Cardinality sequence of MOS scales

It is well-known (among mathematically-inclined music theorists) that there are well-formed scales with cardinality 5 (pentatonic), 7 (diatonic), and 12 (chromatic). ("Cardinality" is the number of notes in the scale.)

But, why these cardinalities, and not others? Why are there not well-formed scales of cardinality 6, 8, 9, etc.? Also, what cardinalities come after 12, that are well-formed?

I asked this of Andrew Milne, who gave a great answer, which I have appended below (with hyperlinks and occasional [editorial comments] added for your convenience...).

The quick answer is that the next couple of well-formed scales fter the chromatic have cardinalities 17 and 19.
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From: Andrew Milne [mailto:andymilne@tonalcentre.org]
Sent: Wednesday, December 30, 2009 11:10 AM
To: 'Jim Plamondon'; 'Bill Sethares'
Subject: RE: well formed scales, cardinality: next after 12?

Hi Jim and Bill

A lot of these concepts – well-formedness, Myhill’s, etc. are mathematically related and their definitions are hard to separate. But this is the method I use to determine the cardinality and tuning range of MOS/well-formed scales.

First you need a Farey sequence of order N. Then any Farey triple (three consecutive terms of the sequence), whose middle member has a denominator of N (i.e., j/k, M/N, p/q) gives the tuning range for an MOS scale of N tones. For example, Farey(12) has the triple 4/7, 7/12, 3/5, which tells us that there is a 12-tone MOS (the chromatic scale) which occurs for tunings where 4/7 < beta/alpha < 3/5. We know this method works, but Bill and I still haven’t worked out quite why.

By generating successively higher-order Farey sequences, you can find triples with higher denominators, and hence the tuning ranges of higher cardinality MOS scales. Note that as the cardinality goes up, the tuning range gets smaller. (This is all illustrated in the MOS labyrinth picture [for which, see Figure 5 in this paper]).

For your example, the next well-formed scale after the 12-tone chromatic is 17 if 7/12 < beta/alpha < 3/5, and 19 if 4/7 < beta/alpha < 7/12. You can either calculate this yourself or, more easily, just read it directly off the MOS labyrinth diagram.

Actually, I’ve just remembered, there’s another easy way to do this. If your MOS has L large steps, and S small, the next higher MOS has cardinality 2L + S with L’ large steps and S’ small where EITHER

• L’=L+S large steps and S’=L small OR
• S’=L+S small steps and L’=L large; and so on. 

For example, the 7-tone diatonic has L=5, S=2; the next higher MOS has 2L + S = 12 tones.

• If this 12-tone MOS has L’ = L+S and S’ = L, then the next higher MOS is 2L’+S’ = 2(L+S)+L = 3L+2S = 19 tones.
• If this 12-tone MOS has S’ = L+S and L’ = L, then the next higher MOS is 2L’+S’=2L+L+S = 3L+S = 17; and so on.

I presume that this rule is a direct result of well-known properties of the Farey sequence.

Every MOS/well-formed scale has a tuning range over which it is also proper (Rothenberg’s term, or “coherent” in Balzano’s terminology). There is a method to find this using the Farey sequence of a higher order (or Stern Brocot tree – which is the same thing but arranged into layers), which was explained by Thomas Noll, in one of the emails he sent when reviewing our JMM paper [see draft here]. I can’t quite remember what he said (I do have a copy, so I can check), but it amounted to something like stepping up to the next higher MOS scale and using that tuning range, or similar. For example, the diatonic scale has a tuning range of 4/7 to 3/5, but is proper only over 4/7 to 7/12.

Presumably there are also ways to generalise these things for 3-D tunings producing “pairwise well-formed” scales, and higher-D tunings. This would certainly form the basis of a groundbreaking paper...

Andy

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 More to follow.