User:Romeolz/Isomorphic layouts/Diatonic layouts
What is a diatonic layout?
A diatonic layout is any layout that has the 5L2s scale structure as its albitonic scale. The albitonic scale can be in any arrangement that doesn't imply another temperament. A layout implies another temperament when for example the layout physically has a key exactly in the middle of the root and fifth. This would imply that the fifth can be split equally in half, which is a property diatonic doesn't have. In other words, every key in the layout can uniquely and consistently be described as an interval in the 3-limit. I will be navigating the intervals of diatonic through fifths-based notation.
Take the Bosanquet-Wilson layout. It's basically the diatonic layout, as it's the most common, and modelled after the piano. The layout highlights the major and minor seconds.
image
We can compress the layout by backtracking by one key at every minor second. This can be thought of as a shear operation (see more).
image
If we keep shearing, we get an upside down Wicki-Hayden layout!
image of upside down and normal
Diatonic layouts and their offsets
To be a diatonic layout, the offsets of said layout have to be able to be represented as fifths-based intervals (part of the 3-limit). Yet, not all layouts that satisfy this condition are valid. A layout is invalid when there are 3-limit intervals that don't exist on it.
To understand this, we can take a look at the offsets in terms of large and small steps. 2L1s 3L1s
todotodotodo (how do i generalize this??? i don't understand why some worka nd why some don't)