Ternary scale theorems

A ternary scale is a scale with three (positive) step sizes, with no other constraints such as maximum variety. This page documents known properties of subtypes of ternary scales and their proofs.

Conventions

• Bolded Latin variables refer to step vectors (linear combinations of step sizes).
• Indices for all words are 1-indexed.
  • If s is a circular word and i < 1 or i > len(s), we first replace i with i % len(s) + 1 before using it as an argument in s[-].
• The notation s(X₁, ..., X_r) is used for an r-ary scale word with variables X₁, ..., X_r possibly standing in for any sizes. If s(X, Y) = XXY then s(A, B) = AAB.
• We leave the distinction between linear words (words in the ordinary sense) and circular words up to context. We usually also elide the distinction between subwords and the dyad sizes that subtend them.
• For a word w and letter x, |w|_x denotes the number of occurrences of the letter x in w. For a step vector size v, |v|_x is similar.

Definitions

• Dyad is used for the musical sense of interval to avoid confusion with the mathematical sense of interval. But strictly speaking, interval in the musical sense is different than dyad. (An octave is an interval but not a dyad, and a 2:3:4 chord is a dyad but not an interval.)
• A circular word s (representing the steps of a periodic scale) of size n is generator-offset if it satisfies the following properties. The following conditions do not imply that g₁ and g₂ are the same number of scale steps. For example, 5-limit blackdye has g₁ = 9/5 (a 9-step) and g₂ = 5/3 (a 7-step).
  1. s is generated by two chains of stacked generators g separated by a fixed offset δ; either both chains are of size n/2, or one chain has size (n + 1)/2 and the second has size (n − 1)/2. Equivalently, s can be built by stacking a single chain of alternants g₁ and g₂, resulting in a circle of the form either g₁ g₂ ... g₁ g₂ g₁ g₃ or g₁ g₂ ... g₁ g₂ g₃.
  2. The scale is well-formed with respect to g, i.e. all occurrences of the generator g are k-steps for a fixed k.
• A scale or scale word is a circular word with a chosen size for its equave. As we're not working with scales with distinct equaves simultaneously, all three terms are effectively synonymous for our purposes.
• A scale is primitive if its period is the same as its equave. A multiMOS or multiperiod MOS is a non-primitive MOS. A MOS aL bs is primitive iff gcd(a, b) = 1. This corresponds to the term single-period in common xen parlance. Any multiMOS can be constructed from a primitive MOS by repeating the MOS pattern multiple times, e.g. if 3L 2s is LLsLs, then 9L 6s is LLsLsLLsLsLLsLs.
• An n-ary scale is a scale with n different step sizes. Binary and ternary are used when n = 2 and 3 respectively.
• For the generator-offset property, see the article.
• A strengthening of the generator-offset property, here the swung-generator-alternant property (SGA), states that the alternants g₁ and g₂ can be taken to always subtend the same number of scale steps, thus both representing "detemperings" of a generator of a primitive MOS scale (otherwise known as a well-formed scale). This is simply the property of having a generator sequence of period 2. All odd generator-offset scales are SGA, and aside from odd generator-offset scales, the only ternary scales to satisfy SGA are (XY)^rXZ, r ≥ 1. The Zarlino and diasem scales above are both SGA. Blackdye is generator-offset but not SGA.
• An odd-step is a k-step where k is odd; an even-step is defined similarly.
• Given a linear or circular word s with a step size X, define E_X(s) as the scale word resulting from deleting all instances of X from s.
• By a subword, substring, or slice of a word s, denoted s[i : j] (j > i), we mean s[i] s[i + 1] ... s[j − 1].
• Given a MOS aX bY, a chunk of X's is a maximal (possibly length 0) substring made of X's, bounded by Y's. We do not include the boundary Y's.
• Length is another term for a scale's size. The length of a scale s is denoted len(s).
• A projection of a ternary scale is the operation of equating two of its step sizes.
• A ternary scale is pairwise-well-formed if all its projections are well-formed (i.e. primitive MOSes).

Theorem 1 (Properties of SGA scales)

Let s be a ternary scale word in L, M, and s of length n, and suppose s is SGA. Then:

1. The length of s is odd, or s is equivalent to (xy)^rxz for some integer r ≥ 1.
2. If n is odd, s is of the form ax by bz for some permutation (x, y, z) of (L, M, s).
3. If n is odd, s is abstractly SV3 (i.e. SV3 for almost all tunings).
4. If n is odd, s is pairwise-MOS. That is, the following operations each result in a MOS: setting L = M, setting L = s, and setting M = s.
5. If n is odd, s = aX bY bZ is obtained from some mode of the (primitive) MOS aX 2bW by replacing all the Ws successively with alternating Ys and Zs (or alternating Zs and Ys for the other chirality, fixing the mode of aX 2bW). The two alternants differ by replacing one Y with a Z.

In particular, odd generator-offset scales always satisfy these properties (see Proposition 2 below).

[Note: This is not true with SGA replaced with generator-offset; blackdye is a counterexample that is MV4.]

Proof

Let e be the equave of s.

Assuming SGA, we have two chains of the aggregate generator g (going right). In the diagrams below, O represents a note and - represents a generator g. The two cases are:

CASE 1: EVEN LENGTH
O-O-...-O (n/2 notes)
O-O-...-O (n/2 notes)

and

CASE 2: ODD LENGTH
O-O-O-...-O ((n+1)/2 notes)
O-O-...-O ((n-1)/2 notes).

Label the notes (1, j) and (2, j), 1 ≤ j ≤ (number of notes in the chain), for notes in the upper and lower chain respectively.

Statement (1)

In case 1, let g₁ = (2, 1) − (1, 1), g₂ = (1, 2) − (2, 1), and g₃ = (1, 1) − (n/2, 2) = ((−n/2 − 1)*g₁ − n/2*g₂) mod e. We assume that g₁, g₂ and e are ℤ-linearly independent. We have the chain g₁ g₂ g₁ g₂ ... g₁ g₃ which visits every note in s.

Since s is generator-offset it is well-formed with respect to the aggregate generator g = (g₂ + g₁). Since g₁ and g₂ subtend the same number of steps by the SGA assumption, each is an odd-step. All multiples of the aggregate generator g must be even-steps, and those dyads that are "offset" by g₁ must be odd-steps. Letting M be the subset consisting of all even-numbered notes (which are generated by g) and considering M as a scale by dividing degree indices in M by two, M is well-formed with respect to g, thus M (and its offset) must be a MOS subset. Hence (g₃ + g₁), the imperfect generator of the MOS generated by g, subtends the same number of steps as g. Thus g₂ and g₃ subtend the same number of steps, a fact we need in order to be able to substitute one instance of g₂ with g₃ in the next part.

Let r be odd and r ≥ 3. Consider the following abstract sizes for the dyad class of k-steps reached by stacking r generators:

1. from g₁ g₂ ... g₁, we get a₁ = (r − 1)/2 * g + g₁ = ceil(r/2) g₁ + floor(r/2) g₂
2. from g₂ g₁ ... g₂, we get a₂ = (r − 1)/2 * g + g₂ = floor(r/2) g₁ + ceil(r/2) g₂
3. from g₂ (...even # of gens...) g₁ g₃ g₁ (...even # of gens...) g₂, we get a₃ = (r − 1)/2 g₁ + (r − 1)/2 g₂ + g₃ ≡ (r − n/2 − 3/2)g₁ + (r − n/2 − 1/2)g₂ mod e.
4. from g₁ (...odd # of gens...) g₁ g₃ g₁ (...odd # of gens...) g₁, we get a₄ = (r + 1)/2 g₁ + (r − 3)/2 g₂ + g₃ ≡ (r − n/2 − 1/2)g₁ + (r − n/2 − 3/2)g₂ mod e.

Since n > 0, these are all distinct by ℤ-linear independence; hence there are at least 4 sizes for k-steps. A 1-step must be reached by stacking an odd number of generators, thus by applying this argument to 1-steps, we see that there must be at least 4 step sizes in some tuning, a contradiction. Thus g₁ and g₂ must themselves be step sizes. Thus we see that an even-length SGA ternary scale must be of the form (xy)^rxz. (Note that (xy)^rxz is not SV3, since it has only two kinds of 2-steps, xy and xz.) This proves (1).

Statement (2)

In case 2, let n ≥ 3 and let (2, 1) − (1, 1) = g₁, (1, 2) − (2, 1) = g₂ be the two alternants. Let g₃ be the closing generator after stacking alternating g₁ and g₂. Then the generator circle is (g₁ g₂)^floor(n/2) g₃. If a step is formed by stacking k generators, we may assume that k is odd, and the combinations of alternants corresponding to a step come in exactly 3 sizes:

1. ceil(k/2)g₁ + floor(k/2)g₂
2. floor(k/2)g₁ + ceil(k/2)g₂
3. floor(k/2)g₁ + floor(k/2) g₂ + g₃

(since the scale size is odd, we can always ensure this by taking octave complements of all the generators). By counting the length-k subwords of the (linear) word (g₁ g₂)^floor(n/2), we see that the first two sizes must both occur (n − k)/2 times. This proves (2).

Statement (3)

We only need to see that if len(s) is odd and s is SGA, s is abstractly SV3. But the argument in case 2 above works when you substitute any odd-step dyad classes in s instead of a 1-step (abstract SV3 wasn't used). To get even-step dyad classes, we can take octave complements. Hence any dyad class in such a scale comes in (abstractly) exactly 3 sizes.

Statement (4)

Odd-numbered SGA scales are Fokker blocks (in the 2-dimensional lattice generated by the generator and the offset). To see this, consider the following lattice depiction of such a scale:

x x x ... x 
x x x ... x x

and use the vectors (-1, 2) and (ceil(n/2), 1) as the Fokker block chromas. A rank-3 Fokker block has the property that tempering out by each of the chromas gives two MOSes. These correspond to two of the temperings X = Y, Y = Z and X = Z. The third tempering follows by symmetry (by taking the other chirality). [math]\square[/math]

Statement (5)

By part (2), we have that s has step signature aX bY bZ, a odd. By part (4), we have that T(X, W) = s(X, W, W) is a MOS scale aX2bW. If b = 1, there's nothing to prove, so assume b > 1.

Consider the two generators in the GS of s, which are detemperings of the generator iX + jW of T(X, W), where gcd(j, 2k) = 1. Assume, possibly after inverting the generator, that the imperfect generator of T has j + 1 Ws and the perfect generator has j Ws.

Claim 1: Deleting Xs from the generator subwords of s gives every j-step subword in the scale E_X(s)(Y, Z), the scale word obtained by deleting all X's from s.

Proof: Consider the subword for the closing generator of s on index p, which is I = s[p : p + i + j], and suppose the result of deleting all X's from I has a j-step subword w. Shifting I one step to the left and one step to the right, s[p − 1: p − 1 + i + j] and s[p + 1 : p + 1 + i + j] are both detemperings of perfect generators of T, and have one fewer non-X step than I by our assumption. Thus the word I must both begin and end in a non-X letter. Removing all the X's from I results in a word that is j + 1 letters long and is the j-step word w with just one extra letter appended. Thus one of the two perfect generators above, namely the one that removes the extra letter, must contain this j-step. The rest of the j-step subwords of s can all be obtained by deleting Xs from detempered perfect generators; take q ≠ p to be the index of the first letter of one such j-step subword (as contained in s) and use s[q : q + i + j].

Claim 2: If a binary necklace U has b Ys and b Zs, gcd(j, 2b) = 1, and consecutively stacked j-steps in U occur in 2 alternating sizes, then U = (YZ)^b.

Proof: Write u and v for the two sizes of j-steps. Since gcd(j, 2b) = 1, there exists m such that stacking m-many j-steps yields scale steps of U, and m is odd because gcd(m, 2b) = 1. Hence the scale steps of U are (uv)^(m−1)/2u mod e and (vu)^(m−1)/2v mod e, and the step sizes alternate because u and v do.

These two claims prove that E_X(S) = (YZ)^b and that the two GS generators' sizes differ by replacing one Y for a Z.

Theorem 2 (Odd generator-offset scales are SGA)

Suppose that a periodic scale satisfies the following:

• is generator-offset
• has odd size n.

Then the scale is SGA.

Proof

Assume that the generator g is a k-step and k is even. (If k is not even, invert the generator.) On some note p we have a chain of (n + 1)/2 notes and on p′ = p + offset we'll have (n − 1)/2) notes.

Assume 1 < gcd(k, n) < n and n ≥ 5. Since n is odd, d = gcd(k, n) is an odd number at least 3, and by well-formedness with respect to the generator, there must be a circle of n/d < floor(n/2) notes formed by g, contrary to the assumption of GO. Thus, gcd(k, n) = 1.

Since n is odd, rk ≡ k/2 mod n iff r ≡ (n + 1)/2 mod n. (Note that both 2 and k are coprime with n, hence multiplicatively invertible mod n.) This proves that the offset, which must be reached after (n + 1)/2 k-steps, is a k/2-step, as desired. (As [k] is a generator of ℤ/n, stacking (n − 1)-many k-steps must visit every note exactly once. Thus if the offset wasn't reached in (n + 1)/2 steps, the two generator chains wouldn't have the assumed lengths.) [math]\square[/math]

Theorem 3 (Properties of even generator-offset ternary scales)

A primitive generator-offset ternary scale s of even size 6 or greater, where the generator g is an even-step, has the following properties:

1. s is a union of two copies of a primitive MOS M of size n/2 generated by g; thus it is a flought scale obtained by taking two offset copies of said primitive MOS.
2. s is not SV3.
3. s is not chiral.
4. If M = M(y, z) is the primitive MOS necklace above, then s = M(XY, XZ) for some assignment of variable names X, Y, and Z to the three letters of s.

Proof

(1) and (2) were proved in the proof of Proposition 1 (the part that we appeal to, from "all multiples of the generator g must be even-steps ..." to "These are all distinct by ℤ-linear independence", does not rely on s having the SGA property). (3) and (4) are easy to check using (1). [math]\square[/math]

Theorem 4 (Classification of pairwise well-formed scales)

Let s(a, b, c) be a scale word in three ℤ-linearly independent step sizes a, b, c. Suppose s is pairwise well-formed (equivalently, all its projections are primitive MOSes). Then s is SV3 and has an odd number of notes. Moreover, s is either generator-offset or equivalent to the scale word abacaba.

Proof

If the generator of a projection of s is a k-step, the word of stacked k-steps in s is pairwise well-formed

Suppose s has n notes (after dealing with small cases, we may assume n ≥ 7) and s projects to primitive MOSes s₁ (via identifying b with c), s₂ (via identifying a with c), and s₃ (via identifying a with b). Suppose s₁'s generator is a k-step, which comes in two sizes: P, the perfect k-step, and I, the imperfect k-step. By stacking n-many k-steps, we get two words of length n of k-steps of s₂ and s₃, respectively. These binary words, which we call Σ₂ and Σ₃, must be MOSes, since m-steps in the new words correspond to mk-steps in the MOS words s₁ and s₂, which come in at most two sizes. Since s₁ is a primitive MOS, gcd(k, n) = 1. Hence when 0 < m < n, mk is not divisible by n and mk-steps come in exactly two sizes; hence both Σ₂ and Σ₃ are primitive MOSes.

index: 1 2 3 4 ...   n
Σ1:    P P P P ... P I
Σ2:    [some MOS]
Σ3:    [some MOS]

Below we write step sizes resulting from identification as a~b, b~c, and a~c.

Two sizes of k-steps in s project to s₁'s perfect generator

We can write sizes of dyads in s as vectors (p, q, r) using the basis (a, b, c).

Suppose for sake of contradiction that only one size of k-step (α, β, γ) in s projects to P in s₁. Then projecting to s₂ shows that s₂'s generator is the k-step (α + γ)*(a~c) + βb, and Σ₂'s imperfect generator is located at index n, like Σ₁'s imperfect generator is. Then s₁ and s₂ are the same mode of the same MOS pattern (up to knowing which step size is the bigger one). Assume the L of s₁ (it could be s, but it doesn't matter) is the result of identifying b and c, and all s steps in s₁ come from a. Then the steps of s₂ corresponding to the L of s₁ must be either all b's or all a~c's, thus these steps are all b's in s (otherwise they would be identified with the a, against the assumption that s₁ and s₂ are the same MOS pattern and mode). So s has only two step sizes (a and b), contradicting the assumption that s is ternary.

Only two sizes of k-steps of s can project to P in s₁, for if there are three sizes of k-steps (α, β, γ), (α, β′, γ′), (α, β′′, γ′′) in s that project to P, then β, β′ and β′′ are three distinct values. Thus these would project to three different k-steps in s₃, contradicting the MOS property of s₃.

n is odd, etc.

Suppose Q = (α, β, γ) ≠ R = (α, β′, γ′) are the two k-steps in s that project to P. Then T = (α′, β′′, γ′′) projects to I. Here the values in each component differ by at most 1, and α ≠ α′. Then the circular word Λ₁ formed by the a-components of the k-steps in P is α...αα′. Since Σ₂ is a primitive MOS pattern of βb + (n − β)(a~c) and β′a + (n − β′)(a~c), the circular word Λ₂ = the pattern of β and β′ must be a primitive MOS. Similarly, Λ₃ = the pattern of γ and γ′ is a primitive MOS.

Suppose Λ₂ is the MOS λβ μβ′. Then Λ₃ is the MOS (λ ± 1)γ (μ ∓ 1)γ′. Since both Λ₂ and Λ₃ are primitive, and at least one of μ and (μ ∓ 1) are even, it is now immediate that n is odd.

Either β′′ = β or β′′ = β′. Assume β′′ = β′. Then γ′′ = γ, and Λ₃ is (λ + 1)γ (μ − 1)γ′. Also assume that the first k-step in Σ is Q. Then we have:

     1 …        n
Σ  = Q W(Q, R)  T
Λ1 = α …      α α′
Λ2 = β W(β, β′) β′
Λ3 = γ W(γ, γ′) γ

where W = W(x, y) is a word in two variables x and y, of length n − 2.

Case analysis

Since, by our assumption, Λ₃ has two γ in a row, Λ₃ must have more γ than γ′, so μ − 1 < n/2. Since Λ₃ is a MOS, μ − 1 ≥ 1. So we have 2 ≤ μ ≤ ceil(n/2).

We have three cases to consider:

Case 1: μ = 2, i.e. Λ₂ is the MOS (n − 2)β 2β′.

For Λ₂ to be a MOS, the first, and only, occurrence of R must be at either f = floor(n/2) or ceil(n/2). We may assume that it is at f; otherwise reverse the chain and reindex the words to start at 2f.

     1 …   f    … 2f n
Σ  = Q … Q R  Q … Q  T
Λ1 = α … α α  α … α  α′
Λ2 = β … β β′ β … β  β′
Λ3 = γ … γ γ′ γ … γ  γ

We need only consider stacks up to f-many k-steps. Either:

1. the stack has only copies of Q and R; or
2. the stack has one T and does not contain any R (since it's more than f − 1 generators away).

These give exactly three distinct sizes for every dyad class. Hence s is SV3.

In this case s has two chains of Q, one with floor(n/2) notes and one offset by Q^(f−1)R with ceil(n/2) notes. Every instance of Q must be a k-step, since by ℤ-linear independence Q = αa + βb + γc is the only way to write Q in the basis (a, b, c); so s is well-formed with respect to Q. Thus s also satisfies the generator-offset property with generator Q.

Case 2: μ ≥ ceil(n/2), i.e. Λ₂ has fewer β than β′.

Since Λ₃ has more β than β′, Λ₂ is floor(n/2)β ceil(n/2)β′, and Λ₃ is ceil(n/2)γ floor(n/2)γ′. There is a unique mode of ceil(n/2)γ floor(n/2)γ′ that both begins and ends with γ, namely γγ′γγ′…γγ′γ. Thus Λ₂ is ββ′ββ′…ββ′β′. It is now easy to see that if the number of k-steps stacked is odd, then there are two sizes that do not contain T and one size that contains T; if the number of k-steps stacked is even, then there is one size that does not contain T and two sizes that contain T. Hence s is SV3.

In this case we have Σ = QRQR...QRT, and s is well-formed with respect to the generator Q + R, thus s satisfies the generator-offset property. By Proposition 1, s is SV3.

Case 3: 3 ≤ μ ≤ floor(n/2).

Λ₂ has a chunk of β (after the first β′) of size x where x = floor(n/μ) ≥ floor(n/floor(n/2)) = 2 or x = ceil(n/μ) = floor(n/μ) + 1. Hence Λ₃ has a chunk of γ of size x. Λ₃ also has a chunk that contains Λ₃[n : 2] as a subword. This chunk must be of size y, where

[math]2 \lfloor\frac{n}{\mu}\rfloor - 1 = 2 \big(\lfloor \frac{n}{\mu} \rfloor - 1\big) + 1 \leq y \leq 2 \big(\lfloor\frac{n}{\mu}\rfloor + 1 \big) + 1 = 2\lfloor\frac{n}{\mu}\rfloor + 3.[/math]

(The lower bound is reached if Λ₃ has chunks of sizes floor(n/μ) − 1 and floor(n/μ), and the upper bound is reached if Λ₃ has chunks of sizes floor(n/μ) and ceil(n/μ).)

The difference between the chunk sizes of Λ₃ is y − x, which must be 1 since Λ₃ is pairwise well-formed. We thus have the following subcases: (In the following, chunk of Λ₂ means chunk of β, and chunk of Λ₃ means chunk of γ.)

Case 3.1: (x, y) = (floor(n/μ), 2*floor(n/μ) − 1).

Since y − x = floor(n/μ) − 1, we have x = floor(n/μ) = 2 and y = 3. The chunk in Λ₃ whose size was defined to be y is made from two consecutive chunks in Λ₂ of size 1. (So Λ₂ has chunks of size 1 and 2, and Λ₃ has chunks of size 2 and 3.) Since chunk sizes of a MOS themselves form a MOS, Λ₂ has more chunks of size 1 than it has chunks of size 2.

Λ₂ has only two chunks of size 1, Λ₂[n − 1] and Λ₂[1], since otherwise Λ₃ would have a chunk of size 1 within Λ₃[1 : n]. Thus Λ₂ has exactly one chunk of size 2. Thus Λ₂ = ββ′βββ′ββ′ and Λ₃ = γγ′γγγ′γγ. Thus we have:

     1 2  3 4 5  6 7
Σ =  Q R  Q Q R  Q T
Λ1 = α α  α α α  α α′
Λ2 = β β′ β β β′ β β′
Λ3 = γ γ′ γ γ γ′ γ γ

Suppose a step of s is reached by stacking t-many k-steps. We have three cases after accounting for equave complements:

1. t = 1: s is equivalent to abacaba.
2. t = 2: s is QR QQ RQ TQ RQ QR QT => s is equivalent to abacaba.
3. t = 3: s is QRQ QRQ TQR QQR QTQ RQQ RQT => s is equivalent to abacaba.

(This also implies s is SV3.)

Case 3.2: (x, y) = (floor(n/μ) + 1, 2*floor(n/μ) − 1) is impossible: Here (x, y) = (4, 5). But then Λ₂ has a chunk of size < 3 because of the β' at index n, contradicting that x is one of the chunk sizes of Λ₂.

Case 3.3: (x, y) = (floor(n/μ) + 1, 2*floor(n/μ)) is impossible: Here (x, y) = (3, 4). But then Λ₂ has a chunk of size 1 because of the β' at index n, and another chunk of size 0 or 2, contradicting that x is one of the chunk sizes of Λ₂.

The remaining cases are all impossible because they imply y − x ≥ 2:

• Case 3.4: (x, y) = (floor(n/μ) + 1, 2*floor(n/μ) + 1)
• Case 3.5: (x, y) = (floor(n/μ) + 1, 2*floor(n/μ) + 2)
• Case 3.6: (x, y) = (floor(n/μ) + 1, 2*floor(n/μ) + 3)
• Case 3.7: (x, y) = (floor(n/μ), 2*floor(n/μ))
• Case 3.8: (x, y) = (floor(n/μ), 2*floor(n/μ) + 1)
• Case 3.9: (x, y) = (floor(n/μ), 2*floor(n/μ) + 2)
• Case 3.10: (x, y) = (floor(n/μ), 2*floor(n/μ) + 3)

[math]\square[/math]

Theorem 5 (PWF scales are balanced)

All pairwise-well-formed scales are balanced.

Proof

Let s be a PWF (thus primitive) scale. The case where s is equivalent to XYXZXYX can be manually verified, so by Theorem 4, the only remaining case is when s can be constructed by stacking two alternating sizes, g₁ and g₂, of k-steps. We assume that s has step signature aX bY bZ where a is odd. This k corresponds to a class of generators of the primitive MOS aX 2bW. This MOS is obtained from s by applying the letterwise substitution function π such that π(X) = X and π(Y) = π(Z) = W. Naturally, π applies to linear words, circular words, and step vector sizes. Additionally, we can choose k so that the two sizes of k-steps in π(s) are:

• the perfect generator g = tX + (k − t)W (note (k − t) is odd by a previous proof), and
• the imperfect generator i = (t + 1)X + (k − t − 1)W.

We have π(g₁) = π(g₂) = g. Let h be the size in s such that π(h) = i. Hence only one k-step subword h has this size in s. By Theorem 1, we also may assume |g₁|_Y = |g₂|_Y + 1 and |g₁|_Z = |g₂|_Z − 1 (the other case corresponds to the opposite chirality).

As s is periodic and gcd(len(s), k) = 1, it suffices to count letters in stacks of k-steps in s. The j-step on any note of s can be computed by reducing a stack of k-step subwords on that note, which alternate in size between g₁ and g₂. Note that:

1. Such a stack has a unique number m = m(j) of k-steps for a given j.
2. Either such a stack has i as one of its k-steps, or it does not.

Depending on point (2), m may be even or odd. If m is odd, then in any stack in s that does not have i, the number of stacked g generators in the projection is odd, hence the number of non-X letters in the corresponding km-step word in s is odd. Each incremental shift of the boundary that does not result in including i results in one Y being swapped for a Z, or vice versa, while the number of X steps remains t. On the other hand, i has an even number of non-X steps, thus the numbers of Y and Z are the same in v and equal to min(|u₁|_Y, |u₂|_Y).

In summary: Let u₁, u₂ be the two sizes that do not include i, and v be the size that does. Say that u₁ has one more g₁ than g₂. Then

• |u₁|_X = |u₂|_X = |v|_X − 1
• |u₁|_Y = |u₂|_Y + 1
• |u₁|_Z = |u₂|_Z − 1
• |v|_Y = |v|_Z = |u₁|_Y = min(|u₁|_Y, |u₂|_Y).

This proves that the set of j-steps is balanced. When m is even, take the equave-complement of the set of j-steps to reduce to the above case. [math]\square[/math]

Theorem 6 (Generator-offset structure of diregular scales)

Definition (Diregular scale)

A primitive ternary scale s is diregular if len(s) is even and s is equivalent to a word constructed from taking the MOS 2aX2cZ with a odd and gcd(a, c) = 1, and replacing every other X with Y. In particular, s has step signature equivalent to aXaYbZ with a odd and b even. For example, LsLsLmsLsLsm (achiral diachrome, 5L2m5s) is a diregular scale.

Theorem

If s = s(X, Y, Z) is diregular, then:

1. s consists of two generator chains, each with len(s)/2 notes;
2. the generator has the same interval class as some generator of the MOS 2aW2cZ;
3. the two generator chains are offset by a len(s)/2-step interval.

Proof

The result of substituting Y with X (let us call this map p) is the MOS M = 2aX2cZ, which has exactly 2 periods since gcd(a, c) = 1. M thus consists of two generator chains separated by the period of M, which has a + c = len(s) steps. It thus suffices for there to exist k, 0 < k < a + c, such that every perfect k-step generator has the same preimage in s, which will be our desired generator. Suppose that the perfect k-step of M is iW + jZ where 0 < i < a. Since a is odd, possibly after taking the period-complement we may assume that i is even. Hence each subword w of s such that its projection p(w) subtends a perfect k-step satisfies |w|_X = |w|_Y (= i/2). It plainly follows that every such w satisfies |w|_X = |w|_Y = i/2 and |w|_Z = j. [math]\square[/math]

Theorem 7 (Classification of MV3 scales)

In the following, equivalent means "is the same circular word after permuting X, Y, and Z." This means that XYXZXYX is equivalent to YZYXYZY, or XZXYXZX, and so on.

Theorem 7.1 (Classification of ternary balanced scales)

1. A primitive balanced MV3 scale s is one of the following:
   1. sporadic balanced: s is equivalent to XYXZXYX, the ternary Fraenkel word, with step signature 4X2Y1Z.
   2. regular: len(s) is odd, and s is equivalent to a word constructed from taking the brightest mode of the MOS cXbZ with c even and gcd(c, b) = 1, and replacing every other X with Y. We assume X > Z when constructing the MOS. In particular, s has step signature aXaYbZ where b is odd (with a = c/2).
   3. diregular: len(s) is even, and s is equivalent to a word constructed from taking the brightest mode of the MOS 2aX2cZ with a odd and gcd(a, c) = 1, and replacing every other X with Y. In particular, s has step signature aXaYbZ with a odd and b even.
2. All primitive balanced ternary scales are MV3.
3. A balanced primitive ternary scale is SV3 if and only if it is not diregular.
4. Regular balanced primitive ternary scales have a generator sequence of period 2.

(Condensed: All single-period balanced ternary scales that are not the Fraenkel word are aX aY bZ. In this case, if b is odd, then the scale is regular. If b is even, then the scale is diregular.)

Theorem 7.2 (Classification of MV3 scales)

A primitive MV3 scale is either

1. balanced (classified by the previous theorem),
2. sporadic non-balanced: equivalent to XYZYX, or
3. twisted: equivalent to a word constructed as follows:
   • Start with the brightest multiMOS word kcXkbZ with c being an even number.
   • Interchange a Z and an X at some (possibly more than one) of the boundaries of these copies of the MOS word w. Here, the boundary of two consecutive copies of w is the last letter of the first word and the first letter of the second word. (At the ends of the whole multiMOS word, the boundaries are just the first and last letters of the word.) For example, let w be the multiMOS word 8X6Z, XXZXZXZXXZXZXZ. Then the border between the copies of the MOS subword XXZXZXZ are w[7]w[8] and w[14]w[1] (using one-based numbering).
   • Replace every other X with Y in w. (Thus in particular, twisted MV3 scales have step signature kaXkaYkbZ)

Proof

Most of this has been proved by Bulgakova, Buzhinsky and Goncharov (2023), "On balanced and abelian properties of circular words over a ternary alphabet"; however, the designations sporadic, regular, and diregular for the classes are ours.

Note: The xen term "brightest MOS word" is equivalent to "Christoffel word" in the paper, and similarly "brightest multiMOS word" is equivalent to "powers of a Christoffel word". Also see Glossary for combinatorics on words for more equivalents between xen community terms and standard academic terminology.

For 7.1.3: The ternary Fraenkel word may be verified as SV3 by inspection, and we have already shown in Theorem 1 that regular balanced scales are SV3. To show that diregular balanced scales are not SV3, observe that (a+c)-steps come in only 2 sizes in such a scale s: floor(a/2)X + ceil(a/2)Y + cZ and ceil(a/2)X + floor(a/2)Y + cZ, since the underlying MOS 2aX2cY only has the (a+c)-step aX + cZ. The construction replaces the Xs in these subwords with alternating Xs and Ys; either of X or Y may occur first, corresponding to the two possible sizes, since a is odd and thus the (a+c)-step subword s[k : k+a+c] becomes the subword s[k+a+c : k+2a+2c] via interchanging X and Y.

Claim 7.1.4 can be verified by noting that such scales are PWF and using Theorem 4. [math]\square[/math]

Open problems

1. Classify all (abstractly) SV3 scales. This is tantamount to classifying all twisted SV3 scales.
2. Conjecture: If a twisted MV3 is not SV3, then it is constructed from kaXkbZ where k is composite.

Conjecture ("MV3 Sequences")

Given any two generators, we can iterate them to any number of notes and see what the maximum-variety of the resulting scale is. In particular, we can look at those scale sizes which are MV3, and thus compute the MV3 sequence for the pair of generators (similar to the "MOS sequence" one can compute for one generator). Thus, for any pair of generators, we can form the associated sequence of increasingly large MV3 scales.

Surprisingly, for almost all pairs of generators, this sequence seems to terminate after some (usually relatively small) scale. That is, if we simply take all possible pairs of generators between 0 and 1200 cents, and for each pair we compute the MV3 sequence for all generator pairs up to some maximum N, such as 1000, we can easily see that most points will have only a few entries in it, after which no MV3 scales are apparently generated. It would seem to be true that as the two generators get closer and closer in size, the MV3 sequence gets longer and longer, until when the two generators are equal you have an infinite-length sequence (corresponding to MOS).

It is pretty easy to see this behavior is true if we simply compute the MV3 sequences up to any very large N, far beyond the scale sizes we typically use in music theory, but it would be good to have a proof.

Open questions

This heading has those open questions for which no conjecture has yet been formed either way. (These can be updated as necessary)

1. Given any arbitrary MOS (or DE, etc) scale with at least three notes per period, is there *always* a MV3 generator-offset scale which can be derived as a "detempering" of that scale? Or is this only true for some MOS's? For instance, the MOS LLsLLLs has the MV3 generator-offset scale LmsLmLs as a detempering. Does a similar MV3 detempering exist for every possible DE scale with at least three notes per period, or at least for strict MOS's with one period per octave (e.g. well-formed scales)?
2. The scale tree is a great way to analyze MOS scales. For any generator, we can compute the various MOS's it forms if we simply look at the scale tree, and indeed MOS "words" like LLsLLLs can be identified with regions on the scale tree (in this situation the interval between 4/7 and 3/5). A similar "scale plane" should exist for generator-offset-MV3 scales, where given some word representing a generator-offset-MV3 scale, we can look at the set of points on the generator plane which generates it; these seem to often be triangles, with the lines corresponding to MOS's and the vertices corresponding to EDOs (though is this always true?). What is the big picture of this scale plane? Can we use Viggo Brun's algorithm for this, generalizing the theory of continued fractions? Is there some simple formula we can use to predict, given some generator-offset-MV3 scale, which region on the scale plane it corresponds to? Can we plot simple generator-size-proportions as points in this space? And so on.
3. In the theory of MOS, there is a second scale tree that is less frequently talked about, which Erv Wilson calls the "Rabbit Sequence" (Erv Wilson's original version, interactive version 1, interactive version 2). This is a tree for which each MOS word has two children, depending on if the MOS is "soft" (with L/s < 2) or "hard" (with L/s > 2). For instance, LsLss has the two children LLsLLLs and ssLsssL. Does a similar scale plane exist for these generator-offset-MV3 scales?