Interval variety

The interval variety of an interval class in a scale is the number of different interval qualities available for that interval class. For example, the interval class "fifth" in the diatonic scale has interval variety 2, because there are two sizes of fifths in that scale: 6 perfect fifths and 1 diminished fifth.

The concept of interval variety can be applied to all interval classes of a scale at once. Here are some such properties:

• Highest interval variety (see also maximum variety)
• Mean interval variety
• Median interval variety
• Lowest interval variety

In addition, strict variety scales, such as single-period MOS scales and trivalent scales, have the same interval variety for all interval classes (except the unison, which always trivially has interval variety 1).

For scales of arity at least 3, there is a critical distinction between free or abstract interval variety and conditional interval variety. Free or abstract means that the interval variety property holds for any tuning of the scale steps, and conditional means that the interval variety property only holds for a particular tuning of the steps.

Terminology

A standard academic counterpart to the xen term variety is the abelian complexity function of a word: a function ρ^ab : N -> N where ρ^ab(n) is the number of distinct sizes (abelianizations, living in a free abelian group over the step sizes) that length-n subwords can have in a word.

Facts

In the following, two letters are to be considered the same if their numerical values are congruent modulo n.

Theorem — For all n ≥ 1, the word 0123...(n-1) is SVn.

Proof

All k-letter subwords of 0123...(n-1) is of the form (i)(i+1)...(i+k-1), and there are exactly n of them. [math]\displaystyle{ \square }[/math]

Theorem — For all n ≥ 1, the word 0123...(n-2)(n-1)(n-2)...3210 is SVn.

Proof

We prove this by dividing this word into four overlapping noncircular subwords which cover all cases.

Consider the subwords 0123...(n-2)(n-1) and (n-1)(n-2)...3210. If we treat these two words as noncircular, then there are n-k distinct k-letter subwords.

Now consider the k-letter noncircular subword (r-1)(r-2)...210012...(s-2)(s-1), where r, s ≥ 1. Note that interchanging r and s yields equivalent subwords. If k is odd, there are (k-1)/2 distinct subwords of this kind; if k is even, there are k/2 subwords, each corresponding to a solution of r + s ≥ k.

A similar thing goes with (n-r-1)...(n-2)(n-1)(n-2)..(n-s-1). If k is odd, there are (k+1)/2 distinct subwords of this kind; if k is even, there are k/2 subwords, each corresponding to a solution of r + s - 1 ≥ k.

Thus there are

(n - k) + (k-1)/2 + (k+1)/2 = n (k odd)

(n - k) + 2(k/2) = n (k even)

distinct k-letter subwords, so the word is SVn. [math]\displaystyle{ \square }[/math]

Abstractly SV4 scale patterns

Abstractly SV4 scale patterns (patterns that are SV4 for any choice of distinct cent values for the four steps):

• 4 notes: 0123
• 5 notes: 01023
• 6 notes: none
• 7 notes: 0123210, 0102013, and 0103102 (The last two patterns are a chiral pair fixing concrete sizes for the steps.)
• 8 notes: 00100232 and 01212103
• 9 notes: none
• 10 notes: 0010020302 and 0102103012
• 11 to 14 notes: none

(Note that abstract SV4-ness implies that a scale pattern is primitive, or single-period.)

Conjecture: there are no SV4 scale patterns with more than 10 notes.

Open questions

• Why are (abstractly) SV4 scale patterns seemingly so rare?
  • Conjecture: There are only finitely many SV4/MV4 circular words.
  • Conjecture: For all n greater than a sufficiently large m, the longest abstractly SVn word is 0123...(n−2)(n−1)(n−2)...3210, with length 2n - 1.
  • Related may be the following conjecture: For a sufficiently long ternary noncircular word, there exists k > 1 such that the interval class of k-steps has at least 3 sizes and the interval class of (k − 1)-steps also has at least 3 sizes.