Just intervals and non-standard bases of the integer lattice

Hi everyone in the group!

I'm experimenting with non-standard bases of the integer lattice with the intent of using them for interval notation, just like monzos use the standard basis. I'd like to share my ideas and possibly receive hints on how to improve them.
You will see that I'm not professional (neither mathematician nor musician), but I think I found something that is worth sharing. Let me give you a somewhat long summary. Thanks for reading it. Any feedback is welcome.

Kalman Keri
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A notation for 5-limit intervals

Years before I was searching for a notation for 5-limit intervals that would make calculations with them easier. My intuition told me to find the smallest possible superparticular ratios in the 5-limit. Finally I found three ratios by trial and error:

81:80, 25:24, 16:15.

It turned out that any 5-limit interval can be built using only these ratios. More precisely, they are a non-standard basis of the free abelian group formed by 5-smooth numbers and their fractions under multiplication (thanks to Wikipedia and the Xenharmonic wiki - I couldn't have worded this a few months ago).
I instinctively developed a vectorial notation from these ratios. The following table lists the most common intervals using my notation, where the numbers in brackets (aka coordinates) represent the exponents of 81/80, 25/24 and 16/15 respectively.

[0 0 0]
1:1
prime
[0 0 1]
16:15
minor second
[0 1 1]
10:9
major second (minor tone)
[1 1 1]
9:8
major second (major tone)
[1 1 2]
6:5
minor third
[1 2 2]
5:4
major third
[1 2 3]
4:3
perfect fourth
[2 3 4]
3:2
perfect fifth
[2 3 5]
8:5
minor sixth
[2 4 5]
5:3
major sixth
[2 4 6]
16:9
minor seventh (complement of major tone)
[3 4 6]
9:5
minor seventh (complement of minor tone)
[3 5 6]
15:8
major seventh
[3 5 7]
2:1
octave

The table exhibits a few interesting patterns:
monotonicity: each coordinate is growing monotonously as the interval is growing
priority: none of the coordinates is smaller than its left neighbour
correspondence of the last coordinate with interval families (e.g. 1 ~ seconds)
These patterns, mainly the last one came as a surprise to me. Of course I chose intervals for my table deliberately. Choosing other intervals, the patterns can be broken but they nevertheless imply an inherent structure.

In the last years I revisited the topic a few times and made some progress. I discovered Stormer's theorem and found that my three ratios correspond to the three greatest 5-limit neighbors. I have since verified this pattern holds only up to 7-limit, but one can find many bases by choosing n instances of the n-smooth superparticular ratios.

Graphical representation

Recently I've been playing with the graphical representation of my notation and found more exciting patterns. Using an online tool I created a 3D model, where each point represents an interval in the table above (accomplished with a diminished fifth and an augmented fourth). Here is a link to the interactive model.

http://technology.cpm.org/general/3dgraph/?graph3ddata=____hHw7kw7kw7kIw7kw7kxmuIw7kxmuxmuIxmuxmuxmuJxmuxmuxBEJxmuxBExBEHxmuxBExQOHxmuxQOxQOGxBExQOxQOGxmuxBEx1YKxBExBEx1YKxBExQOx1YLxBExQOyg4LxBEx1Yyg4MxBEx1YyweMxQOx1YyweMxQOyg4yweHxQOyg4yLo http://technology.cpm.org/general/3dgraph/?graph3ddata=____hHw7kw7kw7kIw7kw7kxmuIw7kxmuxmuIxmuxmuxmuJxmuxmuxBEJxmuxBExBEHxmuxBExQOHxmuxQOxQOGxBExQOxQOGxmuxBEx1YKxBExBEx1YKxBExQOx1YLxBExQOyg4LxBEx1Yyg4MxBEx1YyweMxQOx1YyweMxQOyg4yweHxQOyg4yLo

As the model is rotated, we get various 2D orthographic views of the integer lattice. Views parallel to a base plane are bounded by an n×m rectangle where n and m are either 3, 5 or 7, according to the vector coordinates of the octave that are incidentally these prime numbers.

While there is a limited number of parallel views, there are infinitely many oblique ones. I will call some of them diagonal. Intuitively speaking, one gets a diagonal view when grid lines align and the view "cleans up". Diagonal views are bounded by rectangles (or parallelograms) whose one or both sides comprise a compound number of grid units.

In the following sample images intervals are color encoded. Prime and octave is red, seconds are orange, thirds are yellow and so on. The tool I used supports only six colours, so I used red for fourths too.

Pic.1. (left): a 5×7 view parallel to the yz plane.
Pic.2. (right): the same view rotated by 45° around the z axis yields a 8×7 view.

In specific view angles some points are hidden by others. When the view direction is parallel to the X axis (whose unit is the syntonic comma) this results in "comma hiding". For example in Pic.1 9:8 hides 10:9. Rotating the view by180° one can unhide it in expense of hiding 9:8. In Pic.2 both intervals are visible due to the diagonal view.
I leave the decision to the reader, whether this is an acceptable way of comma elimination, but the fact that the syntonic comma is a unit of the vector space makes this effect possible.

Pic.3. The same 5×7 view rotated and subdivided gives us an idea how to transform diatonic scales into the one dimensional space (aka ET). This view incidentally sorts notes into 12 groups, and even suggests the inclusion of 2 halftone steps into an octave.

Pic.4. Double-diagonal views create triangular grids. The grid shape is familiar, but due to the flattened 3rd dimension, intervals between adjacent tone contours couldn't be predicted by assigning basic steps to edges.

The above pictures are only a few examples of the many views that can be created by rotating the 3D model, what is essentially the 3 dimensional integer lattice decorated with notes of a 5-limit scale.

Comparison with the standard basis

Using the standard basis { 2, 3, 5, ... } for notation is a well established practice in the microtonal community. It facilitates calculation with just intervals the same way as my notation - by replacing multiplication with vectorial addition.

The basis { 81/80, 25/24, 16/15 } looks promising because - informally speaking - it distributes 5-limit intervals in the proximity of an angled line in 3D and the position along the line is proportional to the size of the interval. For reference, see the model of the same intervals on the standard basis that doesn't show such proportional distribution.

http://technology.cpm.org/general/3dgraph/?graph3ddata=____hHw7kw7kw7kIx1YwWawWaIxmuwG0xmuIwrUxBEw7kJxmuxmuwWaJwG0w7kxmuHxBEwWaw7kHwWawG0xBEGv-AxBExmuGywewG0wWaKxBExBEwG0KwWaxmuw7kLxQOw7kwWaLw7kwWaxmuMx1YwG0w7kMw7kxBEwWaMwrUxmuxmuHxmuw7kw7k http://technology.cpm.org/general/3dgraph/?graph3ddata=____hHw7kw7kw7kIx1YwWawWaIxmuwG0xmuIwrUxBEw7kJxmuxmuwWaJwG0w7kxmuHxBEwWaw7kHwWawG0xBEGv-AxBExmuGywewG0wWaKxBExBEwG0KwWaxmuw7kLxQOw7kwWaLw7kwWaxmuMx1YwG0w7kMw7kxBEwWaMwrUxmuxmuHxmuw7kw7k

Pic.5. Common 5-limit intervals on the standard basis. The view is parallel to the xy plane.

Extensibility

An appealing property of the standard basis is its extensibility to higher prime limits. Extensibility of a basis can be defined as the ability of placing a vector on the basis into a higher dimensional space without altering its coordinates. For example [ 1 0 0 ... > denotes the octave in any prime limit. It's not trivial to achieve this with a non-standard basis though.

Conclusion

In this post I have described a notation for 5-limit just intervals, based on the non-standard basis { 81/80, 25/24, 16/15 } of the three dimensional integer lattice. I created an interactive 3D model of the most common intervals in the notation, and basically demonstrated how to use the model as a scale-transformation device. I also informally worded the proportional distribution property of the described basis.

I think the generalized idea is to
find an appropriate basis for a harmonic limit,
represent your scale of choice as a spatial model,
rotate, translate and scale the model to yield projections in a lower dimension, and
define some useful metric on the projection.

This is by no means an end result, rather an area to be discovered. There is huge room for experimentation with various limits, bases, intervals and viewing angles.

Further directions for myself and anyone who is interested:
Examine the proportional distribution property and describe it more precisely. (I already have some insight on this)
Discover more bases of the integer lattice and evaluate their usefulness as notational and transformational devices.
Try to find non-standard extensible bases.
Relate this approach to other methods, e.g. the bra-ket notation to see how expressive it is in terms of defining temperaments.
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Lie algebra on torus comes to mind