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{{cent}} in 12edo) which is added repeatedly to make a chain of scale steps, starting from the unison or 0{{cent}} 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")

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# 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 {{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 {{w|~~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 convergent or semiconvergent to the {{w|continued fraction}} of ''g''/''P'', the ratio of the generator to 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.

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Note that some procedures below depend on others. Special procedure names (i.e. not built into Maple) are shown in <code>'''bold type'''</code>.

<pre<includeonly />>

<pre<includeonly/>>

'''log2''' := proc(x)

<nowiki /># logarithm base 2

<nowiki/># logarithm base 2

evalf(ln(x)/ln(2)) end:

'''nextfarey''' := proc(q, n)

<nowiki /># next in row n of Farey sequence from q, 0 <= q <= 1

<nowiki/># next in row n of Farey sequence from q, 0 <= q <= 1

local a, b, r, s;

if q >= (n-1)/n then RETURN(1) fi;

Line 105:

'''prevfarey''' := proc(q, n)

<nowiki /># previous in row n of Farey sequence from q, 0 <= q <= 1

<nowiki/># previous in row n of Farey sequence from q, 0 <= q <= 1

local a, b, r, s;

if q=0 then RETURN(0) fi;

Line 116:

'''fareypair''' := proc(q)

<nowiki /># Farey pair with q as its mediant

<nowiki/># Farey pair with q as its mediant

local n;

n := denom(q);

Line 122:

'''mediant''' := proc(u, v)

<nowiki /># mediant of two rational numbers u and v

<nowiki/># mediant of two rational numbers u and v

(numer(u) + numer(v))/(denom(u) + denom(v)) end:

'''convergents''' := proc(z)

<nowiki /># convergent list for z

<nowiki/># convergent list for z

local q;

convert(z,confrac,'q');

Line 132:

'''exlist''' := proc(l)

<nowiki /># expansion of a convergent list to semiconvergents

<nowiki/># expansion of a convergent list to semiconvergents

local i, j, s, d;

if nops(l)<3 then RETURN(l) fi;

Line 146:

'''semiconvergents''' := proc(z)

<nowiki /># semiconvergent list for z

<nowiki/># semiconvergent list for z

'''exlist'''('''convergents'''(z)) end:

'''penult''' := proc(q)

<nowiki /># penultimate convergent to q

<nowiki/># penultimate convergent to q

local i, u;

u := '''convergents'''(q);

Line 159:

'''Ls''' := proc(q)

<nowiki /># large-small steps from mediant q

<nowiki/># large-small steps from mediant q

local u;

u := '''fareypair'''(q);

Line 165:

'''medi''' := proc(u)

<nowiki /># mediant from Large-small steps

<nowiki/># mediant from Large-small steps

local q, r;

if u[2]=1 then RETURN(1/(u[1]+1)) fi;

Line 175:

'''Lsgen''' := proc(g, n)

<nowiki /># given generator g and scale size n determines large-small steps

<nowiki/># given generator g and scale size n determines large-small steps

local q, u, w;

q := round(n*g)/n;

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'''revlist''' := proc(l)

<nowiki /># reverse of list

<nowiki/># reverse of list

local i, v, e;

e := nops(l);

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'''invcon''' := proc(l)

<nowiki /># inverse continued fraction

<nowiki/># inverse continued fraction

local d, i, h, k;

h[-2] := 0;

Line 206:

'''quest''' := proc(x)

<nowiki /># Minkowski ? function

<nowiki/># Minkowski ? function

local i, j, d, l, s, t;

l := convert(x, confrac);

Line 220:

'''Box''' := proc(x)

<nowiki /># inverse ? function

<nowiki/># inverse ? function

local d, e, i, n, w, y;

if type(x, integer) then RETURN(x) fi;

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See [[Recursive structure of MOS scales#Proofs]].

=== MOS scales ''aL'' ''bs'' with gcd(a,&~~nbsp~~;~~b) >~~ 1 reduce to single-period MOS scales on a smaller period ===

=== MOS scales ''aL'' ''bs'' with {{nowrap|gcd(a, b) > 1}} reduce to single-period MOS scales on a smaller period ===

Assume that {{nowrap|gcd(''m'', ''n'') > 1}} and assume that ({{frac|''n''|''d''}})-steps came in 2 sizes, {{nowrap|''pa'' + ''qb''}} and {{nowrap|''ra'' + ''sb''}}. Then at least one size, say {{nowrap|''pa'' + ''qb''}}, must differ from {{nowrap|{{frac|''a''|''d''}} L + {{frac|''b''|''d''}} s}}. WLOG {{nowrap|''p'' > {{frac|''a''|''d''}}}} and {{nowrap|''pa'' + ''qb''}} occurs on degree 0. Since the equave is equal to {{nowrap|''aL'' + ''bs''}}, on a degree ''k''({{frac|''n''|''d''}}) for some integer {{nowrap|''k'' > 0}}, there must be another ({{frac|''n''|''d''}})-step with fewer L's than {{frac|''a''|''d''}}. This involves more than two changes from ''L'' to ''s''. Scooting an ({{frac|''n''|''d''}})-step one step at a time from degree 0 to ''k''({{frac|''n''|''d''}}) changes its size one step size substitution at a time, showing that intermediate ({{frac|''n''|''d''}})-steps also exist. This violates the MOS property, whence ({{frac|''n''|''d''}})-steps have only one size, which must be the period.

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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:

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

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

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

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

We thus have:

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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}<nowiki/>}}. 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}<nowiki/>}} there exists at most one {{nowrap|''k'' {{=}} ''k''(''j'') ∈ {1, …, ''n'' − 1}<nowiki/>}} 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{{cent}} in 12edo) which is added repeatedly to make a chain of scale steps, starting from the unison or 0{{cent}} 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")
@@ Line 11: / Line 11: @@
 # 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 {{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 {{w|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 convergent or semiconvergent to the {{w|continued fraction}} of ''g''/''P'', the ratio of the generator to 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 89: / Line 89: @@
 Note that some procedures below depend on others. Special procedure names (i.e. not built into Maple) are shown in <code>'''bold type'''</code>.
-<pre<includeonly />>
+<pre<includeonly/>>
 '''log2''' := proc(x)
-<nowiki /># logarithm base 2
+<nowiki/># logarithm base 2
 evalf(ln(x)/ln(2)) end:
 '''nextfarey''' := proc(q, n)
-<nowiki /># next in row n of Farey sequence from q, 0 <= q <= 1
+<nowiki/># next in row n of Farey sequence from q, 0 <= q <= 1
 local a, b, r, s;
 if q >= (n-1)/n then RETURN(1) fi;
@@ Line 105: / Line 105: @@
 '''prevfarey''' := proc(q, n)
-<nowiki /># previous in row n of Farey sequence from q, 0 <= q <= 1
+<nowiki/># previous in row n of Farey sequence from q, 0 <= q <= 1
 local a, b, r, s;
 if q=0 then RETURN(0) fi;
@@ Line 116: / Line 116: @@
 '''fareypair''' := proc(q)
-<nowiki /># Farey pair with q as its mediant
+<nowiki/># Farey pair with q as its mediant
 local n;
 n := denom(q);
@@ Line 122: / Line 122: @@
 '''mediant''' := proc(u, v)
-<nowiki /># mediant of two rational numbers u and v
+<nowiki/># mediant of two rational numbers u and v
 (numer(u) + numer(v))/(denom(u) + denom(v)) end:
 '''convergents''' := proc(z)
-<nowiki /># convergent list for z
+<nowiki/># convergent list for z
 local q;
 convert(z,confrac,'q');
@@ Line 132: / Line 132: @@
 '''exlist''' := proc(l)
-<nowiki /># expansion of a convergent list to semiconvergents
+<nowiki/># expansion of a convergent list to semiconvergents
 local i, j, s, d;
 if nops(l)<3 then RETURN(l) fi;
@@ Line 146: / Line 146: @@
 '''semiconvergents''' := proc(z)
-<nowiki /># semiconvergent list for z
+<nowiki/># semiconvergent list for z
 '''exlist'''('''convergents'''(z)) end:
 '''penult''' := proc(q)
-<nowiki /># penultimate convergent to q
+<nowiki/># penultimate convergent to q
 local i, u;
 u := '''convergents'''(q);
@@ Line 159: / Line 159: @@
 '''Ls''' := proc(q)
-<nowiki /># large-small steps from mediant q
+<nowiki/># large-small steps from mediant q
 local u;
 u := '''fareypair'''(q);
@@ Line 165: / Line 165: @@
 '''medi''' := proc(u)
-<nowiki /># mediant from Large-small steps
+<nowiki/># mediant from Large-small steps
 local q, r;
 if u[2]=1 then RETURN(1/(u[1]+1)) fi;
@@ Line 175: / Line 175: @@
 '''Lsgen''' := proc(g, n)
-<nowiki /># given generator g and scale size n determines large-small steps
+<nowiki/># given generator g and scale size n determines large-small steps
 local q, u, w;
 q := round(n*g)/n;
@@ Line 185: / Line 185: @@
 '''revlist''' := proc(l)
-<nowiki /># reverse of list
+<nowiki/># reverse of list
 local i, v, e;
 e := nops(l);
@@ Line 193: / Line 193: @@
 '''invcon''' := proc(l)
-<nowiki /># inverse continued fraction
+<nowiki/># inverse continued fraction
 local d, i, h, k;
 h[-2] := 0;
@@ Line 206: / Line 206: @@
 '''quest''' := proc(x)
-<nowiki /># Minkowski ? function
+<nowiki/># Minkowski ? function
 local i, j, d, l, s, t;
 l := convert(x, confrac);
@@ Line 220: / Line 220: @@
 '''Box''' := proc(x)
-<nowiki /># inverse ? function
+<nowiki/># inverse ? function
 local d, e, i, n, w, y;
 if type(x, integer) then RETURN(x) fi;
@@ Line 255: / Line 255: @@
 See [[Recursive structure of MOS scales#Proofs]].
-=== MOS scales ''aL''&nbsp;''bs'' with gcd(a,&nbsp;b) > 1 reduce to single-period MOS scales on a smaller period ===
+=== MOS scales ''aL''&nbsp;''bs'' with {{nowrap|gcd(a, b) &gt; 1}} reduce to single-period MOS scales on a smaller period ===
 Assume that {{nowrap|gcd(''m'', ''n'') &gt; 1}} and assume that ({{frac|''n''|''d''}})-steps came in 2 sizes, {{nowrap|''pa'' + ''qb''}} and {{nowrap|''ra'' + ''sb''}}. Then at least one size, say {{nowrap|''pa'' + ''qb''}}, must differ from {{nowrap|{{frac|''a''|''d''}}&#x200A;L + {{frac|''b''|''d''}}&#x200A;s}}. WLOG {{nowrap|''p'' &gt; {{frac|''a''|''d''}}}} and {{nowrap|''pa'' + ''qb''}} occurs on degree 0. Since the equave is equal to {{nowrap|''aL'' + ''bs''}}, on a degree ''k''({{frac|''n''|''d''}}) for some integer {{nowrap|''k'' &gt; 0}}, there must be another ({{frac|''n''|''d''}})-step with fewer L's than {{frac|''a''|''d''}}. This involves more than two changes from ''L'' to ''s''. Scooting an ({{frac|''n''|''d''}})-step one step at a time from degree 0 to ''k''({{frac|''n''|''d''}}) changes its size one step size substitution at a time, showing that intermediate ({{frac|''n''|''d''}})-steps also exist. This violates the MOS property, whence ({{frac|''n''|''d''}})-steps have only one size, which must be the period.
@@ Line 276: / Line 276: @@
 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:
-<nowiki>{</nowiki>{{nowrap|(−''n'' + 1)''g''}}, …, −''g'', 0, ''g'', …, {{nowrap|(''n'' − 1)''g''}}<nowiki>}</nowiki>,
+<nowiki>{</nowiki>{{nowrap|(−''n'' + 1)''g''}}, …, −''g'', 0, ''g'', …, {{nowrap|(''n'' − 1)''g''}}<nowiki/>},
-and all sizes {0, ''g'', …, {{nowrap|(''n'' − 1)''g''}}<nowiki>}</nowiki> are distinct.
+and all sizes {0, ''g'', …, {{nowrap|(''n'' − 1)''g''}}<nowiki/>} are distinct.
 We thus have:
@@ Line 288: / Line 288: @@
 for appropriate integers ''c, d, e, f'', where {{nowrap|{{abs|''c''}}, {{abs|''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'' − 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'' − 1}<nowiki/>}}. 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'' {{=}} −''a''}} (this corresponds to assuming that ''g'' is the "bright" generator). Let {{nowrap|&chi; {{=}} L − 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'' − 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 &le; ''k'' &le; ''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,&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'' {{=}} −''a''}} (this corresponds to assuming that ''g'' is the "bright" generator). Let {{nowrap|&chi; {{=}} L − 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'' − 1}<nowiki/>}} there exists at most one {{nowrap|''k'' {{=}} ''k''(''j'') ∈ {1, …, ''n'' − 1}<nowiki/>}} 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 &le; ''k'' &le; ''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&nbsp;7s) by stacking one more 10\17 generator.