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| Rank-3 scales described on this page are generalizations of [[MOS scales]], and similar rank-2 scales, which will first be introduced.
| | #REDIRECT [[Rank-3 scale]] |
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| == Rank-2 scales ==
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| MOS scales are the MV2 (Max Variety 2) scales. MOS scales are DE ([[distributionally even]]), along with rank-1 scales, i.e., [[ET]]<nowiki/>s, which are MV1.
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| MOS scales can be generated by stacking a single generator modulo a period (not all generated scales are MOS).
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| MOS scales are [[Scale properties simplified|symmetric]] (not all symmetric scales are MOS) and can be uniquely defined by their ''MOS signature'', i.e. the diatonic scale by 5L 2s.
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| MOS scales consist of ''strict MOS'' and ''Multi-MOS'' scales.
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| 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.
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| ''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.
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| MOS scales are ''[[Constant structure|constant structures]]'', meaning that it's generic interval classes are distinct, meaning that each specific interval is always subtended by the same number of steps. This is also known as the ''[[Scale properties simplified|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.
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| == Rank-3 scales ==
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| === Product words ===
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| 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.
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| 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.
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| Product words have max variety at most 4.
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| Any Fokker block where the unison vectors are smaller than the smallest steps will be constant structures (CS). Not all Fokker blocks are CS.
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| === Pairwise well-formed (PWF) scales ===
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| Pairwise well-formed (PWF) scales, a generalization of WF scales into rank-3, are a subset of product words.
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| 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).''
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| PWF scales not only have MV3, but are also ''trivalent'', where each generic interval comes in 3 sizes.
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| When mappings are considered, PWF scales are rank-3 [[Gallery of wakalixes|wakalixes]].
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| Not all MV3, or trivalent scales are PWF. Only the scale aabcb is trivalent and not PWF.
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| PWF scales can only have odd numbers of notes.
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| PWF scales can be generated by an alternating generator sequence, modulo a period, apart from abacaba.
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| === Pairwise DE/MOS scales ===
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| 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.
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| 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.
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| PWF and pairwise-DE scales include the same number of incidences of steps of 2 of the 3 different step sizes, apart from abacaba.
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| The scale abacaba is the only PWF / PDE / PMOS scale, and the only trivalent scale that is also symmetric.
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| The scales a…ba…c, and the scale abacaba are the only pairwise-DE scales, and the only MV3 scales that are also symmetric.
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| 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.
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| === 3-SN scales ===
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| The scales a...ba...c and abacaba are [[SN scales]], which are symmetric, and can be uniquely defined by a signature.
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| SN scales include MOS scales. MOS scales are the rank-2 SN scale, or 2-SN scales.
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| The only SN scale that is 3-Myhill is abacaba
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| The only SN scales that are MV3 are a…ba…c, and abacaba.
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| The only SN scales that are MV3, and have mean variety < 3 are a…ba…c
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| It follows from the proof of [[omnitetrachordality]] that any SN scale (or any [[MOS Cradle Scales|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.
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| '''Theorem:''' Scales of the form a...ba...c have mean variety (3''N''-4)/(''N''-1)
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| '''Proof:'''
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| Since there are three step sizes, a,b, and c, interval class ''N'' has variety 3
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| 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.
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| 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.
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| The total variety of the scale is then 2+(''N''-2)*3 = 3''N''-4
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| And the mean variety of the scale is (3''N''-4)/(''N''-1)
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| '''Conjecture:''' No SN scales have max variety > 5.
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| '''Conjecture:''' Only the middle interval classes of an SN of odd cardinality may have a variety of 5, and no SN of even cardinality has has variety > 4.
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| 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.
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| '''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 (3''N''-4)/(''N''-1)
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| '''Proof:'''
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| Every second step of the scale gives the WF scale the 3-SN scale is generated from. We can this scale W.
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| 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.
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| 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.
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| 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.
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| Then the total number of specific intervals in the scale is (''N''/2-1)*2 + 2*3 + (''N''/2-2)*4 = 6+''N''-2+2''N''-8 = 3''N''-4, and the mean variety = (3''N''-4)/(''N''-1)
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| '''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.
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| '''Conjecture:''' Scales of the form a...ba...ba...c have variety ((''N''/3-1)*(2*3+4) + 2*2) / (''N''-1) =(10''N''/3-6) / (''N''-1).
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| '''Conjecture:''' Scales with 2 instances of a gen added to a WF scale have variety ((''N''/3-1)*2 + 4*3 + 2(''N''/3-2)*4) / (N-1) = (10''N''/3-6)/(''N''-1)
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| '''Conjecture:''' abacaba and aabaabaac are the only SN scales with mean variety = 3.
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