Ternary scale theorems: Difference between revisions

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*# 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.

* 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.

* 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 ''a'''''L''' ''b'''''s''' is primitive iff gcd(''a'', ''b'') = 1. This corresponds to the term ''single-period'' in common xen parlance. Any multimos can be constructed from a primitive mos by repeating the mos pattern multiple times, e.g. if 3'''L''' 2'''s''' is '''LLsLs''', then 9'''L''' 6'''s''' is '''LLsLsLLsLsLLsLs'''.

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

* For the [[generator-offset property]], see the article.

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* By a ''subword'', ''substring'', or ''slice'' of a word ''S'', denoted ''S''[''i'' : ''j''] (''j'' > ''i''), we mean ''S''[''i''] ''S''[''i'' + 1] ... ''S''[''j'' − 1].

* Given a mos a'''X''' b'''Y''', a ''chunk'' of '''X''''s is a maximal (possibly length 0) substring made of '''X''''s, bounded by '''Y''''s. We do not include the boundary '''Y''''s.

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

* ''Length'' is another term for a scale's size. The length of a scale S is denoted len(S).

* 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. primitive mosses).

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

In the following, ''equivalent'' means "is the same circular word after ~~replacing~~ '''X''', '''Y''', and '''Z''' ~~with σ(~~'''X'''~~), σ(~~'''Y'''), ~~and σ(~~'''Z'''~~) for some permutation σ~~ on ~~{'''X''', '''Y''', '''Z'''}~~."

In the following, ''equivalent'' means "is the same circular word after permuting '''X''', '''Y''', and '''Z'''." This means that '''XYXZXYX''' is equivalent to '''YZYXYZY''', or '''XZXYXZX''', and so on.

=== Theorem 5.1 (Classification of ternary balanced scales) ===

# A primitive [[balanced]] MV3 scale ''s'' is one of the following:

# A primitive [[balanced]] MV3 scale ''S'' is one of the following:

## If ~~|''s''|~~ is odd, then ''s'' is equivalent to XYXZXYX or ~~equivalent~~ to a word constructed from ~~<math>\mathsf{Christoffel}_{~~a,b~~}(\mathbf{X}, \mathbf{~~Z~~})</math>~~ with ''a'' even and gcd(''a'', ''b'') = 1 by replacing every other '''X''' with '''Y'''. ~~Here <math>\mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})</math> is the brightest mode of~~ ''a'''''X''' ''b'''''Z''' ~~taking~~ '''X''' > '''Z'''.

## If len(S) is odd, then ''S'' is equivalent to '''XYXZXYX''', or to a word constructed from taking the brightest mode of the MOS ''a'''''X''' ''b'''''Z''' with ''a'' even and gcd(''a'', ''b'') = 1, and replacing every other '''X''' with '''Y'''. We assume '''X''' > '''Z''' when constructing the MOS.

~~## If |~~''s''~~| is even, it is equivalent to a word constructed from <math>\mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})^2</math>~~ with ''a'' odd and gcd(''a'', ''b'') = 1 by replacing every other '''X''' with '''Y'''.

## If len(S) is even, then ''S'' is equivalent to a word constructed from taking the brightest mode of the MOS 2''a'''''X''' 2''b'''''Z''' with ''a'' odd and gcd(''a'', ''b'') = 1, and replacing every other '''X''' with '''Y'''.

# All primitive balanced ternary scales are MV3.

# Balanced primitive ternary scales not of type (3) are always SV3.<!-- TODO: this needs to be proved:

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=== Theorem 5.2 (Classification of MV3 scales) ===

# A primitive MV3 scale is either (1) balanced, (2) equivalent to XYZYX, or (3) equivalent to a "twisted" word constructed as follows:

# A primitive MV3 scale is either (1) balanced, (2) equivalent to '''XYZYX''', or (3) equivalent to a "twisted" word constructed as follows:

## Start with a multimos word ~~<math>w = \mathsf{Christoffel}_{a,b}(\mathbf{~~X~~}, \mathbf{~~Z~~})^k</math> such that~~ ''a'' is even.

## Start with the brightest multimos word ''ka'''''X''' ''kb'''''Z''' with ''a'' being an even number.

## 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 8'''X'''6'''Z''', '''XXZXZXZXXZXZXZ'''. Then the border between the copies of the MOS subword '''XXZXZXZ''' are ''w''[7]''w''[8] and ''w''[14]''w''[1].

## Interchange a '''Z''' and an '''X''' at some (possibly more than one) of the boundaries of these copies of the mos word ''w''. Here, the boundary of two consecutive copies of ''w'' is the last letter of the first word and the first letter of the second word. (At the ends of the whole multimos word, the boundaries are just the first and last letters of the word.) For example, let ''w'' be the multimos word 8'''X'''6'''Z''', '''XXZXZXZXXZXZXZ'''. Then the border between the copies of the MOS subword '''XXZXZXZ''' are ''w''[7]''w''[8] and ''w''[14]''w''[1] (using one-based numbering).

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

=== Proof ===

Proven by Bulgakova, Buzhinsky and Goncharov (2023), "[https://pdf.sciencedirectassets.com/271538/1-s2.0-S0304397522X0039X/1-s2.0-S0304397522006417/am.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEJv%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJHMEUCICIhaHaQFcuBGL19NZOHJFGS7MwzVcZ0xFToxGOkhW6vAiEA06hbPi%2F2O7jyKxdIEbHTvESiHGyC%2F9N4%2FeNLiQc89tQqswUIZBAFGgwwNTkwMDM1NDY4NjUiDIotbpjT6VGvNgDaIiqQBYD4g30jpZUHk5lLFkEbKwIpoDAp8HjE0XulHGFvU2D1xJHVEfBPADJMfpaYreqDiMXRZefNc553rr%2FCY%2FSvP6XSWN6li2Xu3abYhaiV1uUBiMEBu66ZJ%2F%2F2CDRoy%2Bf3uFVkipUvBBmTAtDodFJ6%2BcyEoBkQeIsgJPL%2B9xI2iUX7brAtUr52yKswGXCPTjLhYcrApMV6XvtbUweF63yeYKA8b6oqp4bcARuxP8ubH3m3LRsqkvDYaht8f%2FDZewiq0oyfIWWktRbAiAA5gRySac9vunwVOgx2YrLy1kIkj0ejZF7haArPTKjeUJmnwswCve74TuiVG9pNCjJya0PjJQZa1xZIErf5Q5dKj%2Bps8ITvjp6YHtoCPL2XIv94A2pTjQyws7DdwLkRpBJL6YCeOboX44IsDc%2B7yiSJIWwEkb%2Bkgj93Mb%2BJSJ7AtbzllZAC0NwPMUOxOYPvjHY6gM1NHardi%2BLuT3rKm5yznsV7YUhqNs0jHd2yrfTyJ0Isoa6J%2Bsv5aARbz9CMkusgx4wenGytsdbpHw49GaE%2F23CwooxEv%2BFH9nMpsSzt2axTOIjLReULOV%2F9aZWbuB8Uy26cq3MnWnzVXzhofPwy5Uhi93T8tiRw8dKDXO2tXSjmPUiqk1n41XwHY9z7kZ5J6yIrgkH1ASWa0YVHEW8Cx2sTSkTLwFtgsebpfw0P0AMM%2B%2Bfdcbr714ZvXPmUi77iMhqux%2Feq%2Frw6urUE%2FwFg8PPp%2F8rmHru3VOLjFHjVB6SX0LzyDw1CVwU6%2FRNVUzelLzyHg1X1eUHz0%2FyeQIEdqoHTK8m1ybCuWa88v2IVwfVOZsR1wOpnIIrR4dnjDZ9X5iw%2B9NoQywSM8diF8WHfM59KWUCFMLXp9K0GOrEBlktkq2t3BAB%2B99W%2Bz8Of6btHd4eum8D%2Fgv%2FhzH71opl%2FHeNRcVEJVFU6tFwz7x%2BHySPNMPU0FfD2G8yCy3Qm6EpIQ2R9AZGWteuDCxV6LrPKL4c6BupkmIAkE8HyyqnH4oOc10YZEEvlucl4UUOf7QYQ8J2uwt6sbJh8haepOI6eRPtb%2BLg3xz1M9LP4xG9C2UhONIYd9kkbMZWa%2Fjs4xM%2Fhb8M5JDsl2zQ1KwPkGR5Y&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20240202T190401Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTY4OSCZ3W7%2F20240202%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=d8cdeefedc146cd9b6ae50dfb047bd170aff67c73b4cc6bad4794b0b9634223e&hash=fcd7ef6d1f8e476a88555788d397d1ff5c5720a68482ed4a2f8ec051685c7849&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=S0304397522006417&tid=pdf-bb78adbe-12da-476f-ac55-dd339ae07c1a&sid=758d15492e7c4647a5993de3aaa4e236e5b1gxrqa&type=client On balanced and abelian properties of circular words over a ternary alphabet]".

Note: The xen term "brightest mos word" is equivalent to "Christoffel word" in the paper, and similarly "brightest multimos word" is equivalent to "powers of a Christoffel word". Also see [[Glossary for combinatorics on words]] for more equivalents between xen community terms and standard academic terminology.

[[Category:Math]]

@@ Line 11: / Line 11: @@
 *# 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.
-* 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.
+* 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 ''a'''''L''' ''b'''''s''' is primitive iff gcd(''a'', ''b'') = 1. This corresponds to the term ''single-period'' in common xen parlance. Any multimos can be constructed from a primitive mos by repeating the mos pattern multiple times, e.g. if 3'''L''' 2'''s''' is '''LLsLs''', then 9'''L''' 6'''s''' is '''LLsLsLLsLsLLsLs'''.
 * An ''n''-''ary'' scale is a scale with ''n'' different step sizes. ''Binary'' and ''ternary'' are used when ''n'' = 2 and 3 respectively.
 * For the [[generator-offset property]], see the article.
@@ Line 19: / Line 19: @@
 * By a ''subword'', ''substring'', or ''slice'' of a word ''S'', denoted ''S''[''i'' : ''j''] (''j'' > ''i''), we mean ''S''[''i''] ''S''[''i'' + 1] ... ''S''[''j'' &minus; 1].
 * Given a mos a'''X''' b'''Y''', a ''chunk'' of '''X''''s is a maximal (possibly length 0) substring made of '''X''''s, bounded by '''Y''''s. We do not include the boundary '''Y''''s.
-* ''Length'' is another term for a scale's size.
+* ''Length'' is another term for a scale's size. The length of a scale S is denoted len(S).
 * 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. primitive mosses).
@@ Line 220: / Line 220: @@
 {{qed}}
 == Theorem 5 (Classification of MV3 scales) ==
-In the following, ''equivalent'' means "is the same circular word after replacing '''X''', '''Y''', and '''Z''' with σ('''X'''), σ('''Y'''), and σ('''Z''') for some permutation σ on {'''X''', '''Y''', '''Z'''}."
+In the following, ''equivalent'' means "is the same circular word after permuting '''X''', '''Y''', and '''Z'''." This means that '''XYXZXYX''' is equivalent to '''YZYXYZY''', or '''XZXYXZX''', and so on.
 === Theorem 5.1 (Classification of ternary balanced scales) ===
-# A primitive [[balanced]] MV3 scale ''s'' is one of the following:
+# A primitive [[balanced]] MV3 scale ''S'' is one of the following:
-## If |''s''| is odd, then ''s'' is equivalent to XYXZXYX or equivalent to a word constructed from <math>\mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})</math> with ''a'' even and gcd(''a'', ''b'') = 1 by replacing every other '''X''' with '''Y'''. Here <math>\mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})</math> is the brightest mode of ''a'''''X''' ''b'''''Z''' taking '''X''' > '''Z'''.
+## If len(S) is odd, then ''S'' is equivalent to '''XYXZXYX''', or to a word constructed from taking the brightest mode of the MOS ''a'''''X''' ''b'''''Z''' with ''a'' even and gcd(''a'', ''b'') = 1, and replacing every other '''X''' with '''Y'''. We assume '''X''' > '''Z''' when constructing the MOS.
-## If |''s''| is even, it is equivalent to a word constructed from <math>\mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})^2</math> with ''a'' odd and gcd(''a'', ''b'') = 1 by replacing every other '''X''' with '''Y'''.
+## If len(S) is even, then ''S'' is equivalent to a word constructed from taking the brightest mode of the MOS 2''a'''''X''' 2''b'''''Z''' with ''a'' odd and gcd(''a'', ''b'') = 1, and replacing every other '''X''' with '''Y'''.
 # All primitive balanced ternary scales are MV3.
 # Balanced primitive ternary scales not of type (3) are always SV3.<!-- TODO: this needs to be proved:
@@ Line 230: / Line 230: @@
 === Theorem 5.2 (Classification of MV3 scales) ===
-# A primitive MV3 scale is either (1) balanced, (2) equivalent to XYZYX, or (3) equivalent to a "twisted" word constructed as follows:
+# A primitive MV3 scale is either (1) balanced, (2) equivalent to '''XYZYX''', or (3) equivalent to a "twisted" word constructed as follows:
-## Start with a multimos word <math>w = \mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})^k</math> such that ''a'' is even.
+## Start with the brightest multimos word ''ka'''''X''' ''kb'''''Z''' with ''a'' being an even number.
-## 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'' = [<i>w'</i>] where <i>w'</i> is a linear word, replace ''w'' with <i>w'</i> in the previous sentence.) For example, let ''w'' be the multimos word 8'''X'''6'''Z''', '''XXZXZXZXXZXZXZ'''. Then the border between the copies of the MOS subword '''XXZXZXZ''' are ''w''[7]''w''[8] and ''w''[14]''w''[1].
+## Interchange a '''Z''' and an '''X''' at some (possibly more than one) of the boundaries of these copies of the mos word ''w''. Here, the boundary of two consecutive copies of ''w'' is the last letter of the first word and the first letter of the second word. (At the ends of the whole multimos word, the boundaries are just the first and last letters of the word.) For example, let ''w'' be the multimos word 8'''X'''6'''Z''', '''XXZXZXZXXZXZXZ'''. Then the border between the copies of the MOS subword '''XXZXZXZ''' are ''w''[7]''w''[8] and ''w''[14]''w''[1] (using one-based numbering).
 ## Replace every other '''X''' with '''Y''' in ''w''.
 === Proof ===
 Proven by Bulgakova, Buzhinsky and Goncharov (2023), "[https://pdf.sciencedirectassets.com/271538/1-s2.0-S0304397522X0039X/1-s2.0-S0304397522006417/am.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEJv%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJHMEUCICIhaHaQFcuBGL19NZOHJFGS7MwzVcZ0xFToxGOkhW6vAiEA06hbPi%2F2O7jyKxdIEbHTvESiHGyC%2F9N4%2FeNLiQc89tQqswUIZBAFGgwwNTkwMDM1NDY4NjUiDIotbpjT6VGvNgDaIiqQBYD4g30jpZUHk5lLFkEbKwIpoDAp8HjE0XulHGFvU2D1xJHVEfBPADJMfpaYreqDiMXRZefNc553rr%2FCY%2FSvP6XSWN6li2Xu3abYhaiV1uUBiMEBu66ZJ%2F%2F2CDRoy%2Bf3uFVkipUvBBmTAtDodFJ6%2BcyEoBkQeIsgJPL%2B9xI2iUX7brAtUr52yKswGXCPTjLhYcrApMV6XvtbUweF63yeYKA8b6oqp4bcARuxP8ubH3m3LRsqkvDYaht8f%2FDZewiq0oyfIWWktRbAiAA5gRySac9vunwVOgx2YrLy1kIkj0ejZF7haArPTKjeUJmnwswCve74TuiVG9pNCjJya0PjJQZa1xZIErf5Q5dKj%2Bps8ITvjp6YHtoCPL2XIv94A2pTjQyws7DdwLkRpBJL6YCeOboX44IsDc%2B7yiSJIWwEkb%2Bkgj93Mb%2BJSJ7AtbzllZAC0NwPMUOxOYPvjHY6gM1NHardi%2BLuT3rKm5yznsV7YUhqNs0jHd2yrfTyJ0Isoa6J%2Bsv5aARbz9CMkusgx4wenGytsdbpHw49GaE%2F23CwooxEv%2BFH9nMpsSzt2axTOIjLReULOV%2F9aZWbuB8Uy26cq3MnWnzVXzhofPwy5Uhi93T8tiRw8dKDXO2tXSjmPUiqk1n41XwHY9z7kZ5J6yIrgkH1ASWa0YVHEW8Cx2sTSkTLwFtgsebpfw0P0AMM%2B%2Bfdcbr714ZvXPmUi77iMhqux%2Feq%2Frw6urUE%2FwFg8PPp%2F8rmHru3VOLjFHjVB6SX0LzyDw1CVwU6%2FRNVUzelLzyHg1X1eUHz0%2FyeQIEdqoHTK8m1ybCuWa88v2IVwfVOZsR1wOpnIIrR4dnjDZ9X5iw%2B9NoQywSM8diF8WHfM59KWUCFMLXp9K0GOrEBlktkq2t3BAB%2B99W%2Bz8Of6btHd4eum8D%2Fgv%2FhzH71opl%2FHeNRcVEJVFU6tFwz7x%2BHySPNMPU0FfD2G8yCy3Qm6EpIQ2R9AZGWteuDCxV6LrPKL4c6BupkmIAkE8HyyqnH4oOc10YZEEvlucl4UUOf7QYQ8J2uwt6sbJh8haepOI6eRPtb%2BLg3xz1M9LP4xG9C2UhONIYd9kkbMZWa%2Fjs4xM%2Fhb8M5JDsl2zQ1KwPkGR5Y&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20240202T190401Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTY4OSCZ3W7%2F20240202%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=d8cdeefedc146cd9b6ae50dfb047bd170aff67c73b4cc6bad4794b0b9634223e&hash=fcd7ef6d1f8e476a88555788d397d1ff5c5720a68482ed4a2f8ec051685c7849&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=S0304397522006417&tid=pdf-bb78adbe-12da-476f-ac55-dd339ae07c1a&sid=758d15492e7c4647a5993de3aaa4e236e5b1gxrqa&type=client On balanced and abelian properties of circular words over a ternary alphabet]". <!--The claim "Balanced primitive ternary scales of type (2) have a generator sequence of period 2" is proven using Theorem 2 and 4 on this page, since such scales are odd GO scales.-->
+Note: The xen term "brightest mos word" is equivalent to "Christoffel word" in the paper, and similarly "brightest multimos word" is equivalent to "powers of a Christoffel word". Also see [[Glossary for combinatorics on words]] for more equivalents between xen community terms and standard academic terminology.
 [[Category:Math]]