A scale satisfies the '''alternating generator property''', or the '''AG''' property for short, if it satisfies the following equivalent properties:
A scale satisfies the '''generator-offset property''' if it satisfies the following properties:
* the scale can be built by stacking alternating generators
# The scale is generated by two chains of stacked copies of an interval called the ''generator''.
* the scale is generated by two chains of generators separated by a fixed interval, and the lengths of the chains differ by at most one.
# 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.
[[Diasem]] is an example of an AG scale, because it is built by stacking alternating 7/6 and 8/7 for [[chirality|left-handed]] diasem, or 8/7 and 7/6 for right-handed diasem.
[[File:MV3-Labeled.png|thumb|240px|Plot of at-most-decatonic MV3 generator-offset scale regions in the plane.]]
More formally, a cyclic word ''S'' (representing a [[periodic scale]]) of size ''n'' is '''AG''' if it satisfies the following equivalent properties:
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.
# ''S'' can be built by stacking a single chain of alternating generators ''g''<sub>1</sub> and ''g''<sub>2</sub>, resulting in a circle of the form either ''g''<sub>1</sub> ''g''<sub>2</sub> ... ''g''<sub>1</sub> ''g''<sub>2</sub> ''g''<sub>1</sub> ''g''<sub>3</sub> or ''g''<sub>1</sub> ''g''<sub>2</sub> ... ''g''<sub>1</sub> ''g''<sub>2</sub> ''g''<sub>3</sub>.
# ''S'' is generated by two chains of generators separated by a fixed interval; either both chains are of size ''n''/2, or one chain has size (''n'' + 1)/2 and the second has size (''n'' − 1)/2.
These are equivalent, since the separating interval can be taken to be ''g''<sub>1</sub> and the generator of each chain = ''g''<sub>1</sub> + ''g''<sub>2</sub>.
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.
== Theorems ==
=== Theorem 1 ===
If a 3-step-size scale word ''S'' in L, M, and s is both AG and unconditionally [[MV3]] (i.e. MV3 regardless of tuning), then the scale is of the form ''ax by bz'' for some permutation (''x'', ''y'', ''z'') of (L, M, s); and the scale's cardinality is either odd, or 4 (and is of the form ''xyxz''). Moreover, any odd-cardinality AG scale is unconditionally MV3.
==== Proof ====
Assuming both AG and unconditionally MV3, we have two chains of generator ''g''<sub>0</sub> (going right). The two cases are:
CASE 1: EVEN CARDINALITY
O-O-...-O (n/2 notes)
O-O-...-O (n/2 notes)
and
CASE 2: ODD CARDINALITY
O-O-O-...-O ((n+1)/2 notes)
O-O-...-O ((n-1)/2 notes).
Label the notes (1, ''k'') and (2, ''k''), 1 ≤ ''k'' ≤ (chain length), for notes in the upper and lower chain respectively.
Note: In Inthar's contribution to [[aberrismic theory]], this term has been superseded by [[guide frame]]s.
== Mathematical definition ==
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:
# ''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'' − 1)/2 (implying ''n'' is odd).
# The scale is ''well-formed'' with respect to g, i.e. all occurrences of the generator g are ''k''-steps for a fixed ''k''.
In case 1, let ''g''<sub>1</sub> = (2, 1) − (1, 1) and ''g''<sub>2</sub> = (1, 2) − (2, 1). We have the chain ''g''<sub>1</sub> ''g''<sub>2</sub> ''g''<sub>1</sub> ''g''<sub>2</sub> ... ''g''<sub>1</sub> ''g''<sub>3</sub>.
Let ''r'' be odd and ''r'' ≥ 3. Consider the following abstract sizes for the interval class reached by stacking ''r'' generators:
[[Category:Scale]]
# from ''g''<sub>1</sub> ... ''g''<sub>1</sub>, we get ''a''<sub>1</sub> = (''r'' − 1)/2*''g''<sub>0</sub> + ''g''<sub>1</sub> = (''r'' + 1)/2 ''g''<sub>1</sub> + (''r'' − 1)/2 ''g''<sub>2</sub>
# from ''g''<sub>2</sub> ... ''g''<sub>2</sub>, we get ''a''<sub>2</sub> = (''r'' − 1)/2*''g''<sub>0</sub> + ''g''<sub>2</sub> = (''r'' − 1)/2 ''g''<sub>1</sub> + (''r'' + 1)/2 ''g''<sub>2</sub>
# from ''g''<sub>2</sub> (...even # of gens...) ''g''<sub>1</sub> ''g''<sub>3</sub> ''g''<sub>1</sub> (...even # of gens...) ''g''<sub>2</sub>, we get ''a''<sub>3</sub> = (''r'' − 1)/2 ''g''<sub>1</sub> + (''r'' − 1)/2 ''g''<sub>2</sub> + ''g''<sub>3</sub>
# from ''g''<sub>1</sub> (...odd # of gens...) ''g''<sub>1</sub> ''g''<sub>3</sub> ''g''<sub>1</sub> (...odd # of gens...) ''g''<sub>1</sub>, we get ''a''<sub>4</sub> = (''r'' + 1)/2 ''g''<sub>1</sub> + (''r'' − 3)/2 ''g''<sub>2</sub> + ''g''<sub>3</sub>.
Choose a tuning where ''g''<sub>1</sub> and ''g''<sub>2</sub> are both very close to but not exactly 1/2*''g''<sub>0</sub>, resulting in a scale very close to the mos generated by 1/2 ''g''<sub>0</sub>. (i.e. ''g''<sub>1</sub> and ''g''<sub>2</sub> differ from 1/2*''g''<sub>0</sub> by ε, a quantity much smaller than the chroma of the ''n''/2-note mos generated by ''g''<sub>0</sub>, which is |''g''<sub>3</sub> − ''g''<sub>2</sub>|). Thus we have 4 distinct sizes for k-steps:
# ''a''<sub>1</sub>, ''a''<sub>2</sub> and ''a''<sub>3</sub> are clearly distinct.
# ''a''<sub>4</sub> − ''a''<sub>3</sub> = ''g''<sub>1</sub> − ''g''<sub>2</sub> != 0, since the scale is a non-trivial AG.
# ''a''<sub>4</sub> − ''a''<sub>1</sub> = ''g''<sub>3</sub> − ''g''<sub>2</sub> = (''g''<sub>3</sub> + ''g''<sub>1</sub>) − (''g''<sub>2</sub> + ''g''<sub>1</sub>) != 0. This is exactly the chroma of the mos generated by ''g''<sub>0</sub>.
By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus ''g''<sub>1</sub> and ''g''<sub>2</sub> must themselves be step sizes. Thus we see that an even-cardinality, unconditionally MV3, AG scale must be of the form ''xy...xyxz''. But this pattern is not unconditionally MV3 if ''n'' ≥ 6, since 3-steps come in 4 sizes: ''xyx'', ''yxy'', ''yxz'' and
''xzx''. Thus ''n'' = 4 and the scale is ''xyxz''.
In case 2, let (2, 1) − (1, 1) = ''g''<sub>1</sub>, (1, 2) − (2, 1) = ''g''<sub>2</sub> be the two alternating generators. Let ''g''<sub>3</sub> be the leftover generator after stacking alternating ''g''<sub>1</sub> and ''g''<sub>2</sub>. Then the generator circle looks like ''g''<sub>1</sub> ''g''<sub>2</sub> ''g''<sub>1</sub> ''g''<sub>2</sub> ... ''g''<sub>1</sub> ''g''<sub>2</sub> ''g''<sub>3</sub>. Then the generators corresponding to a step are:
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.)
Now we only need to see that AG + odd cardinality => unconditionally MV3. But the argument in case 2 above works for any interval class (unconditional MV3 wasn't used), hence any interval class comes in at most 3 sizes regardless of tuning.
== Conjectures ==
=== Conjecture 2 ===
If a non-multiperiod 3-step size scale word is
# MV3,
# has odd cardinality,
# is not of the form ''mx my mz'',
# and is not of the form ''xyxzxyx'',
then it is AG.
[[Category:Theory]]
[[Category:AG scales| ]]<!--Main article-->
Latest revision as of 13:19, 9 August 2024
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) a1 and a2. Note that a1 does not need to subtend, i.e. occur as, the same number of steps as a2.)
The generator always occurs as the same number of steps. For example, the generator is never both a 2-step and a 3-step.
Plot of at-most-decatonic MV3 generator-offset scale regions in the plane.
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.
Generator-offset scales generalize the notion of dipentatonic and 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.
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:
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 − 1)/2 (implying n is odd).
The scale is well-formed with respect to g, i.e. all occurrences of the generator g are k-steps for a fixed k.
