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Rank-3 scale theorems: Difference between revisions

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* Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)
* Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)
* Triple Fokker blocks form a [http://en.wikipedia.org/wiki/Trihexagonal_tiling trihexagonal tiling] on the lattice.
* Triple Fokker blocks form a [http://en.wikipedia.org/wiki/Trihexagonal_tiling trihexagonal tiling] on the lattice.
* A scale imprint is that of a Fokker block if and only if it is the [[product word]] of two DE scale imprints with the same number of notes. See [https://link.springer.com/chapter/10.1007/978-3-642-21590-2_24 Introduction to Scale Theory over Words in Two Dimensions | SpringerLink]
* A scale imprint is that of a Fokker block if and only if it is the [[product word|product]] of two DE scale imprints with the same number of notes. See [https://link.springer.com/chapter/10.1007/978-3-642-21590-2_24 Introduction to Scale Theory over Words in Two Dimensions | SpringerLink]
* If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L > m > n > s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s
* If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L > m > n > s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s
* Any convex object on the lattice can be converted into a hexagon.
* Any convex object on the lattice can be converted into a hexagon.
* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.
* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.
 
* An MV3 scale always has two of the step sizes occurring the same number of times, except powers of abacaba. Except multi-period MV3's, such scales are always either pairwise-well-formed, a power of abcba, or a "twisted" word constructed from the mos 2qX rY. A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba. The PWF scales are exactly the single-period rank-3 [[billiard scales]].
== Unproven Conjectures ==
== Conjectures ==
* Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes.
* Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes.
== MV3 proofs ==
Under construction
=== Definitions and theorems ===
Throughout, let ''S'' be a scale word in steps ''x'', ''y'', ''z'' (and assume all three of these letters are used).
==== Definition: PMOS ====
''S'' is ''pairwise MOS'' (PMOS) if the result of equating any two of the step sizes is a MOS.
==== Definition: AG ====
''S'' satisfies the ''alternating generator property'' (AG) if it satisfies the following equivalent properties:
# ''S'' can be built by stacking alternating generators, resulting in a circle of the form  either g1 g2 ... g1 g2 g1 g3 or g1 g2 ... g1 g2 g3.
# ''S'' is generated by two chains of generators separated by a fixed interval; either both chains are of size ''m'', or one chain has size ''m'' and the second has size ''m-1''.
==== Definitions: LQ ====
First attempt at a definition: "A scale word ''S'' with ''k'' step sizes X_1, ..., X_k (with a_1 X_1's, ..., and a_k X_k's) is ''line-quantizing'' (LQ) if ''S'', when viewed as a set of instructions tracing a path in Z^k from the origin (in which each X_i means "go 1 step in the positive x_i direction"), results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k."
The problem with this definition is that it is unclear what "closest approximation" means. We take several different definitions of that phrase and define different versions of the LQ property.
Let n = a+b+c be the scale size, w = aX bY cZ be the scale word, let R be the corresponding path following the word w (R(k) = your location after taking k steps according to w), and put n+1 equally spaced points p_n on the line segment L = {(a,b,c)t : 0 <= t <= n}, i.e. the points {L(k) = (a,b,c) k : k ∈ {0, ..., n}}.
* Assume ''S'' is a 2-step scale. Then ''S'' is ''slope-LQ'' if the slope between any two pair of points (representing a ''k''-mosstep) is one of the two nearest possible slopes (in the set {k/0,...,0/k}) to b/a.
* Say that a 2-step scale ''S'' is ''floor-LQ'' if some mode of ''S'' gives the graph of floor(b/a*x).
===== MV2 is equivalent to floor-LQ in 2-step scales (WIP) =====
Assume wlog there are more L's than s's.
Take the graph of the brightest mode of the mos, M_b(x) (right = L, up = s). We claim that this is the required graph of F(x) = floor(b/a*x).
M_b <= F: Prove that F(x) describes a mos.
Say F has #s s's and #L L's across interval [m, m'] in R/nZ = circle of circumference n. Say there is #s+t small steps and #L-t large steps on some k step [r, r'], t >= 2. This implies that the slope of the line b/a* x itself satisfies
(F(m')-F(m)-1)/(m'-m) <= b/a <= (F(m')-F(m)+1)/(m'-m).
(bounded by "floor minus ceiling" and "ceiling minus floor" slopes; this is because x-x' <= x-floor(x') <= floor(x)+1-floor(x').)
Rearranging,
F(m') - F(m) - 1 <= b/a(m'-m) <= F(m') - F(m) + 1
But F(m') -F(m) = #s and m'-m = #L. So #s -1 <= b/a*#L <= #s+1. Do the same thing for the "bad" interval [r', r] and you get
#s+t-1 <= b/a(#L-t) <= #s+t+1.
Thus b/a#L <= b/a(#L-t), a contradiction.
M_b >= F: (bc it's a mos) Suppose there is an x-value x_0 where M_b(x_0) < F(x_0). Let m = min(n_0, n-n_0), n = scale size. Then find three different m-mossteps by taking one interval before n_0, one interval containing n_0 and one interval after n_0.
==== MV3 Theorem 1 (WIP) ====
''The following are equivalent for a non-multiperiod scale word S with steps x, y, z:''
# ''S is MV3.''
# ''S is PMOS, or S is of the form x'y'z'y'x' or its repetitions.''
# ''S is AG, or S is of the form x'y'z'y'x' or its repetitions, or x'y'x'z'x'y'x' or its repetitions.''
====== Lemma 1: S is pairwise MOS (PMOS) except in the case "xyzyx" (WIP) ======
TODO: account for case xyzyx.
To eliminate xyzyx we manually check all words up to length 5... (todo)
Now assume len(S) >= 6.
WOLOG consider chunks of x. Use q for any occurrence of either y and z.
say you have some intvl class (k steps) with 3 variants in x's and q's:
* S1 = a1x + b1q, represented by the word s1 in the MV3 scale
* S2 = a2x + b2q, word s2
* S3 = a3x + b3q, word s3
(si to be chosen later)
Say that the mos formed by the Ys and Zs is rY sZ, wolog r > s.
The # of chunks in rY sZ is s. The min chunk size is floor(r/s).
The idea is to keep extending si to the right until the chunk size in the MOS guarantees an extra variety.
First we prove that chunk sizes can't differ by 2 or more.
have some length (say that of biggest chunk of x's) word with no q's and >=2 q's.
now y[biggest]z => contradiction bc two kinds of "one q"
so some non biggest chunk has to have y[chunk]z (or z[chunk]y)
then by using size of [y[non-biggest chunk]z] you get a contradiction bc you can scoot to get an x (since consecutive q's cant happen if there are consecutive x's)
u get [xyxxx...x]z, x[yxxxx...xz]x, y[x...xz], and all x's from the max chunk
If we have more q's than x's then we can't have "xx", so we're done.
This proves the claim about chunk sizes.
''To be continued...''
====== PMOS implies AG (except in the case xyxzxyx) (WIP) ======
We now prove that except in the case xyxzxyx, if the scale is pairwise MOS, then it is AG.
To eliminate xyxzxyx we manually check all words up to length 7... (todo)
Now assume len(S) >= 8.
PMOS -> Consider mos temperings
* in x, ξ (ξ = y or z), with gen g1 -> g1g1...g1g1' (g1' = imperfect gen)
* in y, η (η = x or z), with gen g2
* in z, ζ (ζ = x or y), with gen g3.
Denote their detemperings as G11, G12, G13, G21, G22, G23, G31, G32, G33.
''To be continued...''
<!--
A gen chain g1...g1g1' [assuming this word is not multiperiod] detempers to G1i(1)G1i(2)...G1i(n-1)G13, where i(t) is in {1,2} and n = len(S). This word must be MV3, since otherwise the original scale wouldn't be MV3. Similarly, g2...g2g2' detempers to G2j(1)...G2j(n-1)G23, and g3...g3g3' detempers to G3k(1)...G3k(n-1)G33.-->
====== AG implies "ax by bz" ======
'''Assuming the alternating generator property''', we have two chains of generator g0 (going right). The two cases are:
O-O-...-O (m notes)
O-O-...-O (m notes)
and
O-O-O-...-O (m notes)
O-O-...-O (m-1 notes).
Label the notes (1,k) and (2,k), 1 ≤ k ≤ m or m-1, for notes in the upper and lower chain respectively.
In case 1, let g1 = (2,1)-(1,m) and g2 = (1,1)-(2,m). The circle of stacked g0 generators is (starting from top left): (m-1 g0's) g1 (m-1 g0's) g2. A scale step is always a same number k (which must be odd) of such generators gi. Assume (after taking octave complement) that a single step takes less than half of the generators. So a word corresponding to the scale step is formed by one of:
* k g0
* (k-1) g0 + g1
* (k-1) g0 + g2.
It is clear that the last two sizes must occur the same number of times.
In case 2, let (2,1)-(1,1) = g1, (1,2)-(2,1) = g2 be the two alternating generators. Let g3 be the leftover generator after stacking alternating g1 and g2. Then the generator circle looks like g1 g2 g1 g2 ... g1 g2 g3. Then the generators corresponding to a step are:
* k g1 + (k-1) g2
* (k-1) g1 + k g2
* (k-1) g1 + (k-1) g2 + g3
if a step is an odd number of generators (since the scale size is odd, we can always ensure this by taking octave complements of all the generators). The first two sizes must occur the same number of times. QED.
==== 3-DE implies MV3 (WIP) ====
We prove that 3-DE + not abcba implies PMOS, which is known to imply MV3.
==== MV3 Theorem 2 ====
''Once you have chosen a rank-3 temperament and a specific generator interval, there is a mechanical procedure to generate all max-variety-3 scales of a certain size (of which there are, however, infinitely many).''


[[Category:Fokker block]]
[[Category:Fokker block]]
[[Category:Math]]
[[Category:Math]]
[[Category:Rank 3]]
[[Category:Rank 3]]
[[Category:Scales]]
[[Category:Scale]]
[[Category:Theory]]
[[Category:Pages with open problems]]
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