Rank-3 scale: Difference between revisions

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.

Strict MOS scales are the MOS scales that possess [[Scale properties simplified#Properties|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.

|Product words = rank-3 Fokker blocks (superset of Pairwise DE/MOS 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''.  

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.

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.

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.

The scales ''a…ba…c'' and ''abacaba'' are [[step-nested scale|step-nested]] (SN) scales, which are mirror-symmetric, and can be uniquely defined by a signature.

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

'''Conjecture''': The only SN scales that are MV3 are ''abacaba'', and scales of the form ''a…ba…c''.

'''Conjecture''': The only SN scales that are MV3, and have mean variety < 3 are those of the form ''a…ba…c''.

'''Conjecture''': The only SN scales that are [[Balanced word|balanced]] are the ''Power SNS'', which are equivalent to the [[Fraenkel word|Fraenkel words]].

It follows from the proof of [[omnitetrachordality]] that any SN scale (or any [[MOS Cradle|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.

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

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.

'''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.

'''Conjecture:''' SN scales only of the form ''a…ba…c'', or generated by a single instance of a third gen at the top or bottom of each step of a WF scale, 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) =(10''N''/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) = (10''N''/3-6) / (''N''-1)

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