Mathematics of MOS - Revision history

ArrowHead294: De-escape as <nowiki/> works

2026-01-26T00:49:54Z

De-escape as <nowiki/> works

Inthar: /* Maximally even scales are MOS */

2026-01-25T16:41:26Z

Maximally even scales are MOS

← Older revision		Revision as of 16:41, 25 January 2026
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	=== Maximally even scales are MOS ===		=== Maximally even scales are MOS ===
	Let {{nowrap\|1 > ''m'' > ~~''n''~~\|gcd(''m'', ''n'') {{=}} d}}, and ME(''m'', ''n'') be as in [[Maximal evenness]]. An arbitrary ''k''-step in ME(''m'', ''n'') has size {{nowrap\|''u'' {{=}} &lfloor;(''i'' + ''k'') {{frac\|''n''\|''m''}}&rfloor; − &lfloor;{{frac\|''in''\|''m''}}&rfloor;}}, and		Let {{nowrap\|''n'' > ''m'' > 1\|gcd(''m'', ''n'') {{=}} d}}, and ME(''m'', ''n'') be as in [[Maximal evenness]]. An arbitrary ''k''-step in ME(''m'', ''n'') has size {{nowrap\|''u'' {{=}} &lfloor;(''i'' + ''k'') {{frac\|''n''\|''m''}}&rfloor; − &lfloor;{{frac\|''in''\|''m''}}&rfloor;}}, and

	<math>\lfloor in/m \rfloor + \lfloor kn/m\rfloor - \lfloor in/m\rfloor = \lfloor kn/m\rfloor \leq u		<math>\lfloor in/m \rfloor + \lfloor kn/m\rfloor - \lfloor in/m\rfloor = \lfloor kn/m\rfloor \leq u

ArrowHead294: /* Classification via the question-mark function */

2026-01-21T18:01:41Z

Classification via the question-mark function

← Older revision		Revision as of 18:01, 21 January 2026
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	Yet another way of classifying MOS is via {{w\|Minkowski's question-mark function}}. Here ?(''x''), {{nowrap\|? : ℝ → ℝ}} is a continuously increasing function which has some peculiar properties, one being that it sends rational numbers to {{w\|dyadic rational}}s. Hence if ''q'' is a rational number {{nowrap\|0 < ''q'' < 1}} in use in the mediant system of classifying MOS scales, {{nowrap\|''r'' {{=}} ?(''q'')}} = {{sfrac\|''A''\|2<sup>''n''</sup>}} will be a dyadic rational number which can also be used. Note that the ? function is invertible, and it and its inverse function, the Box function, have code given for them in the algorithms section at the bottom of the article.		Yet another way of classifying MOS is via {{w\|Minkowski's question-mark function}}. Here ?(''x''), {{nowrap\|? : ℝ → ℝ}} is a continuously increasing function which has some peculiar properties, one being that it sends rational numbers to {{w\|dyadic rational}}s. Hence if ''q'' is a rational number {{nowrap\|0 < ''q'' < 1}} in use in the mediant system of classifying MOS scales, {{nowrap\|''r'' {{=}} ?(''q'')}} = {{sfrac\|''A''\|2<sup>''n''</sup>}} will be a dyadic rational number which can also be used. Note that the ? function is invertible, and it and its inverse function, the Box function, have code given for them in the algorithms section at the bottom of the article.

	The integer ''n'' in the denominator of ''r'' (with ''A'' assumed to be odd) is the order (or {{nowrap\|''n'' + 1}} is, according to some sources) of ''q'' in the {{w\|Stern–Brocot tree}}. The two neighboring numbers of order {{nowrap\|''n'' + 1}}, which define the range of propriety, can also be expressed in terms of the ? and Box functions as {{nowrap\|Box(''r'' ± 2<sup>−''n'' − 1</sup>)}}. If ''r'' represents a MOS, the range of possible values for a generator of the MOS will be {{nowrap\|Box(''r'') < ''g''}} {{nowrap\|< Box(''r'' + 2<sup>−''n''</sup>)}}, and the proper generators will be {{nowrap\|Box(''r'') < ''g'' < Box(''r'' + 2<sup>−''n'' − 1</sup>)}}. So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that {{nowrap\|{{frac\|2\|21}} < ''g'' < {{frac\|1\|10}}}}, and will be proper if {{nowrap\|{{frac\|2\|21}} < ''g''}} < {{frac\|3\|31}}. Hence 7/72, a generator for miracle temperament, will define a MOS but it will not be proper since {{nowrap\|{{frac\|7\|72}} > {{frac\|3\|31}}}} {{nowrap\|{{=}} Box({{frac\|3\|2048}} + {{frac\|1\|4096}})}}.		The integer ''n'' in the denominator of ''r'' (with ''A'' assumed to be odd) is the order (or {{nowrap\|''n'' + 1}} is, according to some sources) of ''q'' in the {{w\|Stern–Brocot tree}}. The two neighboring numbers of order {{nowrap\|''n'' + 1}}, which define the range of propriety, can also be expressed in terms of the ? and Box functions as {{nowrap\|Box(''r'' ± 2<sup>−''n'' − 1</sup>)}}. If ''r'' represents a MOS, the range of possible values for a generator of the MOS will be {{nowrap\|Box(''r'') < ''g''}} {{nowrap\|< Box(''r'' + 2<sup>−''n''</sup>)}}, and the proper generators will be {{nowrap\|Box(''r'') < ''g''}} {{nowrap\|< Box(''r'' + 2<sup>−''n'' − 1</sup>)}}. So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that {{nowrap\|{{frac\|2\|21}} < ''g''}} {{nowrap\|< {{frac\|1\|10}}}}, and will be proper if {{nowrap\|{{frac\|2\|21}} < ''g''}} < {{frac\|3\|31}}. Hence 7/72, a generator for miracle temperament, will define a MOS but it will not be proper since {{nowrap\|{{frac\|7\|72}} > {{frac\|3\|31}}}} {{nowrap\|{{=}} Box({{frac\|3\|2048}} + {{frac\|1\|4096}})}}.

	== MOS in equal tunings ==		== MOS in equal tunings ==

ArrowHead294 at 12:41, 6 December 2025

2025-12-06T12:41:00Z

← Older revision		Revision as of 12:41, 6 December 2025
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	Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base ''P'' when ''P'' is the period.		Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base ''P'' when ''P'' is the period.

	Suppose the fractions {{frac\|''a''\|''b''}} and {{frac\|''c''\|''d''}} are a [[Wikipedia:Farey sequence#Farey neighbours\|Farey pair]], meaning that {{nowrap\|{{frac\|''a''\|''b''}} < {{frac\|''c''\|''d''}}}} and {{nowrap\|''bc'' − ''ad'' {{=}} 1}}. If {{nowrap\|''g'' {{=}} (1 − ''t'') {{frac\|''a''\|''b''}} + (''t'') {{frac\|''c''\|''d''}}}} for {{nowrap\|0 ≤ ''t'' ≤ 1}}, then when {{nowrap\|''t'' {{=}} 0}}, the scale generated by ''g'' will consist of an equal division of 1 (representing P) into steps of size {{frac\|1\|''b''}}, and when {{nowrap\|''t'' {{=}} 1}} into steps of size {{frac\|1\|''d''}}. In between, when {{nowrap\|''t'' {{=}} {{sfrac\|''b''\|''b'' + ''d''}}}}, we obtain a generator equal to the ~~[[Wikipedia:Mediant_%28mathematics%29~~\|mediant]] {{nowrap\|''m'' {{=}} {{sfrac\|''a'' + ''c''\|''b'' + ''d''}}}} and which will divide the period into {{nowrap\|''b'' + ''d''}} equal steps. For all other values {{nowrap\|{{frac\|''a''\|''b''}} < ''g'' < {{frac\|''c''\|''d''}}}} we obtain two different sizes of steps, the small steps ''s'', and the large steps ''L'', with the total number of steps {{nowrap\|''b'' + ''d''}}, and these scales are the MOS associated to the Farey pair. When ''g'' is between {{frac\|''a''\|''b''}} and ''m'', there will be ''b'' large steps and ''d'' small steps, and when it is between ''m'' and {{frac\|''c''\|''d''}}, ''d'' large steps and ''b'' small ones.		Suppose the fractions {{frac\|''a''\|''b''}} and {{frac\|''c''\|''d''}} are a [[Wikipedia:Farey sequence#Farey neighbours\|Farey pair]], meaning that {{nowrap\|{{frac\|''a''\|''b''}} < {{frac\|''c''\|''d''}}}} and {{nowrap\|''bc'' − ''ad'' {{=}} 1}}. If {{nowrap\|''g'' {{=}} (1 − ''t'') {{frac\|''a''\|''b''}} + (''t'') {{frac\|''c''\|''d''}}}} for {{nowrap\|0 ≤ ''t'' ≤ 1}}, then when {{nowrap\|''t'' {{=}} 0}}, the scale generated by ''g'' will consist of an equal division of 1 (representing P) into steps of size {{frac\|1\|''b''}}, and when {{nowrap\|''t'' {{=}} 1}} into steps of size {{frac\|1\|''d''}}. In between, when {{nowrap\|''t'' {{=}} {{sfrac\|''b''\|''b'' + ''d''}}}}, we obtain a generator equal to the {{w\|mediant (mathematics)\|mediant}} {{nowrap\|''m'' {{=}} {{sfrac\|''a'' + ''c''\|''b'' + ''d''}}}} and which will divide the period into {{nowrap\|''b'' + ''d''}} equal steps. For all other values {{nowrap\|{{frac\|''a''\|''b''}} < ''g'' < {{frac\|''c''\|''d''}}}} we obtain two different sizes of steps, the small steps ''s'', and the large steps ''L'', with the total number of steps {{nowrap\|''b'' + ''d''}}, and these scales are the MOS associated to the Farey pair. When ''g'' is between {{frac\|''a''\|''b''}} and ''m'', there will be ''b'' large steps and ''d'' small steps, and when it is between ''m'' and {{frac\|''c''\|''d''}}, ''d'' large steps and ''b'' small ones.

	While all the scales constructed by generators ''g'' with {{nowrap\|{{frac\|''a''\|''b''}} < ''g'' < {{frac\|''c''\|''d''}}}} with the exception of the mediant which gives an equal tuning are MOS, not all the scales are [[Wikipedia:Rothenberg_propriety\|proper]] in the sense of Rothenberg. The ''range of propriety'' for MOS is {{nowrap\|{{sfrac\|2''a'' + ''c''\|2''b'' + ''d''}} ≤ ''g''}} ≤ {{sfrac\|''a'' + 2''c''\|''b'' + 2''d''}}, where MOS coming from a Farey pair ({{frac\|''a''\|''b''}}, {{frac\|''c''\|''d''}}) are proper when in this range, and improper (unless the MOS has only one small step) when out of it. If {{nowrap\|{{sfrac\|2''a'' + ''c''\|2''b'' + ''d''}} < ''g''}} < {{sfrac\|''a'' + 2''c''\|''b'' + 2''d''}}, then the scales are strictly proper. Hence, the diatonic scale in 12et, with generator 7/12, is proper but not strictly proper since starting from the pair ({{frac\|1\|2}}, {{frac\|3\|5}}) we find the range of propriety for these seven-note MOS to be [{{frac\|5\|9}}, {{frac\|7\|12}}].		While all the scales constructed by generators ''g'' with {{nowrap\|{{frac\|''a''\|''b''}} < ''g'' < {{frac\|''c''\|''d''}}}} with the exception of the mediant which gives an equal tuning are MOS, not all the scales are [[Wikipedia:Rothenberg_propriety\|proper]] in the sense of Rothenberg. The ''range of propriety'' for MOS is {{nowrap\|{{sfrac\|2''a'' + ''c''\|2''b'' + ''d''}} ≤ ''g''}} ≤ {{sfrac\|''a'' + 2''c''\|''b'' + 2''d''}}, where MOS coming from a Farey pair ({{frac\|''a''\|''b''}}, {{frac\|''c''\|''d''}}) are proper when in this range, and improper (unless the MOS has only one small step) when out of it. If {{nowrap\|{{sfrac\|2''a'' + ''c''\|2''b'' + ''d''}} < ''g''}} < {{sfrac\|''a'' + 2''c''\|''b'' + 2''d''}}, then the scales are strictly proper. Hence, the diatonic scale in 12et, with generator 7/12, is proper but not strictly proper since starting from the pair ({{frac\|1\|2}}, {{frac\|3\|5}}) we find the range of propriety for these seven-note MOS to be [{{frac\|5\|9}}, {{frac\|7\|12}}].

ArrowHead294 at 15:15, 5 December 2025

2025-12-05T15:15:07Z

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	{{nowrap\|s {{=}} ''eg'' + ''fp''}}		{{nowrap\|s {{=}} ''eg'' + ''fp''}}

	for appropriate integers ''c, d, e, f'', where {{nowrap\|{{~~!}}~~''c''~~{{!~~}}, {{~~!}}~~''e''~~{{!~~}} < ''n''}}.		for appropriate integers ''c, d, e, f'', where {{nowrap\|{{abs\|''c''}}, {{abs\|''e''}} < ''n''}}.

	Now we assume that ''g'' and ''p'' are linearly independent. By assumption {{nowrap\|''a''L + ''b''s {{=}} (''ac'' + ''be'')''g'' + (''ad'' + ''bf'')''p''}} = ''p''. Since {{nowrap\|''a''L + ''b''s}} occurs on the "brightest" mode, from generatedness we have {{nowrap\|''ac'' + ''be'' ∈ {{(}}0, …, ''n'' − 1{{)}}}}. Hence we must have {{nowrap\|''ac'' + ''be'' {{=}} 0}}, and thus {{nowrap\|''c'' {{=}} ±''b''}} and {{nowrap\|''e'' {{=}} ∓''a''}}, from the assumption that ''a'' and ''b'' are coprime.		Now we assume that ''g'' and ''p'' are linearly independent. By assumption {{nowrap\|''a''L + ''b''s {{=}} (''ac'' + ''be'')''g'' + (''ad'' + ''bf'')''p''}} = ''p''. Since {{nowrap\|''a''L + ''b''s}} occurs on the "brightest" mode, from generatedness we have {{nowrap\|''ac'' + ''be'' ∈ {{(}}0, …, ''n'' − 1{{)}}}}. Hence we must have {{nowrap\|''ac'' + ''be'' {{=}} 0}}, and thus {{nowrap\|''c'' {{=}} ±''b''}} and {{nowrap\|''e'' {{=}} ∓''a''}}, from the assumption that ''a'' and ''b'' are coprime.

ArrowHead294 at 22:38, 4 December 2025

2025-12-04T22:38:14Z

← Older revision		Revision as of 22:38, 4 December 2025
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	# Binary and [[distributionally even]]		# Binary and [[distributionally even]]
	# Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or {{nowrap\|L − s {{=}} c}})		# Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or {{nowrap\|L − s {{=}} c}})
	# Mode of a Christoffel word. (If gcd(''p'',~~ ~~''q'') = 1, a ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line {{nowrap\|''y'' {{=}} (''p''/''q'')''x''}} that stays as close to the line as possible without crossing it.)		# Mode of a Christoffel word. (If {{nowrap\|gcd(''p'', ''q'') {{=}} 1}}, a ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line {{nowrap\|''y'' {{=}} (''p''/''q'')''x''}} that stays as close to the line as possible without crossing it.)

	While each characterization has a generalization to scale structures with more step sizes, the generalizations are no longer equivalent:		While each characterization has a generalization to scale structures with more step sizes, the generalizations are no longer equivalent:
	# [[Maximum variety]] n		# [[Maximum variety]] ''n''
	# [[Generator-offset property]], [[generator sequence]]		# [[Generator-offset property]], [[generator sequence]]
	# [[Distributional evenness]]		# [[Distributional evenness]]

Inthar: /* Characterizations */

2025-12-04T21:10:08Z

Characterizations

← Older revision		Revision as of 21:10, 4 December 2025
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	# Binary and [[distributionally even]]		# Binary and [[distributionally even]]
	# Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or {{nowrap\|L − s {{=}} c}})		# Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or {{nowrap\|L − s {{=}} c}})
	# Mode of a Christoffel word. (If gcd(''p'', ''q'') = 1, a ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line {{nowrap\|''y'' {{=}} (''p''/''q'')''x''}} that stays as close to the line without crossing it.)		# Mode of a Christoffel word. (If gcd(''p'', ''q'') = 1, a ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line {{nowrap\|''y'' {{=}} (''p''/''q'')''x''}} that stays as close to the line as possible without crossing it.)

	While each characterization has a generalization to scale structures with more step sizes, the generalizations are no longer equivalent:		While each characterization has a generalization to scale structures with more step sizes, the generalizations are no longer equivalent:

Inthar: /* Characterizations */

2025-12-04T21:09:39Z

Characterizations

← Older revision		Revision as of 21:09, 4 December 2025
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	# Binary and [[distributionally even]]		# Binary and [[distributionally even]]
	# Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or {{nowrap\|L − s {{=}} c}})		# Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or {{nowrap\|L − s {{=}} c}})
	# Mode of a Christoffel word. (If gcd(p, q) = 1, a ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line {{nowrap\|''y'' {{=}} ''p''/''q''~~ ~~''x''}} that stays as close to the line without crossing it.)		# Mode of a Christoffel word. (If gcd(''p'', ''q'') = 1, a ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line {{nowrap\|''y'' {{=}} (''p''/''q'')''x''}} that stays as close to the line without crossing it.)

	While each characterization has a generalization to scale structures with more step sizes, the generalizations are no longer equivalent:		While each characterization has a generalization to scale structures with more step sizes, the generalizations are no longer equivalent:

Inthar: /* Characterizations */

2025-12-04T21:08:52Z

Characterizations

← Older revision		Revision as of 21:08, 4 December 2025
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	# Binary and [[distributionally even]]		# Binary and [[distributionally even]]
	# Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or {{nowrap\|L − s {{=}} c}})		# Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or {{nowrap\|L − s {{=}} c}})
	# Mode of a Christoffel word. (A ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line {{nowrap\|''y'' {{=}} ''p''/''q'' ''x''}} that stays as close to the line without crossing it.)		# Mode of a Christoffel word. (If gcd(p, q) = 1, a ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line {{nowrap\|''y'' {{=}} ''p''/''q'' ''x''}} that stays as close to the line without crossing it.)

	While each characterization has a generalization to scale structures with more step sizes, the generalizations are no longer equivalent:		While each characterization has a generalization to scale structures with more step sizes, the generalizations are no longer equivalent:

ArrowHead294: /* Binary generated scales with independent period and generator are MOS */ De-escape

2025-12-04T00:18:28Z

Binary generated scales with independent period and generator are MOS: De-escape

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	Suppose that such a scale ''S'' (with {{nowrap\|''n'' ≥ 2}} notes) has ''a'' L steps and ''b'' s steps per period ''p'', and has generator ''g''. Since ''S'' is generated, the interval sizes modulo ''p'' that occur in ''S'' are:		Suppose that such a scale ''S'' (with {{nowrap\|''n'' ≥ 2}} notes) has ''a'' L steps and ''b'' s steps per period ''p'', and has generator ''g''. Since ''S'' is generated, the interval sizes modulo ''p'' that occur in ''S'' are:

	{~~{(}}~~{{nowrap\|(−''n'' + 1)''g''}}, …, −''g'', 0, ''g'', …, {{nowrap\|(''n'' − 1)''g''}}~~{{)}~~},		<nowiki>{</nowiki>{{nowrap\|(−''n'' + 1)''g''}}, …, −''g'', 0, ''g'', …, {{nowrap\|(''n'' − 1)''g''}}<nowiki>}</nowiki>,

	and all sizes {0, ''g'', …, {{nowrap\|(''n'' − 1)''g''}}~~{{)}~~} are distinct.		and all sizes {0, ''g'', …, {{nowrap\|(''n'' − 1)''g''}}<nowiki>}</nowiki> are distinct.

	We thus have:		We thus have:

← Older revision		Revision as of 15:15, 5 December 2025
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	{{nowrap\|s {{=}} ''eg'' + ''fp''}}		{{nowrap\|s {{=}} ''eg'' + ''fp''}}

	for appropriate integers ''c, d, e, f'', where {{nowrap\|{{~~!}}~~''c''~~{{!~~}}, {{~~!}}~~''e''~~{{!~~}} < ''n''}}.		for appropriate integers ''c, d, e, f'', where {{nowrap\|{{abs\|''c''}}, {{abs\|''e''}} < ''n''}}.

	Now we assume that ''g'' and ''p'' are linearly independent. By assumption {{nowrap\|''a''L + ''b''s {{=}} (''ac'' + ''be'')''g'' + (''ad'' + ''bf'')''p''}} = ''p''. Since {{nowrap\|''a''L + ''b''s}} occurs on the "brightest" mode, from generatedness we have {{nowrap\|''ac'' + ''be'' ∈ {{(}}0, …, ''n'' − 1{{)}}}}. Hence we must have {{nowrap\|''ac'' + ''be'' {{=}} 0}}, and thus {{nowrap\|''c'' {{=}} ±''b''}} and {{nowrap\|''e'' {{=}} ∓''a''}}, from the assumption that ''a'' and ''b'' are coprime.		Now we assume that ''g'' and ''p'' are linearly independent. By assumption {{nowrap\|''a''L + ''b''s {{=}} (''ac'' + ''be'')''g'' + (''ad'' + ''bf'')''p''}} = ''p''. Since {{nowrap\|''a''L + ''b''s}} occurs on the "brightest" mode, from generatedness we have {{nowrap\|''ac'' + ''be'' ∈ {{(}}0, …, ''n'' − 1{{)}}}}. Hence we must have {{nowrap\|''ac'' + ''be'' {{=}} 0}}, and thus {{nowrap\|''c'' {{=}} ±''b''}} and {{nowrap\|''e'' {{=}} ∓''a''}}, from the assumption that ''a'' and ''b'' are coprime.