Ternary scale theorems: Difference between revisions

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* A scale is ''primitive'' if its period is the same as its equave. A ''multimos'' or ''multiperiod mos'' is a non-primitive mos. A mos aLbs is primitive iff gcd(a, b) = 1. This corresponds to the term ''single-period'' in common xen parlance.

* An ''n''-''ary'' scale is a scale with ''n'' different step sizes. ''Binary'' and ''ternary'' are used when ''n'' = 2 and 3 respectively.

* A strengthening of the generator-offset property, tentatively named the ''swung-generator-alternant property'' (SGA), states that the alternants g1 and g2 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). All odd generator-offset scales are SGA, and aside from odd generator-offset scales, the only ternary scales to satisfy SGA are (xy)''r''xz, ''r'' ≥ 1. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is generator-offset but not SGA.

* A strengthening of the generator-offset property, tentatively named the ''swung-generator-alternant property'' (SGA), states that the alternants g1 and g2 can be taken to always subtend the same number of scale steps, thus both representing "detemperings" of a generator of a primitive [[mos]] scale (otherwise known as a well-formed scale). All odd generator-offset scales are SGA, and aside from odd generator-offset scales, the only ternary scales to satisfy SGA are (xy)''r''xz, ''r'' ≥ 1. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is generator-offset but not SGA.

* An ''odd-step'' is a ''k''-step where ''k'' is odd; an ''even-step'' is defined similarly.

* Given a linear or circular word ''S'' with a step size X, define ''E''X(''S'') as the scale word resulting from deleting all instances of X from ''S''.

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* ''Length'' is another term for a scale's size.

* A ''projection'' of a ternary scale is the operation of equating two of its step sizes.

* A ternary scale is ''pairwise-well-formed'' if all its projections are well-formed (i.e. ~~single-period~~ mosses).

* A ternary scale is ''pairwise-well-formed'' if all its projections are well-formed (i.e. primitive mosses).

== Proposition 1 (Properties of SGA scales) ==

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# If ''n'' is odd, ''S'' is of the form ''a''x ''b''y ''b''z for some permutation (x, y, z) of (L, M, s).

# If ''n'' is odd, ''S'' is abstractly SV3 (i.e. SV3 for almost all tunings).

# If ''n'' is odd, ''S'' = ''a''X ''b''Y ''b''Z is obtained from some mode of the (~~single-period~~) mos ''a''X 2''b''W by replacing all the W's successively with alternating Y's and Z's (or alternating Z's and Y's for the other chirality, fixing the mode of ''a''X 2''b''W). The two alternants differ by replacing one Y with a Z.

# If ''n'' is odd, ''S'' = ''a''X ''b''Y ''b''Z is obtained from some mode of the (primitive) mos ''a''X 2''b''W by replacing all the W's successively with alternating Y's and Z's (or alternating Z's and Y's for the other chirality, fixing the mode of ''a''X 2''b''W). The two alternants differ by replacing one Y with a Z.

# If ''n'' is odd, ''S'' is pairwise-mos. That is, the following operations each result in a [[mos]]: setting L = M, setting L = s, and setting M = s.

# If ''n'' is odd, ''S'' is elimination-mos. That is, ''E''X(''S''), ''E''Y(''S''), ''E''Z(''S'') are all mosses.

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== Theorem 4 (Classification of pairwise well-formed scales) ==

Let ''S''(a, b, c) be a scale word in three '''Z'''-linearly independent step sizes a, b, c. Suppose ''S'' is pairwise well-formed (equivalently, all its projections are ~~single-period~~ mosses). Then ''S'' is SV3 and has an odd number of notes. Moreover, ''S'' is either generator-offset or equivalent to the scale word abacaba.

Let ''S''(a, b, c) be a scale word in three '''Z'''-linearly independent step sizes a, b, c. Suppose ''S'' is pairwise well-formed (equivalently, all its projections are primitive mosses). Then ''S'' is SV3 and has an odd number of notes. Moreover, ''S'' is either generator-offset or equivalent to the scale word abacaba.

=== Proof ===

==== If the generator of a projection of ''S'' is a ''k''-step, the word of stacked ''k''-steps in ''S'' is pairwise well-formed ====

Suppose ''S'' has ''n'' notes (after dealing with small cases, we may assume ''n'' ≥ 7) and ''S'' projects to ~~single-period~~ mosses ''S''1 (via identifying b with c), ''S''2 (via identifying a with c) and ''S''3 (via identifying a with b). Suppose ''S''1's generator is a ''k''-step, which comes in two sizes: P, the perfect ''k''-step, and I, the imperfect ''k''-step. By stacking ''n''-many ''k''-steps, we get two words of length ''n'' of ''k''-steps of ''S''2 and ''S''3, respectively. These binary words, which we call Σ2 and Σ3, must be mosses, since ''m''-steps in the new words correspond to ''mk''-steps in the mos words ''S''1 and ''S''2, which come in at most two sizes. Since ''S''1 is a ~~single-period~~ mos, gcd(''k'', ''n'') = 1. Hence when 0 < ''m'' < ''n'', ''mk'' is ''not'' divisible by ''n'' and ''mk''-steps come in ''exactly'' two sizes; hence both Σ2 and Σ3 are ~~single-period~~ mosses.

Suppose ''S'' has ''n'' notes (after dealing with small cases, we may assume ''n'' ≥ 7) and ''S'' projects to primitive mosses ''S''1 (via identifying b with c), ''S''2 (via identifying a with c) and ''S''3 (via identifying a with b). Suppose ''S''1's generator is a ''k''-step, which comes in two sizes: P, the perfect ''k''-step, and I, the imperfect ''k''-step. By stacking ''n''-many ''k''-steps, we get two words of length ''n'' of ''k''-steps of ''S''2 and ''S''3, respectively. These binary words, which we call Σ2 and Σ3, must be mosses, since ''m''-steps in the new words correspond to ''mk''-steps in the mos words ''S''1 and ''S''2, which come in at most two sizes. Since ''S''1 is a primitive mos, gcd(''k'', ''n'') = 1. Hence when 0 < ''m'' < ''n'', ''mk'' is ''not'' divisible by ''n'' and ''mk''-steps come in ''exactly'' two sizes; hence both Σ2 and Σ3 are primitive mosses.

index: 1 2 3 4 ... ''n''

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==== ''n'' is odd, etc. ====

Suppose Q = (α, β, γ) ≠ R = (α, β′, γ′) are the two ''k''-steps in ''S'' that project to P. Then T = (α′, β′′, γ′′) projects to I. Here the values in each component differ by at most 1, and α ≠ α′. Then the circular word Λ1 formed by the a-components of the ''k''-steps in P is α...αα′. Since Σ2 is a ~~single-period~~ mos pattern of βb + (''n'' − β)(a~c) and β′a + (''n'' − β′)(a~c), the circular word Λ2 = the pattern of β and β′ must be a ~~single-period~~ mos. Similarly, Λ3 = the pattern of γ and γ′ is a ~~single-period~~ mos.

Suppose Q = (α, β, γ) ≠ R = (α, β′, γ′) are the two ''k''-steps in ''S'' that project to P. Then T = (α′, β′′, γ′′) projects to I. Here the values in each component differ by at most 1, and α ≠ α′. Then the circular word Λ1 formed by the a-components of the ''k''-steps in P is α...αα′. Since Σ2 is a primitive mos pattern of βb + (''n'' − β)(a~c) and β′a + (''n'' − β′)(a~c), the circular word Λ2 = the pattern of β and β′ must be a primitive mos. Similarly, Λ3 = the pattern of γ and γ′ is a primitive mos.

Suppose Λ2 is the mos λβ μβ′. Then Λ3 is the mos (λ ± 1)γ (μ ∓ 1)γ′. Since neither Λ2 nor Λ3 are multimosses, and at least one of μ and (μ ∓ 1) are even, it is now immediate that ''n'' is odd.

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== Theorem 5 (Classification of MV3 scales) ==

# A ~~single-period~~ MV3 is either (1) equivalent to XYZYX, (2) equivalent to XYXZXYX, (3) constructed from ''a''X ''b''Z with ''a'' even and gcd(''a'', ''b'') = 1 by replacing every other X with Y, (4) constructed from 2''a''X 2''b''Z with ''a'' odd and gcd(''a'', ''b'') = 1 by replacing every other X with Y, or (5) a "twisted" word constructed as follows:

# A primitive MV3 is either (1) equivalent to XYZYX, (2) equivalent to XYXZXYX, (3) constructed from ''a''X ''b''Z with ''a'' even and gcd(''a'', ''b'') = 1 by replacing every other X with Y, (4) constructed from 2''a''X 2''b''Z with ''a'' odd and gcd(''a'', ''b'') = 1 by replacing every other X with Y, or (5) a "twisted" word constructed as follows:

## Start with a multimos word ''w''(X, Z) = ''ka''X ''kb''Z such that ''a'' is even and each ''a''X ''b''Z subword of ''w'' is of the form X''P''(X, Z)Z where ''P''(X, Z) is a palindrome.

## Interchange some of the Z's and X's at some of the boundaries of these copies of the mos word ''w''. Here, if ''w'' is a word and ''w'' = u'uvv', the ''boundary'' between ''u'' and ''v'' is ''u''[len(''u'')]''v''[1] within w. (If w is a circular word and w = [w'] where w' is a linear word, replace w with w' in the previous sentence.) For example, let ''w'' be the multimos word 8X6Z, XXZXZXZXXZXZXZ. Then the border between the copies of the MOS subword XXZXZXZ are w[7]w[8] and w[14]w[1].

## Replace every other X with Y in ''w''.

# ~~Single-period~~ MV3 scales not of type (4) are always SV3, and those of type (4) are SV3 with the exception of the ''n''/2-step (''n'' = scale length) which is variety 2.

# primitive MV3 scales not of type (4) are always SV3, and those of type (4) are SV3 with the exception of the ''n''/2-step (''n'' = scale length) which is variety 2.

# ~~Single-period~~ MV3 scales not of type (5) and not of the form XYZYX are ''balanced'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size.

# primitive MV3 scales not of type (5) and not of the form XYZYX are ''balanced'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size.

=== Proof ===

Proven by Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]" (and Theorem 4). Note that odd GO scales are type (3) in this classification.

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 * A scale is ''primitive'' if its period is the same as its equave. A ''multimos'' or ''multiperiod mos'' is a non-primitive mos. A mos aLbs is primitive iff gcd(a, b) = 1. This corresponds to the term ''single-period'' in common xen parlance.
 * An ''n''-''ary'' scale is a scale with ''n'' different step sizes. ''Binary'' and ''ternary'' are used when ''n'' = 2 and 3 respectively.
-* A strengthening of the generator-offset property, tentatively named the ''swung-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). All odd generator-offset scales are SGA, and aside from odd generator-offset scales, the only ternary scales to satisfy SGA are (xy)<sup>''r''</sup>xz, ''r'' &ge; 1. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is generator-offset but not SGA.
+* A strengthening of the generator-offset property, tentatively named the ''swung-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 primitive [[mos]] scale (otherwise known as a well-formed scale). All odd generator-offset scales are SGA, and aside from odd generator-offset scales, the only ternary scales to satisfy SGA are (xy)<sup>''r''</sup>xz, ''r'' &ge; 1. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is generator-offset but not SGA.
 * An ''odd-step'' is a ''k''-step where ''k'' is odd; an ''even-step'' is defined similarly.
 * Given a linear or circular word ''S'' with a step size X, define ''E''<sub>X</sub>(''S'') as the scale word resulting from deleting all instances of X from ''S''.
@@ Line 20: / Line 20: @@
 * ''Length'' is another term for a scale's size.
 * A ''projection'' of a ternary scale is the operation of equating two of its step sizes.
-* A ternary scale is ''pairwise-well-formed'' if all its projections are well-formed (i.e. single-period mosses).
+* A ternary scale is ''pairwise-well-formed'' if all its projections are well-formed (i.e. primitive mosses).
 == Proposition 1 (Properties of SGA scales) ==
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 # If ''n'' is odd, ''S'' is of the form ''a''x ''b''y ''b''z for some permutation (x, y, z) of (L, M, s).
 # If ''n'' is odd, ''S'' is abstractly SV3 (i.e. SV3 for almost all tunings).
-# If ''n'' is odd, ''S'' = ''a''X ''b''Y ''b''Z is obtained from some mode of the (single-period) mos ''a''X 2''b''W by replacing all the W's successively with alternating Y's and Z's (or alternating Z's and Y's for the other chirality, fixing the mode of ''a''X 2''b''W). The two alternants differ by replacing one Y with a Z.
+# If ''n'' is odd, ''S'' = ''a''X ''b''Y ''b''Z is obtained from some mode of the (primitive) mos ''a''X 2''b''W by replacing all the W's successively with alternating Y's and Z's (or alternating Z's and Y's for the other chirality, fixing the mode of ''a''X 2''b''W). The two alternants differ by replacing one Y with a Z.
 # If ''n'' is odd, ''S'' is pairwise-mos. That is, the following operations each result in a [[mos]]: setting L = M, setting L = s, and setting M = s.
 # If ''n'' is odd, ''S'' is elimination-mos. That is, ''E''<sub>X</sub>(''S''), ''E''<sub>Y</sub>(''S''), ''E''<sub>Z</sub>(''S'') are all mosses.
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 == Theorem 4 (Classification of pairwise well-formed scales) ==
-Let ''S''(a, b, c) be a scale word in three '''Z'''-linearly independent step sizes a, b, c. Suppose ''S'' is pairwise well-formed (equivalently, all its projections are single-period mosses). Then ''S'' is SV3 and has an odd number of notes. Moreover, ''S'' is either generator-offset or equivalent to the scale word abacaba.
+Let ''S''(a, b, c) be a scale word in three '''Z'''-linearly independent step sizes a, b, c. Suppose ''S'' is pairwise well-formed (equivalently, all its projections are primitive mosses). Then ''S'' is SV3 and has an odd number of notes. Moreover, ''S'' is either generator-offset or equivalent to the scale word abacaba.
 === Proof ===
 ==== If the generator of a projection of ''S'' is a ''k''-step, the word of stacked ''k''-steps in ''S'' is pairwise well-formed ====
-Suppose ''S'' has ''n'' notes (after dealing with small cases, we may assume ''n'' ≥ 7) and ''S'' projects to single-period mosses ''S''<sub>1</sub> (via identifying b with c), ''S''<sub>2</sub> (via identifying a with c) and ''S''<sub>3</sub> (via identifying a with b). Suppose ''S''<sub>1</sub>'s generator is a ''k''-step, which comes in two sizes: P, the perfect ''k''-step, and I, the imperfect ''k''-step. By stacking ''n''-many ''k''-steps, we get two words of length ''n'' of ''k''-steps of ''S''<sub>2</sub> and ''S''<sub>3</sub>, respectively. These binary words, which we call Σ<sub>2</sub> and Σ<sub>3</sub>, must be mosses, since ''m''-steps in the new words correspond to ''mk''-steps in the mos words ''S''<sub>1</sub> and ''S''<sub>2</sub>, which come in at most two sizes. Since ''S''<sub>1</sub> is a single-period mos, gcd(''k'', ''n'') = 1. Hence when 0 < ''m'' < ''n'', ''mk'' is ''not'' divisible by ''n'' and ''mk''-steps come in ''exactly'' two sizes; hence both Σ<sub>2</sub> and Σ<sub>3</sub> are single-period mosses.
+Suppose ''S'' has ''n'' notes (after dealing with small cases, we may assume ''n'' ≥ 7) and ''S'' projects to primitive mosses ''S''<sub>1</sub> (via identifying b with c), ''S''<sub>2</sub> (via identifying a with c) and ''S''<sub>3</sub> (via identifying a with b). Suppose ''S''<sub>1</sub>'s generator is a ''k''-step, which comes in two sizes: P, the perfect ''k''-step, and I, the imperfect ''k''-step. By stacking ''n''-many ''k''-steps, we get two words of length ''n'' of ''k''-steps of ''S''<sub>2</sub> and ''S''<sub>3</sub>, respectively. These binary words, which we call Σ<sub>2</sub> and Σ<sub>3</sub>, must be mosses, since ''m''-steps in the new words correspond to ''mk''-steps in the mos words ''S''<sub>1</sub> and ''S''<sub>2</sub>, which come in at most two sizes. Since ''S''<sub>1</sub> is a primitive mos, gcd(''k'', ''n'') = 1. Hence when 0 < ''m'' < ''n'', ''mk'' is ''not'' divisible by ''n'' and ''mk''-steps come in ''exactly'' two sizes; hence both Σ<sub>2</sub> and Σ<sub>3</sub> are primitive mosses.
   index: 1 2 3 4 ...   ''n''
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 ==== ''n'' is odd, etc. ====
-Suppose Q = (α, β, γ) ≠ R = (α, β′, γ′) are the two ''k''-steps in ''S'' that project to P. Then T = (α′, β′′, γ′′) projects to I. Here the values in each component differ by at most 1, and α ≠ α′. Then the circular word Λ<sub>1</sub> formed by the a-components of the ''k''-steps in P is α...αα′. Since Σ<sub>2</sub> is a single-period mos pattern of βb + (''n'' &minus; β)(a~c) and β′a + (''n'' &minus; β′)(a~c), the circular word Λ<sub>2</sub> = the pattern of β and β′ must be a single-period mos. Similarly, Λ<sub>3</sub> = the pattern of γ and γ′ is a single-period mos.
+Suppose Q = (α, β, γ) ≠ R = (α, β′, γ′) are the two ''k''-steps in ''S'' that project to P. Then T = (α′, β′′, γ′′) projects to I. Here the values in each component differ by at most 1, and α ≠ α′. Then the circular word Λ<sub>1</sub> formed by the a-components of the ''k''-steps in P is α...αα′. Since Σ<sub>2</sub> is a primitive mos pattern of βb + (''n'' &minus; β)(a~c) and β′a + (''n'' &minus; β′)(a~c), the circular word Λ<sub>2</sub> = the pattern of β and β′ must be a primitive mos. Similarly, Λ<sub>3</sub> = the pattern of γ and γ′ is a primitive mos.
 Suppose Λ<sub>2</sub> is the mos λβ μβ′. Then Λ<sub>3</sub> is the mos (λ ± 1)γ (μ ∓ 1)γ′. Since neither Λ<sub>2</sub> nor Λ<sub>3</sub> are multimosses, and at least one of μ and (μ ∓ 1) are even, it is now immediate that ''n'' is odd.
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 == Theorem 5 (Classification of MV3 scales) ==
-# A single-period MV3 is either (1) equivalent to XYZYX, (2) equivalent to XYXZXYX, (3) constructed from ''a''X ''b''Z with ''a'' even and gcd(''a'', ''b'') = 1 by replacing every other X with Y, (4) constructed from 2''a''X 2''b''Z with ''a'' odd and gcd(''a'', ''b'') = 1 by replacing every other X with Y, or (5) a "twisted" word constructed as follows:
+# A primitive MV3 is either (1) equivalent to XYZYX, (2) equivalent to XYXZXYX, (3) constructed from ''a''X ''b''Z with ''a'' even and gcd(''a'', ''b'') = 1 by replacing every other X with Y, (4) constructed from 2''a''X 2''b''Z with ''a'' odd and gcd(''a'', ''b'') = 1 by replacing every other X with Y, or (5) a "twisted" word constructed as follows:
 ## Start with a multimos word ''w''(X, Z) = ''ka''X ''kb''Z such that ''a'' is even and each ''a''X ''b''Z subword of ''w'' is of the form X''P''(X, Z)Z where ''P''(X, Z) is a palindrome.
 ## Interchange some of the Z's and X's at some of the boundaries of these copies of the mos word ''w''. Here, if ''w'' is a word and ''w'' = <i>u'uvv'</i>, the ''boundary'' between ''u'' and ''v'' is ''u''[len(''u'')]''v''[1] within w. (If w is a circular word and w = [w'] where w' is a linear word, replace w with w' in the previous sentence.) For example, let ''w'' be the multimos word 8X6Z, XXZXZXZXXZXZXZ. Then the border between the copies of the MOS subword XXZXZXZ are w[7]w[8] and w[14]w[1].
 ## Replace every other X with Y in ''w''.
-# Single-period MV3 scales not of type (4) are always SV3, and those of type (4) are SV3 with the exception of the ''n''/2-step (''n'' = scale length) which is variety 2.
+# primitive MV3 scales not of type (4) are always SV3, and those of type (4) are SV3 with the exception of the ''n''/2-step (''n'' = scale length) which is variety 2.
-# Single-period MV3 scales not of type (5) and not of the form XYZYX are ''balanced'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size.
+# primitive MV3 scales not of type (5) and not of the form XYZYX are ''balanced'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size.
 === Proof ===
 Proven by Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]" (and Theorem 4). Note that odd GO scales are type (3) in this classification.