L'Antica Musica - Revision history

Kili: /* On Practice V: on the Archicembalo Instrument */ sentnce spelling

2026-05-13T13:17:11Z

On Practice V: on the Archicembalo Instrument: sentnce spelling

← Older revision		Revision as of 13:17, 13 May 2026
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	== On Practice V: on the Archicembalo Instrument ==		== 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.		Because ''On Practice IV'' describes banal 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.		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.

Unque at 11:57, 30 April 2026

2026-04-30T11:57:48Z

← Older revision		Revision as of 11:57, 30 April 2026
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	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 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.		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.
	{\| class="wikitable"		{\| class="wikitable"
	\|+Intervals of the 31-form		\|+Intervals of the 31-form
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	=== Diatonic modes ===		=== 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.		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.
	{\| class="wikitable"		{\| class="wikitable"
	\|+Diatonic Modes		\|+Diatonic Modes
Line 471:		Line 471:

	==== Vicentino's potential preferences ====		==== Vicentino's potential preferences ====
	In the first paradigm, with a fifth between B♯-^G♭, it is likely that a tuning around [[Meantone#Quarter-comma Meantone\|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 first paradigm, with a fifth between B♯-^G♭, it is likely that a tuning around [[Meantone#Quarter-comma Meantone\|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 [[Meantone#1/3-comma Meantone\|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.		In the second paradigm, with a fraction of a comma between C-^C, the most likely preferred tuning would have been [[Meantone#1/3-comma Meantone\|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 ====		==== 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.		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.

	=== Interval ratios ===		=== Interval ratios ===
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	\|7		\|7
	\|Minimal Third		\|Minimal Third
	\| colspan="3" \|Unstated		\| colspan="3" \|Unstated, can be taken as 7/6 based on relations to other intervals
	\|-		\|-
	\|8		\|8

Inthar: Reverted edit by Inthar (talk) to last revision by Unque

2026-03-31T01:18:48Z

Reverted edit by Inthar (talk) to last revision by Unque

← Older revision		Revision as of 01:18, 31 March 2026
Line 459:		Line 459:
	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.		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.		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.

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	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♯.		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♯.

	~~In the second tuning~~, it is ~~likely~~ that ~~Vicentino preferred tunings of~~ the ~~fifth between 1~~/3-comma ~~and~~ 2/7-comma; in the ~~first tuning~~, ~~it is likely that Vicentino preferred tunings near 1/4-comma~~.		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.

	~~Alternatively~~, ~~one can tune~~ it ~~such~~ that ^C is ~~an untempered perfect~~ fifth ~~above F; 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 [[Meantone#Quarter-comma Meantone\|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 [[Meantone#1/3-comma Meantone\|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 ===		=== 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:		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:
	{\| class="wikitable"		{\| class="wikitable"
	\|+Ratios of the 31-form		\|+Ratios of the 31-form

Inthar: Reverted edit by Unque (talk) to last revision by Hkm

2026-03-31T01:17:52Z

Reverted edit by Unque (talk) to last revision by Hkm

← Older revision		Revision as of 01:17, 31 March 2026
Line 459:		Line 459:
	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.		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.		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.

Line 466:		Line 465:
	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♯.		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~~.		In the second tuning, it is likely that Vicentino preferred tunings of the fifth between 1/3-comma and 2/7-comma; in the first tuning, it is likely that Vicentino preferred tunings near 1/4-comma.

	~~=== Tuning ===~~		Alternatively, one can tune it such that ^C is an untempered perfect fifth above F; 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.

	~~==== Vicentino's potential preferences ====~~
	~~In the first paradigm~~, ~~with a fifth between B♯-~~^~~G♭, it~~ is ~~likely that a tuning around [[Meantone#Quarter-comma Meantone\|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 [[Meantone#1/3-comma Meantone\|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 ===		=== 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~~:		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:
	{\| class="wikitable"		{\| class="wikitable"
	\|+Ratios of the 31-form		\|+Ratios of the 31-form

Unque at 22:32, 29 March 2026

2026-03-29T22:32:29Z

← Older revision		Revision as of 22:32, 29 March 2026
Line 459:		Line 459:
	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.		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.		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.

Line 465:		Line 466:
	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♯.		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♯.

	~~In the second tuning~~, it is ~~likely~~ that ~~Vicentino preferred tunings of~~ the ~~fifth between 1~~/3-comma ~~and~~ 2/7-comma; in the ~~first tuning~~, ~~it is likely that Vicentino preferred tunings near 1/4-comma~~.		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.

	~~Alternatively~~, ~~one can tune~~ it ~~such~~ that ^C is ~~an untempered perfect~~ fifth ~~above F; 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 [[Meantone#Quarter-comma Meantone\|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 [[Meantone#1/3-comma Meantone\|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 ===		=== 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:		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:
	{\| class="wikitable"		{\| class="wikitable"
	\|+Ratios of the 31-form		\|+Ratios of the 31-form

Hkm: /* On Practice V: on the Archicembalo Instrument */

2026-03-22T05:39:08Z

On Practice V: on the Archicembalo Instrument

← Older revision		Revision as of 05:39, 22 March 2026
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	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.		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 [http://tonalsoft.com/enc/h/halberstadt.aspx Halberstadt] layout, Vicentino instructs that they be tuned as if they were an organ or clavichord. The fifth size throughout ~~may have been variable, but it seems like most of the fifths had sizes between those of 1/3-comma and 2/7-comma meantone~~. 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♯.		Because the first two ranks form a familiar [http://tonalsoft.com/enc/h/halberstadt.aspx 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♯, ~~though if tuned~~ near ~~quarter~~-comma~~, this wolf fifth can be nearly the same size as the regular tempered fifth (hence 31edo)~~.		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♯.

			In the second tuning, it is likely that Vicentino preferred tunings of the fifth between 1/3-comma and 2/7-comma; in the first tuning, it is likely that Vicentino preferred tunings near 1/4-comma.

	Alternatively, one can tune it such that ^C is an untempered perfect fifth above F; 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.		Alternatively, one can tune it such that ^C is an untempered perfect fifth above F; 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.

Hkm: /* On Practice V: on the Archicembalo Instrument */

2026-03-22T05:34:00Z

On Practice V: on the Archicembalo Instrument

← Older revision		Revision as of 05:34, 22 March 2026
Line 461:		Line 461:
	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.		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 [http://tonalsoft.com/enc/h/halberstadt.aspx Halberstadt] layout, Vicentino instructs that they be tuned as if they were an organ or clavichord~~; it is unclear what tuning this entails, if any in particular~~, but ~~the~~ most ~~popular frameworks~~ of the ~~time were meantone tunings~~ between 1/4-comma ~~(~697c fifth)~~ and 1/3-comma ~~(~694c fifth)~~. 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♯.		Because the first two ranks form a familiar [http://tonalsoft.com/enc/h/halberstadt.aspx Halberstadt] layout, Vicentino instructs that they be tuned as if they were an organ or clavichord. The fifth size throughout may have been variable, but it seems like most of the fifths had sizes between those of 1/3-comma and 2/7-comma meantone. 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♯, though if tuned near quarter-comma, this wolf fifth can be nearly the same size as the regular tempered fifth (hence 31edo).		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♯, though if tuned near quarter-comma, this wolf fifth can be nearly the same size as the regular tempered fifth (hence 31edo).

	Alternatively, one can tune it such that ^C is an untempered perfect fifth above F; if the lower keyboard were tuned to ~~quarter~~-comma meantone, then ^C in this tuning would be a ~~quarter~~ 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.		Alternatively, one can tune it such that ^C is an untempered perfect fifth above F; 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.

	=== Interval ratios ===		=== Interval ratios ===

Unque at 17:12, 19 March 2026

2026-03-19T17:12:58Z

← Older revision		Revision as of 17:12, 19 March 2026
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	'''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 [[31edo\|31 equal divisions of the octave]]<sup>†</sup>, a tuning for which Vicentino had designed [[wikipedia:Archicembalo\|his own custom instruments]].		'''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 [[31edo\|31 equal divisions of the octave]]<sup>†</sup>, a tuning for which Vicentino had designed [[wikipedia:Archicembalo\|his own custom instruments]].

	<small><sup>†</sup>Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS\|MOS scale]] based on [[meantone]] temperament~~. See [[L'Antica Musica#On Practice V: on the Archicembalo Instrument\|Practice V]] for details on tuning~~.</small>		<small><sup>†</sup>Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS\|MOS scale]] based on [[meantone]] temperament.</small>
	== On Music Theory ==		== 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.		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\|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 [[Perfect fifth\|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 Greek musical tradition, [[Just Intonation\|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 [[Perfect fifth\|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.		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.
Line 465:		Line 465:
	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♯, though if tuned near quarter-comma, this wolf fifth can be nearly the same size as the regular tempered fifth (hence 31edo).		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♯, though if tuned near quarter-comma, this wolf fifth can be nearly the same size as the regular tempered fifth (hence 31edo).

	Alternatively, one can tune it such that ^F is an untempered perfect fifth above F; if the lower keyboard were tuned to quarter-comma meantone, then ^F in this tuning would be a quarter comma higher in pitch than C. This tuning yields two wolf fifths, one ~~on each~~ keyboard between B♯ and G♭.		Alternatively, one can tune it such that ^C is an untempered perfect fifth above F; if the lower keyboard were tuned to quarter-comma meantone, then ^C in this tuning would be a quarter 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.

	=== Interval ratios ===		=== 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~~. These ratio targets may inform how the instrument is tuned.~~		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:
	{\| class="wikitable"		{\| class="wikitable"
	\|+Ratios of the 31-form		\|+Ratios of the 31-form
Line 481:		Line 481:
	\|81/80		\|81/80
	\|126/125		\|126/125
	\|105/104		\|105/104, 385/384
	\|-		\|-
	\|1		\|1
Line 493:		Line 493:
	\|25/24		\|25/24
	\|21/20		\|21/20
	\|~~117~~/~~112~~		\|65/63
	\|-		\|-
	\|3		\|3

Unque at 17:01, 19 March 2026

2026-03-19T17:01:42Z

← Older revision		Revision as of 17:01, 19 March 2026
Line 1:		Line 1:
	'''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 [[31edo\|31 equal divisions of the octave]]<sup>†</sup>, a tuning for which Vicentino had designed [[wikipedia:Archicembalo\|his own custom instruments]].		'''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 [[31edo\|31 equal divisions of the octave]]<sup>†</sup>, a tuning for which Vicentino had designed [[wikipedia:Archicembalo\|his own custom instruments]].

	<small><sup>†</sup>Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS\|MOS scale]] based on [[meantone]] temperament.</small>		<small><sup>†</sup>Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS\|MOS scale]] based on [[meantone]] temperament. See [[L'Antica Musica#On Practice V: on the Archicembalo Instrument\|Practice V]] for details on tuning.</small>
	== On Music Theory ==		== 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.		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.
Line 468:		Line 468:

	=== Interval ratios ===		=== 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:		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. These ratio targets may inform how the instrument is tuned.
	{\| class="wikitable"		{\| class="wikitable"
	\|+Ratios of the 31-form		\|+Ratios of the 31-form

Unque at 19:56, 18 March 2026

2026-03-18T19:56:39Z

@@ Line 1: / Line 1: @@
 '''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 [[31edo|31 equal divisions of the octave]]<sup>†</sup>, a tuning for which Vicentino had designed [[wikipedia:Archicembalo|his own custom instruments]].
 <small><sup>†</sup>Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS|MOS scale]] based on [[meantone]] temperament.</small>
 == 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.
@@ Line 454: / Line 451: @@
 |}
-'''''Todo: finish Practice III, document rest of books in treatise'''''
+== On Practice V: on the Archicembalo Instrument ==