User:Romeolz/Isomorphic layouts

Note: I am very much a math person and this is very theoretical. I mostly work in a Regular Temperament Theory (RTT) framework (sorry Domin).

Terminology I came up with

If there are established names for these concepts please let me know!!

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.

Interval math with interval vectors

I only figured this out recently, 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)

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

p₆ = p_2*3 = p₂ + p₃,

p_6/5 = p_2*3*(1/5) = p₂ + p₃ + (-p₅) = p₂ + p₃ - p₅

This lines up perfectly with Monzo notation. 6/1 = [1 1 0>, 6/5 = [1 1 -1>

81/80 = [-4 4 -1>, p_81/80 = 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...

Aural bias??? vs. harmonic bias

This concept refers to how the notes of a layout are situated relative to one another.

Layouts with an aural bias lay notes close in pitch near each other physically on the layout.

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

(add examples)

Linear, linearity (non-linear, nonlinearity), (rank-2 not supported, unison vector)

The linearity of a layout refers to how edostepwise motion looks on a layout. This is a notable trait when talking about aurally biased layouts.

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). Usually this direction is the same as of the octave vector. 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.

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.

Jaggedness, (rank-2 not supported, unison vector)

This trait mostly concerns aurally biased layouts. This layout property is heavily context dependent. For its evaluation you need to have an idea of what (relative) intervals you're going to be using to traverse up and down the layout.

A layout is not jagged when you can easily traverse it along the playing direction using your preferred intervals such that you only need to use one shape per interval without veering too far from the playing direction.

A layout is jagged when the aforementioned traversal requires regular use of two shapes per interval to stay in the playing direction.

Jaggedness, legibility and ergonomics

Any jaggedness makes a layout immediately less legible, because of the added complexity of two shapes per interval. Though having many different options for paths to take when traversing the layout can make playing a lot more ergonomic, as seen with layouts like "slanted 27edo Bosanquet-Wilson" regularly used by stalefleas.