Mathematics of MOS: Difference between revisions

Line 5:

A MOS scale consists of:

# A period "P" (of any size but most commonly the octave or a 1/N fraction of an octave)

# A period ''P'' (of any size but most commonly the octave or a 1/N fraction of an octave)

# A generator "g" (of any size, for example 700 cents in 12edo) which is added repeatedly to make a chain of scale steps, starting from the unison or 0 cents scale step, and then reducing to within the period

# A generator ''g'' (of any size, for example 700 cents in 12edo) which is added repeatedly to make a chain of scale steps, starting from the unison or 0 cents scale step, and then reducing to within the period

# No more than two sizes of scale steps (Large and small, often written "L" and "s")

# Where ''each'' number of scale steps, or generic interval, within the scale occurs in no more than two different sizes, and in exactly two if the interval is not a multiple of the period except in such cases as an ET.

# The unison or starting point of the scale is then allowed to be transferred to any scale degree--all the modes of an MOS are legal.

Condition 4 is ~~[[Wikipedia:Myhill's property~~|Myhill's property]] where, as a [[periodic scale]], the scale has every generic interval aside from the initial unison interval and intervals some number of periods from it having exactly two specific intervals. Another characterization of when a generated scale is a MOS is that the number of scale steps is the denominator of a [[Wikipedia:Continued_fraction|convergent or semiconvergent]] of the ratio g/P of the generator and the period.

Condition 4 is {{w|Myhill's property}}, where, as a [[periodic scale]], the scale has every generic interval aside from the initial unison interval and intervals some number of periods from it having exactly two specific intervals. Another characterization of when a generated scale is a MOS is that the number of scale steps is the denominator of a [[Wikipedia:Continued_fraction|convergent or semiconvergent]] of the ratio ''g''/''P'' of the generator and the period.

These conditions entail that the generated scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and fail to be a MOS, meaning that not all non-unison classes have only two specific intervals.

Line 31:

== Properties ==

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.

Line 39:

Given a generator ''g'', we can find an MOS for ''g'' with period 1 by means of the [[Wikipedia:Continued_fraction#Semiconvergents|semiconvergents]] to ''g''. A pair of successive semiconvergents have the property that they define a Farey pair, and when ''g'' is contained in the pair (that is, {{nowrap|{{frac|''a''|''b''}} < ''g'' < {{frac|''c''|''d''}}}}), we have defined a MOS for ''g'' with {{nowrap|''b'' + ''d''}} as the number of notes in the MOS, with ''b'' notes of one size and ''d'' of the other.

For example, suppose we want an MOS for 1/4-comma meantone. The generator will then be {{sfrac|log<sub>2</sub>(5)|4}}, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/~~112...~~ If we settle on 31 as a good size for our MOS, we see 18/31 is the mediant between the Farey pair 11/19 and 7/12, for which the range of strict propriety is {{nowrap|29/50 < ''x'' < 25/43}}. Since ''g'' is in that range and not equal to 18/31, we will get a strictly proper MOS.

For example, suppose we want an MOS for 1/4-comma meantone. The generator will then be {{sfrac|log<sub>2</sub>(5)|4}}, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112… If we settle on 31 as a good size for our MOS, we see 18/31 is the mediant between the Farey pair 11/19 and 7/12, for which the range of strict propriety is {{nowrap|29/50 < ''x'' < 25/43}}. Since ''g'' is in that range and not equal to 18/31, we will get a strictly proper MOS.

== Visualizing MOS: Generator chains, pitch spaces, and hierarchies ==

Line 275:

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''}}{{)}},

{{(}}{{nowrap|(−''n'' + 1)''g''}}, …, −''g'', 0, ''g'', …, {{nowrap|(''n'' − 1)''g''}}{{)}},

and all sizes {0, ''g'', ~~...~~, {{nowrap|(''n'' − 1)''g''}}{{)}} are distinct.

and all sizes {0, ''g'', …, {{nowrap|(''n'' − 1)''g''}}{{)}} are distinct.

We thus have:

Line 287:

for appropriate integers ''c, d, e, f'', where {{nowrap|{{!}}''c''{{!}}, {{!}}''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.

In fact, {L, s} is another valid basis for the abelian group with basis {''p'', ''g''}, since by binarity we have {{nowrap|''p, g'' ∈ span(L, s)}}. Assume {{nowrap|''c'' {{=}} ''b''}} and {{nowrap|''e'' {{=}} −''a''}} (this corresponds to assuming that ''g'' is the "bright" generator). Let {{nowrap|χ {{=}} L − s}} > 0; then χ is ''p''-equivalent to +''ng''. Now by generatedness and binarity, any interval class that has at least two sizes must have sizes separated by ''ng'' (the separation corresponding to changing an L step to an s step). Since ''g'' and ''p'' are linearly independent, for each {{nowrap|''j'' ∈ {{(}}1, ~~...~~, ''n'' − 1{{)}}}} there exists at most one {{nowrap|''k'' {{=}} ''k''(''j'') ∈ {{(}}1, ~~...~~, ''n'' − 1{{)}}}} such that ''jg'' is ''p''-equivalent to one size of ''k''-step. Hence if the class of ''k''-steps has ''at least'' two sizes, the sizes must be ''j''(''k'')''g'' and {{nowrap|(''j''(''k'') − ''n'')''g''}}; any other size must leave the range {{nowrap|(1 − ''n'')''g''}}, ~~...~~, 0, ~~...~~, {{nowrap|(''n'' − 1)''g''}}. Thus the class of ''k''-steps has at most two sizes for {{nowrap|1 ≤ ''k'' ≤ ''n'' − 1}}. Each non-''p''-equivalent class must have ''exactly'' two sizes, since the inverse of the ''k''-step that is equivalent to ''jg'' is an {{nowrap|(''n'' − ''k'')}}-step equivalent to −''jg'', which by linear independence must be distinct from an {{nowrap|(''n'' − ''k'')}}-step equivalent to a positive number of ''g'' generators. (Note that the latter {{nowrap|(''n'' − ''k'')}}-step does occur in the "brightest" mode of ''S'', i.e. the mode with the most ''g'' generators stacked ''up'' rather than ''down'' from the tonic.) This completes the proof.

In fact, {L, s} is another valid basis for the abelian group with basis {''p'', ''g''}, since by binarity we have {{nowrap|''p, g'' ∈ span(L, s)}}. Assume {{nowrap|''c'' {{=}} ''b''}} and {{nowrap|''e'' {{=}} −''a''}} (this corresponds to assuming that ''g'' is the "bright" generator). Let {{nowrap|χ {{=}} L − s}} > 0; then χ is ''p''-equivalent to +''ng''. Now by generatedness and binarity, any interval class that has at least two sizes must have sizes separated by ''ng'' (the separation corresponding to changing an L step to an s step). Since ''g'' and ''p'' are linearly independent, for each {{nowrap|''j'' ∈ {{(}}1, …, ''n'' − 1{{)}}}} there exists at most one {{nowrap|''k'' {{=}} ''k''(''j'') ∈ {{(}}1, …, ''n'' − 1{{)}}}} such that ''jg'' is ''p''-equivalent to one size of ''k''-step. Hence if the class of ''k''-steps has ''at least'' two sizes, the sizes must be ''j''(''k'')''g'' and {{nowrap|(''j''(''k'') − ''n'')''g''}}; any other size must leave the range {{nowrap|(1 − ''n'')''g''}}, …, 0, …, {{nowrap|(''n'' − 1)''g''}}. Thus the class of ''k''-steps has at most two sizes for {{nowrap|1 ≤ ''k'' ≤ ''n'' − 1}}. Each non-''p''-equivalent class must have ''exactly'' two sizes, since the inverse of the ''k''-step that is equivalent to ''jg'' is an {{nowrap|(''n'' − ''k'')}}-step equivalent to −''jg'', which by linear independence must be distinct from an {{nowrap|(''n'' − ''k'')}}-step equivalent to a positive number of ''g'' generators. (Note that the latter {{nowrap|(''n'' − ''k'')}}-step does occur in the "brightest" mode of ''S'', i.e. the mode with the most ''g'' generators stacked ''up'' rather than ''down'' from the tonic.) This completes the proof.

The previous argument cannot be used for the non-linearly independent case. This is because not all binary generated scales with rational step ratios are limit points of binary generated scales with irrational step ratios. A counterexample is the 13-note scale ssLsLssLsLss ({{nowrap|s {{=}} 1\17|L {{=}} 2\17}}), which is obtained from LsLsLssLsLss (5L 7s) by stacking one more 10\17 generator.

@@ Line 5: / Line 5: @@
 A MOS scale consists of:
-# A period "P" (of any size but most commonly the octave or a 1/N fraction of an octave)
+# A period ''P'' (of any size but most commonly the octave or a 1/N fraction of an octave)
-# A generator "g" (of any size, for example 700 cents in 12edo) which is added repeatedly to make a chain of scale steps, starting from the unison or 0 cents scale step, and then reducing to within the period
+# A generator ''g'' (of any size, for example 700 cents in 12edo) which is added repeatedly to make a chain of scale steps, starting from the unison or 0 cents scale step, and then reducing to within the period
 # No more than two sizes of scale steps (Large and small, often written "L" and "s")
 # Where ''each'' number of scale steps, or generic interval, within the scale occurs in no more than two different sizes, and in exactly two if the interval is not a multiple of the period except in such cases as an ET.
 # The unison or starting point of the scale is then allowed to be transferred to any scale degree--all the modes of an MOS are legal.
-Condition 4 is [[Wikipedia:Myhill's property|Myhill's property]] where, as a [[periodic scale]], the scale has every generic interval aside from the initial unison interval and intervals some number of periods from it having exactly two specific intervals. Another characterization of when a generated scale is a MOS is that the number of scale steps is the denominator of a [[Wikipedia:Continued_fraction|convergent or semiconvergent]] of the ratio g/P of the generator and the period.
+Condition 4 is {{w|Myhill's property}}, where, as a [[periodic scale]], the scale has every generic interval aside from the initial unison interval and intervals some number of periods from it having exactly two specific intervals. Another characterization of when a generated scale is a MOS is that the number of scale steps is the denominator of a [[Wikipedia:Continued_fraction|convergent or semiconvergent]] of the ratio ''g''/''P'' of the generator and the period.
 These conditions entail that the generated scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and fail to be a MOS, meaning that not all non-unison classes have only two specific intervals.
@@ Line 31: / Line 31: @@
 == Properties ==
-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''}} &lt; {{frac|''c''|''d''}}}} and {{nowrap|''bc'' &minus; ''ad'' {{=}} 1}}. If {{nowrap|''g'' {{=}} (1 &minus; ''t''){{frac|''a''|''b''}} + (''t''){{frac|''c''|''d''}}}} for {{nowrap|0 &le; ''t'' &le; 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''}} &lt; ''g'' &lt; {{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.
@@ Line 39: / Line 39: @@
 Given a generator ''g'', we can find an MOS for ''g'' with period 1 by means of the [[Wikipedia:Continued_fraction#Semiconvergents|semiconvergents]] to ''g''. A pair of successive semiconvergents have the property that they define a Farey pair, and when ''g'' is contained in the pair (that is, {{nowrap|{{frac|''a''|''b''}} &lt; ''g'' &lt; {{frac|''c''|''d''}}}}), we have defined a MOS for ''g'' with {{nowrap|''b'' + ''d''}} as the number of notes in the MOS, with ''b'' notes of one size and ''d'' of the other.
-For example, suppose we want an MOS for 1/4-comma meantone. The generator will then be {{sfrac|log<sub>2</sub>(5)|4}}, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see 18/31 is the mediant between the Farey pair 11/19 and 7/12, for which the range of strict propriety is {{nowrap|29/50 &lt; ''x'' &lt; 25/43}}. Since ''g'' is in that range and not equal to 18/31, we will get a strictly proper MOS.
+For example, suppose we want an MOS for 1/4-comma meantone. The generator will then be {{sfrac|log<sub>2</sub>(5)|4}}, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112… If we settle on 31 as a good size for our MOS, we see 18/31 is the mediant between the Farey pair 11/19 and 7/12, for which the range of strict propriety is {{nowrap|29/50 &lt; ''x'' &lt; 25/43}}. Since ''g'' is in that range and not equal to 18/31, we will get a strictly proper MOS.
 == Visualizing MOS: Generator chains, pitch spaces, and hierarchies ==
@@ Line 275: / Line 275: @@
 Suppose that such a scale ''S'' (with {{nowrap|''n'' &ge; 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|(&minus;''n'' + 1)''g''}}, ..., &minus;''g'', 0, ''g'', ..., {{nowrap|(''n'' &minus; 1)''g''}}{{)}},
+{{(}}{{nowrap|(&minus;''n'' + 1)''g''}}, …, &minus;''g'', 0, ''g'', …, {{nowrap|(''n'' &minus; 1)''g''}}{{)}},
-and all sizes {0, ''g'', ..., {{nowrap|(''n'' &minus; 1)''g''}}{{)}} are distinct.
+and all sizes {0, ''g'', …, {{nowrap|(''n'' &minus; 1)''g''}}{{)}} are distinct.
 We thus have:
@@ Line 287: / Line 287: @@
 for appropriate integers ''c, d, e, f'', where {{nowrap|{{!}}''c''{{!}}, {{!}}''e''{{!}} &lt; ''n''}}.
-Now we assume that ''g'' and ''p'' are linearly independent. By assumption {{nowrap|''a''L + ''b''s {{=}} (''ac'' + ''be'')''g'' + (''ad'' + ''bf'')''p''}} =&nbsp;''p''. Since {{nowrap|''a''L + ''b''s}} occurs on the "brightest" mode, from generatedness we have {{nowrap|''ac'' + ''be'' ∈ {{(}}0, ..., ''n'' &minus; 1{{)}}}}. Hence we must have {{nowrap|''ac'' + ''be'' {{=}} 0}}, and thus {{nowrap|''c'' {{=}} &#177;''b''}} and {{nowrap|''e'' {{=}} &#x2213;''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''}} =&nbsp;''p''. Since {{nowrap|''a''L + ''b''s}} occurs on the "brightest" mode, from generatedness we have {{nowrap|''ac'' + ''be'' ∈ {{(}}0, …, ''n'' &minus; 1{{)}}}}. Hence we must have {{nowrap|''ac'' + ''be'' {{=}} 0}}, and thus {{nowrap|''c'' {{=}} &#177;''b''}} and {{nowrap|''e'' {{=}} &#x2213;''a''}}, from the assumption that ''a'' and ''b'' are coprime.
-In fact, {L,&nbsp;s} is another valid basis for the abelian group with basis {''p'',&nbsp;''g''}, since by binarity we have {{nowrap|''p, g'' ∈ span(L, s)}}. Assume {{nowrap|''c'' {{=}} ''b''}} and {{nowrap|''e'' {{=}} &minus;''a''}} (this corresponds to assuming that ''g'' is the "bright" generator). Let {{nowrap|&chi; {{=}} L &minus; s}} &gt;&nbsp;0; then &chi; is ''p''-equivalent to +''ng''. Now by generatedness and binarity, any interval class that has at least two sizes must have sizes separated by ''ng'' (the separation corresponding to changing an L step to an s step). Since ''g'' and ''p'' are linearly independent, for each {{nowrap|''j'' ∈ {{(}}1, ..., ''n'' &minus; 1{{)}}}} there exists at most one {{nowrap|''k'' {{=}} ''k''(''j'') ∈ {{(}}1, ..., ''n'' &minus; 1{{)}}}} such that ''jg'' is ''p''-equivalent to one size of ''k''-step. Hence if the class of ''k''-steps has ''at least'' two sizes, the sizes must be ''j''(''k'')''g'' and {{nowrap|(''j''(''k'') &minus; ''n'')''g''}}; any other size must leave the range {{nowrap|(1 &minus; ''n'')''g''}}, ..., 0, ..., {{nowrap|(''n'' &minus; 1)''g''}}. Thus the class of ''k''-steps has at most two sizes for {{nowrap|1 &le; ''k'' &le; ''n'' &minus; 1}}. Each non-''p''-equivalent class must have ''exactly'' two sizes, since the inverse of the ''k''-step that is equivalent to ''jg'' is an {{nowrap|(''n'' &minus; ''k'')}}-step equivalent to &minus;''jg'', which by linear independence must be distinct from an {{nowrap|(''n'' &minus; ''k'')}}-step equivalent to a positive number of ''g'' generators. (Note that the latter {{nowrap|(''n'' &minus; ''k'')}}-step does occur in the "brightest" mode of ''S'', i.e. the mode with the most ''g'' generators stacked ''up'' rather than ''down'' from the tonic.) This completes the proof.
+In fact, {L,&nbsp;s} is another valid basis for the abelian group with basis {''p'',&nbsp;''g''}, since by binarity we have {{nowrap|''p, g'' ∈ span(L, s)}}. Assume {{nowrap|''c'' {{=}} ''b''}} and {{nowrap|''e'' {{=}} &minus;''a''}} (this corresponds to assuming that ''g'' is the "bright" generator). Let {{nowrap|&chi; {{=}} L &minus; s}} &gt;&nbsp;0; then &chi; is ''p''-equivalent to +''ng''. Now by generatedness and binarity, any interval class that has at least two sizes must have sizes separated by ''ng'' (the separation corresponding to changing an L step to an s step). Since ''g'' and ''p'' are linearly independent, for each {{nowrap|''j'' ∈ {{(}}1, …, ''n'' &minus; 1{{)}}}} there exists at most one {{nowrap|''k'' {{=}} ''k''(''j'') ∈ {{(}}1, …, ''n'' &minus; 1{{)}}}} such that ''jg'' is ''p''-equivalent to one size of ''k''-step. Hence if the class of ''k''-steps has ''at least'' two sizes, the sizes must be ''j''(''k'')''g'' and {{nowrap|(''j''(''k'') &minus; ''n'')''g''}}; any other size must leave the range {{nowrap|(1 &minus; ''n'')''g''}}, …, 0, …, {{nowrap|(''n'' &minus; 1)''g''}}. Thus the class of ''k''-steps has at most two sizes for {{nowrap|1 &le; ''k'' &le; ''n'' &minus; 1}}. Each non-''p''-equivalent class must have ''exactly'' two sizes, since the inverse of the ''k''-step that is equivalent to ''jg'' is an {{nowrap|(''n'' &minus; ''k'')}}-step equivalent to &minus;''jg'', which by linear independence must be distinct from an {{nowrap|(''n'' &minus; ''k'')}}-step equivalent to a positive number of ''g'' generators. (Note that the latter {{nowrap|(''n'' &minus; ''k'')}}-step does occur in the "brightest" mode of ''S'', i.e. the mode with the most ''g'' generators stacked ''up'' rather than ''down'' from the tonic.) This completes the proof.
 The previous argument cannot be used for the non-linearly independent case. This is because not all binary generated scales with rational step ratios are limit points of binary generated scales with irrational step ratios. A counterexample is the 13-note scale ssLsLssLsLss ({{nowrap|s {{=}} 1\17|L {{=}} 2\17}}), which is obtained from LsLsLssLsLss (5L&nbsp;7s) by stacking one more 10\17 generator.