Ternary scale theorems: Difference between revisions

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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'''}."

=== Classification of ternary balanced scales ===

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

# A primitive [[balanced]] MV3 scale is either

## equivalent to XYXZXYX

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# Balanced primitive ternary scales of type (2) have a generator sequence of period 2.

=== Classification of MV3 scales ===

=== 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:

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

Line 237:

=== 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]" (and ~~Theorem~~ 4). ~~{{qed}}~~

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.

[[Category:Math]]

@@ Line 221: / Line 221: @@
 == 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'''}."
-=== Classification of ternary balanced scales ===
+=== Theorem 5.1 (Classification of ternary balanced scales) ===
 # A primitive [[balanced]] MV3 scale is either
 ## equivalent to XYXZXYX
@@ Line 230: / Line 230: @@
 # Balanced primitive ternary scales of type (2) have a generator sequence of period 2.
-=== Classification of MV3 scales ===
+=== 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:
 ## Start with a multimos word <math>\mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})^k</math> such that ''a'' is even.
@@ Line 237: / Line 237: @@
 === 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]" (and Theorem 4). {{qed}}
+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.
 [[Category:Math]]