(De Chordis, Vocibus, ac Notis Musicis, earumque nomenclatura, & distributione in Systemate Harmonico seu in Scala Musica)

[I.] Three chief accidents are wont to be considered in the modulation of voices or sounds.

First, the natural quality of the voice or sound — in which respect there are four notable species of voices: namely Discantus (or Cantus), Altus (or Contratenor), Tenor, and Basis — in Italian Soprano, Contralto, Tenore, and Basso (for this last Julius Pollux calls Basis).

Second, the continuous quantity in duration or time-delay, by which we dwell on pronouncing some voice or sustaining a sound; and to this serve those notes which Johannes de Muris of Paris is said to have invented about the year of the Lord 1320 — whose order is such that each preceding [note], with respect to the following, requires a delay twice as long. The value, name, and properties of these notes see in the following table, [reckoned] in the time of the Prime Mobile [i.e. in seconds], supposing the ordinary regulation of the hand raised and lowered [the beat], measured by us with the aid of a pendulum:

But before such notes, neither Guido of Arezzo nor others had notes distinguishing the time-delays, but used mere thickish points or little circles — whence also the Art of Counterpoint (Contrapunctum) got its name; for just as now Note is set against Note, so then point was marked against point (punctum contra punctum).

[II.] Third, in sounds and voices is considered intension and remission — from gravity [low] toward acuteness [high], or from acute toward grave — that is, the change of voices through certain degrees, by which the voice as it were ascends or descends. To discern these degrees, the ancients employed either diverse parts of the same string (some longer, some shorter) or diverse strings of diverse tension, on which they imposed certain names; and they so ordered them that, out of 15, or 16, or 18 strings, they nevertheless established five tetrachords — because the last string of the first tetrachord was the first of the second, and so on — except the first of the fourth tetrachord, which does not share with the last of the third (wherefore the fourth tetrachord is called of the disjunct strings); and thus they constituted the greatest Diatonic System [the Greater Perfect System].

The names of the five Tetrachords Vitruvius supplies us (bk. 5, ch. 4), saying: “There are five tetrachords: the first, the lowest, called in Greek ὕπατον (hypaton); the second, the middle, μέσον (meson); the third, the conjunct, συνημμένον (synemmenon); the fourth, the disjunct, διεζευγμένον (diezeugmenon); the fifth, which is the highest, ὑπερβόλαιον (hyperbolaion).” With whom agree Martianus Capella (bk. 9, in the chapter on Tones) and Boethius (bk. 1, ch. 20). The names of the strings, the explanations of those names, and their number according to various [authors], you have in the following table — in which we have written ὑπατῶν (hypatōn) with Glareanus, and not ὑπάτων, as some write corruptly.

[The catchword “CHOR-” points to p. 517 (PDF 552), which opens with the great table of the Strings (Chordae) of the Harmonic System — their Greek and Latin names, letters, and numbers — still within Chapter VI.]

(printed p. 517 — Chapter VI continued. The Table of the Strings sets out the eighteen notes of the Greek Greater Perfect System with their Greek and Latin names, with notes on the authorities who count fifteen, eighteen, or only eight strings. Then Guido of Arezzo’s reform is recounted: his five-line staff, the six solmization syllables drawn from the hymn of St. John the Baptist, the enlargement of the scale to twenty strings with Gamma prefixed, and variant solmization systems, introducing the great Musical Scale.)

Table of the Strings

(Chordarum Nomina secundum tropos Musicos — “Names of the Strings according to the Musical tropes”)

Now from the aforesaid strings, in surveying the Greek System, some enumerate only fifteen — such as Ptolemy (Harmonics 2, chs. 5 and 11), Glareanus (Dodecachordon 1, ch. 19), Ludovico Fogliano (Musica Theorica, sect. 3, last chapter), Zarlino (Harmonic Institutions 2, ch. 28), Mersenne (on Genesis 4, verse 24, p. 1670), and Kepler (Harmonics 3, ch. 11) — all of whom omit strings 9, 10, and 11 (the third, penultimate, and last of the conjunct strings), and after the Mese place immediately the Paramese, numbering the rest in the order given above. But Vitruvius (bk. 5, ch. 4; and there Daniele Barbaro and Philander), Euclid (and with him Herigone, vol. 5 of the Mathematical Course, in Euclid’s Music), Martianus Capella (bk. 9, in the chapter on the Tropes), and Kircher (Musurgia 3, ch. 13, p. 144) reckon 18 strings in the same order as I — although Capella in their nomenclature differs in some places, as is clear from the following little table; in the others he agrees with us.

Capella’s peculiar nomenclature (Nomenclatura peculiaris Martiani Capellae)

But Blancanus (on the Mathematical passages of Aristotle, at sect. 19 of the Problems) and Bettini (Apiarium 10, Proludium 1, prop. 1) do not enumerate all the strings, but only the chief eight in the common octochord: Hypate, Parhypate, Lichanos, Mese, Paramese, Trite, Paranete, Nete. But before we adjoin the Aretinian [Guidonian] notes to the aforesaid strings, and order the Greatest System according to the three genera — namely Diatonic, Chromatic, and Enharmonic — a few things must be premised about the notes of the strings which Guido of Arezzo devised, and increased up to the number of 22 strings.

[Margin: Guido of Arezzo, restorer of Music.]

[III.] For before Guido of Arezzo, most Europeans were wont in Ecclesiastical chant to use eight straight lines, as eight strings, whose beginnings were marked with Greek letters; and on those lines points were marked (as now notes), but not in the intermediate spaces — as Vincenzo Galilei teaches in his Dialogue on Music, and our Kircher shows from manuscripts of the Vatican and Messina libraries (Musurgia 5, ch. 1). Afterward Guido of Arezzo — a Benedictine monk from Arezzo in Etruria, and prefect of the monastic choir — when he was at Pomposa, a town of the Duchy of Ferrara, in the year of the Lord 1024, devised and discovered a new manner of singing, easy and pleasant, which Europe now also uses; and with harpsichords devised by him he commended it, and taught it throughout all Italy up to the year 1028, with the approval of the Popes John XX and Benedict VIII, by whom he was summoned to Rome and honorably received.

First, when he saw that, among the eight lines used by his predecessors, the spaces were idle and empty of notes (for every gradation was made from line to line), he restricted the lines to five, but inserted notes in the spaces, so that with fewer lines he might comprehend more intervals.

Second, to distinguish the three chief tetrachords, he substituted for the Greek strings these six syllables — Ut, re, mi, fa, sol, la — by which the ascent is made from the lowest Ut to the highest La, choosing them from that strophe of the hymn of St. John the Baptist:

UT queant laxis RE-sonare fibris, MI-ra gestorum FA-muli tuorum; SOL-ve polluti LA-bii reatum, Sancte Ioannes. [“That with loosened voices thy servants may resound the wonders of thy deeds, loose the guilt of our polluted lip, O holy John.”]

Of which the first tetrachord they express by Ut, re, mi, fa; the second by Re, mi, fa, sol; the third by Mi, fa, sol, la. And mi–fa (or fa–mi) is the semitone; but the rest, being next to one another, are greater and lesser tones.

Third, he so distributed the aforesaid syllables in five tetrachords — corresponding to the five fingers of the hand, and to as many keys — that he constituted 20 strings, and by an admirable compendium represented every difference of tones and semitones; yet retaining the seven letters earlier devised by St. Gregory the Great (A B C D E F G), which being completed, a return is made to A. But before the first A he placed Γ (Gamma), the capital Greek letter, to signify that the Greeks were the inventors of Music, and to add a tone toward completing the diapason, which two conjunct tetrachords do not fill. Concerning this hand or Musical Scale he himself wrote a book called the Micrologus or Introductorium, and dedicated it to Theobald, Bishop of Arezzo, promising in the dedicatory epistle that as much skill in singing could be drawn from it within a month as scarcely anyone could acquire in many years by the old method; and at the end he adds this clause: “The end of the Micrologus of Guido, aged 34 years, under Pope John XX,” etc.

But although the greatest part of Europe received those six syllables, some were content with only four — Ut, re, mi, fa — as Mersenne reports (on Genesis 4, p. 1679). Others proposed seven — Ut, re, mi, fa, sol, la, bi — as Erycius Puteanus in his Musathena, that he might distinguish seven phthongi (which properly make up the diapason [octave]), just as the Greeks are said of old to have distinguished [theirs] by their seven vowels α ε υ ι η ο ω. But some Belgians (by Kepler’s testimony, Harmonics 3, ch. 9) use these seven: Bo, ce, di, ga, lo, ma, ni; although in the year 1547 (by Maillard’s testimony, ch. 10 on tones) these eight were celebrated in Belgium: Ut, re, mi, fa, sol, la, sy, o.

These things premised, behold now the Musical Scale, or greatest “mute” System [shown silently in a diagram], corresponding to the Greek strings and tetrachords, with the proportions of the intervals — which system indeed com-…

[The catchword “com-” points to p. 518 (PDF 553), which presents the great full-width scale-table of the whole Harmonic System (the Greek strings, Guido’s letters and syllables, and the interval-proportions of the three genera), still within Chapter VI.]

(printed p. 518 — Chapter VI continued. The sources compiled for the great scale-diagram are named, and it is noted that Guido’s scale is fitted in practice only to the Diatonic genus, enlarged to twenty-two strings beyond the ancients’ disdiapason. The page presents the Greatest Diatonic System — the full medieval gamut of twenty-two notes with string-names, monochord proportions, clef-letters, solmization, and intervals — followed by an examination finding the chief consonance-species repeated in it and a corollary on why the consonances are named Octave, Fifth, Fourth, Twelfth, and Fifteenth.)

(…completing ¶III from p. 517, on the great scale-diagram:) …which we have compiled from those things which Guido himself hands down in the Introductorium; Barbaro and Philander (on Vitruvius bk. 5, ch. 4); Fogliano (last chapter of sect. 3); Zarlino (Harmonic Institutions 2, chs. 30, 33, and 36); Glareanus (Dodecachordon 1, chs. 5 and 19); Herigone (vol. 5 of the Mathematical Course, in Euclid’s Music); Mersenne (on Genesis 4, from p. 1668); and Kircher (Musurgia 3, chs. 8, 9, and 13, and bk. 4, chs. 2 and 3). We have supplied from others the things which some omitted, and corrected what more than one confuses.

And although Guido’s Scale could be adapted to the Chromatic and Enharmonic genus, it is not wont to be adapted except to the Diatonic; and although among the Ancients the greatest System consisted within the disdiapason (double octave) — as in the following scale, from the Proslambanomenos string to the Nete hyperbolaeon string, since the number 9216 to 2304 is quadruple, which constitutes the disdiapason, just as 9216 to 4608, or 4608 to 2304, is double and constitutes the diapason [octave] — nevertheless Guido, adding other strings, increased this system up to 22 strings.

THE GREATEST DIATONIC SYSTEM

(Systema Maximum Diatonicum, cum Divisione Monochordi Diatonici, & cum Typo Scalae Musicae Guidonis Aretini — “with the Division of the Diatonic Monochord, and with the figure of Guido of Arezzo’s Musical Scale”)

Read top (highest pitch) to bottom (lowest). Columns: the Greek string-name · its number (Greek/Boethian count | Guido’s count) · the monochord-proportion (string-length) · the clef-letter · the solmization syllables · the interval down to the next string. The five top rows (ee–bb) and the bottom row (Γ) are Guido’s additions, having no Greek name or Greek-number. “durum” = (natural) (hard/square b, sung mi*); “molle” = (flat) (soft/round b, sung* fa*).*

[The four tetrachords are bracketed in the original margin: Hyperbolaeon (Nete–Trite hyperbolaeon), Diezeugmenon (Nete–Trite diezeugmenon, with the note “hîc fit disiunctio chordarum” — “here occurs the disjunction of the strings” — at the Mese/Paramese whole-tone gap), Meson (Mese–Hypate meson), and Hypaton (Lichanos–Hypate hypaton). The hypaton tetrachord E·D·C·B = 6144 : 6912 : 7776 : 8192 is exactly the Pythagorean Diatonic tetrachord of ch. V (tone, tone, limma). All 22 proportions are mutually consistent: each “Tone” is 9 : 8, each “lesser Semitone” the limma 256 : 243, each “greater Semitone” the apotome 2187 : 2048.]

[IV.] Let us now examine this system, and in it, besides the tones and semitones, we shall find repeated the first five species of Consonances — as will stand from the following table:

The Whole String of Boethius

(Tota chorda Boetii) — as the ratio is, so is the whole string (9216) to the part:

[Margin: 3rd Corollary.]

From which, First, it is clear that the Diapason terminates at the eighth chord, the Diapente at the fifth, and the Diatessaron at the fourth chord, in either scale [Boethius’s and Guido’s]; but the Diapason-diapente at the twelfth of Boethius and at the thirteenth of Guido, and the Disdiapason at the fifteenth of Boethius but at the sixteenth of Guido. From which it is clear why the said consonances are called the Octave, Fifth, Fourth, Twelfth, and Fifteenth: namely because the chords consonant with the whole (which is placed first) are so situated that they are numbered in the eighth, fifth, [fourth, twelfth, fifteenth] seat.

It is clear, Secondly, that between the chords making the Diapason there are five tones [with two semitones], that is, 6 tones [in effect]; and between the chords making the Diapente, three tones with [a semitone]; … chords 1 and 4, which make the Diatessaron, … [two tones and] a semitone; and between the chords [the 12th of Boethius or] 13th of Guido, which make the Diapason-[diapente] …

[The page’s text ends here, mid-sentence; the tone-content analysis of the consonances continues on p. 519 (PDF 554), still within Chapter VI. (The right-hand column runs to the page edge and its final word is not legibly captured by the scan.)]

(printed p. 519 — Chapter VI continued. The Third Corollary concludes with the tone-content of the consonances and the observation that eleven of the fourteen consonances are not found exactly in the diatonic scale, as a subjoined table shows. Then follow the parallel gamut-tables of the Chromatic and Enharmonic systems, Martianus Capella’s eight octave-species, and the introduction of Kepler’s “Greatest System,” the perfect disdiapason system of Ptolemy, to be tabulated on p. 520.)

(…concluding the tone-content analysis from p. 518:) …and a semitone; finally, between [chords] 1 and 15 of Boethius, or 1 and 16 of Guido, which make the Disdiapason, there are twelve tones. Which whole doctrine concerning the Tones and semitones contained in the aforesaid consonances Macrobius sets forth expressly (bk. 2 on the Dream of Scipio, ch. 1), and in great part Censorinus (On the Birthday, ch. 11) and Pliny (bk. 2, ch. 22).

Thirdly, it is established that the remaining eleven consonances out of the 14 numbered by us in ch. 4 (in Table 1) are not found exactly in this scale — because in it the ratio of tones and semitones is observed, and it is a division not so much of a single string as a comparison of several different strings. But if in Guido’s Scale the remaining consonances ought to be found, the numbers which you see in the subjoined table would have to be found in it:

The inexact consonances in Guido’s 10368-string

(as the ratio is, so is the whole string 10368 to the part — but the parts do not come out as whole numbers)

[Translator’s note: the values are transcribed exactly as printed (verified at 400 dpi). Riccioli’s point is that these consonances yield no clean integer in Guido’s diatonic scale: by the exact arithmetic the parts would be 8294.4, 8640, 6220.8, 6480, 4147.2, 4320, 3888, 3110.4, 3240. Several printed figures are the nearest roundings (8294, 6220, 3112), but a few appear to be printer’s slips — notably 6485 for the minor Hexachord (8:5 gives 6480), 6147 for the major Tenth (5:2 gives ≈4147), and 3891 for the Eleventh (8:3 gives 3888). In the Decima-minor row the whole string is misprinted “18368” (for 10368); the intended 10368 is shown above.]

THE CHROMATIC SYSTEM

(Systema Chromaticum, seu Divisio Monochordi Chromatici — each tetrachord descends by a Trihemitone (19:16) and two Semitones; the proportions are the interval-numbers)

Synemmenon (conjunct) branch, replacing Paramese: Nete synemmenon 3456 (Trihemitone) — Paranete synemmenon 4104 (Semitone) — Trite synemmenon 4374 (Semitone) — Mese 4608.

THE ENHARMONIC SYSTEM

(Systema Enharmonicum, seu Divisio Monochordi Enharmonici — each tetrachord descends by a Ditone (81:64) and two Dieses)

Synemmenon (conjunct) branch: Nete synemmenon 3456 (Ditone) — Paranete synemmenon 4374 (Diesis) — Trite synemmenon 4491 (Diesis) — Mese 4608.

[These tables are internally consistent: in the Chromatic each Trihemitone = 19:16 and each tetrachord spans the Diatessaron 4:3; in the Enharmonic each Ditone = 81:64 (the Pythagorean ditone) and the hypaton tetrachord 6144 / 7776 / 7984 / 8192 matches the “Ancient Enharmonic” tetrachord of ch. V. The disjunctive Tone (Hypate hypaton → Proslambanomenos, 8192 → 9216) is 9:8.]

In the aforesaid systems — but chiefly in the Diatonic — Martianus Capella (bk. 9, ch. What a System is) considers eight perfect species of systems [octave-species]:

1. from Proslambanomenos to Mese;
2. from Hypate hypaton to Paramese;
3. from Parhypate hypaton to Trite diezeugmenon;
4. from Lichanos hypaton to Paranete diezeugmenon;
5. from Hypate meson to Nete diezeugmenon;
6. from Parhypate meson to Trite hyperbolaeon;
7. from Lichanos meson to Paranete hyperbolaeon;
8. from Mese to Nete hyperbolaeon;

— so that each Species comprises an Octochord [an octave of eight notes].

[V.] But because Kepler constitutes his consonances otherwise, it is pleasing to subjoin here, from his Book 3 of the Harmonics, ch. 11, the Greatest System — containing the perfect and imperfect consonances with their intervals, through two diapasons (or through the disdiapason), which Ptolemy also called the perfect System; and it contains consonances both perfect and imperfect, for the discerning of which we have added from our own [resources] another little table.

[The catchword “SYSTE-” points to p. 520 (PDF 555), which presents Kepler’s “Greatest System” table (the disdiapason perfect system, with the perfect and imperfect consonances) and Riccioli’s added discerning-table, still within Chapter VI.]

(printed p. 520 — Chapter VI continued: Kepler’s System, an almost entirely tabular page. The Systema Keplerianum sets out Kepler’s chromatic two-octave scale of twenty-five strings, flanked by the principal diatonic gamut-notes with Guidonian solmization. Below, Riccioli’s added table shows that the whole string of 2160, struck with each of the others, renders all fourteen consonances exactly — the point being that, unlike Guido’s diatonic scale, Kepler’s system contains every consonance in exact whole numbers.)

THE KEPLERIAN SYSTEM

(Systema Keplerianum — “Strings with Clefs and Intervals,” Chordae cum Clavibus et Intervallis)

Kepler’s chromatic two-octave scale (the disdiapason 540 : 2160 = 4 : 1), read top (highest) to bottom (lowest). The clef-letters are Kepler’s note-names (doubled letters = the higher register; a “g”-suffix marks the chromatic/sharp note). Each printed interval is verified: Semitone = 16:15, Limma = 256:243, Diesis = 25:24. (“B” appears at the very top of the column, apparently a register-label above the highest string gg.)

Beside this runs the column of principal (diatonic) strings with their Guidonian solmization: ee la · dd la-sol · cc sol-fa · bb fa / b mi · aa la-mi-re · g sol-re-ut · f fa-ut · e la-mi · d sol-re · c fa-ut · b fa / b mi · a la-mi-re · G sol-re-ut · F fa-ut · E la-mi · D sol-re · C fa-ut · B mi · A re · Γ ut — “here not only the lines, but also the spaces, signify the principal strings, after the manner of the more recent Diagrams.”

The discerning-table

(The whole string BG = 2160, struck together with each string written below, renders the consonance written below, by the proportions appended)

[All proportions check exactly: 2160 / 1080 = 2, / 1440 = 3/2, / 1620 = 4/3, / 720 = 3, / 540 = 4, / 1728 = 5/4, / 1800 = 6/5, / 1296 = 5/3, / 1350 = 8/5, / 864 = 5/2, / 900 = 12/5, / 810 = 8/3, / 648 = 10/3, / 675 = 16/5, / 576 = 15/4. The last (15:4) has no compact consonance-name (hence the three crosses in the original) and is reckoned the “most imperfect Fourteenth.”]

[The catchword “Idem” points to p. 521 (PDF 556), still within Chapter VI.]

(printed p. 521 — Chapter VI ends with Kepler’s two remaining tables, giving the just major and minor octave-scales and the least chromatic intervals; then Chapter VII begins, on whether and in what order the Muses and the strings ought to be accommodated to the celestial spheres. The world-harmony theme opens with Macrobius on the Sirens of the orbs, the Muses as the song of the world, strophe and antistrophe figuring the two heavenly motions, and the disputed number and names of the Muses, closing with Apollo Musagetes.)

(conclusion — Kepler’s two remaining tables)

But the same Kepler (Harmonics 3, ch. 7), the whole string being divided into 720 equal parts, determined the quantity of the remaining strings up to the Octave, for the hard and soft song [major and minor], as you see in the first of the following little tables. And the whole string being divided into 2160 parts (ch. 8), he determined the least intervals within one Diapason, as in the latter table.

Table I — Of the Length of the Strings

(Tabula I. Longitudinis Chordarum)

(The two scales differ only at the 3rd and 6th degrees. The major 720·640·576·540·480·432·405·360 has the step-pattern T-t-s-T-t-s-T; the minor 720·640·600·540·480·450·405·360 has T-s-t-T-s-t-T — the just diatonic scales, where T = 9:8, t = 10:9, s = 16:15.)

Table II — For the least Intervals

(Tabula II. Pro minimis Intervallis) — the chromatic notes within one Diapason (1080 : 2160)

(Verified: every Semitone = 16:15, every Limma = 256:243, every Diesis = 25:24; 1080 : 2160 = the octave 2:1.)