Ternary scale theorems: Difference between revisions

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== Definitions ==

* ''Dyad'' is used for the musical sense of ''interval'' to avoid confusion with the mathematical sense of ''interval''. But strictly speaking, ''interval'' in the musical sense is different than ''dyad''. (An octave is an interval but not a dyad, and a 2:3:4 chord is a dyad but not an interval.)

* A circular word ''s'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''generator-offset''' if it satisfies the following properties. The following conditions do not imply that '''g'''1 and '''g'''2 are the same number of scale steps. For example, 5-limit [[blackdye]] has {{nowrap|'''g'''1 {{=}} 9/5}} (a 9-step) and {{nowrap|'''g'''2 {{=}} 5/3}} (a 7-step).

* A circular word ''s'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''generator-offset''' if it satisfies the following properties. The following conditions do not imply that '''g'''1 and '''g'''2 are the same number of scale steps. For example, 5-limit [[blackdye]] has {{nowrap|'''g'''1 {{=}} {{frac|9|5}}}} (a 9-step) and {{nowrap|'''g'''2 {{=}} {{frac|5|3}}}} (a 7-step).

*# ''s'' is generated by two chains of stacked generators g separated by a fixed offset δ; either both chains are of size ''n''/2, or one chain has size {{sfrac|''n'' + 1|2}} and the second has size {{sfrac|''n'' − 1|2}}. Equivalently, ''s'' can be built by stacking a single chain of alternants '''g'''1 and '''g'''2, resulting in a circle of the form either '''g'''1 '''g'''2 ... '''g'''1 '''g'''2 '''g'''1 '''g'''3 or '''g'''1 '''g'''2 ... '''g'''1 '''g'''2 '''g'''3.

*# ''s'' is generated by two chains of stacked generators g separated by a fixed offset δ; either both chains are of size {{frac|''n''|2}}, or one chain has size {{sfrac|''n'' + 1|2}} and the second has size {{sfrac|''n'' − 1|2}}. Equivalently, ''s'' can be built by stacking a single chain of alternants '''g'''1 and '''g'''2, resulting in a circle of the form either '''g'''1 '''g'''2 ... '''g'''1 '''g'''2 '''g'''1 '''g'''3 or '''g'''1 '''g'''2 ... '''g'''1 '''g'''2 '''g'''3.

*# The scale is ''well-formed'' with respect to g, i.e. all occurrences of the generator g are ''k''-steps for a fixed ''k''.

* A ''scale'' or ''scale word'' is a circular word with a chosen size for its equave. As we're not working with scales with distinct equaves simultaneously, all three terms are effectively synonymous for our purposes.

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Assuming SGA, we have two chains of the aggregate generator '''g''' (going right). In the diagrams below, O represents a note and - represents a generator '''g'''. The two cases are:

CASE 1: EVEN LENGTH

O-O-...-O (n/2 notes)

O-O-...-O ({{frac|''n''|2}} notes)

O-O-...-O (n/2 notes)

O-O-...-O ({{frac|''n''|2}} notes)

and

CASE 2: ODD LENGTH

O-O-O-...-O ((n+1)/2 notes)

O-O-O-...-O ({{frac|''n'' + 1|2}} notes)

O-O-...-O ((n-1)/2 notes).

O-O-...-O ({{frac|''n'' − 1|2}} notes).

Label the notes (1, ''j'') and (2, ''j''), {{nowrap|1 ≤ ''j'' ≤ ''N''}} where ''N'' is the number of notes in the chain, for notes in the upper and lower chain, respectively.

==== Statement (1) ====

In case 1, let {{nowrap|'''g'''1 {{=}} (2, 1) − (1, 1)|'''g'''2 {{=}} (1, 2) − (2, 1)}}, and {{nowrap|'''g'''3 {{=}} (1, 1) − (''n''/2, 2)}} {{nowrap|{{=}} ((−''n''/2 − 1)*'''g'''1 − ''n''/2*'''g'''2) (mod '''e''')}}. We assume that '''g'''1, '''g'''2 and '''e''' are ℤ-linearly independent. We have the chain '''g'''1 '''g'''2 '''g'''1 '''g'''2 ... '''g'''1 '''g'''3 which visits every note in ''s''.

In case 1, let {{nowrap|'''g'''1 {{=}} (2, 1) − (1, 1)|'''g'''2 {{=}} (1, 2) − (2, 1)}}, and {{nowrap|'''g'''3 {{=}} (1, 1) − ({{frac|''n''|2}}, 2)}} {{nowrap|{{=}} (({{frac|−''n''|2}} − 1)*'''g'''1 − {{frac|''n''|2}}*'''g'''2) (mod '''e''')}}. We assume that '''g'''1, '''g'''2 and '''e''' are ℤ-linearly independent. We have the chain '''g'''1 '''g'''2 '''g'''1 '''g'''2 ... '''g'''1 '''g'''3 which visits every note in ''s''.

Since ''s'' is generator-offset it is well-formed with respect to the aggregate generator {{nowrap|'''g''' {{=}} ('''g'''2 + '''g'''1)}}. Since '''g'''1 and '''g'''2 subtend the same number of steps by the SGA assumption, each is an odd-step. All multiples of the aggregate generator '''g''' must be even-steps, and those dyads that are "offset" by '''g'''1 must be odd-steps. Letting ''M'' be the subset consisting of all even-numbered notes (which are generated by '''g''') and considering ''M'' as a scale by dividing degree indices in ''M'' by two, ''M'' is well-formed with respect to '''g''', thus ''M'' (and its offset) must be a MOS subset. Hence {{nowrap|('''g'''3 + '''g'''1)}}, the imperfect generator of the MOS generated by '''g''', subtends the same number of steps as '''g'''. Thus '''g'''2 and '''g'''3 subtend the same number of steps, a fact we need in order to be able to substitute one instance of '''g'''2 with '''g'''3 in the next part.

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It is pretty easy to see this behavior is true if we simply compute the MV3 sequences up to any very large ''N'', far beyond the scale sizes we typically use in music theory, but it would be good to have a proof.

=== Open questions ===

This heading has those open questions for which no conjecture has yet been formed either way. (These can be updated as necessary)

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 == Definitions ==
 * ''Dyad'' is used for the musical sense of ''interval'' to avoid confusion with the mathematical sense of ''interval''. But strictly speaking, ''interval'' in the musical sense is different than ''dyad''. (An octave is an interval but not a dyad, and a 2:3:4 chord is a dyad but not an interval.)
-* A circular word ''s'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''generator-offset''' if it satisfies the following properties. The following conditions do not imply that '''g'''<sub>1</sub> and '''g'''<sub>2</sub> are the same number of scale steps. For example, 5-limit [[blackdye]] has {{nowrap|'''g'''<sub>1</sub> {{=}} 9/5}} (a 9-step) and {{nowrap|'''g'''<sub>2</sub> {{=}} 5/3}} (a 7-step).
+* A circular word ''s'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''generator-offset''' if it satisfies the following properties. The following conditions do not imply that '''g'''<sub>1</sub> and '''g'''<sub>2</sub> are the same number of scale steps. For example, 5-limit [[blackdye]] has {{nowrap|'''g'''<sub>1</sub> {{=}} {{frac|9|5}}}} (a 9-step) and {{nowrap|'''g'''<sub>2</sub> {{=}} {{frac|5|3}}}} (a 7-step).
-*# ''s'' is generated by two chains of stacked generators g separated by a fixed offset δ; either both chains are of size ''n''/2, or one chain has size {{sfrac|''n'' + 1|2}} and the second has size {{sfrac|''n'' − 1|2}}. Equivalently, ''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>.
+*# ''s'' is generated by two chains of stacked generators g separated by a fixed offset δ; either both chains are of size {{frac|''n''|2}}, or one chain has size {{sfrac|''n'' + 1|2}} and the second has size {{sfrac|''n'' − 1|2}}. Equivalently, ''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>.
 *# The scale is ''well-formed'' with respect to g, i.e. all occurrences of the generator g are ''k''-steps for a fixed ''k''.
 * A ''scale'' or ''scale word'' is a circular word with a chosen size for its equave. As we're not working with scales with distinct equaves simultaneously, all three terms are effectively synonymous for our purposes.
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 Assuming SGA, we have two chains of the aggregate generator '''g''' (going right). In the diagrams below, O represents a note and - represents a generator '''g'''. The two cases are:
   CASE 1: EVEN LENGTH
-  O-O-...-O (n/2 notes)
+  O-O-...-O ({{frac|''n''|2}} notes)
-  O-O-...-O (n/2 notes)
+  O-O-...-O ({{frac|''n''|2}} notes)
 and
   CASE 2: ODD LENGTH
-  O-O-O-...-O ((n+1)/2 notes)
+  O-O-O-...-O ({{frac|''n'' + 1|2}} notes)
-  O-O-...-O ((n-1)/2 notes).
+  O-O-...-O ({{frac|''n'' &minus; 1|2}} notes).
 Label the notes (1, ''j'') and (2, ''j''), {{nowrap|1 &le; ''j'' &le; ''N''}} where ''N'' is the number of notes in the chain, for notes in the upper and lower chain, respectively.
 ==== Statement (1) ====
-In case 1, let {{nowrap|'''g'''<sub>1</sub> {{=}} (2, 1) &minus; (1, 1)|'''g'''<sub>2</sub> {{=}} (1, 2) &minus; (2, 1)}}, and {{nowrap|'''g'''<sub>3</sub> {{=}} (1, 1) &minus; (''n''/2, 2)}} {{nowrap|{{=}} ((&minus;''n''/2 &minus; 1)*'''g'''<sub>1</sub> &minus; ''n''/2*'''g'''<sub>2</sub>) (mod '''e''')}}. We assume that '''g'''<sub>1</sub>, '''g'''<sub>2</sub> and '''e''' are ℤ-linearly independent. 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> which visits every note in ''s''.
+In case 1, let {{nowrap|'''g'''<sub>1</sub> {{=}} (2, 1) &minus; (1, 1)|'''g'''<sub>2</sub> {{=}} (1, 2) &minus; (2, 1)}}, and {{nowrap|'''g'''<sub>3</sub> {{=}} (1, 1) &minus; ({{frac|''n''|2}}, 2)}} {{nowrap|{{=}} (({{frac|&minus;''n''|2}} &minus; 1)*'''g'''<sub>1</sub> &minus; {{frac|''n''|2}}*'''g'''<sub>2</sub>) (mod '''e''')}}. We assume that '''g'''<sub>1</sub>, '''g'''<sub>2</sub> and '''e''' are ℤ-linearly independent. 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> which visits every note in ''s''.
 Since ''s'' is generator-offset it is well-formed with respect to the aggregate generator {{nowrap|'''g''' {{=}} ('''g'''<sub>2</sub> + '''g'''<sub>1</sub>)}}. Since '''g'''<sub>1</sub> and '''g'''<sub>2</sub> subtend the same number of steps by the SGA assumption, each is an odd-step. All multiples of the aggregate generator '''g''' must be even-steps, and those dyads that are "offset" by '''g'''<sub>1</sub> must be odd-steps. Letting ''M'' be the subset consisting of all even-numbered notes (which are generated by '''g''') and considering ''M'' as a scale by dividing degree indices in ''M'' by two, ''M'' is well-formed with respect to '''g''', thus ''M'' (and its offset) must be a MOS subset. Hence {{nowrap|('''g'''<sub>3</sub> + '''g'''<sub>1</sub>)}}, the imperfect generator of the MOS generated by '''g''', subtends the same number of steps as '''g'''. Thus '''g'''<sub>2</sub> and '''g'''<sub>3</sub> subtend the same number of steps, a fact we need in order to be able to substitute one instance of '''g'''<sub>2</sub> with '''g'''<sub>3</sub> in the next part.
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 It is pretty easy to see this behavior is true if we simply compute the MV3 sequences up to any very large ''N'', far beyond the scale sizes we typically use in music theory, but it would be good to have a proof.
 === Open questions ===
 This heading has those open questions for which no conjecture has yet been formed either way. (These can be updated as necessary)