Talk:Hexany

AfD discussion request for help

Combination product sets are common parlance amongst modern microtonal theorists interested in just intonation tunings and musical lattices. I'm sure there must be more articles published on them.

Can anyone help add more references?

One of the basic notions of the field along with Moment of Symmetry etc.

Robertinventor (talk) 19:08, 7 February 2008 (UTC)

geometrical name for figure with the Eikosany vertices

Does anyone know the name for the geometrical figure whose vertices are the vertices of the Eikosany (five dimensional figure with 20 vertices)?

Robertinventor (talk) 19:08, 7 February 2008 (UTC)

possibly birectified 5-simplex. —Tamfang (talk) 19:14, 16 February 2008 (UTC)

Thanks! That's obviously right, I'll edit the page, accordingly. Robert Walker (talk) 10:07, 17 February 2008 (UTC)

On second thought I'd prefer a more abstract birectified hexatope. —Tamfang (talk) 18:48, 6 June 2010 (UTC)

huh?

I think I understand what the article is getting at, but for a reader without a strong grasp of combinatorics it must be baffling.

• It's not really about the "hexany" but about its generalizations and supersets.
• If frequencies are to be wrapped into an octave, why have 2 as an axis of the hypercube?
• If we must have the hypercube, wouldn't it be simpler to choose (0,0,0,0) aka "1" as one of the corners of the octahedron? If not, say why not. Perhaps you could explicitly list the four octahedra derived from the hypercube.
• Nobody needs an explicit list of all the strings of n bits.
• Are the higher-dimensional examples for harmonies with more primes, or what?
• The passage about Pascal's triangle is obscure.

—Tamfang (talk) 03:37, 26 January 2009 (UTC)

Yes the higher dimensions introduce more primes. Each dimension of the hypercube is one of the primes. The 2 is usually omitted, but that makes the connection with pascal's triangle and the hypercube etc. not so easy to see geometrically.

The reason Erv Wilson did the hexany as the middle slice of the hypercube is because to get all the triad relationships for the faces, you need to have a separate prime (including 2) for each of the dimensions of 4D space - then you need to do it so all the vertices of the hexany are at all permutations of 1,1,0,0. That's the insight that makes the whole thing work so is an essential point of the construction.

If you used 0,0,0,0 as one of the vertices then you wouldn't get the right pitches for the vertices to get a scale with so many pure triads in it. Indeed you wouldn't even have the pure diads for the edges, never mind triads for the faces. E.g. if 1 (0,0,0,0) and 3*7 (0,1,0,1) are two of the vertices of an octahedron then the diad is 3*7 or 21, or reduced to the octave 21/16 which is rather a complex interval not a low ratio pure harmony. So it just wouldn't work musically if you did that.

Hope that is a bit clearer, at least maybe cleared up a few points.

BTW this is "RobertInventor" writing this but forgotten my login on this machine - I wrote the original version of the article and quite a lot of it still is as it was originally, especially the more mathematical bits.

You see things better when you come back after a few years. So, perhaps I can clean the maths up a bit some time with a fresh look at it - but very very busy right now, don't have the time - maybe later this year.

I suppose the point in the article is that the hexany is like the "first interesting CPS set" - the tetrahedra aren't that interesting as they are just tetrads - and the higher 5-dimensional CPS sets are too complex to visualise very easily - the hexany as an octahedron is easy to understand - and it's embedding in the hypercube is not that hard to follow if you have some idea of 4D space, only six points after all, and you can see that all the diagonals of the squares are the same length so it has to be a perfect octahedron, easy to see that.

You could separate it out into a whole series of articles, one on CPS sets and connection with Pascal's triangle and hypercube - and then separate articles for each of the more interesting CPS sets. But I thought it worked quite well as a way of making a short article, if you want to avoid getting too technical and specialised, to focus on the hexany, and then treat the other CPS sets as a way of putting the hexany in context. If someone did a separate article on CPS sets then much of that could be put in there instead, but not sure if that is the best way to take things - and if you don't have a separate article on CPS sets, then you need quite a bit of substance in the hexany article instead, wouldn't be a good idea to just focus on the hexany and never mention the higher dimensional versions.

Perhaps you could call it Hexany and higher dimensional analogues?? or Hexany as introduction to CPS sets, or some such,...

Nice to see some of the recent improvements in parts of the article - first time I looked at it for a year or two, thanks everyone for improving my original article :).

--- — Preceding unsigned comment added by 86.53.57.23 (talk) 23:27, 22 July 2011 (UTC)

This scale seems to be more or less C, Db-, Eb+, F#-, G, Bb-- (using + as up a comma, - down a comma, -- down two commas), and seems to be mostly a subset of the octotonic scale, with some notes missing (E, A) and others altered (commas up or down). Or alternatively, using a different starting note, C, Db+, E-, F#+, G, Bb++. —Preceding unsigned comment added by 24.200.155.32 (talk) 08:44, 8 January 2011 (UTC)

Proof that it is a regular octahederon when embedded in the hypercube

I just realised that informal proof isn't quite complete. Also - perhaps it counts as "original research".

It could still be a "folded octahedron" - a 3D object folded in 4D - like folding a perfect 2D square along its diagonal in 3D.

You can see that the octahedron's three equatorial "squares" have opposite sides parallel which means that they can be drawn within a 2D space - still that doesn't prove that the angles at the vertices have to be all the same, but by the symmetry of the hypercube then there is no reason for any angle to be larger than any other - so I think they have to be squares.

So - with equatorial squares (surely), and all edges exactly the same length, it seems pretty likely that it is a perfect octahedron, to me - but still not sure how you prove it rigorously. If it is a perfect octahedron then you should be able to show that there is a 3D slice of the hypercube which has all of its vertices included within it. Vice versa if you can show that then it proves it really is a perfect octahedron.

Part of it is that I am a bit out of practise with maths, have been programming for many years now and only done occasional maths for perhaps about 15 years or more. Perhaps some of you reading this who are active mathematicians can see this more easily than me. Also is there any reference - that's best of all if someone somewhere has published a proof that it is a perfect octahedron so it is not OR though for simple maths that anyone could follow not sure it really counts as OR.

Robert Walker (talk) 21:58, 27 November 2012 (UTC)

You can prove it is perfectly symmetrical because all the points are equally distant from the 0 point so they are all in the same 3D plane and then, all are equidistant from each other so must be a perfect octahedron in 3D. Robert Walker (talk) 18:23, 5 February 2017 (UTC)

Doesn't mention that the hexany also occurs in 3D e.g. in the cubic lattice

I wrote this as an article about Combination Product Sets so what I describe here is the hexany as a 2)4 CPS.

But the 3D hexany also occurs in 3D of course as someone has just commented off wiki about this page I talk about the 3D hexany here on my own website :) in cubic lattices [1]

So just leaving this as a note for now. It obviously needs to be updated to mention this. I just forgot to say and nobody else seems to have noticed the omission in all this time. But the article should continue to present the hexany as a 2)4 (2 out of 4) CPS set. After all that's where the idea originated and it's part of the progression of CPS sets to higher dimensions, the dekanies etc. Robert Walker (talk) 17:27, 5 February 2017 (UTC)

Extraneous material

the two later section after tuning do not relate to the hexany and i deleted some time back only to see it put back in. It does not even make logical sense. it speaks about the dekany in language that is impsosible to understand. the writer does not understand what they are talking about. It is slander and an attempt to undermine the understanding of it which is quite simple. — Preceding unsigned comment added by Banaphshu (talk • contribs) 05:26, 17 February 2024 (UTC)

I agree that the last two sections (relating to Pascal's triangle) are impossible to understand and add nothing useful to the article. DaveSeidel (talk) 14:14, 17 February 2024 (UTC)

Removed. The rest of the article is still somewhat lacking in intelligibility. —David Eppstein (talk) 17:54, 17 February 2024 (UTC)