User:Romeolz/Isomorphic layouts: Difference between revisions

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=== Interval vector ===

This refers to the shape (or shapes) of an interval in an isomorphic layout. A unison vector is the same as what the folks at Lumatone call repeating note patterns.

If you don't know what vectors are: A vector is basically an arrow with a length and a direction. I'm working with 2-dimensional vectors, so they're comprised of an x-component and a y-component. Vectors can be added and subtracted together, and they can be multiplied or divided by numbers. [https://www.geogebra.org/classic/u3aatst8 Here] you can have a look at what vectors and the operations done on them look like.

This refers to the shape (or shapes) of an interval in an isomorphic layout. A unison vector (aka redundancy) is the same as what the folks at Lumatone call repeating note patterns.

[[File:19edo bosanquet unison octave offset vectors terpstra temp.png|none|thumb|600x600px|19edo Bosanquet-Wilson layout with drawn on vectors, using the Terpstran convention]]

==== Offset vectors ====

Every isomorphic layout can be defined by two offset vectors. I will call them ov and oh, offset vertical and offset horizontal~~. Usually they have a length (magnitude) of 1~~. They correspond to a key next to the origin. ~~It is possible for the offsets to have different magnitudes. Magnitudes of 1 and 1 and an~~ angle of 60 or 120 degrees ~~correspond~~ to a hexagonal layout. An angle of 90 degrees corresponds to a square grid. Every interval's shape can be described as a combination of ov:s and oh:s in the form of pa/b = x*oh + y*ov.

Every isomorphic layout can be defined by two offset vectors. I will call them ov and oh, offset vertical and offset horizontal. They correspond to a key next to the origin. An angle of 60 or 120 degrees correspons to a hexagonal layout. An angle of 90 degrees corresponds to a square grid.

[[File:19edo bosanquet octave constructed with offset vectors.png|none|thumb|600x600px|The octave vector expressed as a combination of offset vectors]]

Every interval's shape can be described as a combination of ov:s and oh:s in the form of pa/b = x*oh + y*ov.

==== Offset conventions for hexagonal keyboards ====

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** ⇒ Terpstran: ov ⇒ oh - ov

** ⇒ Workshop: ov ⇒ -ov

[[File:19edo bosanquet interval vector math.png|thumb|600x600px|19edo Bosanquet-Wilson: p3 - p2 = p3/2, p5 - 2*p2 = p5/4.]]

==== Interval math with interval vectors ====

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Let's define vectors p1, p2, p3, p5... as the shape of *1/1, 2/1, 3/1, 5/1... on a given isomorphic layout. p1 might not always exist at all: this is the case with rank-2 layouts. p1 is the "unison vector", and there is no obvious choice which way to point it, because there will always be two options equally far away. I have decided to always choose the one that has a smaller angle between it and the "playing direction", more on that later. Every other interval vector should be defined as the physically closest option to the origin. (except for a few edge cases up to interpretation like 2/1 in 12edo Bosanquet-Wilson)

Now we can derive the shapes of new intervals by adding and subtracting these vectors like so:

Now we can derive the shapes of new intervals by adding and subtracting these vectors like so: (examples not pictured)

p6 = p2*3 = p2 + p3,

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==== Lumatone: Classic mode, melodic mode, harmonic mode ====

Though intended for 12edo, these terms can be generalized to other tunings.

[[File:Lumatone classic mode with playing direction and scale.png|none|thumb|600x600px|add caption]]

Classic mode refers to a 12edo Bosanquet-Wilson layout originally, but I will expand the meaning to include other layouts that have specific properties. Their equave vectors and playing directions are the same, that being horizontal. They're non-jagged, and have some easily accessible albitonic scale as their basis. It's a meandering line that closely follows the playing direction. Classic mode is analogous to aural bias.

[[File:Lumatone melodic mode with playing direction and scale.png|none|thumb|add caption]]

Melodic mode refers to the Wicki-Hayden layout in 12edo. I will add layouts to the term that adhere to the following. The equave vector is close to vertical, and so is the playing direction. They also have a very clear albitonic scale as their basis, but rather it's laid out in a block-like shape much more densely. It zigzags around the playing direction. Melodic mode can be aurally biased or harmonically biased depending on the harmonic content of the albitonic scale.

[[File:Lumatone harmonic mode with 3 major scale constructions.png|none|thumb|750x750px|Three different ways to construct a major scale: 5-limit, pythagorean with augmented octave, pythagorean with diminished octave]]

Harmonic mode is the way Lumatone calls the 12edo Harmonic Table layout. It, and others like it, put harmonically related notes very close to each other. Pitch distance is not well represented in these layouts. There may not always be a clear playing direction, or they may be very jagged. The concept of an albitonic scale is often completely disregarded. Harmonic mode maps very nicely to the concept of harmonic bias.

@@ Line 19: / Line 19: @@
 === Interval vector ===
-This refers to the shape (or shapes) of an interval in an isomorphic layout. A unison vector is the same as what the folks at Lumatone call repeating note patterns.
+If you don't know what vectors are: A vector is basically an arrow with a length and a direction. I'm working with 2-dimensional vectors, so they're comprised of an x-component and a y-component. Vectors can be added and subtracted together, and they can be multiplied or divided by numbers. [https://www.geogebra.org/classic/u3aatst8 Here] you can have a look at what vectors and the operations done on them look like.
+This refers to the shape (or shapes) of an interval in an isomorphic layout. A unison vector (aka redundancy) is the same as what the folks at Lumatone call repeating note patterns.
+[[File:19edo bosanquet unison octave offset vectors terpstra temp.png|none|thumb|600x600px|19edo Bosanquet-Wilson layout with drawn on vectors, using the Terpstran convention]]
 ==== Offset vectors ====
-Every isomorphic layout can be defined by two offset vectors. I will call them o<sub>v</sub> and o<sub>h</sub>, offset vertical and offset horizontal. Usually they have a length (magnitude) of 1. They correspond to a key next to the origin. It is possible for the offsets to have different magnitudes. Magnitudes of 1 and 1 and an angle of 60 or 120 degrees correspond to a hexagonal layout. An angle of 90 degrees corresponds to a square grid. Every interval's shape can be described as a combination of o<sub>v</sub>:s and o<sub>h</sub>:s in the form of p<sub>a/b</sub> = x*o<sub>h</sub> + y*o<sub>v</sub>.
+Every isomorphic layout can be defined by two offset vectors. I will call them o<sub>v</sub> and o<sub>h</sub>, offset vertical and offset horizontal. They correspond to a key next to the origin. An angle of 60 or 120 degrees correspons to a hexagonal layout. An angle of 90 degrees corresponds to a square grid.
+[[File:19edo bosanquet octave constructed with offset vectors.png|none|thumb|600x600px|The octave vector expressed as a combination of offset vectors]]
+Every interval's shape can be described as a combination of o<sub>v</sub>:s and o<sub>h</sub>:s in the form of p<sub>a/b</sub> = x*o<sub>h</sub> + y*o<sub>v</sub>.
 ==== Offset conventions for hexagonal keyboards ====
@@ Line 43: / Line 48: @@
 ** ⇒ Terpstran: o<sub>v</sub> ⇒ o<sub>h</sub> - o<sub>v</sub>
 ** ⇒ Workshop: o<sub>v</sub> ⇒ -o<sub>v</sub>
+[[File:19edo bosanquet interval vector math.png|thumb|600x600px|19edo Bosanquet-Wilson: p<sub>3</sub> - p<sub>2</sub> = p<sub>3/2</sub>, p<sub>5</sub> - 2*p<sub>2</sub> = p<sub>5/4</sub>.]]
 ==== Interval math with interval vectors ====
@@ Line 50: / Line 56: @@
 Let's define vectors p<sub>1</sub>, p<sub>2</sub>, p<sub>3</sub>, p<sub>5</sub>... as the shape of *1/1, 2/1, 3/1, 5/1... on a given isomorphic layout. p<sub>1</sub> might not always exist at all: this is the case with rank-2 layouts. p<sub>1</sub> is the "unison vector", and there is no obvious choice which way to point it, because there will always be two options equally far away. I have decided to always choose the one that has a smaller angle between it and the "playing direction", more on that later. Every other interval vector should be defined as the physically closest option to the origin. (except for a few edge cases up to interpretation like 2/1 in 12edo Bosanquet-Wilson)
-Now we can derive the shapes of new intervals by adding and subtracting these vectors like so:
+Now we can derive the shapes of new intervals by adding and subtracting these vectors like so: (examples not pictured)
 p<sub>6</sub> = p<sub>2*3</sub> = p<sub>2</sub> + p<sub>3</sub>,
@@ Line 73: / Line 79: @@
 ==== Lumatone: Classic mode, melodic mode, harmonic mode ====
 Though intended for 12edo, these terms can be generalized to other tunings.
+[[File:Lumatone classic mode with playing direction and scale.png|none|thumb|600x600px|add caption]]
 Classic mode refers to a 12edo Bosanquet-Wilson layout originally, but I will expand the meaning to include other layouts that have specific properties. Their equave vectors and playing directions are the same, that being horizontal. They're non-jagged, and have some easily accessible albitonic scale as their basis. It's a meandering line that closely follows the playing direction. Classic mode is analogous to aural bias.
+[[File:Lumatone melodic mode with playing direction and scale.png|none|thumb|add caption]]
 Melodic mode refers to the Wicki-Hayden layout in 12edo. I will add layouts to the term that adhere to the following. The equave vector is close to vertical, and so is the playing direction. They also have a very clear albitonic scale as their basis, but rather it's laid out in a block-like shape much more densely. It zigzags around the playing direction. Melodic mode can be aurally biased or harmonically biased depending on the harmonic content of the albitonic scale.
+[[File:Lumatone harmonic mode with 3 major scale constructions.png|none|thumb|750x750px|Three different ways to construct a major scale: 5-limit, pythagorean with augmented octave, pythagorean with diminished octave]]
 Harmonic mode is the way Lumatone calls the 12edo Harmonic Table layout. It, and others like it, put harmonically related notes very close to each other. Pitch distance is not well represented in these layouts. There may not always be a clear playing direction, or they may be very jagged. The concept of an albitonic scale is often completely disregarded. Harmonic mode maps very nicely to the concept of harmonic bias.