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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. There are good viewers for these on the p.c., but unfortunately, I haven't heard of any that show these models okay on the mac. (especially, beware of the Cosmo Player beta for the mac - while the Cosmo player is one of the best viewers for the P.c., I've been told that the mac beta can erase your address book when you install it!! If you want to try it, back up your address book first.).
Two exellent VRML viewers for the p.c: Cosmo Player, and Cortona. It is harder to find a good one for the Mac - the Cosmo player beta isn't recommended as I've heard of people having problems with it. Cortona have recently made a release of their browser for the Mac, so maybe this is the one we've been waiting for??
The idea of a 2 3 5 7 hexany is that you take all pairs of numbers from 2 3 5 7 and multiply them together:
2*3 2*5 2*7 3*5 3*7 5*7
That gives six numbers. You can then place them on the vertices of an octahedron, and if you wish, octave reduce them. The octahedron used this way is known as a hexany because it has six vertices.
(program uses 2 instead of 1, but up to octave reduction, it's the same).
To make this into a conventional scale, note that 1*1 isn't in it, so we don't yet have a 1/1.
So choose one of these notes as the 1/1, 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).
The interesting thing about this scale is that if you put the notes on the vertices of an octahedron, and if each face plays the three notes for its vertices as a chord, then all the chords are consonant triads. So it is a great way of exploring triads and making a scale with many interesting triads in it.
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.
Here is a 3D model (or open in new window).
It is best to wait for everything to load before you start clicking on the vertices - 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 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.
Any just intonation triad can be written in two ways, e.g. 1 3 5 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. If simplest with them on the bottom, its a utonal triad - expressed most simply using the undertone series.
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 such as 1/1 3/2 7/4 and 1/1 2/3 4/7.
Otonal and utonal chords with the same factors are on opposite faces of the hexany.
The hexany is a common component of 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 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.
Add another factor, and 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 is to make a projection into a smaller number of dimensions.
We do this 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.
Here is a 3D model (or open in new window)
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.
Alternatively, you can use the zip [200KB] 278 files. Unpacks to 971 Kb.
The 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.
This model uses a particularly symmetrical projection that I adapted from one that Paul Erlich made. It's a perspective 3D view on a 4D solid.
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.
Here is a 3D model (or open in new window)
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 7*11
Then selecting by a single factor gives an otonal tetrad. Selecting by 11 gives the bluesy dominant seventh 1*11 3*11 5*11 7*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.
2)6 Pentadekany (or open in new window)
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. 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.