A scale satisfies the '''generator-offset property''' (also '''GO''', '''alternating generator''' or '''AG''') if it satisfies the following equivalent properties:
A scale satisfies the '''generator-offset property''' if it satisfies the following properties:
* the scale is generated by two chains of stacked copies of a ''generator'', the two chains are separated by an ''offset'', and the lengths of the chains differ by at most one.
# The scale is generated by two chains of stacked copies of an interval called the ''generator''.
* the scale can be built by stacking two alternating generators (called ''alternants''), which do not necessarily take up the same number of steps
# 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.
The [[Zarlino]] (3L 2M 2S) JI scale is an example of a GO scale, because it is built by stacking alternating 5/4 and 6/5 generators. 7-limit [[diasem]] (5L 2M 2S) is another example, with generators 7/6 and 8/7.
[[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 the steps of a [[periodic scale]]) of size ''n'' is '''GO''' 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'' is generated by two chains of stacked 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.
# ''S'' can be built by stacking a single chain of alternants ''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>.
These are equivalent, since the offset can be taken to be ''g''<sub>1</sub> and the generator then equals ''g''<sub>1</sub> + ''g''<sub>2</sub>. This doesn't imply that ''g''<sub>1</sub> and ''g''<sub>2</sub> are the same number of scale steps. For example, 5-limit [[blackdye]] has ''g''<sub>1</sub> = 9/5 (a 9-step) and ''g''<sub>2</sub> = 5/3 (a 7-step).
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.
More generally, we say that a scale is ''m'''''-GO''' if the scale consists of ''m'' + 1 chains of stacked generator ''g'' (implying ''m'' offsets δ<sub>1</sub>, ...,δ<sub>''m''</sub> from a fixed chain), each chain having either ''k'' or ''k'' + 1 notes. (Thus 1-GO is the same thing as GO.) An ''m''-GO scale can be interpreted as a scale in a rank-(''m'' + 2) [[regular temperament]] (though the specific tuning used may be of lower rank), with basis ''p'' (period), ''g'', δ<sub>1</sub>, ...,δ<sub>''m''</sub>.
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''.
== Other definitions ==
* A strengthening of the generator-offset property, tentatively named the ''split-generator-alternant property'' (SGA), states that the alternants ''g''<sub>1</sub> and ''g''<sub>2</sub> can be taken to always subtend the same number of scale steps, thus both representing "detemperings" of a generator of a single-period [[mos]] scale (otherwise known as a well-formed scale). Only odd GO scales and xyxz can satisfy this property. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is GO but not SGA.
* We say that a particular interval size in a scale ''is subtended constantly'' if the interval size always occurs as the same number of steps. (This alludes to the term [[constant structure]].)
== Theorems ==
[[Category:Scale]]
=== Proposition 1 ===
Let ''S'' be a 3-step-size scale word in L, M, and s, and suppose ''S'' is SGA. Then:
# ''S'' is unconditionally MV3 (i.e. MV3 regardless of tuning).
# ''S'' is of the form ''ax by bz'' for some permutation (''x'', ''y'', ''z'') of (L, M, s).
# The cardinality (size) of ''S'' is either odd, or 4 (and ''S'' is of the form ''xyxz'').
[Note: This is not true with SGA replaced with GO; [[blackdye]] is a counterexample that is MV4.]
==== Proof ====
Assuming SGA, 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, ''j'') and (2, ''j''), 1 ≤ ''j'' ≤ (chain length), for notes in the upper and lower chain respectively.
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 (''k''-steps) reached by stacking ''r'' generators:
# 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-degenerate AG scale.
# ''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 alternants. 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 combinations of alternants 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 SGA + 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. <math>\square</math>
=== Proposition 2 ===
Suppose that a periodic scale satisfies the following:
* is generator-offset (chain lengths off by 1)
* has odd length ''n''
* the generator is subtended constantly.
Then the scale is SGA.
Note: The third condition is required. Without it, this is false for the 5-note scale 0 250 300 700 750 1200 with alternants 700 and 50: the 1-steps form the chain 250 50 400 50 450 and the 2-steps form the chain 300 450 700 250 500.
==== Proof ====
Assume that the generator is a ''k''-step and ''k'' is even. (If ''k'' is not even, invert the generator.) On some tonic p we have a chain of ceil(''n''/2) notes and on some other note ''p' = p'' + offset (not on the first chain) we'll have floor(''n''/2) notes.
We must have gcd(''k'', ''n'') = 1. If not, since ''n'' is odd, gcd(''k'', ''n'') is an odd number at least 3, and the ''k''-steps must form more than 2 parallel chains.
By modular arithmetic we have ''rk'' mod ''n'' = ''k''/2 iff ''r'' = ceil(''n''/2) mod ''n''. (Since gcd(2, ''n'') = 1, 2 is multiplicatively invertible mod ''n'', and we can multiply both sides by 2 to check this.) This proves that the offset, which must be reached after ceil(''n''/2) generator steps, is a ''k''/2-step, as desired. (If the offset wasn't reached in ceil(''n''/2) steps, the two generator chains either wouldn't be disjoint or wouldn't have the assumed lengths.) <math>\square</math>
== Open conjectures ==
=== Conjecture 3 ===
If a non-multiperiod 3-step size scale word is
# unconditionally MV3,
# has odd cardinality,
# is not of the form ''mx my mz'', ''xyzyx'' or ''xyxzxyx'',
then it is SGA. (a converse to Theorem 1)
=== Conjecture 4 ===
An SGA scale is always pairwise-well-formed. That is, the result of identifying any two step sizes of an SGA scale is always a non-multiperiod mos.
== Falsified conjectures ==
[[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.
