Rank-3 scale

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A rank-n scale is a scale whose intervals (in cents, or any other logarithmic interval size measure) generate a rank-n group. Alternatively, a rank-n scale is a finite set of notes of a rank-n tuning, which is an infinite set of notes that can be generated by n generators, one of which is taken to be the period, at which any scale of the tuning repeats.

Rank-1 tunings and scales are equal tunings (ET). Edos are rank-1 because the generator achieves the octave by default. Thus, the octave is not counted as a generator.

Rank-2 scales include MOS scales and other generated scales, MODMOS scales, and other more complex scales that we are not as interested in.

Rank-3 scales described on this page are generalizations of rank-2 scales (MOS scales and permutations thereof, and other scales that have a single generator), which will first be introduced.

Rank-2 scales

MOS scales are the maximum variety 2 (MV2) scales. MOS scales are distributionally even (DE), along with rank-1 scales (ETs) which are MV1.

MOS scales can be generated by stacking a single generator modulo a period. Not all generated scales are MOS.

MOS scales are mirror-symmetric, or achiral, wherein the scale is symmetric about a point. In mirror-symmetric scales of odd cardinality, the axis of symmetric lies on a note of the scale, and so the scale has a symmetric mode, wherein the inverse of each interval (about the period) also exists in the mode. The step arrangement of the scale in such a mode is a palindrome - e.g., the diatonic scale in Dorian mode has step pattern LsLLLsL. For mirror-symmetric scales of even cardinality, the axis of symmetric lies exactly half-way between two notes of the scale, and no such mode exists. Mirror-symmetric scales of odd cardinality are symmetric about a note, and mirror-symmetric scales of even cardinality are symmetric about an interval. Mirror-symmetric scales of even cardinality can be written in a mode for which the inverse of every interval in the scale about the largest interval of the scale bar the period also exists in the mode. We will call such a mode the even-symmetric mode. The step pattern of such a mode is a palindrome, followed by a single step size. For example, Magic[10] in the even-symmetric mode has step pattern sLssLssLss. Mirror-symmetric scales may alternatively be defined as scales for which the inverse of every mode is also a mode of the scale. Clearly the symmetric mode is an inverse of itself.

MOS scales and can be uniquely defined by their MOS signature, i.e. the diatonic scale by 5L 2s.

MOS scales consist of strict MOS, which are the MOS scales as originally defined by Erv Wilson, and multi-MOS scales.

Strict MOS scales are the MOS scales that possess Myhill’s property, in which all generic intervals come in 2 sizes, and are also known as well-formed (WF) scales. The two sizes of each interval class in a WF or strict MOS scale differ by the chroma – the difference between the large and small steps of the scale. We will refer to these scales as WF scales for the remainder of this page. WF scales are defined (Carey & Clampitt, 1989) as generated scales for which the generator is of invariant generic interval size.

Multi-MOS scales are MOS scales that are multiple periods of a WF scale. The interval class represented by any multiple of a period of a WF scale comes in only a single size, hence multi-MOS scales do not possess Myhill’s property.

Rank-3 scales are introduced from here, as generalizations of MOS scales.

The following table summarizes properties of MOS scales, and the classes of scale that generalize them into rank-3 (not an exhaustive list).

Generalizations of MOS scale properties into rank-3
Property Rank-3 scale class
Maximum variety 2 (implies distributional evenness) MV3 scales
Distributional evenness Pairwise DE/MOS scales (subset of MV3 scales, superset of PWF scales)
Generated Generator-offset scales
Myhill's property (implies MV2) SV3 scales (subset of MV3 scales)
Well-formed (implies Myhill's property and Generated) (only strict MOS scales are well-formed) Pairwise well-formed scales (intersection of SV3 scales and Pairwise DE/MOS scales)
Can tessellate the entire lattice of pitch classes that it lives in Product words = rank-3 Fokker blocks (superset of Pairwise DE/MOS scales)
Recursive structure, Uniquely defined by step signature and mapping (implies mirror-symmetric) 3-SN scales (rank-3 step-nested scales)
Uniquely defined by mapping and cardinality

+ comprises least complex intervals in temperament defined by the mapping

Rank-3 hobbit scales

MV3 scales

Maximum variety 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. We often speak of strict-variety 3 (SV3) instead, meaning that every interval class has exactly three sizes. SV3 scales are also called trivalent.

Conjecture: For all odd-cardinality SV3 scales apart from the scales abacaba, and its repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times.

All GO scales of odd cardinality are MV3. The only GO scale of even cardinality is abac.

Conjecture: The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a…ba…c (and their repetitions). Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-SN scales (introduced below).

Product words

Two MOS scales can be combined into a rank-3 scale as a product word, which reduces back to the two MOS scales when two of the three pairs of interval sizes are equated.

When associated with a mapping, product words are the rank-3 Fokker blocks. Fokker blocks have unison vectors, which generalize the concept of the chroma of MOS scales to higher ranks. If these intervals are plotted onto a plane representing rank-3 octave equivalent pitch space, they tile the space into Fokker blocks which differ by combinations of these unison vectors. Rank-2 Fokker blocks are the MOS scales, so Fokker blocks can be considered a generalization of MOS scales into higher ranks.

Product words have maximum variety at most 4. The scale steps can be readily notated, sorted by size, as L, l, S, s, and they satisfy L - l = S - s.

Any Fokker block where the unison vectors are smaller than the smallest steps will be constant structures (CS). Not all Fokker blocks are CS.

Pairwise well-formed scales

Pairwise well-formed (PWF) scales, another generalization of WF scales into rank-3, are a subset of product words.

If equating any pair of step sizes (tempering out their difference, if we involve mappings) or a rank-3 scale leads to 3 WF scales, the rank-3 scale is pairwise well-formed (PWF).

PWF scales are MV3, and even SV3.

When mappings are considered, PWF scales are rank-3 wakalixes - Fokker blocks which are Fokker blocks in more than one way.

Not all SV3 scales are PWF. Only a single scale - abcba - is SV3 and not PWF.

Only a single PWF scale is mirror-symmetric - abacaba.

Apart from abacaba, PWF scales can be generated by an alternating generator sequence of two generators, modulo the period.

PWF scales can only have odd numbers of notes.

Pairwise DE/MOS scales

A similar generalization, a larger subset of product words, and a superset of PWF scales are pairwise DE (PDE) scales, defined for rank-3 scales such that equating any pair of steps (tempering out their difference, if we involve mappings), leads to 3-DE scales, or equivalently, MOS scales. We may also call these pairwise MOS (PMOS) scales.

Pairwise DE scales have MV3. Pairwise DE scales that are not PWF are not SV3; and one of the DE scales / MOS scales found by equating a pair of steps of such scales is a multi-MOS, which is DE / MV2, but does not demonstrate Myhill's property.

PWF and pairwise-DE scales include the same number of instances of steps of 2 of the 3 different step sizes, apart from abacaba.

The scale abacaba is the only mirror-symmetric PWF / PDE / PMOS, and the only mirror-symmetric SV3 scale.

The scales a…ba…c, and the scale abacaba are the only mirror-symmetric pairwise-DE scales, and the only mirror-symmetric MV3 scales.

There is only one way to arrange the steps of these scales such that they are pairwise-DE. This means that they can be uniquely described by a signature, like MOS scales.

3-SN scales

The scales abac and abacaba are step-nested (SN) scales, which are mirror-symmetric, and can be uniquely defined by a signature.

SN scales are generated iteratively by placing an instance of a new or the existing smallest step at the top or bottom of every larger step.

SN scales include MOS scales. MOS scales are the rank-2 SN scales, or 2-SN scales.

3-SN scales are generated from MOS scales, and 4-SN scales are generated from 3-SN scales, etc.

Conjecture 1: The only SN scale that is SV3 is abacaba.

Conjecture 2: The only SN scales that are MV3 are abacaba, and scales of the form abac.

Conjecture 3: The only SN scales that are MV3, and have mean variety < 3 are those of the form abac. This follows from Conjecture 1 and 2.

Conjecture 4: The only 3-SN scales that are balanced are abacaba, and scales of the form a…ba…c. Given that scales of the form a…ba…c are balanced (proof of this is left as an exercise for the reader), this follows from Fraenkel's conjecture, and Conjecture 2.

See Gallery of 3-SN scales for examples of 3-SN scales.

SN scales, OTC scales, and MOS Cradle Scales

It follows from the proof of omnitetrachordality that any SN scale (or any MOS Cradle scale) generated from an approximation of the Pythagorean trichord 4/3 4/3 9/8 is omnitetrachordal (OTC), and any SN scale generated from an approximation of the Pythagorean pentatonic is strongly OTC. All SN scales generated from OTC scales are OTC, and all SN scales generated from strongly OTC scales are strongly OTC. Further, the episturmium morphism that generates SN scales (in which an instance of a new or the existing smallest step is added to the top or bottom of every larger step, see SN scales) can be applied to generate larger OTC and strongly OTC scales from OTC and strongly OTC scales, or a more general morphism, in which an instance of a new or the existing smallest step is added to the top or bottom of every instance of any larger step size or subset of larger step sizes. Even more broadly, any MOS Cradle scales generated from an OTC scale is OTC, and any MOS Cradle scale generated from a strongly OTC scale is strongly OTC.

SN scales, OTC scales and MOS Cradle scales can have any rank.

Theorems, Proofs and Conjectures on 3-SN scales

Theorem: Scales of the form a...ba...c have mean variety (3N-4) / (N-1).

Proof:

Since there are three step sizes, a, b, and c, interval class N has variety 3.

Scale segments of length 1 ≤ length ≤ N/2-1 comprise either all a’s, all a’s but for a single b, or all a’s but for a single c, and therefore interval classes of length 1 ≤ length ≤ N/2-1 have variety 3. Interval classes of length N/2+1 ≤ length ≤ N-1 also have variety 3 by symmetry (given that scale segments of length N/2+1 ≤ length ≤ N-1 are the complement of scale segments of length 1 ≤ length ≤ N/2-1.

Finally, scale segments of length N/2 contain all a’s but for one b, or all a’s but for one c, and so interval class N/2 has variety 2.

The total variety of the scale is then 2+(N-2)*3 = 3N-4, and the mean variety of the scale is (3N-4) / (N-1).

Conjecture: No 3-SN scales have max variety > 5.

Conjecture: Only the two interval classes of a 3-SNS of odd cardinality may have a variety of 5, and no 3-SNS of even cardinality has max variety > 4.

Many SN scales have max variety 4 and mean variety < 3, including all scales for which the number of instances of one step size is equal to the sum of the numbers of instances of the remaining 2 step sizes. It follows that such scales are of even cardinality (possess an even number of notes), and can be generated by adding to any WF scale an instance of a third generator, smaller than the small step of the WF scale, above or below each step of the WF scale. A reasonably well-known example of this is MET-24, which can be generated by adding an instance of a generator around 57c above or below each step of the Pythagorean chromatic scale (and then tempering). MET-24 has three sizes of 2nd and 24th, 2 sizes of 3rd, 5th, 7th, …, 23rd etc. (the parapythagorean chromatic scale), and 4 sizes of 4th, 6th, 8th, …, 22nd.

Theorem: The mean variety of scales X with cardinality N, generated by a single instance of a third generator G at the top or bottom of each step of a WF scale W is equal to (3N-4)/(N-1).

Proof:

X consists of two instances of W, separated by G, such that in any mode of X, removing every second step leads to W.

Call the small and large steps of W 'S' and 'L', respectively, with G<S<L, and the period of the scale 'P'. There are 3 sizes of second (interval class 1): G, S-G, and L-G. It follows immediately that there are also three sizes of the largest interval class of the scale, interval class N­-1, i.e., the difference between P and the 3 sizes of second.

W has cardinality N/2, so we have N/2-1 interval classes with 2 step sizes.

For interval class 1+2C, for 1 ≤C ≤ (N/2)-2, from the 2 sizes A and A+L-S of interval class 2C, may be added the steps G, S-G, or L-G, leading to the possible interval sizes A+G, A+S-G, A+L-G, A+L-S+G, A+L-S+S-G=A+L-G, and A+L-S+L-G. However, since A<A+L-S, if we have both A+S-G, and A+L-S+L-G, then, after adding G, the next step of the scale, to both, to get to an interval class of W, we have step sizes differing by 2S-2L, and W would not be WF, and so we can have only one of these, reducing our set of possible interval sizes to 4.

Then the total number of specific intervals in X is (N/2-1)*2 + 2*3 + (N/2-2)*4 = 6+N-2+2N-8 = 3N-4, and the mean variety = (3N-4) / (N-1)

Conjecture: SN scales only of the form a…ba…c, or generated by a single instance of a third generator at the top or bottom of each step of a WF scale (SN flought scales) have mean variety < 3.

Conjecture: Scales of the form a...ba...ba...c have mean variety ((N/3-1)*(2*3+4) + 2*2) / (N-1) =(10N/3-6) / (N-1).

Conjecture: Scales with 2 instances of a generator added to a WF scale have mean variety ((N/3-1)*2 + 4*3 + 2(N/3-2)*4) / (N-1) = (10N/3-6) / (N-1).

Conjecture: abacaba and aabaabaac are the only SN scales with mean variety = 3.

Conjecture: The only SN scales that are balanced are the Power SNS, which are equivalent to the Fraenkel words, and SNS wherein two step sizes occur only once.