and Diatessaron; then follows the Table of Consonances in 23 rows.)

(…continuing the definition of dissonant intervals broken off on p. 508:) …which, although in themselves they are not Consonances, are nevertheless degrees by which one ascends or descends from one consonance into another, and serve for discerning them and for composing them aptly — just as the cartilages [serve] the bones.

[Margin: How the difference of Consonances is to be investigated.]

Further, that the difference between two Consonances may be found, and that its proportion-species may be assigned to it, the greater number of the one consonance is to be multiplied by the lesser of the other, and again the greater number of the other consonance by the lesser of the first; for the two numbers thus produced will be the terms of the proportion sought, that is, the difference of those consonances. For example, let the difference between the Diapente, that is 3 : 2, and the Diatessaron, that is 4 : 3, be sought: 3 multiplied by 3 makes 9, and 2 multiplied by 4 makes 8; this proportion, then, of 9 to 8 — that is, the sesquioctave — is the difference between the Diapente and the Diatessaron, and the one which below we shall teach to be called the greater Tone; and so of the rest.

[Margin: Explanation of the following Tables.]

[VI.] It remains that we bring together into a Table the Consonances, perfect and imperfect, and then the Dissonant but concinnous Intervals — the Unison being omitted, since it is the beginning of consonances and not a consonance. In which Table we shall set down the names — Graecolatin, Greek, Italolatin; then the numbers of the proportions in their least and radical terms (which, if each be multiplied by one and the same number, will yield other consonances of the same species, [extendable] to infinity — as if you multiply the Diapente’s terms 3, 2 by 10, there result 30, 20, between which is likewise the same species of proportion and the same Consonance); and at last we shall add both the Denomination of the species of whatever proportion, and the Genus of the proportion itself, indicated by these five marks M. S. s. MS. Ms. — of which M signifies Multiplex; S (capital) Superparticular; s (small) Superpartient; MS Multiplex-superparticular; and Ms Multiplex-superpartient.

[Margin: Things to be noted in the nomenclature of the Consonances.]

But one must beware of taking the words Diapason, Diapente, Diatessaron as nominative cases of a singular noun, and inflecting or declining them, as certain unskilled persons do — thus: Diapason, Diapasondos; Diatessaron, diatessarondos; Diapente, diapentes. For in Greek they are genitive plural cases with the preposition διά; and so in Greek they are written διὰ πασῶν, διὰ πέντε, διὰ τεσσάρων — of which the first signifies through all (or about all), the second through Five, and the third through Four. For the Diapason goes through all the concords, and contains in itself radically all the more perfect consonances; and because after seven chords we return to it [as the eighth], it is called by the Practical musicians the Octave. The Diapente is called the Fifth, because it is between voices distant by the fifth degree of the chords, as will be clear from the Diatonic system, which we shall set down in chapter 6, number 3 and 5 — just as the Diatessaron [is called] the Fourth, and the Disdiapason the Fifteenth, etc. Finally, by the opinion of Macrobius (bk. 2 On the Dream of Scipio, ch. 1), the Diapason consists of six tones; the Diapente of three tones and a hemitone; the Diatessaron of two tones and a semitone; the Diapason-diapente of nine tones and a hemitone; and the Disdiapason of twelve tones. With whom agrees Censorinus (On the Birthday, ch. 11) — which will be shown to be true from what is to be said in ch. 6, num. 3, toward the end of the Diatonic System. These things explained, let the Table now stand.

I. TABLE OF CONSONANCES

(I. Tabula Consonantiarum)

Columns: Order · Nomenclature (Graecolatin / Greek / Italolatin) · Proportion in least terms · Species · Genus (M = Multiplex, S = Superparticular, s = Superpartient, MS = Multiplex-superparticular, Ms = Multiplex-superpartient).

PERFECT consonances

IMPERFECT consonances

CONTROVERSIAL, and admitted by few

[Translator’s notes on the Table: (a) the marks in the rightmost column are Riccioli’s five genus-abbreviations as explained in ¶VI. (b) The ”✠ ✠” in the Species column for № 15 (4096 : 2187, the Pythagorean diminished octave = 2¹² : 3⁷) and № 17 (45 : 32, the tritone) stand in the original where no compact species-name could be given — their superpartient denominations would be impossibly long. (c) The Italolatin names of № 6–7 and 8–9 carry the Italian “dura / molle” (hard / soft) for major / minor. (d) Rows 20–21 (Ptolemy’s 7 : 6 and 8 : 7) and 22–23 (added by Zarlino and Kepler) have no traditional Greek or Italian names, hence the dashes.]

[Below the table, the catchword “II. Tabu-” points to p. 510 (PDF 545), which opens with the Second Table — “Intervalla Dissona, sed Concinnitati servientia” (Dissonant Intervals serving Concinnity: the Tones, Semitones, Diesis, Limma, Comma, Schisma, etc.), followed by Question 1 (which consonances each authority — Pythagoreans, Aristoxenus, Vitruvius, Barbaro, Ptolemy, Euclid, Zarlino, Fogliano, Mersenne, Kircher, Kepler — accepted), Question 2 (why consonances arise rather from ratios of greater than of lesser inequality), and Question 3 (whether a Tone can be divided into two equal semitones).]

(printed p. 510 — Chapter IV continued. The Table of Dissonant Intervals lists fourteen sub-consonant intervals — the greater and lesser Tone, five semitones, the enharmonic Diesis, the Platonic Limma and Apotome, the commas, Schisma, and Diaschisma — with their ratios and genera. Three Questions follow: which consonances each authority accepted; why consonances arise from ratios of greater rather than lesser inequality; and whether a Tone can be split into two equal semitones, which most deny.)

II. TABLE OF DISSONANT INTERVALS

(II. Tabula. Intervalla Dissona, sed Concinnitati servientia continens — “Containing the Dissonant Intervals, yet such as serve Concinnity”)

Columns: Order · Varied Nomenclature (with the defining difference) · Proportion in least terms · Species · Genus (S = Superparticular, s = Superpartient).

From what has been said it is clear that “diesis” is a very equivocal name, since some have used it for one and others for another species of semitone. Macrobius, however (bk. 2 on the Dream of Scipio), says — from the usage of more recent writers — that the Diesis is a sound less than a semitone; and the Practical musicians now, where they wish a half-voice to be employed, append the sign of the diesis.

[Translator’s notes on Table II: (a) Several entries are printed in doubled terms (e.g. № 2’s “20 : 18,” № 3’s “54 : 50”) — the un-reduced products that arise when the interval is computed as a difference by cross-multiplication; the reduced value follows in parentheses. As a result № 3 (Semitonium maius) and № 7 (Semitonium maximum) coincide at 27 : 25, as printed. (b) The blank Species cells (№ 9, 10, 12, 13, 14) are blank in the original — these ratios (256:243, 2187:2048, 10240:10125, 4352:4330, 162:160) have no compact superpartient name. (c) Latin species-names are kept as in the original; e.g. “Superbipartiens vigesimasquintas” = 27:25 (1 + 2/25), “Superseptupartiens centesimas-vigesimas-octavas” = 135:128 (1 + 7/128).]

Question 1. Which Consonances did the Authors listed below accept?

(Quaestio 1. Quasnam Consonantias acceptarint Authores infrascripti?)

[VII.] That we may answer the question more briefly, in place of the names we shall set down the index-numbers of the consonances, to be sought in Table I, column 1. Besides the Consonances, moreover, nearly all admitted the Tone, the Semitones, the Diesis or Limma, and not a few the Apotome.

…as will be clear from what is to be said in ch. 6, num. 5 — granting that among the perfect he had earlier numbered the Unison and six other consonances, namely 1. 2. 3. 4. 5. 8. 22, as is plain from his bk. 3, ch. 2.

Question 2. Why do Consonances arise rather from Proportions of Greater than of Lesser inequality?

(Quaestio 2. Cur Consonantiae oriantur magis à Proportionibus Maioris, quàm Minoris inaequalitatis?)

[VIII.] For it is commonly said by Musicians that the Diapason arises from the double proportion rather than from the subduple, and the Diapente from the sesquialtera rather than from the subsesquialtera — although between the same terms 2 and 1 there is both the proportion of 2 to 1 (double) and of 1 to 2 (subduple), and so on. Lodovico Fogliano (sect. 2 of the Musica Theorica, ch. 3) and Zarlino (part 2 of the Harmonic Institutions, ch. 50) answer that the cause is this: that in a proportion of greater inequality the greater term is compared with the lesser in a more perfect way — namely as the container, which has the character of form, with the contained, which has the character of matter; and because the relation of excess or majority is positive and real, whereas the relation of defect or minority is nothing but a privation, and is not a real relation but [a relation] of reason.

Question 3. Whether a Tone can be divided into two equal Semitones?

(Quaestio 3. An Tonus dividi possit in duo Semitonia aequalia?)

[IX.] Most deny it — chiefly Macrobius (bk. 2 on the Dream of Scipio, ch. 5), Boethius (at the beginning of bk. 3), Glareanus (bk. 1 of the Dodecachordon, ch. 10), Daniel Barbaro (on bk. 5 of Vitruvius, ch. 4), and Kircher (bk. 3 of the Musurgia, axiom 10). For just as a semivowel is not so called because it divides a vowel into two equal parts, so the semitone is so called because it is indeed between the extremes — yet not because it is the exact half of a tone. And the reason is that the Tone is in the superparticular genus, since its proportion is 9 to 8; but no superparticular proportion can be divided into two equal parts. Wherefore Barbaro reproves Aristoxenus, who — following the judgment of the ears rather than the reasons of numbers — divided the Tone into two equal parts; and Barbaro says it is cut into two unequal parts, of which one is called the greater hemitone or apotome, the other the lesser hemitone or Diesis. But Carolus Valgulius, in [his commentary] on Plutarch’s Music, defends Aristoxenus, and teaches that, although the unit lying between 9 and 8 cannot be cut into two…

[…continues on p. 511 (PDF 546) with the catchword “duas” (duas [partes] — “into two [equal parts]”): the completion of Valgulius’s defense of Aristoxenus, still within Question 3 of Chapter IV.]

(printed p. 511 — Chapter IV continued. Question 3 closes (the string, being continuous, can be halved, though the ratio cannot be rationally divided); Question 4 explains why the Diapason contains the whole diversity of consonances and is their queen; and Question 5 asks by what artifice the consonances are distinguished by ear, introducing the Monochord and its first method of division by the sum of the ratio’s terms, closing with a two-part table of the division of the monochord.)

(…completing ¶IX from p. 510, on whether a Tone can be cut into two equal semitones:) …into two whole parts; yet [Valgulius teaches] that the string or chord itself — on which, as on a ruler, by partitions duly made the concords are formed — since its magnitude is continuous, can be cut anywhere, and so into two equal parts; and that Aristoxenus was not so unskilled in Arithmetic, nor ignorant of the doctrines of the Pythagoreans, since he wrote whole volumes on Arithmetic and had as his teacher Xenophanes, a noble Pythagorean. But Fr. Athanasius Kircher (bk. 3 of the Musurgia, ch. 12, propositions 1 and 2) teaches that a Tone can be divided into two equal parts by irrational numbers, but cannot by rational ones. And he adds that whoever, with Philolaus and Boethius, posits the half of a Comma — namely the Schisma — cannot deny the half of a tone; whom consult, if you desire more on this matter.

Question 4. Why does the Diapason contain in itself, or terminate, the whole diversity of the Consonances, and is the Queen of Consonances?

(Quaestio 4. Quare Diapason totam diversitatem Consonantiarum in se contineat, aut terminet, & sit Regina Consonantiarum?)

[X.] The Diapason — in Greek διὰ πασῶν, that is, through all, or to all, or about all — is so called because all consonances are either contained in it or terminate at it. For although outside it various harmonic proportions are found, yet according to sense, and considering the judgment of the ears, they seem to have almost no diversity from those contained below the diapason — indeed, not even from the diapason itself; nor is any wholly new consonance found in which something of the Diapason is not perceived. For the ditone seems to affect the sense in the same way as the diapason-plus-ditone; and the semiditone as the diapason-plus-semiditone; so the diatessaron and the diapason-diatessaron, and so of the other consonances.

[Margin: The Diapason, the most beautiful of Consonances.]

Hence it comes about that, once the seven simple consonances — discrete from one another by a sensible difference — are completed (namely Diapason, Diapente, Diatessaron, Ditone, Semiditone, the greater and the lesser Hexachord), a repetition of the consonances is made, and all their diversity terminates at the diapason. And because all easily discern it, and are refreshed by it, it is called by Aristotle (sect. 19, problem 35) καλλίστη συμφωνία — that is, the most beautiful Consonance. And this too belongs to it singularly: that, added to a like consonance, it generates other consonances to infinity — namely disdiapason, trisdiapason, quaterdiapason, etc. But if to a Diapente you add a Diapente, no consonance is begotten; just as neither [is one begotten] from other [consonances] of the same species added together, as Aristotle teaches (sect. 19, problem 42). The rest, to the praise of the Diapason, read in Aristotle (sect. 19, problems 17 and 32).

Question 5. By what artifice can all the Consonances be distinguished by the hearing itself; and how is the Division of the Monochord to be made?

(Quaestio 5. Quo Artificio Disterni possint ipso auditu Consonantiae omnes; & quomodo Monochordi Divisio sit facienda?)

[XI.] We have already learned above, from Macrobius, that Pythagoras hunted out the consonances from the sound of the various hammers, and then from strings — that is, from sinews and gut-strings stretched and hung with various weights, in that proportion which he had detected in the hammers — so that the tensions of the strings, [though] of the same kind and thickness, would turn out different according to the diversity of the weight hung on them.

[Margin: The notion and definition of the Monochord.]

But in all respects the Monochord is preferable, because it is difficult to find several strings of exactly the same kind, thickness, and uniformity, or to stretch them with precisely such a diversity [of tension] as the exact distinction of the consonances requires. Now the Monochord is so named from μόνος, which means single or solitary, and from χορδή, which [means] string — because it is an instrument containing a single stretched string, which Ptolemy and Boethius call the Harmonic Rule (Regula Harmonica). This instrument Suidas calls Magadea, from the nominative Μαγάς; and he says it is a squared wood, hollow within, containing strings fit for grasping the variety of tones. For it can consist of two strings, of which one is always [taken] as the whole, and the other exhibits the part due to the consonance; but because the reckoning by a single string is simpler, Guido of Arezzo defined it thus: “A Monochord is a long squared wood, hollow within, with a string drawn over it, by whose sound we apprehend the varieties of tones.”

[Margin: What the Magas is.]

But Magas — or, as others have it (in Barbaro above, and Kircher bk. 4, ch. 1), Magadis — is usually taken either for a double support or little bridge, hemispherical and immobile, which bounds and sustains the extremities of the whole string; or for a third little bridge or movable stool, which is placed under the string at that point at which, by designation, the division of the string has been made according to the proportion due to the consonance sought — which more recent writers call the cursor (runner), because it is moved forward and backward while we hunt the consonances. The Pecten or plectrum is that with which we strike the string, that by its vibration it may give forth sound. And from Magas is formed the verb μαγαδίζειν (magadizein), which means to play upon a string, forming various consonances on it.

[Margin: The first method of dividing the Monochord.]

Now the first use of the Monochord is that through it we may explore the consonances singly, one by one, or also the dissonant intervals, by a simple division of the string into only two parts — a division, I say, not real, but one equivalent to a real one: namely by a marking-out on a straight line, placed under the stretched string itself and of the same length, so that the place of the movable Magas may be known. And the division of the string is to be made into as many equal parts as the sum made from both numbers [of the ratio] contains, [the sum] determining the proportion due to the consonance. Thus, if you wish to explore the Diapason, whose proportion is between 2 and 1 — since 2 and 1 make 3 — you must divide the string into three equal parts: for if you strike the part of the string containing two of those parts, and immediately the other containing a single one, you will hear this consonance. So if you wish to discern the Diapente, whose proportion is as 3 to 2 — since 3 and 2 make 5 — you will divide the whole string into 5 equal parts; and, the Magas being placed where two parts end on the one side and three on the other, from striking each you will learn this consonance. And so of the rest, as the following table will show — in which the first column contains the number of equal parts into which (by the aid of a compass, or of folded paper) the whole string is to be divided; the second, the number of parts of the longer partial string; the third, of the shorter — which, struck together or one immediately after the other, will exhibit to purged ears the consonance sought, or the dissonant interval that serves concinnity.

Division of the Monochord

(Monochordi Divisio)

For the Consonances (Pro Consonantijs)

For the Dissonant Intervals (Pro Dissonis Intervallis)

(All values are as printed and verified — in each row the longer + shorter portions equal the “equal parts,” and the longer : shorter ratio is the interval. Note that, as printed, Semitonium maius (54:50) and Semitonium maximum (27:25) reduce to the same ratio, 27:25 — an apparent slip in the original, since the two are meant to be distinct semitones.)

[The catchword “RE-” points to p. 512 (PDF 547), which continues Chapter IV.]

(printed p. 512 — Chapter IV continued. The monochord-division table from p. 511 is completed; then the second method divides the string into 120 parts so one division serves all seven simple consonances, with a note on the far finer divisions required for a fully chromatic and enharmonic monochord. The third method presents Ptolemy’s Helicon, a rectangle with a movable diagonal from whose intersections every consonance can be read off, set out in its own table.)

Remainder of the Preceding Table — Division of the Monochord

(Residuum Tabulae Praecedentis — Monochordi Divisio)

For the Consonances (Pro Consonantijs) — continued (8–23):

For the Dissonant Intervals (Pro Dissonis Intervallis) — continued (8–14):

[Margin: The Second Method.]

The Second Method is when a single string is to be divided into equal parts in such a way that the same division may serve, if not for all, at least for several consonances — say the seven simple ones; and then you will divide the string into 120 equal parts. For if you strike the whole string, and then its half — namely the partial string consisting of 60 parts — you will hear the Diapason; and you will hear the same in another way if you strike on the one side a string having 80 parts, on the other [a string] having 40. But if you strike the whole string, and then the partial one having 80 parts, you will feel the Diapente. And if you strike the whole string, and at once the partial one having 90 parts, you will hear the Diatessaron. And if you strike the whole string, and then the partial one having 96 parts, you will hear the Ditone. But if you strike the whole string, and at once the partial one having 100 parts, you will feel the Semiditone. But if you strike the whole string, and then that partial one which has 72 parts, you will hear the greater Hexachord. Finally, if you strike the whole string, and at once that of the partials which has 75 parts, you will feel the lesser Hexachord. And thus, once trained, your ear becomes [a judge] of the seven simple consonances, which Virgil calls the Seven distinctions of tones (Septem discrimina vocum); for which see the synopsis in the following table.

Division of the Monochord for the 7 Simple Consonances

(Divisio Monochordi pro 7 Consonantijs Simplicibus) — let the whole string be divided into 120 equal parts

The whole string is struck, and immediately (or just after it) the partial string is struck, having the parts shown:

*Translator’s note: the original prints “75 — Hexachordum maius, 72 — Hexachordum minus,” but this is transposed relative to its own prose just above (and to the arithmetic): 120 : 72 = 5:3 = the greater Hexachord (major sixth), and 120 : 75 = 8:5 = the lesser Hexachord (minor sixth). The prose order (72 → greater, 75 → lesser) is the correct one.

But concerning Monochords in the Diatonic, Chromatic, and Enharmonic genus we shall speak below, after the difference of these genera has been explained — namely in ch. 6, num. 3 and 4 — where you will learn that the diatonic Monochord, fit for the consonances and the dissonant-but-concinnous intervals, is to be divided into 9216 equal parts according to Boethius, Glareanus, and Zarlino; or into 3600 parts according to Fogliano; or into 2160 or 720 parts according to Kepler.

[Margin: The Third Method.]

The Third Method is by the figure which Ptolemy (bk. 1 of the Harmonics, ch. 11) calls the HELICON — that is, by a rectangle ABCD, whose side BD divide in half at F, and CD in half at E; then divide the side BD into four equal parts BG, GF, FH, HD; again divide the same side BD into three equal parts BK, KI, ID. And from the points of the divisions of side BD, draw to the opposite side AC [lines] perpendicular [to it] and parallel to the [upper and lower] sides: GL, KM, FN, IO, HP; and at last from B draw the straight line BE to E, the midpoint of side CD. For if the aforesaid parallels — together with the sides to which they are parallel — be strings of the same kind and tension, and the line BE make a perpetual chord-cutter (chordotomum) or Magas, you will detect the Consonances, or the concinnous intervals, written below.

[Figure — the HELICON of Ptolemy] A rectangle A (top-left) B (top-right) C (bottom-left) D (bottom-right). The right side BD is divided (from B downward) at G, K, F, I, H (into halves at F, thirds at K & I, quarters at G, F, H); the left side AC carries the matching feet L, M, N, O, P. Five horizontal parallels cross the figure: GL, KM, FN, IO, HP. The diagonal BE runs from B down to E, the midpoint of the bottom side CD, cutting the five parallels at the points Q (on GL), R (on KM), S (on FN), T (on IO), and V (on HP). Each interval is then read off as the ratio between two of these segments (see the Table).

Table for the Helicon

(Tabella pro Helicone)

The remaining methods, as fit for fewer consonances, I pass over — namely the Mesolabium (on which Zarlino, part 2, ch. 25), and the Square divided into eight parallelograms with an oblique line such as BE, on which Atha…

[…continues on p. 513 (PDF 548) with the catchword “Atha-” (Athanasius Kircher): the close of the survey of monochord-dividing devices, still within Chapter IV.]

(printed p. 513 — Chapter IV concludes with the last monochord-dividing devices and a note on sympathetic vibration, then its Scholia begins, expounding Kepler’s doctrine of the origin of the consonances from sections of the circle in the Harmonice Mundi. Book 1’s grades of knowability of polygon-sides are summarized, showing which figures admit proper demonstration; Book 2 concludes only twelve figures are congruent; and Book 3’s axioms determine consonant versus dissonant arcs, yielding the harmonic section of the string into only seven consonant parts.)

(conclusion, and the SCHOLIA on Kepler’s circle-division theory of the consonances)

(…concluding the survey of monochord-dividing instruments from p. 512:) …on which [see] Athanasius Kircher (bk. 4 of the Musurgia, ch. 5, Lemma 2), and others indicated by Glareanus (bk. 1 of the Dodecachordon, chs. 17–18), or by Ludovico Fogliano (sect. 2, chs. 14–15).

I only add that, when one string is struck, the other — though not struck, but in unison with it or akin in consonance — is set vibrating, if it lies within the sphere of the vibrations of the first; which you may test by laying upon the other a little bent straw, for you will see it leap up. This effect was known to Macrobius (bk. 2 on the Dream of Scipio), and Kepler attempts to give its cause (bk. 3 of the Harmonics, p. 14), as do Fracastoro (On Sympathy, ch. 11) and the Physiologists in the question on action at a distance.

SCHOLIA

[Margin: Kepler’s doctrine on the origin of the Consonances.]

[I.] Kepler, therefore, about to investigate the origin of the consonances from the sections of the circle, premises in Book 1 of the Harmonics that the demonstration of a figure in a circle is the deduction of a quantity to be known (measured, or described) from the diameter through the possible intermediates.

[Margin: Proper and improper demonstration of figures in a circle.]

The proper demonstration of such a figure is when the number of the angles of that figure, or of a cognate figure — by being double or half [the number of] sides — becomes the middle term for determining the proportion of the side to the diameter. The demonstration is improper when the proportion of the side to the diameter cannot be determined geometrically from the number of angles immediately applied, unless the side of another figure be brought in, one having neither double nor half the number of sides.

[Margin: The several grades of knowability — of the effable and of the ineffable.]

Further, he establishes several grades of knowledge (scibilitas). The First Grade is when some line can be demonstrated equal to the diameter, or a plane equal to the square of the diameter. The Second Grade is when, the diameter being divided into some certain number of equal parts (or its square likewise), the proposed line or plane is demonstrated equal to such a part or parts; and then that plane is called Effable [expressible], and the line Effable in length — for number is, as it were, the speech of Geometers. The Third Grade is when a line is ineffable in length but its square is effable; and then the line is said to be Effable in power. The remaining grades he calls simply ineffable, rather than irrational or surd. Among these, the Fourth Grade — the first of the ineffables — is when neither the line nor its square is effable, yet the square is transformable into such a rectangle whose sides are at least effable in power; and such a line is called Mese, because it is the mean proportional between two [lines] commensurable in power alone, while its square is called Meson, whether of square form or transformed into a rectangle — from which kind of Plane, and from the effable Plane, arise the other species and grades. The Fifth Grade is when two lines, neither both effable nor mese, and plainly incommensurable with each other, yet make an effable sum of squares and a common Rectangle. The Sixth Grade is when two lines, neither effable, nor mese, nor commensurable, yet constitute the one effable and the other meson (that is, either the sum of squares or the common Rectangle). The Seventh Grade is when, of two ineffable and incommensurable lines, neither is effable (neither the sum of squares nor the common rectangle), yet each is Meson. Of the remaining grades arising from combinations of these, consult [Kepler] himself, up to proposition 34, from which he begins to determine the knowable sides of the figures according to the aforesaid grades.

[Margin: Square, Octagon; 16-gon, Triangle, Hexagon; Dodecagon; the 24-sided figure; Decagon.]

And he shows that: the diameter is knowable in the 1st grade; the side of the square inscribed in a circle in the 3rd, but its square in the 2nd; the side of the octagon and the octagonal star not knowable except in the 8th, but the sides of both joined in the 6th; the side of the 16-gon knowable in grades far below the 8th; the side of the triangle inscribed in the circle in the 3rd, and of the hexagon in the 2nd (but their planes are Mese and in double proportion to one another); the sides of the dodecagon and the dodecagonal star, if joined, knowable in the 5th, if separated, in the 8th (the dodecagon’s plane being effable); likewise the regular 24-sided figure, and all arising from it by continuously doubling the number of sides (and their stars), have knowable sides, but in a grade below the 8th; the sides of the decagon and decagonal star, joined, knowable in the 5th, separated in the 8th, but joined with the diameter in the 4th; the sides of the pentagon and the pentagonal star, separately, knowable in the 8th, but joined, both in the 4th and the 6th; the planes of the decagon, pentagon, icosagon (20-gon) and the rest of this class fall into more remote grades.

[Margin: 15-gon; Heptagon.]

But the sides of the 15-gon and the star arising from it, like those of the 30-gon, have no proper demonstration; and moreover the heptagon (7-gon) is not knowable and has no proper demonstration. Hence the section of the circle into 3, 5, 7, etc. equal parts — and by any ratio that is not a continuous doubling of those already demonstrated — cannot constitute knowable sides of figures.

These things demonstrated, he concludes that there are four classes of knowable figures: three having a proper demonstration, and a fourth having an improper one. And in the division of the circle the order is: in the 1st place the diameter (effable in length); in the 2nd, the hexagon (its side, being equal to the semidiameter, effable in length); in the 3rd, the square and triangle (their sides effable in power only); in the 4th, the sides of the dodecagon and decagon and of their stars (being from things ineffable in power, and Composites of the first species); in the 5th, the sides of the pentagon and octagon, and the pentagonal and octagonal stars, etc.

[Margin: Congruent figures.]

But in Book 2 the Harmonics treats of the geometric congruence or incongruence (unsociability) of figures, and concludes that the figures congruent by plane or solid angles, or by both — that is, those whose angles so meet in one point as to leave no gap between the sides — are only the twelve written below:

[Margin: Kepler’s doctrine on the origin and number of the Consonances.]

[II.] But in Book 3 the Harmonics investigates the origin of the harmonic proportions, and the nature and differences of the things pertaining to song, and premises these axioms:

1. The diameter of the circle, and the sides of the radical figures fully explained in Book 1 (those having a proper demonstration), determine a part of the circle consonant with the whole circle — whether the circumference be stretched into a ring or into a straight line as a single string. And therefore the consonances are infinite, because the demonstrable figures are infinite, though some of them are [to be] chosen.
2. By whatever grade the demonstration of the side stands distant from the first, by the same grade the Consonance of the part of the circle cut off by that side recedes, with respect to the whole circle, from the most perfect consonance of the unison.
3. The indemonstrable sides of figures and stars determine a part of the circle dissonant from the whole — as do also the sides that are demonstrable, but not by a proper demonstration.
4. Figures that have cognate demonstrations of their sides beget cognate harmonies.
5. Strings or arcs of equal tension, having among themselves (by reason of length) the same proportion as between a part (or the residue) of the circle and the whole circle, have also the same consonance or dissonance, though it be contained between other terms or sounds.
6. When two strings have given forth identical sounds, a third voice consonant with the one is consonant with the other; or dissonant with the one, dissonant with the other.
7. When two strings or voices have given forth identical sounds, a third voice identisonant with one of them is also identically consonant with the other.

These laid down, (1.) he affirms that, after the Unison (which he calls the most perfect consonance), the half [of the circle] with the whole begets the most perfect and simplest consonance in the first grade, identical from either side, because it arises from the diameter. (2.) If the greater part of the circle does not stand in continuously-double proportion to the whole, and is consonant with the whole, the lesser part will be dissonant with the whole. (3.) Strings in continuously-double proportion all consonate identically among themselves. (4.) A string consonant with either term of a continuous double-multiple proportion is consonant with the other also; and if dissonant with one, dissonant with the other. Further, in Chapter 2 he treats of the harmonic section of the string — when the whole string is cut into such parts as are consonant both with the whole and singly among themselves — and teaches that such [parts] are only Seven, namely those whose proportion is indicated by the numbers written below, and of which he had treated in the Mysterium Cosmographicum, ch. 12.

[The catchword “Sectio-” points to p. 514 (PDF 549), which continues into Chapter V.]

(printed p. 514 — the Scholia of Chapter IV concludes with the table of Kepler’s seven Harmonic Sections and a synopsis of Harmonice Book 3; then Chapter V begins, “On Music, and the various Divisions and Genera of Melodies and Songs.” The divisions of music are given — theoretical versus practical, Boethius’s mundane, human, and instrumental music, natural versus artificial organic music with its wind, stringed, and percussion instruments — followed by the division into the twelve or fifteen Modes, each characterized in turn.)

(SCHOLIA — conclusion)

The Harmonic Sections

(Sectiones Harmonicae — if one Part of the String be to the other Part as:)

After these, in chapter 3 [Kepler] treats of the trinity of concordant sounds, or of the harmonic means; in chapter 4, of the origin of the concinnous intervals; in chapter 5, of the natural section of the consonant intervals into concinnous ones; in chapter 6, of hard and soft song [major and minor]; in chapter 7, of the section of a single octave in each genus of song; in chapter 8, of the number and order of the least intervals; in chapters 9 and 10, of the notes and characters of voices and strings, and of the syllables Ut, re, mi, fa, sol, la; in chapter 11, of the composition of systems; in chapter 13, of naturally concinnous song; in chapters 14 and 15, of the Modes of melodies, or Tones; and in chapter 16, of figured song.