User:Romeolz/Isomorphic layouts: Difference between revisions

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

I only figured this out recently as of September 2025, and it's a beautiful way of thinking about intervals, commas and RTT.

I only figured this out recently as of September 2025, and it's a beautiful way of thinking about intervals, commas and RTT.

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)~~

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 could be defined as the physically closest option to the playing direction when it exists. Defining the vectors ultimately comes down to personal preference, but it's handy to have a mathematical way of defining them when there are too many to define by hand.

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

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81/80 = [-4 4 -1〉, p81/80 = p2-4*34*5-1 = -4*p2 + 1*p3 - 1*p5

The really neat thing about this is that the exponents and multiplications in the subscript turn into multiplications and additions respectively. The reason why is because we're going from linear frequency space (Hz) to logarithmic pitch space (cents)! That fact that all isomorphic layouts are logarithmic in nature and the math we do with them reflects that, was groundbreaking to me. Though it's obvious in hindsight...

The really neat thing about this is that the exponents and multiplications in the subscript turn into multiplications and additions respectively. The reason why is because we're going from linear frequency space (Hz) to logarithmic pitch space (cents)! That fact that all isomorphic layouts are logarithmic in nature and the math we do with them reflects that, was groundbreaking to me. Though it's obvious in hindsight...

=== Aural bias vs. harmonic bias ===

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Layouts with a '''harmonic bias''' lay notes with a '''small harmonic distance''' near each other physically on the layout.

~~(add examples)~~

==== 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|Bosanquet-Wilson: the albitonic major scale in blue overlaid with the playing direction/octave vector in purple]]

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.

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-[[User:Romeolz/Isomorphic layouts#Jaggedness, (rank-2 not supported, unison vector)|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|Wicki-Hayden: the albitonic major scale in blue overlaid with the playing direction/octave vector in purple]]

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.

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Linearity implies that the layout has a specific direction along which it is most natural to play. On a Lumatone, for example, to an extent you're locked to a horizontal playing style (if you need to access all of the keys that is). This direction is the same as the octave/equave vector on non-jagged layouts (see below). I call this the '''playing direction'''.

A layout is the most linear when the angle between the unison vector and playing direction is 90°. When this is the case, every ascending edostep is ~~incresingly~~ far away from a line parallel to the unison vector situated at the starting point.

A layout is the most linear when the angle between the unison vector and playing direction is 90°. When this is the case, every ascending edostep is increasingly far away from a line parallel to the unison vector situated at the starting point.

Linearity can be given a numerical value based on the smaller angle (α) between the unison vector and playing direction. α/90° * 100% gives us a value from 0-100% describing the linearity of the layout.

@@ Line 51: / Line 51: @@
 ==== Interval math with interval vectors ====
-I only figured this out recently as of September 2025, and it's a beautiful way of thinking about intervals, commas and RTT.
+<small>I only figured this out recently as of September 2025, and it's a beautiful way of thinking about intervals, commas and RTT.</small>
-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)
+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 could be defined as the physically closest option to the playing direction when it exists. Defining the vectors ultimately comes down to personal preference, but it's handy to have a mathematical way of defining them when there are too many to define by hand.
 Now we can derive the shapes of new intervals by adding and subtracting these vectors like so: (examples not pictured)
@@ Line 66: / Line 67: @@
 /80 = [-4 4 -1〉, p<sub>81/80</sub> = p<sub>2</sub><sup>-4</sup><sub>*3</sub><sup>4</sup><sub>*5</sub><sup>-1</sup> = -4*p<sub>2</sub> + 1*p<sub>3</sub> - 1*p<sub>5</sub>
-The really neat thing about this is that the exponents and multiplications in the subscript turn into multiplications and additions respectively. The reason why is because we're going from linear frequency space (Hz) to logarithmic pitch space (cents)! That fact that all isomorphic layouts are logarithmic in nature and the math we do with them reflects that, was groundbreaking to me. Though it's obvious in hindsight...
+<small>The really neat thing about this is that the exponents and multiplications in the subscript turn into multiplications and additions respectively. The reason why is because we're going from linear frequency space (Hz) to logarithmic pitch space (cents)! That fact that all isomorphic layouts are logarithmic in nature and the math we do with them reflects that, was groundbreaking to me. Though it's obvious in hindsight...</small>
 === Aural bias vs. harmonic bias ===
@@ Line 74: / Line 75: @@
 Layouts with a '''harmonic bias''' lay notes with a '''small harmonic distance''' near each other physically on the layout.
-(add examples)
 ==== 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|Bosanquet-Wilson: the albitonic major scale in blue overlaid with the playing direction/octave vector in purple]]
-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.
+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-[[User:Romeolz/Isomorphic layouts#Jaggedness, (rank-2 not supported, unison vector)|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|Wicki-Hayden: the albitonic major scale in blue overlaid with the playing direction/octave vector in purple]]
 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.
@@ Line 91: / Line 90: @@
 Linearity implies that the layout has a specific direction along which it is most natural to play. On a Lumatone, for example, to an extent you're locked to a horizontal playing style (if you need to access all of the keys that is). This direction is the same as the octave/equave vector on non-jagged layouts (see below). I call this the '''playing direction'''.
-A layout is the most linear when the angle between the unison vector and playing direction is 90°. When this is the case, every ascending edostep is incresingly far away from a line parallel to the unison vector situated at the starting point.
+A layout is the most linear when the angle between the unison vector and playing direction is 90°. When this is the case, every ascending edostep is increasingly far away from a line parallel to the unison vector situated at the starting point.
 Linearity can be given a numerical value based on the smaller angle (α) between the unison vector and playing direction. α/90° * 100% gives us a value from 0-100% describing the linearity of the layout.