Rank 3 scale

Rank-3 scales described on this page are generalizations of MOS scales, and similar rank-2 scales, which will first be introduced.

Rank-2 scales

MOS scales are the MV2 (maximum variety 2) scales. MOS scales are DE (distributionally even), along with rank-1 scales, i.e., 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 symmetric (not all symmetric scales are MOS) and can be uniquely defined by their MOS signature, i.e. the diatonic scale by 5L 2s.

MOS scales consist of strict MOS 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.

MOS scales are constant structures, meaning that its generic interval classes are distinct, meaning that each specific interval is always subtended by (i.e. occurs as) the same number of steps. This is also known as the partitioning property. MODMOS scales are also constant structures - which may be arrived at by raising or lower an interval or intervals of a MOS scale by a chroma. Constant structures exist in any rank. ETs are the rank-1 constant structures.

Rank-3 scales

MV3 scales

Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank-3.

For all MV3 scales apart from the scale abacaba, and its repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times, and may be generated by an alternating generator sequence. Because of this, there always exists a generator for a MV3 scale (other than the one exception) such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.

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

Trivalent scales

WF scales may be generalized into rank-3 via a generalization of Myhill's property into rank-3. We call a scale where each generic interval comes in 3 sizes trivalent. Trivalent scales are clearly a subset of MV3 scales. Trivalent scales can only have odd numbers of notes.

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 max variety at most 4.

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

Conjecture: Product words cannot be symmetric.

Pairwise well-formed (PWF) 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 not only have MV3, but are trivalent.

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

Not all trivalent scales are PWF. Only a single scale - aabcb - is trivalent and not PWF.

Only a single PWF scale is 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 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 scales.

Pairwise-DE scales have MV3. Pairwise-DE scales that are not PWF are not trivalent; 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 incidences of steps of 2 of the 3 different step sizes, apart from abacaba.

The scale abacaba is the only PWF / PDE / PMOS scale, and the only trivalent scale that is also symmetric.

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

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 a...ba...c and abacaba are SN scales, which are symmetric, and can be uniquely defined by a signature.

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

The only SN scale that is trivalent is abacaba

The only SN scales that are MV3 are a…ba…c, and abacaba.

The only SN scales that are MV3, and have mean variety < 3 are a…ba…c

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, and any SN scale generated from an approximation of the Pythagorean pentatonic is strongly omnitetrachordal.

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 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 SN scales have max variety > 5.

Conjecture: Only the two interval classes of an SN of odd cardinality may have a variety of 5, and no SN 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 incidences of one step size is equal to the sum of the numbers of incidences 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 incidence 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 incidence 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 generated by a single incidence of a third generator at the top or bottom of each step of a WF, with cardinality N, is equal to (3N-4)/(N-1)

Proof:

Every second step of the scale gives the WF scale the 3-SN scale is generated from. We can this scale W.

Call the small and large steps of the WF scale 'S' and 'L', respectively, the size of the new generator 'G', where 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, and 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.

For a scale of cardinality N, The WF scale it is generated by 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 the scale 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 incidence of a third gen at the top or bottom of each step of a WF, 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.