L'Antica Musica
L'Antica Musica was a treatise published in 1555 by Nicola Vicentino, which explores how Renaissance-era advancements in musical tuning could be used to adapt the lost traditions of Ancient Greek musicians in such a way that blends them with the sensibilities of the 16th century. It is perhaps most revered in xenharmonic communities for its attestation and argument for 31 equal divisions of the octave†, a tuning for which Vicentino had designed his own custom instruments.
†Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a well temperament or MOS scale based on meantone temperament.
On Music Theory
The first book of L'Antica Musica, On Music Theory, is intended as a sort of "recap" of the principles of Ancient Greek music theory, and the means by which those principles arose.
According to Greek musical tradition, just intonation was discovered by the mathematician Pythagoras when he noticed that the hammers used by blacksmiths would produce more consonant sounds with one another if the sound that they displaced created certain frequency ratios; notable was the 3/2 ratio, which was considered the most consonant of all. Ancient Greek musicians considered the usage of low-complexity just intonation intervals as the primary way of tuning music.
According to Vicentino, Pythagoras also went on to construct the first tetrachord, by stacking two concordant whole tones within the span of a perfect fourth. This construction, Vicentino asserts, would pave the way for more such tetrachords to be devised, and they were split into three "genera" based on the step sizes. Vicentino provides diagrams to display the forms of these genera, with their steps arranged from the largest to the smallest.
| Genus | Step Sizes | Comments | ||
|---|---|---|---|---|
| L | M | s | ||
| Diatonic | Tone | Tone | Limma | Vicentino seems to consider the two "tones" necessarily equal. |
| Chromatic | Minor Third | Apotome | Limma | |
| Enharmonic | Major Third | Diesis | Diesis | The diesis is said by Vicentino to divide the limma into two equal parts. |
Vicentino additionally sorts the genera by their "sweet" (Italian: dolce) quality of sound. This appears to be an attempt to translate the Greek term μαλακός (literally meaning "gentle" or "soft"), which was used by Boethius to describe scales; however, most other Romance scholars used the word mollis (Italian molle) for this. As Boethius used it, the term specifically describes melodies and scale forms that are said to create a sense of intimacy and resolution by the inclusion of exceedingly small intervals in specific places. Vicentino gives little elaboration on what precisely is meant by dolce, but the term is always used alongside a note of a genus's smaller available step sizes. On the rest of this page, the term will be left untranslated to avoid the ambiguity with English senses of "sweet."
Vicentino goes on to discuss the various permutations of these tetrachords that can be constructed by putting the same step sizes in a different order; these permutations constitute the "species" of tetrachords. These permutations can be used to construct eight church modes, which Vicentino enumerates as follows:
| Mode | Pattern | Comments |
|---|---|---|
| Dorian | | LMs sLM | |
| Hypodorian | sLM | LMs | |
| Phrygian | | MsL MsL | In diatonic genus, resembles modern Mixolydian mode |
| Hypophrygian | MsL | MsL | In diatonic genus, resembles modern Dorian mode |
| Lydian | | sLM LMs | In diatonic genus, resembles modern Melodic Minor scale |
| Hypolydian | LMs | sLM | |
| Mixolydian | sLM | sLM | In diatonic genus, resembles modern Phrygian mode |
| Hypermixolydian | sLM sLM | | In diatonic genus, resembles modern Locrian mode |
The | symbol is here used to represent a step of precisely 9/8, which separates the two tetrachords from one another.
Note that these modes are constructed by Vicentino's enumeration of the tetrachords, and do not represent rotations of a single scale pattern (though the familiar "rotated" modes are indeed discussed in a later book). Some of the modes listed here are thus rather odd, such as Dorian having two consecutive semitones, or Lydian having four consecutive whole tones. Some translators suggest that this may represent multiple conflicting schemes of numbering the tetrachords.
On Practice I
Following the book on ancient theory is the first book on contemporary practice of the Renaissance era when the treatise it was published. This first practice book is centered primarily around melodic composition, with the harmonic considerations not being featured until the following book.
The first several chapters summarize musical notation; to accommodate for the microtones of Vicentino's proposed tuning, he suggests a circular accidental which represents raising a note by the interval of a diesis (one step of the 31-form).
The majority of the book, however, is spent the intervals of the 31-form, gives each a name, and describes their unique nature and usage; Vicentino specifically uses properties such as consonance potential, subjective emotional content, and dolce qualities to describe the nature of the intervals used as steps or leaps in a melody.
| Step | Name | Melodic Quality | Comments | |
|---|---|---|---|---|
| Ascending | Descending | |||
| 0 | Comma | N/A (tempered out) | Of note because the archicembalo was well-tempered | |
| 1 | Diesis | Concordant, lax, and dolce | To be used in alternating succession | |
| 2 | Chromatic Semitone | Tense, yet cheerful | Lax and sad | |
| 3 | Diatonic Semitone | |||
| 4 | Minor Tone | Lax or tense | Assimilates tension to nearby intervals | |
| 5 | Natural Tone | Powerful and tense | Lax | 64/63 "is not discernible in singing, but in the tuning of instruments their difference is indispensable." |
| 6 | Major Tone | |||
| 7 | Minimal Third | Lax | Somewhat tense and very sad | |
| 8 | Minor Third | |||
| 9 | Supraminor Third | Also known as "proximate" third; "neutral" had not yet been coined | ||
| 10 | Major Third | Tense and imperious | Somewhat tense; sad if accidental | Notably distinct despite being separated by 64/63 |
| 11 | Supermajor Third | Extremely tense | Extremely sad and lax | |
| 12 | N/A | N/A | Vicentino skips this interval in his analysis | |
| 13 | Perfect Fourth | Tense; points to the thirds | Lax | Level of tension when ascending depends on type of tetrachord used |
| 14 | Superfourth | Lively | Sad and lax | Also known as "proximate" fourth |
| 15 | Augmented Fourth | Vivacious and forceful | Funereal and sad | Vicentino advocates for liberal usage of tritones for "marvelous effect" |
| 16+ | Octave Inversions | Inverse qualities of their inversions | ||
On Practice II
In the second book on musical practice, Vicentino discusses the three methods of writing effective music. The first is the usage of melodic and harmonic intervals to mimic the character of sung lyrics; the second is the usage of melodic and harmonic intervals to create clear senses of tension and release; and the third is to balance different types of characteristics to create a clear sense of motion from start to finish. The bulk of the second book focuses on the principles of writing melodies and harmonies that play into these methods.
Vicentino applies the principles of counterpoint regardless of the method: all voices must begin and end a melodic line on the tonic pitch, contrary motion is preferred between voices, parallel perfect consonances are dispreferred between voices, and resolutions must be approached through stepwise motion.
In the following chapters, the natures of harmonic consonances are discussed, much the same as the melodic natures in the preceding book. Vicentino advocates specifically for the usage of syncopation, whereby two lines have the same rhythm offset by some consistent amount until the resolution; this syncopation will create dissonances between voices that would otherwise have been consonant, which allows for these dissonances to resolve. Vicentino notes that the ear can hear these dissonant intervals such as tritones as more concordant when approached via small steps such as dieses and semitones that create a more dolce melody.
| Step | Name | Harmonic Function | Comments |
|---|---|---|---|
| 1 | Diesis | N/A | Does not occur between voices |
| 2 | Chromatic Semitone | Resolves to minor third or unison;
Bass voice moves down while top voice moves up or remains fixed |
Vicentino prefers semitone suspensions over wholetones. |
| 3 | Diatonic Semitone | ||
| 4 | Minor Tone | Resolves to third or unison;
Bass voice moves down while top voice moves up or remains fixed | |
| 5 | Natural Tone | ||
| 6 | Major Tone | ||
| 7 | Minimal Third | Imperfect consonance approached by seconds | |
| 8 | Minor Third | ||
| 9 | Supraminor Third | ||
| 10 | Major Third | Imperfect consonance approached by fourths
Set up a more dulce sound to approach cadences by |
|
| 11 | Supermajor Third | ||
| 12 | N/A | N/A | Vicentino skips this interval in his analysis |
| 13 | Perfect Fourth | Resolves to major third;
Bass voice remains fixed while top voice moves down |
|
| 14 | Superfourth | ||
| 15 | Augmented Fourth | Resolves by contrary motion to third or sixth | |
| 16 | Diminished Fifth | Fudging of either an augmented fourth or imperfect fifth | |
| 17 | Imperfect Fifth | Half-resolves to perfect fourth or superfourth;
or resolves to perfect fifth or minimal sixth |
|
| 18 | Perfect Fifth | Perfect consonance approached by fourths | |
| 19 | Superfifth | N/A | Does not occur between voices |
| 20 | Minimal Sixth | Imperfect consonance approached by diminished fifth
Leads to perfect fifth |
Concordant but directed;
Vicentino compares fifths and sixths to the sun and moon |
| 21 | Minor Sixth | Imperfect consonances leading to and from fifths | |
| 22 | Supraminor Sixth | ||
| 23 | Major Sixth | Imperfect consonances approached by sevenths
Can create strong directed motion anywhere if set up well |
|
| 24 | Supermajor Sixth | ||
| 25 | Minimal Seventh | Resolves by oblique motion to sixth | |
| 26 | Minor Seventh | ||
| 27 | Supraminor Seventh | ||
| 28 | Major Seventh | Resolves by oblique motion to sixth or octave | |
| 29 | Supermajor Seventh | ||
| 30 | Imperfect Octave | Used in place of octaves to evade parallel motion | |
| 31 | Octave | Perfect consonance approached by ninths or sevenths |
On Practice III
The third book on musical practice comprises demonstrations of the three tetrachord genera, with their associated octave species. The diatonic modes addressed in this book resemble the modern conception rather than the unconventional combination of tetrachord presented in the Theory book, further supporting the suggestion that the prior construction may have been an error.
Diatonic modes
Each diatonic mode has steps spanning 5\31 and 3\31, respectively the natural whole tone and the diatonic semitone; the pattern of the modes can thus be written using L and s to represent these two step sizes. The eight modes represent eight rotations of the Meantone[8] scale, with the final mode being a superfluous repetition of the first and the penultimate mode being noted as lacking distinct reason by proxy.
| Number | Name | Pattern | Modern Equivalent | Notes |
|---|---|---|---|---|
| 1 | Dorian | LsLLLsL | Dorian | Said to sound devout and virtuous |
| 2 | Hypodorian | LsLLsLL | Aeolian | More "modest" version of Dorian |
| 3 | Phrygian/
Trojan |
sLLLsLL | Phrygian | Naturally "lonely," but can be made cheerful when mixed with other modes |
| 4 | Hypophrygian/
Hypotrojan |
sLLsLLL | Locrian | Said to sound sad and funereal |
| 5 | Lydian | LLLsLLs | Lydian | Said to sound haughty and cheerful |
| 6 | Hypolydian | LLsLLLs | Ionian | Somewhat both sad and cheerful |
| 7 | Mixolydian | LLsLLsL | Mixolydian | Said to be a simple mix of Dorian and Lydian, hence the name |
| 8 | Hypomixolydian | LsLLLsL | Dorian | Even Vicentino notes the superfluidity of this mode, as the modes represent 7 rotations of a 7-note scale |
Chromatic modes
The chromatic modes are more complex in construction than the diatonic, each having eight notes per octave compared to the seven notes of each diatonic mode. Each chromatic mode has three distinct types of steps instead of two (the incomposite third of 8\31, the diatonic semitone of 3\31, and the chromatic semitone of 2\31) which can be noted as L, m, and s; additionally, each mode has precisely 2 L steps, 3 m steps, and 3 s steps. These modes are not given unique names, instead simply being numbered.
The first mode is made by taking the fourth and fifth above the tonic degree, then adding a chromatic semitone below each of those three degrees. Finally, the supermajor third intervals which occur over the tonic and fifth are split into sequences of a diatonic semitone and an incomposite minor third, rounding out the scale to 8 notes. The second mode (or perhaps the hypofirst mode) can be found by rotating the scale to begin on the perfect fifth.
The third mode is formed in much the same way as the first, but the supermajor third is split into a minor third followed by a semitone, the inverse of the first mode. The fourth mode (hypothird) can once again be found by rotating the scale to begin on the perfect fifth.
To construct the fifth mode, we begin with an upper tetrachord of P5 - m6 - M6 - 1, which terminates on the upper tonic, and then add a copy of the tetrachord that terminates on the fifth. A natural whole tone can be found above the root, which can be broken up into two unequal semitones to once again round out the scale to eight notes. The sixth mode (hypofifth) can once again be found by rotating the scale to begin on the perfect fifth.
To construct the seventh mode, take the root, fourth, and fifth as in the first two mode pairs; and then add a diatonic semitone above each degree. Just like in the first mode, this mode creates two large steps, though this time they are major thirds rather than supermajor; these can similarly be split into a semitone and minor third to round out the scale to eight notes, though the semitone is chromatic in this mode. The eighth (hyposeventh) and final mode, as with the previous, is found by rotating the scale to begin on the perfect fifth.
| Number | Pattern | Comments |
|---|---|---|
| 1 | mLsmsmLs | Inverted pattern of modes 3 - 6 |
| 2 | mLsmLsms | |
| 3 | msLmsmsL | All rotations of same pattern |
| 4 | msLmsLms | |
| 5 | msmsLmsL | |
| 6 | msLmsmsL | |
| 7 | msLmsmLs | |
| 8 | mLsmsLms |
Enharmonic modes
On Practice V: on the Archicembalo Instrument
Because On Practice IV is spend describing notational paradigms such as clefs and note lengths, it will not be summarized here; any other source on musical notation will provide the same information.
Practice V is perhaps the most important book for understanding the rest of the treatise, as it describes in detail the mechanics and tuning of the Archicembalo. While Vicentino never gives outright values for any of the intervals, this book provides insight on how to play and understand the instrument, which in turn helps inform its tuning.
Archicembalo Layout
The Archicembalo's keys are split into six "ranks," with the lower three forming the lower keyboard and the upper three ranks forming the upper keyboard. The first rank outlines the diatonic scale, with the lowest note being the root of the Lydian mode; the next rank above it contains the notes which are a chromatic semitone above those ordinals, and the third rank contains notes which are a chromatic semitone below them. The fourth, fifth, and sixth rank are related to one another as the first, second, and third ranks are, but 1-4, 2-5, and 3-6 are all offset by some particular amount which varies based on the tuning.
Because the first two ranks form a familiar Halberstadt layout, Vicentino instructs that they be tuned as if they were an organ or clavichord. The fifth size was likely variable throughout. Vicentino specifically says that from the lowest key (the root of the Lydian mode, or F on the Halberstadt layout), all intervals in the first two ranks should be with fifths upwards, which means the wolf fifth occurs between A♯ and F. Vicentino then instructs that the third rank be tuned by fifths downwards from the lowest key; ranks 1 and 3, without considering rank 2, will thus form a Halberstadt layout with its wolf between G♭ and B. There are two additional keys, B♯ and E♯, which are considered by Vicentino to be part of rank 3; however, they are tuned with respect to rank 2, such that E♯ is the tempered fifth above A♯.
Finally, Vicentino notes two possible tunings for the upper keyboard. The first is tuned such that ^G♭ is a tempered fifth above B♯ (therefore making it equivalently F𝄪), and the rest of the upper keyboard can be treated as a copy of the lower which has been transposed by the diesis (the amount by which ^G♭ exceeds G♭). This tuning yields a wolf fifth between Gb and ^C♯.
Alternatively, one can tune it such that ^C is an untempered perfect fifth above F; so that if the lower keyboard were tuned to 2/7-comma meantone, then ^C in this tuning would be a 2/7 of a syntonic comma higher in pitch than C. This tuning yields two wolf fifths, one between B♯ and G♭ in the lower keyboard, and one between ^B♯ and ^G♭ in the upper keyboard.
Tuning
Vicentino's potential preferences
In the first paradigm, with a fifth between B♯-^G♭, it is likely that a tuning around 1/4-comma meantone would be preferred, providing a justly tuned 5/4 as the major third. In the 31-note MOS scale of this tuning, the wolf fifth between G♭ ^C♯ is approximately the size of a Pythagorean fifth, which would ensure that the wolf forms of intervals would fall between the sizes of the Pythagorean tuning and the meantone tuning, making it a favorable tuning for providing consonant tunings for those wolves. Additionally, very little size adjustment from 1/4-comma is needed to provide a wolf fifth which is precisely the same size as the generating fifth, thus closing the circle by yielding precisely 31edo.
In the second paradigm, with a fraction of a comma between C-^C, the most likely preferred tuning would have been 1/3-comma meantone, providing a justly tuned 6/5 as the minor third. Since the two keyboards represent two 19-note circles that don't overlap, the wolf fifths are between B♯-G♭ and their comma-altered counterparts, and 1/3-comma meantone tunes this wolf fifth to a size almost precisely the same as the generating fifth (thus approximately 19edo per keyboard). Additionally, Vicentino offers that a root on the lower keyboard may be paired with a fifth and major third from the upper keyboard, a suggestion that would only make sense if the major third were significantly flat of 5/4; in 1/3-comma tuning, we find a major third of approximately 379 cents. Sharpening this third by 1/3 of a syntonic comma in fact yields a tuning which is almost precisely justly tuned 5/4.
Modern tuning implications
By noticing that 1/3-comma meantone roughly closes at 19 steps along the circle of fifths, we can create a more rotationally-symmetrical version of the 38-note paradigm using the MOS scale 19L 19s, which is two instances of 19edo with their roots offset by a tiny amount. A purely-tuned 3/2 will yield a hardness of approximately 9:1, thus allowing us to cleanly approximate this paradigm by considering it a subset of 190edo. While the number is unwieldy, it provides an additional means of extending the paradigm beyond the two keyboards, such that ten keyboards (thirty ranks) would yield a closed circle of 1/3-commas. This would be incredibly inefficient on an analog keyboard instrument of the 16th century, but could be quite simple and useful on isomorphic keyboards of the modern era, such as the Lumatone.
Interval ratios
In addition to establishing an order of the circle of fifths, Vicentino provides a list of JI ratios represented by the 31 interval classes of the Archicembalo's first paradigm:
| Step | Name | Ratios (5-limit) | Ratios (7-limit) | Ratios (higher) |
|---|---|---|---|---|
| 0 | Comma | 81/80 | 126/125 | 105/104, 385/384 |
| 1 | Diesis | 128/125, 648/625 | 36/35, 50/49, 64/63 | 40/39 |
| 2 | Chromatic Semitone | 25/24 | 21/20 | 65/63 |
| 3 | Diatonic Semitone | 16/15 | 15/14 | 14/13 |
| 4 | Minor Tone | N/A | 13/12 | |
| 5 | Natural Tone | 9/8, 10/9 | ||
| 6 | Major Tone | N/A | 8/7 | |
| 7 | Minimal Third | Unstated | ||
| 8 | Minor Third | 6/5 | ||
| 9 | Proximate Third | N/A | 11/9 | |
| 10 | Major Third | 5/4 | ||
| 11 | Supramajor Third | N/A | 9/7 | |
| 12 | N/A (Vicentino skips this interval in his analysis) | |||
| 13 | Perfect Fourth | 4/3 | ||
| 14 | Superfourth | Unstated | ||
| 15 | Augmented Fourth | 25/18 | 7/5 | 39/28 |
| 16+ | Octave Complements | Inverse ratios of their octave complements | ||
