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MOS Cradle

refers to a technique of embedding one MOS scale inside another, to create a new hybrid scale, a MOS Cradle Scale. I (Andrew Heathwaite) invite you to experiment & share the results here.

Check out & add to a growing repository of MOS Cradle Scales here.

For this tutorial, I assume basic knowledge of Moment of Symmetry scale design. To summarize, you can design scales by building a chain of one interval (the generator) within a period of another interval -- often, but not always, the octave. When the resulting set of notes has exactly two step sizes, we call the scale a Moment of Symmetry, or MOS, scale. A prime example: the Pythagorean Scale, built using the octave as the period & the perfect fifth as the generator.

For this tutorial, I will limit us to MOS scales as subsets of edos, because we can easily show the steps as degrees in the superscale. But do keep in mind that you can apply these ideas to nonoctave & JI scales just as easily & with just as interesting results!

The "Parent"


We begin with a classic MOS scale. So, just to get us started, we'll use 11/31 of an octave as our generator, & the octave as our period. At five notes, we close on a pentatonic scale, a subset of 31edo. Throughout this tutorial, I will show the scales as step degrees of the superscale, like this:

9 2 9 2 9

A nice little scale. Tune your synth up to it & give it a whirl. The MOS Cradle technique will give us a new way to elaborate on this basic structure. We'll use it as the "parent" scale.

The "Cradle"


Our parent scale has two different step sizes. The large step = L = 9. The small step = s = 2. We will select one of these step sizes to use as a "cradle" for new pitches.

Using L


Let's use L = 9. We take those 9 degrees & look at ways of making new MOS scales within that, just as we'd do if we wanted MOS scales in 9edo. So let's try a few:

generator 1/9:
1 8
1 7 1
1 1 5 1 1
1 1 1 3 1 1 1

generator 2/9:
2 7
2 5 2
2 2 1 2 2

generator 3/9:
3 6

generator 4/9

4 5
4 1 4
1 3 1 3 1
1 2 1 1 1 2 1

Now that we have some MOS shapes, we can cut up our original L's back in the parent scale using any of these shapes. I'll show just a few, with the orignal L = 9 in bold & underlined:

4 5 2 4 5 2 4 5
1 7 1 2 1 7 1 2 1 7 1
1 3 1 3 1 2 1 3 1 3 1 2 1 3 1 3 1

Using s


Let's see what happens if we use s = 2 as the cradle. We have only one way to break down 2:

1 1

So if we insert 1 1 for 2, we get:

9 1 1 9 1 1 9

Using both


Let's insert 4 5 for 9 & 1 1 for 2:

4 5 1 1 4 5 1 1 4 5

Some Observations


Using this method, you arrive at new scales which contain the parent scale, plus a few extra notes. You can consider the extra notes "ornamental," secondary to the notes of the parent scale, or you can think of the whole scale as a brand new entity.

Often, the new scale will contain three step sizes, instead of the original two. So in addition to L & s, you'd have M. You can design your scale so that the three step sizes have interesting ratios to one another, if you like. I think it sounds nice when the step sizes don't add or multiply together to make each other.

Sometimes this technique will produce a scale you might have gotten to another way -- like a classic MOS scale.

Doubling/Tripling the edo


If you want to use MOS Cradle to elaborate on a scale in a small edo, consider doubling or tripling, etc., the number of notes. Say you want to use the pentatonic scale in 7edo:

1 2 1 2 1

You can't use L or s as a cradle here to get a new scale. But, if you double the number of pitches, going into the territory of 14edo, you get:

2 4 2 4 2

& this scale you can easily alter with MOS Cradle:

2 3 1 2 3 1 2
1 1 4 1 1 4 1 1

A Cradle in a Cradle


One can, of course, perform MOS Cradle on MOS Cradle scales & produce scales w/ four step sizes. Let's start with Swooning Rushes, a subset of 11edo:

2 3 1 3 2

A fine little scale, I think. Now let's double it:

4 6 2 6 4

& apply MOS Cradle to it:

3 1 6 2 6 1 3

This new scale, a subset of 22edo, has four step sizes (1, 2, 3, 6) & contains both the original MOS & the Cradle Scale Swooning Rushes. Not bad!

(This can go on forever, in theory. If we double it again, we might get this scale, a subset of 44edo: 6 2 7 5 4 5 7 2 6!)

Now I think I've given more than enough examples for you to get started on your own! If you discover other neat properties of these scales, feel free to edit this page & add your findings. & when you design lovely new MOS Cradle Scales, do add them to the repository!