We can take the pitches in a tuning, represent them as dots, and arrange them in a space so that where any two pitches are seperated by a consonant interval, that interval can be shown as a straight line connecting the two dots (pitches) involved.

We can define any set of intervals we want as being consonant, and we can always arrange the dots of any tuning so that _all_ our consonances are shown as straight lines, with none of the lines crossing, if our space has enough dimensions. Sound like Dr. Who yet? :)

Anyway, these spaces are called tonespaces -- usually, we consider the intervals of just intonation (at some limit) consonant, but really we can use any intervals we want.

Let's say we use JI. Consonant intervals are written as fractions in JI. In order to keep our lines from crossing, we need one dimension for every number appearing in a numerator or denominator in our list of consonances. Usually, we abide by octave-equivalence, where factors of 2 are ignored, so that a dot represents all octave copies of its pitch.

One of the simplest and most common tonespaces is the two- factor space of the 5-limit. Our list of consonances is: (1:1, 5:4, 8:5, 3:2, 4:3, 6:5, 5:3). Ignoring the 2's, we only need the numbers 3 and 5 to write all these fractions. So we need two dimensions, x and y. Adding a 3 to a numerator will move us + on x, adding a 3 to a denominator will be a - move on x. Same for the 5's and y. So:

    5/4-----15/8---45/32
    / \     / \    /
   /   \   /   \  /
  /     \ /     \/
1/1----3/2-----9/8

Every line on this diagram is in our list of consonances, and no lines cross. The dots are named with a fraction denoting the _pitch_ in the _tuning_ that's being shown. In the above case, the tuning is [1/1, 9/8, 5/4, 45/32, 3/2, 15/8]. I wrote those fractions with /'s, to show that they are _pitches_. Notice that I wrote the fractions in our list of consonant intervals with :'s to show they are _intervals_. That's a convention developed on this list, to keep intervals straight from pitches. For example, starting at 3/2, we move a 5:4 to 15/8. See?

Now say we wanted to add 15:8 to our list of consonances. We'd add a dimension for 15, and then a step from 1/1 to 15/8 would be a single line on our 15-axis, rather than two lines (one on the 3 and one on the 5). See?

Note!

Now look at this Wilson diagram: http://www.anaphoria.com/dal12.html

It depicts two structures in 6-factor tonespace (1 3 5 7 9 11). There appear to be lines crossing, but the lines only cross in 2-D (on the paper). The thing on the paper is only a shadow of a 6-dimensional object. Cool, huh.

Anywho, the beat is that if we desire tunings with a good number of consonances for a good economy of tones, they look like connected structures in tonespace. One type of very highly-connected structure is the tonality diamond (see Partch -- the figure on the right of the above diagram is the 11-limit diamond). Another type is the Combination Product Set, or CPS.

To make a CPS tuning, start with your list of consonances, and take, say, two of them at a time, follow them out on the lattice, and mark a dot where you stop. For example, taking our 11-limit consonance list: (1 3 5 7 9 11) two at-a-time, we get...

(1 3)   (3 5)   (5 7)  (7 9)  (9 11)
(1 5)   (3 7)   (5 9)  (7 11)
(1 7)   (3 9)   (5 11)
(1 9)   (3 11)
(1 11)

Order doesn't matter, since moving a 5:4 and then a 3:2 away from 1/1 gets us to 15/8 the same as moving a 3:2 and then a 5:4. You can see that all this moving amounts to multiplying the numbers. So our pitches are...

3/2   15/8    35/32  63/32   99/64
5/4   21/16   45/32  77/64
7/4   27/16   55/32
9/8   33/32
11/8

Notice there's no 1/1! All those pitches are measured from 1/1, but it's not in the tuning itself. That's one cool thing about CPSs -- they have no "center" tonality (unlike tonality diamonds).

Note also that while the numbers in these fractions are rather large, the tuning has a lot of low-numbered consonant relationships.

Lastly, note that we could have taken the factors three at a time (that would give the figure on the left of the Wilson diagram above), four at a time, etc. One at a time gives a tuning the same as the consonant list; six at a time gives only one pitch.

You're ready to explore CPSs!

-Carl