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== Theorems ==
A scale satisfies the '''generator-offset property''' if it satisfies the following properties:
# The scale is generated by two chains of stacked copies of an interval called the ''generator''.
# The two chains are separated by a different interval called the ''offset'' (the difference between the first note of the second chain and the first note of the first chain).
# The lengths of the chains differ by at most one. (1-3 can be restated as: The scale can be built by stacking two alternating generators (called ''alternants'') a<sub>1</sub> and a<sub>2</sub>. Note that a<sub>1</sub> does not need to [[subtend]], i.e. occur as, the same number of steps as a<sub>2</sub>.)
# The generator always occurs as the same number of steps. For example, the generator is never both a 2-step and a 3-step.


=== AG + unconditionally MV3 implies "ax by bz" and that the scale's cardinality is odd or 4 ===
[[File:MV3-Labeled.png|thumb|240px|Plot of at-most-decatonic MV3 generator-offset scale regions in the plane.]]
'''Assuming both AG and unconditional MV3''', 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.
The [[Zarlino]] (3L 2M 2S) JI scale is an example of a generator-offset scale, because it is built by stacking alternating 5/4 and 6/5 generators. The 7-limit version of [[diasem]] (5L 2M 2S) is another example, with generators 7/6 and 8/7.


In case 1 (even scale size n = 2^t r where r is odd), let g1 = (2,1) - (1,1) and g2 = (1,2) - (2,1). We have the chain g1 g2 g1 g2... g1 g3. Suppose the k-step is the class generated by r generators (which is an odd number of generator steps):
Generator-offset scales generalize the notion of [[dipentatonic scale|dipentatonic]] and [[diheptatonic scale|diheptatonic]] scales where the pentatonic and heptatonic are [[MOS scales]]. A related but distinct notion is [[alternating generator sequence]]. While scales produced using the generator-offset procedure can be seen as a result of an alternating generator sequence of 2 alternants, the generator-offset perspective views the sum of the two alternants as the "canonical" generator, and the alternants as rather being possible choices of the offset which are effectively equivalent up to chirality. While a well-formed AGS scale requires each alternant in the AGS to subtend the same number of steps, the generator-offset property only requires each (aggregate) generator to subtend the same number of steps.
# from g1 ... g1, get a1 = (r-1)/2*g0 + g1 = (r+1)/2 g1 + (r-1)/2 g2
# from g2 ... g2, get a2 = (r-1)/2*g0 + g2 = (r-1)/2 g1 + (r+1)/2 g2
# from g2 (even) g1 g3 g1 (even) g2, get a3 = (r-1)/2 g1 + (r-1)/2 g2 + g3
# from g1 (odd) g1 g3 g1 (odd) g1, get a4 = (r+1)/2 g1 + (r-3)/2 g2 + g3.  


Choose a tuning where g1 and g2 are both very close to but not exactly 1/2*g0, resulting in a scale very close to the mos generated by 1/2 g0. (i.e. g1 and g2 differ from 1/2*g0 by ε, a quantity much smaller than the chroma of the n/2-note mos generated by g0, which is |g3 - g2|). Assuming n > 4, we have 4 distinct sizes for k-steps, a contradiction to unconditional-MV3:
Note: In Inthar's contribution to [[aberrismic theory]], this term has been superseded by [[guide frame]]s.
# a1, a2 and a3 are clearly distinct.
== Mathematical definition ==
# a4 - a3 = g1 - g2 != 0, since the scale is a non-trivial AG.  
More formally, a cyclic word ''S'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''generator-offset''' if it satisfies the following properties:
# a4 - a1 = g3 - g2 = (g3 + g1) - (g2 + g1) != 0. This is exactly the chroma of the mos generated by g0.
# ''S'' is generated by two chains of stacked generators g separated by a fixed offset δ; either both chains are of size ''n''/2 (implying ''n'' is even), or one chain has size (''n'' + 1)/2 and the second has size (''n''&nbsp;&minus;&nbsp;1)/2 (implying ''n'' is odd).  
# a4 - a2 = g1 - 2 g2 + g3 = (g3 - g2) + (g1 - g2) = (chroma ± ε) != 0 by choice of tuning.
# The scale is ''well-formed'' with respect to g, i.e. all occurrences of the generator g are ''k''-steps for a fixed ''k''.


(For n = 4, the above argument doesn't work because a3 = a4, and xyxz is a counterexample.)


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:
[[Category:Scale]]
# 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.
 
(The above holds for any odd n >= 3.)
 
This proof shows that AG and unconditionally-MV3 scales must have cardinality odd or 4.
 
=== An AG scale is unconditionally MV3 iff its cardinality is odd or 4 ===
We only need to see that AG + odd cardinality => MV3. But the argument in case 2 above works for any interval class (MV3 wasn't used), hence any interval class comes in at most 3 sizes regardless of tuning.
 
=== An even-cardinality unconditional MV3 is of the form W(x,y,z)W(y,x,z) (WIP) ===
 
=== 3-DE implies MV3 (WIP) ===
We prove that 3-DE + not abcba implies PMOS, which is known to imply MV3.
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