Lattice - Revision history

Sintel: /* Isomorphic keyboards */ typo

2025-04-29T18:08:15Z

Isomorphic keyboards: typo

← Older revision		Revision as of 18:08, 29 April 2025
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	Examples of isomorphic keyboard designs include the {{w\|Jankó keyboard}}, the {{w\|Wicki–Hayden note layout}} and digital controllers like the [[Lumatone]] and [[Linnstrument]].		Examples of isomorphic keyboard designs include the {{w\|Jankó keyboard}}, the {{w\|Wicki–Hayden note layout}} and digital controllers like the [[Lumatone]] and [[Linnstrument]].
	Isomorphic layouts are not ~~restricted~~ to keyboards, [[skip fretting]] systems are also examples ~~isomorphic~~.		Isomorphic layouts are not limited to keyboards, [[skip fretting]] systems are also examples.

	[[File:Lattice_bosanquet.png\|300px\|thumb\|none\|An isomorphic [[meantone]] layout (Bosanquet).]]		[[File:Lattice_bosanquet.png\|300px\|thumb\|none\|An isomorphic [[meantone]] layout (Bosanquet).]]

Sintel: /* Isomorphic keyboards */ +skip fretting

2025-04-29T18:06:20Z

Isomorphic keyboards: +skip fretting

← Older revision		Revision as of 18:06, 29 April 2025
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	Examples of isomorphic keyboard designs include the {{w\|Jankó keyboard}}, the {{w\|Wicki–Hayden note layout}} and digital controllers like the [[Lumatone]] and [[Linnstrument]].		Examples of isomorphic keyboard designs include the {{w\|Jankó keyboard}}, the {{w\|Wicki–Hayden note layout}} and digital controllers like the [[Lumatone]] and [[Linnstrument]].
			Isomorphic layouts are not restricted to keyboards, [[skip fretting]] systems are also examples isomorphic.

	[[File:Lattice_bosanquet.png\|300px\|thumb\|none\|An isomorphic [[meantone]] layout (Bosanquet).]]		[[File:Lattice_bosanquet.png\|300px\|thumb\|none\|An isomorphic [[meantone]] layout (Bosanquet).]]

Sintel: Expand "higher dimensions" section. Add section for mathematical definition.

2025-04-29T00:07:54Z

Expand "higher dimensions" section. Add section for mathematical definition.

← Older revision		Revision as of 00:07, 29 April 2025
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	== Tonnetz ==		== Tonnetz ==
	{{Wikipedia\|Tonnetz}}		{{Wikipedia\|Tonnetz}}
	The Tonnetz is a prominent example of a lattice representing [[5-limit]] intervals. In its conventional hexagonal arrangement, the perfect fifth (3/2) and major third (5/4) intervals are positioned 60 degrees apart, with the ~~major sixth (~~[[5/3]]) placed 60 degrees from the major third and 120 degrees from the perfect fifth. This arrangement results in [[5-limit]] major triads ([[4:5:6]]) and minor triads ([[10:12:15]]) forming triangular patterns within the lattice structure.		The Tonnetz is a prominent example of a lattice representing [[5-limit]] intervals. In its conventional hexagonal arrangement, the [[3/2\|perfect fifth]] (3/2) and [[5/4\|major third]] (5/4) intervals are positioned 60 degrees apart, with the [[5/3\|major sixth]] (5/3) placed 60 degrees from the major third and 120 degrees from the perfect fifth. This arrangement results in [[5-limit]] major triads ([[4:5:6]]) and minor triads ([[10:12:15]]) forming triangular patterns within the lattice structure.

	The Tonnetz can be generalized to other subgroups with 3 primes, such as [[2.3.7]], where a respective fundamental chord such as 6:7:8 can be assigned to a triangle.		The Tonnetz can be generalized to other subgroups with 3 primes, such as [[2.3.7]], where a respective fundamental chord such as 6:7:8 can be assigned to a triangle.
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	== Higher dimensions ==		== Higher dimensions ==
	Lattices ~~can be extended to incorporate~~ more dimensions. ~~For instance,~~ {{w\|Vogel's Tonnetz}} extends Euler's [[5-limit]] Tonnetz to the [[7-limit]], resulting in a 3-dimensional diagram. ~~Similarly~~, ~~other~~ lattices can ~~be constructed for different subgroups or temperaments~~.		Lattices naturally extend into higher dimensions as more prime factors are incorporated into the tuning system. After applying octave reduction, a 5-limit system requires two dimensions, a 7-limit system requires three dimensions, and each additional prime factor adds another dimension to the full representation.

			{{w\|Vogel's Tonnetz}} extends Euler's 5-limit Tonnetz to the 7-limit, resulting in a three-dimensional diagram.
			This additional dimension allows for the representation of relationships involving the prime 7, adding intervals like the [[harmonic seventh]] (7/4) and the [[septimal minor third]] (7/6) alongside the 5-limit intervals.

			Higher-dimensional lattices have been used by composers like [[Ben Johnston]], and [[Kyle Gann]] to organize their harmonic materials.
			The lattice structure can guide everything from chord progressions to large-scale form, with paths through the lattice corresponding to harmonic progressions.

	[[File:Lattice_7lim.png\|300px\|thumb\|none\|A 3-dimensional lattice for the [[7-limit]]. Made using [[Scale Workshop]].]]		[[File:Lattice_7lim.png\|300px\|thumb\|none\|A 3-dimensional lattice for the [[7-limit]]. Made using [[Scale Workshop]].]]

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	{{Wikipedia\|Isomorphic keyboard}}		{{Wikipedia\|Isomorphic keyboard}}
	Isomorphic keyboards are a practical application of lattices in instrument design.		Isomorphic keyboards are a practical application of lattices in instrument design.
	An isomorphic keyboard is an instrument with a two-dimensional grid of buttons or keys, arranged so that any given sequence or combination of musical intervals maintains the "same shape" regardless of where it occurs.		An isomorphic keyboard is an instrument with a two-dimensional grid of buttons or keys, arranged so that any given sequence or combination of musical intervals maintains the same shape, regardless of where it occurs.
	This corresponds directly to a lattice: the keyboard layout is essentially a lattice mapped to physical space.		This corresponds directly to a lattice: the keyboard layout is essentially a lattice mapped to physical space.
	This mapping creates consistent fingering patterns for musicians, where a particular chord shape or scale pattern can be transposed by simply moving the same finger configuration to a different position on the keyboard.		This mapping creates consistent fingering patterns for musicians, where a particular chord shape or scale pattern can be transposed by simply moving the same finger configuration to a different position on the keyboard.
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	Examples of isomorphic keyboard designs include the {{w\|Jankó keyboard}}, the {{w\|Wicki–Hayden note layout}} and digital controllers like the [[Lumatone]] and [[Linnstrument]].		Examples of isomorphic keyboard designs include the {{w\|Jankó keyboard}}, the {{w\|Wicki–Hayden note layout}} and digital controllers like the [[Lumatone]] and [[Linnstrument]].

	[[File:Lattice_bosanquet.png\|300px\|thumb\|none\|An isomorphic meantone layout (Bosanquet).]]		[[File:Lattice_bosanquet.png\|300px\|thumb\|none\|An isomorphic [[meantone]] layout (Bosanquet).]]

			== In mathematics ==
			{{Wikipedia\|Lattice (group)}}

			The lattices used in music theory correspond to mathematical structures of the same name.<ref group="note">Not to be confused with a different structure in order theory, which is also called a {{w\|Lattice (order)\|lattice}}.</ref>
			In mathematical terms, a lattice is defined as a free abelian group embedded into {{w\|Euclidean space}}.
			As expected, each point in the lattice represents a pitch class (or pitch), and the vectors between points represent musical intervals.
			The lattice is generated by a set of basis vectors corresponding to prime harmonics, or other fundamental intervals.
			The embedding into Euclidean space then also induces a norm onto intervals, which serves as a way of measuring [[complexity]].

			An example is the construction of [[Tenney-Euclidean]] interval space, which provides a metric for measuring harmonic distances between pitches.
			We map a ''p''-limit JI space into R^n by representing each ratio as a [[monzo\|vector]], according to its prime decomposition.
			The axes are then scaled according to the logarithms of their respective primes, so prime 2 has length log<sub>2</sub>(2) = 1, prime 3 has length log<sub>2</sub>(3), and prime 5 has length log<sub>2</sub>(5), etc.

			For the 5 limit, we get the embedding:
			:<math>
			2^x \cdot 3^y \cdot 5^z \to \left[x, y \log_2(3), z \log_2(5) \right] \in \mathbb{R}^3
			</math>
			When distances are measured in this space, we get the [[Tenney–Euclidean_metrics#TE_norm\|Tenney-Euclidean distance]], a useful measure which is often used for finding optimal temperaments.

	== Gallery ==		== Gallery ==
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	* [http://www.huygens-fokker.org/docs/fokkerpb.html "Unison Vectors and Periodicity Blocks in the Three-Dimensional (3-5-7-) Harmonic Lattice of Notes"] by [[Adriaan Fokker]] (1969)		* [http://www.huygens-fokker.org/docs/fokkerpb.html "Unison Vectors and Periodicity Blocks in the Three-Dimensional (3-5-7-) Harmonic Lattice of Notes"] by [[Adriaan Fokker]] (1969)
	* [http://tonalsoft.com/monzo/lattices/lattices.htm "Harmonic Lattice Diagrams"] by [[Joseph Monzo\|Joseph L. Monzo]] (1998)		* [http://tonalsoft.com/monzo/lattices/lattices.htm "Harmonic Lattice Diagrams"] by [[Joseph Monzo\|Joseph L. Monzo]] (1998)

			== Notes ==
			<references group="note" />

	[[Category:Lattice\| ]] <!-- main article -->		[[Category:Lattice\| ]] <!-- main article -->

Sintel: Add images. Write some stuff about isomorphic keyboards

2025-04-28T23:16:00Z

Add images. Write some stuff about isomorphic keyboards

← Older revision		Revision as of 23:16, 28 April 2025
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	== Tonnetz ==		== Tonnetz ==
	{{Wikipedia\|Tonnetz}}		{{Wikipedia\|Tonnetz}}
	The Tonnetz is a prominent example of a lattice representing [[5-limit]] intervals. In its conventional hexagonal arrangement, the perfect fifth (3/2) and major third (5/4) intervals are positioned 60 degrees apart, with the major sixth ([[5/3]]) placed 60 degrees from the major third and 120 degrees from the perfect fifth. This arrangement results in [[5-limit]] major triads ([[4:5:6]]) and minor triads ([[10:12:15]]) forming triangular patterns within the lattice structure.		The Tonnetz is a prominent example of a lattice representing [[5-limit]] intervals. In its conventional hexagonal arrangement, the perfect fifth (3/2) and major third (5/4) intervals are positioned 60 degrees apart, with the major sixth ([[5/3]]) placed 60 degrees from the major third and 120 degrees from the perfect fifth. This arrangement results in [[5-limit]] major triads ([[4:5:6]]) and minor triads ([[10:12:15]]) forming triangular patterns within the lattice structure.

	The Tonnetz can be generalized to other subgroups with 3 primes, such as [[2.3.7]], where a respective fundamental chord such as 6:7:8 can be assigned to a triangle.		The Tonnetz can be generalized to other subgroups with 3 primes, such as [[2.3.7]], where a respective fundamental chord such as 6:7:8 can be assigned to a triangle.
			[[File:Lattice_5lim.png\|300px\|thumb\|none\|The [[5-limit]] Tonnetz lattice.]]

			== Higher dimensions ==
	Lattices can be extended to incorporate more dimensions. For instance, {{w\|Vogel's Tonnetz}} extends Euler's [[5-limit]] Tonnetz to the [[7-limit]], resulting in a 3-dimensional diagram. Similarly, other lattices can be constructed for different subgroups or temperaments.		Lattices can be extended to incorporate more dimensions. For instance, {{w\|Vogel's Tonnetz}} extends Euler's [[5-limit]] Tonnetz to the [[7-limit]], resulting in a 3-dimensional diagram. Similarly, other lattices can be constructed for different subgroups or temperaments.
			[[File:Lattice_7lim.png\|300px\|thumb\|none\|A 3-dimensional lattice for the [[7-limit]]. Made using [[Scale Workshop]].]]

			== Isomorphic keyboards ==
			:''See also: [[Keyboard#Alternative keyboards\|Alternative keyboards]]''

			{{Wikipedia\|Isomorphic keyboard}}
			Isomorphic keyboards are a practical application of lattices in instrument design.
			An isomorphic keyboard is an instrument with a two-dimensional grid of buttons or keys, arranged so that any given sequence or combination of musical intervals maintains the "same shape" regardless of where it occurs.
			This corresponds directly to a lattice: the keyboard layout is essentially a lattice mapped to physical space.
			This mapping creates consistent fingering patterns for musicians, where a particular chord shape or scale pattern can be transposed by simply moving the same finger configuration to a different position on the keyboard.

			Unlike lattice diagrams that often reduce dimensionality through octave equivalence, isomorphic keyboards must physically represent all pitches.
			This effectively limits them to rank 2 systems, where the two-dimensional physical layout must accommodate the full range of playable notes.
			Quite often these are still tuned to [[EDOs]], which are rank-1 systems.
			In such cases, there is an additional interval on the keyboard that is a [[unison]], although it may be out of reach on the keyboard.

			Examples of isomorphic keyboard designs include the {{w\|Jankó keyboard}}, the {{w\|Wicki–Hayden note layout}} and digital controllers like the [[Lumatone]] and [[Linnstrument]].

			[[File:Lattice_bosanquet.png\|300px\|thumb\|none\|An isomorphic meantone layout (Bosanquet).]]

	== ~~Examples~~ ==		== Gallery ==
	<gallery>		<gallery>
	File:Lattice Marvel.png\|A square lattice representing [[marvel]] temperament.		File:Lattice Marvel.png\|A square lattice representing [[marvel]] temperament.
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	== See also ==		== See also ==
	* [[Fokker block]]		* [[Fokker block]]
			* [[Keyboard]]

	== External links ==		== External links ==

Sintel: /* See also */ fokker block

2025-04-25T17:04:54Z

Sintel: Rewrite (more work needed though!)

2025-04-25T17:01:55Z

Rewrite (more work needed though!)

← Older revision		Revision as of 17:01, 25 April 2025
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	{{Wikipedia\|Lattice (music)}}		{{Wikipedia\|Lattice (music)}}
	A '''~~harmonic~~ lattice ~~diagram~~''' (~~'''lattice''') is an instrument to visualize relations of tones, which projects the multidimensional~~ [[~~just intonation~~\|JI]] ~~interval space onto~~ a ~~plane in such a way that intervals~~ that differ by ~~one particular JI relation will always~~ appear at fixed relative positions to each other. ~~Most commonly, the dimensionality of JI [[subgroup]]s is reduced~~ by ~~one by means of [[octave equivalence]]~~, ~~so that intervals an octave apart are mapped to the same point~~.		A '''lattice''' is a geometric construction that organizes pitches (or [[pitch class\|pitch classes]]) according to their intervallic relationships.
			In a lattice, pitches are represented by points, and tones that differ by a specific interval consistently appear in fixed relative positions to each other.
			These points can be connected by lines to highligh specific consonances.
			This creates a structure where pitch relationships can be analyzed through their geometric arrangement, which can be visualized in a lattice diagram.

			Most lattice diagrams reduce the dimensionality by one through [[octave equivalence]], meaning intervals separated by octaves are treated as equivalent points.
			This reduction makes a lattice diagram more comprehensible when projected onto a two-dimensional plane.

			While lattices are often used to represent [[just intonation]], they can also be applied to tempered spaces, in particular [[rank-3]] systems generated by two distinct intervals (aside from the [[period]]) and which thus function equivalently to three-prime [[JI subgroup]]s.

			== Tonnetz ==
	{{Wikipedia\|Tonnetz}}		{{Wikipedia\|Tonnetz}}
	~~{{Wikipedia\|Vogel's Tonnetz}}~~		The Tonnetz is a prominent example of a lattice representing [[5-limit]] intervals. In its conventional hexagonal arrangement, the perfect fifth (3/2) and major third (5/4) intervals are positioned 60 degrees apart, with the major sixth ([[5/3]]) placed 60 degrees from the major third and 120 degrees from the perfect fifth. This arrangement results in [[5-limit]] major triads ([[4:5:6]]) and minor triads ([[10:12:15]]) forming triangular patterns within the lattice structure.
	The ~~'''~~Tonnetz~~'''~~ is ~~the lattice that represents~~ a ~~tuning system's mapping~~ of [[5-limit]] intervals, ~~conventionally arranged in a hexagonal fashion such that [[~~5/4~~]] and [[3/2]]~~ are 60 degrees apart, with [[5/3]] ~~being~~ 60 degrees from ~~5/4~~ and 120 degrees from ~~3/2, so that~~ the 5-limit [[4:5:6~~\|major (4:5:6)~~]] and [[10:12:15~~\|minor (10:12:15)~~]] ~~chords form triangles~~. ~~Such lattices~~ can ~~also~~ be ~~extended~~ to other subgroups with 3 primes ~~in them~~, where a respective fundamental chord such as ~~[[4~~:5:~~7]] or [[~~8~~:11:14]]~~ can be assigned to a triangle~~. '''Vogel's Tonnetz''' is a 7-limit extension of Euler's 5-limit Tonnetz~~.		The Tonnetz can be generalized to other subgroups with 3 primes, such as [[2.3.7]], where a respective fundamental chord such as 6:7:8 can be assigned to a triangle.

	~~It is also possible~~ to ~~create lattices for [[regular temperament~~\|~~tempered]] systems, in particular~~ [[~~rank-3 temperament\|rank~~-3]] ~~systems generated by two distinct intervals (aside from~~ the [[~~equave~~]] ~~or fraction thereof) and which thus function equivalently to three~~-~~prime JI~~ subgroups.		Lattices can be extended to incorporate more dimensions. For instance, {{w\|Vogel's Tonnetz}} extends Euler's [[5-limit]] Tonnetz to the [[7-limit]], resulting in a 3-dimensional diagram. Similarly, other lattices can be constructed for different subgroups or temperaments.

	== Examples ==		== Examples ==
	<gallery>		<gallery>
	File:Lattice Marvel.png\|A square lattice representing [[marvel]] ~~temperament~~.		File:Lattice Marvel.png\|A square lattice representing [[marvel]] temperament.
	File:41equal lattice 5-limit.png\|A hexagonal lattice representing [[41edo]]'s mapping of [[5-limit]] intervals, using [[ups and downs notation]].		File:41equal lattice 5-limit.png\|A hexagonal lattice representing [[41edo]]'s mapping of [[5-limit]] intervals, using [[ups and downs notation]].
	File:Lattice32.png\|A hexagonal lattice representing a subset of [[7-limit]] intervals, using [[color notation]].		File:Lattice32.png\|A hexagonal lattice representing a subset of [[7-limit]] intervals, using [[color notation]].

Sintel: Sintel moved page Harmonic lattice diagram to Lattice over redirect: Expand scope

2025-04-25T16:14:02Z

Sintel moved page Harmonic lattice diagram to Lattice over redirect: Expand scope

← Older revision	Revision as of 16:14, 25 April 2025
(No difference)

Lériendil at 00:27, 24 April 2025

2025-04-24T00:27:59Z

← Older revision		Revision as of 00:27, 24 April 2025
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	{{Wikipedia\|Lattice (music)}}		{{Wikipedia\|Lattice (music)}}
	A '''harmonic lattice diagram''' ('''lattice''') is an instrument to visualize relations of tones, which projects the multidimensional [[just intonation\|JI]] interval space onto a plane in such a way that intervals that differ by one particular JI relation will always appear at fixed relative positions to each other. Most commonly, the dimensionality of JI [[~~subgroups~~]] is reduced by one by means of [[octave equivalence]], so that intervals an octave apart are mapped to the same point.		A '''harmonic lattice diagram''' ('''lattice''') is an instrument to visualize relations of tones, which projects the multidimensional [[just intonation\|JI]] interval space onto a plane in such a way that intervals that differ by one particular JI relation will always appear at fixed relative positions to each other. Most commonly, the dimensionality of JI [[subgroup]]s is reduced by one by means of [[octave equivalence]], so that intervals an octave apart are mapped to the same point.

	{{Wikipedia\|Tonnetz}}		{{Wikipedia\|Tonnetz}}

Lériendil: expanded the tonnetz writeup

2025-04-24T00:27:47Z

expanded the tonnetz writeup

← Older revision		Revision as of 00:27, 24 April 2025
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	{{Wikipedia\|Lattice (music)}}		{{Wikipedia\|Lattice (music)}}
	A '''harmonic lattice diagram''' ('''lattice''') is an instrument to visualize relations of tones. ~~It's a projection~~ of ~~a multi-dimensional structure onto a 2D screen~~.		A '''harmonic lattice diagram''' ('''lattice''') is an instrument to visualize relations of tones, which projects the multidimensional [[just intonation\|JI]] interval space onto a plane in such a way that intervals that differ by one particular JI relation will always appear at fixed relative positions to each other. Most commonly, the dimensionality of JI [[subgroups]] is reduced by one by means of [[octave equivalence]], so that intervals an octave apart are mapped to the same point.

	{{Wikipedia\|Tonnetz}}		{{Wikipedia\|Tonnetz}}
	{{Wikipedia\|Vogel's Tonnetz}}		{{Wikipedia\|Vogel's Tonnetz}}
	The '''Tonnetz''' is the lattice that represents ~~[[12edo]]~~'s mapping of [[5-limit]] intervals. '''Vogel's Tonnetz''' is a 7-limit ~~version~~ of Euler's 5-limit Tonnetz.		The '''Tonnetz''' is the lattice that represents a tuning system's mapping of [[5-limit]] intervals, conventionally arranged in a hexagonal fashion such that [[5/4]] and [[3/2]] are 60 degrees apart, with [[5/3]] being 60 degrees from 5/4 and 120 degrees from 3/2, so that the 5-limit [[4:5:6\|major (4:5:6)]] and [[10:12:15\|minor (10:12:15)]] chords form triangles. Such lattices can also be extended to other subgroups with 3 primes in them, where a respective fundamental chord such as [[4:5:7]] or [[8:11:14]] can be assigned to a triangle. '''Vogel's Tonnetz''' is a 7-limit extension of Euler's 5-limit Tonnetz.

			It is also possible to create lattices for [[regular temperament\|tempered]] systems, in particular [[rank-3 temperament\|rank-3]] systems generated by two distinct intervals (aside from the [[equave]] or fraction thereof) and which thus function equivalently to three-prime JI subgroups.

	== Examples ==		== Examples ==

VectorGraphics at 02:26, 4 March 2025

2025-03-04T02:26:42Z

← Older revision		Revision as of 02:26, 4 March 2025
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	File:41equal lattice 5-limit.png\|A hexagonal lattice representing [[41edo]]'s mapping of [[5-limit]] intervals, using [[ups and downs notation]].		File:41equal lattice 5-limit.png\|A hexagonal lattice representing [[41edo]]'s mapping of [[5-limit]] intervals, using [[ups and downs notation]].
	File:Lattice32.png\|A hexagonal lattice representing a subset of [[7-limit]] intervals, using [[color notation]].		File:Lattice32.png\|A hexagonal lattice representing a subset of [[7-limit]] intervals, using [[color notation]].
			File:Torus.png\|A '''torus''' of notes in 15edo, notated by porcupine notation with LH-NiceIonian as the base scale.
	</gallery>		</gallery>

← Older revision		Revision as of 00:27, 24 April 2025
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	{{Wikipedia\|Lattice (music)}}		{{Wikipedia\|Lattice (music)}}
	A '''harmonic lattice diagram''' ('''lattice''') is an instrument to visualize relations of tones, which projects the multidimensional [[just intonation\|JI]] interval space onto a plane in such a way that intervals that differ by one particular JI relation will always appear at fixed relative positions to each other. Most commonly, the dimensionality of JI [[~~subgroups~~]] is reduced by one by means of [[octave equivalence]], so that intervals an octave apart are mapped to the same point.		A '''harmonic lattice diagram''' ('''lattice''') is an instrument to visualize relations of tones, which projects the multidimensional [[just intonation\|JI]] interval space onto a plane in such a way that intervals that differ by one particular JI relation will always appear at fixed relative positions to each other. Most commonly, the dimensionality of JI [[subgroup]]s is reduced by one by means of [[octave equivalence]], so that intervals an octave apart are mapped to the same point.

	{{Wikipedia\|Tonnetz}}		{{Wikipedia\|Tonnetz}}

← Older revision		Revision as of 00:27, 24 April 2025
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	{{Wikipedia\|Lattice (music)}}		{{Wikipedia\|Lattice (music)}}
	A '''harmonic lattice diagram''' ('''lattice''') is an instrument to visualize relations of tones. ~~It's a projection~~ of ~~a multi-dimensional structure onto a 2D screen~~.		A '''harmonic lattice diagram''' ('''lattice''') is an instrument to visualize relations of tones, which projects the multidimensional [[just intonation\|JI]] interval space onto a plane in such a way that intervals that differ by one particular JI relation will always appear at fixed relative positions to each other. Most commonly, the dimensionality of JI [[subgroups]] is reduced by one by means of [[octave equivalence]], so that intervals an octave apart are mapped to the same point.

	{{Wikipedia\|Tonnetz}}		{{Wikipedia\|Tonnetz}}
	{{Wikipedia\|Vogel's Tonnetz}}		{{Wikipedia\|Vogel's Tonnetz}}
	The '''Tonnetz''' is the lattice that represents ~~[[12edo]]~~'s mapping of [[5-limit]] intervals. '''Vogel's Tonnetz''' is a 7-limit ~~version~~ of Euler's 5-limit Tonnetz.		The '''Tonnetz''' is the lattice that represents a tuning system's mapping of [[5-limit]] intervals, conventionally arranged in a hexagonal fashion such that [[5/4]] and [[3/2]] are 60 degrees apart, with [[5/3]] being 60 degrees from 5/4 and 120 degrees from 3/2, so that the 5-limit [[4:5:6\|major (4:5:6)]] and [[10:12:15\|minor (10:12:15)]] chords form triangles. Such lattices can also be extended to other subgroups with 3 primes in them, where a respective fundamental chord such as [[4:5:7]] or [[8:11:14]] can be assigned to a triangle. '''Vogel's Tonnetz''' is a 7-limit extension of Euler's 5-limit Tonnetz.

			It is also possible to create lattices for [[regular temperament\|tempered]] systems, in particular [[rank-3 temperament\|rank-3]] systems generated by two distinct intervals (aside from the [[equave]] or fraction thereof) and which thus function equivalently to three-prime JI subgroups.

	== Examples ==		== Examples ==

← Older revision		Revision as of 17:04, 25 April 2025
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	File:Torus.png\|A '''torus''' of notes in 15edo, notated by porcupine notation with LH-NiceIonian as the base scale.		File:Torus.png\|A '''torus''' of notes in 15edo, notated by porcupine notation with LH-NiceIonian as the base scale.
	</gallery>		</gallery>

			== See also ==
			* [[Fokker block]]

	== External links ==		== External links ==