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; see section on tuning for details.

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.

Tetrachord Genera

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:

Eight Modes

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 discussing 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.

Intervals of the 31-form

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.

Intervals of the 31-form

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 character by proxy.

Diatonic Modes

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 (hypo-) Lydian, hence the name
8	Hypomixolydian	LsLLLsL	Dorian	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.

Chromatic Modes

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

The enharmonic modes are the most complex of the scales, with half of them having ten steps to the octave, and the other half having twelve. There are three step sizes from which the enharmonic modes are constructed: the incomposite third of 10\31, the chromatic semitone of 2\31, and the diesis of 1\31, which can be labelled L, m, and s like before; in addition to the aforementioned structural elements, the L step occurs precisely twice in each enharmonic mode.

The first mode is made by taking the diesis above and below the tonic, plus the chroma above the tonic; then including the fourth and fifth above each degree. This mode has twelve steps per octave, the most of any mode Vicentino writes about. The second mode (hypofirst), of course, is found by rotating the scale to begin on the perfect fifth.

The third mode is formed by stepping down from the tonic by a diesis, then lower by a chromatic semitone; the fourth and fifth are added above each of these degrees. Because the steps of this scale do not adhere to the step sizes available in the enharmonic modes, the 15\31 tritone is removed and replaced with the nearby intervals of 14\31 and 16\31, yielding a final count of ten notes per octave. The fourth mode (hypothird) can again be found via starting on the perfect fifth.

Where the other enharmonic modes used the fourth and fifth as spacers, the fifth mode uses the second and fifth instead. From each degree, step up by a diesis and then by a chromatic semitone, and thus can be considered a sort of counterpart to the third mode which steps by this same pattern downwards. Because this scale provides two consecutive semitone steps, Vicentino offers that the latter be split into two dieses, in order to divide up this disorientating segment; this rounds out the scale to ten notes per octave.

The seventh mode, representing final pair of modes, uses 16\31 as a generator to yield a four-note non-MOS scale, and then adds notes a perfect fourth above and a perfect fifth above each degree. This rounds out the mode to twelve notes just like the first pair of modes.

Enharmonic Modes

Number	Pattern	Comments
1	sLsssmssLss
2	LsssLsssmss
3	LmssmssLms	Nearly the inverse of mode 6
4	LmsLmssmss	Nearly the inverse of mode 5
5	smsssmLsmL	Nearly the inverse of mode 4
6	smLsmsssmL	Nearly the inverse of mode 3
7	smLsmsssLss
8	sLsssmLsmss

On Practice V: on the Archicembalo Instrument

Because On Practice IV describes basic 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; this 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.

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:

Ratios of the 31-form

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, can be taken as 7/6 based on relations to other intervals
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

Modern Application

While Vicentino's ideas about modes and intervals can be applied to composition in 31edo, many more ideas in other tunings may be considered through his work.

Diatonic third categorization

Vicentino states that the archicembalo offers four types of thirds that are unheard of in the traditional Halberstadt layout, totaling six types of thirds; however, he only states five of these six qualities: minimal, minor, proximate (supraminor), major, and supermajor. Because the interval at 6\31 is a major tone and not a third, and because the interval at 7\31 is described as minimal, we may deduce that the sixth type of third is above the other five, hence a "maximal" third; indeed, the interval at 12\31 is the only interval not analyzed in Vicentino's treatise, and falls between the supermajor third and the perfect fourth. Because it is the fifth complement of the major tone, we may assign this maximal third the ratio 21/16.

The descriptions given for the melodic and harmonic properties of these thirds may be transferred to other tunings as well, especially tunings with comparable third organization, such as 29edo or 34edo.

38-tone tuning

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 near-pure 3/2 will yield a hardness of around 8:1, thus allowing us to cleanly approximate this paradigm by considering it a subset of 171edo. While the number is unwieldy, it provides an additional means of extending the paradigm beyond the two keyboards, such that nine keyboards (twenty-seven 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.

30edo Enharmonic modes

With the exception of modes seven and eight, the enharmonic modes all contain either 15\31 or 16\31 over the root, never both; in fact, mode 3 actively replaces a 15\31 degree with 16\31 while making no adjustment to accommodate how this impacts the structure of the mode. If we take 15\31 and 16\31 as representing a singular semioctave rather than two separate types of tritones, we might translate the first six enharmonic modes cleanly into 30edo rather than 31.

30edo enharmonic modes

Number	Pattern	Comments
1	sLsssssssLss	The singular m step is now s, making the scale binary (though not MOS)
2	LsssLsssssss
3	LmsssssLms	Contains two Lms tetrachords
4	LmsLmsssss
5	smssmLsmL	Contains two smL tetrachords and one sms tetrachord
6	smLsmssmL

Valentine temperament

Vicentino notes "major" and "minor" forms of the wholetone; these intervals are separated from each other by a chroma, just as major and minor thirds, sixths, and sevenths, but neither of them are the diatonic major second. However, the major tone differs from the minor third by a chroma as well, as does the minor tone differ from the chroma itself by a chroma, hence we may use a chain of chromatic semitones to represent a singular system which categorizes Vicentino's primary intervals within the tetrachord. The temperament which produces this chain -- that is, equally spacing 25/24 - 35/32 - 8/7 - 6/5 - 5/4 -- is known as Valentine. It is also quite comparable to the "Alpha Scale" of Wendy Carlos.