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Newbies to this field may want to look at the Musical Geometry page first. The section Into the third dimensison - musical sculpture at the end of the page introduces the hexany and shows its location in a 3 5 7 lattice.
The Wilson CPS sets provide a wonderful opportunity for exploring the various just intonation generalisations of minor / major chords. CPS stands for."combination product set" and Erv Wilson who devised them is a renowned theorist and inventor of scales.
The models on this page are in VRML. I recomend Cortona as the VRML viewer for the models with mp3 audio clips - I think it may be the only current VRML browser that can play them. If you have another viewer such as Cosmo Player, you can view the models with midi clips.
However Cortona doesn't seem to be able to play the midi clips any more - early versions of Cortona could but from 4.0 it no longer can. It may be because the clips are short. The bug has been reported. They plan to fix it for future versions of their browser. However, 4.2 doesn't seem to play them yet.
You can also make these models yourself if you have Windows, using my Virtual Flower program in collaboration with Fractal Tune Smithy. You do it from Virtual Flower | Output | Musical Geometry. You can then make e.g. .Wav clips, or use any voice you like for the clips. Be sure to ask me if you have any questions about this. Robert Walker, support@virtualflower.co.uk
The idea of a 1 3 5 7 hexany is that you take 1 3 5, and 7 and multiply them together in pairs to give six numbers: 1*3, 1*5, 1*7, 3*5, 3*7, and 5*7
You can then place them on the vertices of an octahedron (a shape with six vertices and triangular faces), and if you wish, reduce them all into the octave so that they are all around the same pitch. In this musical geometry context the octahedron gets called a hexany because it has six vertices.
(Fractal Tune Smithy uses 2 instead of 1 in these numbers, for convenience of the programming - but up to octave reduction, 2 and 1 are the same).
To make this into a conventional scale, note that 1*1 isn't included in the list, so the scale doesn't have a 1/1 yet. So choose one of these notes as the 1/1. It doesn't matter which you choose, lets say, 5*7. Divide all the notes by 5*7 and you get: 1/1 8/7 6/5 48/35 8/5 12/7 2/1 (up to octave reduction). For those new to this notation - the ratios notation for pitches just shows the ratio of the frequencies of the notes. So for instance if the 1/1 is 500 hertz, then 6/5 is 600 Hertz, and so forth.
The interesting thing about this way of arranging the pitches on the hexany is that you can then make it so that each face plays the three notes for its vertices as a chord. Then all the face chords are consonant triads, and each pair of adjacent faces shares a diad.
Hint: when improvising in this scale, it is worthwhile to find the geodesic squares of the octahedron. See the notes to my improvisation in the 1 3 5 7 Hexany.
I've made a java applet of this one here: - Java hexany. You won't need anything installed to see and hear that one - most browsers will support java already.
Then for the VRML model to show in e.g. Cortona .
Hexany with mp3s (or open in new window).
If your browser downloads the clips automatically, wait a few minutes for them all to download before you try it. If you can only see the model and there is no indication of any activity from the browser, then you probably need to get the zip, unzip that and browse it off-line to hear the clips.
Here is the zip: Hexany with mp3s zip
Hexany with midi (or open in new window) - doesn't work currently in Cortona!
It is best to wait for everything to load before you start clicking on the vertices for the one with the midi clips - this one shouldn't take that long to load.
Click on any sphere above the centre of a face to hear the triad. Click on any vertex to hear the note, and on any of the spheres in the middle of the edges to hear a diad.
If your VRML browser has a choice of Study mode, you will find this the most useful for looking at these models, as it keeps the model centred. You may also be able to use the drop list of viewpoints which I've provided for the model.
You may need to wait for sound to stop before clicking on another note / chord in the model. If you don't do that, you may find a note or chord stops sounding, and have to reload the model again in order to hear it. But your browser may let you play several notes or chords at once. The latest version of Cortona can do that with the mp3s.
The triads are in two forms - major sounding and minorsounding - for opposite faces of the shape.
To explain that a bit more, any triad which you can write out as ratios can be written in two ways, e.g. 1/1 3/1 5/1 can be written as 15/15 15/5 15/3, = 1/3 1/5 1/15.
If a triad is simplest with the numbers on the top (ignoring any powers of two), it is an otonal triad - expressed most simply using the overtone series 1 2 3 4 .... If simplest with them on the bottom, its a utonal triad - expressed most simply using the undertone series 1/1 1/2 1/3 1/4 ....
The major chord 1 5/4 3/2 is an example of an otonal triad, and the minor chord 1 2/3 4/5 is an example of a utonal triad. The hexany has both of these, and also has other otonal and utonal chords with a septimal flavour (i.e. using ratios that include 7s in them) such as 1/1 3/2 7/4 and 1/1 2/3 4/7.
The otonal and utonal chords with the same prime factors are on opposite faces of the hexany.
Hexany are components of many of the larger CPS sets.
Now lets look at the dekany, which is what you get if you add one more factor:
3)5 Dekany 1 3 5 7 11 = octave reduced 1*3*5, 1*3*7, 1*3*11, 1*5*7, 1*5*11, 1*7*11, 3*5*7, 3*5*11, 3*7*11, and 5*7*11
The Dekany really needs four space dimensions to draw it properly, so this is a perspective view:
1 3 = cycle of fifths, needs only one dimension
1 3 5 = the 3 5 lattice - needs two dimensions.
1 3 5 7 = needs three dimensions.
1 3 5 7 11 = needs four dimiensions
You need a fourth space dimension at right angles to the three we are familiar with. It is not the dimension of time though, just another space dimension, if one can imagine such a thing, or more likely, fail to imagine it!
There are records of people who have said they can get an inkling of the idea of a fourth space dimension, and even solve problems in 4D by imagining the shapes in periods of concentration. But most are happy to view projections of it, and just rely on the maths to get it right.
What one can do however, to be able to at least work with these models geometrically, is to make a projection into a smaller number of dimensions. We do this anyway whenever we draw a 3D figure on a sheet of paper. In the same way, one can draw a 4D figure in 2D, or indeed, in 3D.
So, this shape is really a 3D drawing of a 4D figure. So though it looks quite symmetrical anyway, if one could only see it in 4D it would be far more symmetrical than this - all the 3D solid faces would be seen as arranged in a perfectly symmetrical pattern about the centre of the figure - if only one could see that, but unfortunately we can't. This model uses a particularly symmetrical projection that I adapted from one that Paul Erlich made. It's done in perspective which is why the shapes apparently in the centre are smaller - this is a perspective 3D view on a 4D solid.
Here it is with the mp3s:
Dekany 3 with mp3s (or open in new window) Wait a few minutes for all the clips to download. If you hear no sound, use the zip instead. (337 Kb of clips).
Here is the zip: Dekany 3 with mp3s zip
Dekany 3 with midi clips (or open in new window) - doesn't work currently in Cortona!
It will take a while to load the sound clips, as they all need to be transferred, and it is best to wait for them all to load before you start clicking on them.
If your browser can play the mp3s, you can use this zip instead [200KB] 278 files. Unpacks to 971 Kb.
The midi zip has this one, the next one, and the dekanies using the factor 9 instead of 11. (Because of number of files in it, though the total size of all the files is 971 KB, it will actually use 9 - 10 Mb if your hard disk has 32K clusters ).
Click on anything and it will sound a note or a chord. The utonal triads are shown as transparent triangles, and the otonal ones are solid. You can click on the otonal triangles to hear those.
The outer hexany of the model is apparently larger and outside - that's because it is nearer in the fourth spatial dimension, and one is looking through it to the tetrahedron gap in the centre. The tetrahedron in the centre is smaller because it is further away. All the triangles you see in this entire model are actually part of the outside of the four dimensional shape!
The octahedron and tetrahedron are nested neatly within each other because one is looking at it from directly opposite the outer octahedron in the fourth space dimension. Compare the method of drawing a cube as two concentric squares joined to each other by radial diagonal lines. This symmetrical view makes it easier to look at in 3D perspective.
Dekany 2 with mp3s (or open in new window) - this uses the Harmonic Flyght voice on the FM7 :-).
Wait a few minutes for all the clips to download. If you hear no sound, use the zip instead. (887 Kb of clips).
Here is the zip: Dekany 2 with mp3s zip
Dekany 2 midi clips (or open in new window)- doesn't work currently in Cortona!
This has otonal tetrads - this time, click on any of the tetrahedra to hear them. The tetrahedra are squashed because this is a 3D view on a 4D object.
The faces you can see through are the utonal triads, and you will find red or magenta spheres in the middle of each which you can click to hear the triads.
This is made by taking pairs of factors from 1 3 5 7 11:
1*3, 1*5, 1*7, 1*11, 3*5, 3*7, 3*11, 5*7, 5*11, and 7*11
Then selecting by a single factor gives an otonal tetrad. Selecting by 11 gives the bluesy dominant seventh 1*1,1 3*11, 5*11, and7*11 = 1 3 5 7.
Choosing two factors at a time out of a list of six, such as 1 3 5 7 11 13, gives the pentadekany, which is made out of six overlapping 2)5 dekanies.
You can also select any four factors at a time to get another pentadekany made out of six overlapping 3)5 dekanies. The pentadekany is a five dimensinoal figure, so hard clearly to show in a 3D projection.
This web page links to some models of the constituent dekanies of the 2)6 pentadekany, and a 3D model that shows most (but not all) of the chords of the complete figure. The two versions of the page show the same chords, but one is played as unison chords on a violin, and the other as broken chords on the 'cello midi voice.
There's no mp3 version of this page at present.
2)6 Pentadekany (or open in new window)- doesn't work currently in Cortona!
The 2)6 pentadekany has otonal pentads which you get by selecting one factor and the 4)6 pentadekany has utonal pentads which you get by skipping one factor. I don't yet have a model for the 4)6 pentadekany.
The Eikosany is even richer in chords than the two dekanies, and by selecting any one of the factors you will get the 2)5 dekany, and by skipping one of them you get a 3)5 dekany.
So, the Eikosany is made out of twelve overlapping dekanies. Admittedly with a lot of overlap - but imagine how many chords that makes!
I don't have a model of one, and it may be a bit too complex for a complete 3D model. However the constituent dekanies are all the ones on the Pentadekany page. The Eikosany doesn't have the otonal or utonal pentads of the pentadekanies.