Mathematics of MOS: Difference between revisions

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# [[Binary]] and has a generator

# 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.)

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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.

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 [[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.

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}}].

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As MOS Scales are generated by repeated iterations of a single interval, the generator, it is useful to visualize a contiguous "generator chain" that organizes the scale. For example, if the generator is some kind of perfect fifth, then the generator chain is a chain of fifths: {{dash|F, C, G, D, A, E, B|med}}. We know that adjacent tones in the chain are a perfect fifth apart (possibly with octave-displacement), such as F to C, and that tones two spaces away are some kind of second or ninth apart, such as F to G. It is clear from the chain that B to F is *not* a perfect fifth, but must be something else (unless the chain closes to form a circle, as would be the case in [[7edo]]). The generator chain shows us that every interval of the MOS scale represents a move on the generator chain some number of generators up or down.

Another common way to view the tones of an MOS scale is as points in logarithmic pitch space, with larger gaps between points representing larger intervals and smaller gaps between points representing smaller intervals. Then we see that our scale has large steps and small steps and intervals that are composed of some stackings of large and small steps. It is not obvious, looking at the generator chain or looking at the tones in pitch space, what the relationship is. Indeed, it is different for different MOS ~~scales—an~~ "L" will not always represent the same number of generators up or down when we move to a different scale.

Another common way to view the tones of an MOS scale is as points in logarithmic pitch space, with larger gaps between points representing larger intervals and smaller gaps between points representing smaller intervals. Then we see that our scale has large steps and small steps and intervals that are composed of some stackings of large and small steps. It is not obvious, looking at the generator chain or looking at the tones in pitch space, what the relationship is. Indeed, it is different for different MOS scales—an "L" will not always represent the same number of generators up or down when we move to a different scale.

Since the generator chain and logarithmic pitch space are both 1-dimensional, it may be helpful to graph them together in 2 dimensions. Here is a diagram for sensi[8], an octatonic [[3L 5s]] MOS scale with a generator of between 442 and 445{{nbhsp}}¢. The x-axis shows the generator chain and the y-axis shows the nine tones (eight plus octave) in logarithmic pitch space. You can see that the vertical lines are evenly-spaced (since every generator is the same), while the horizontal lines have large and small gaps, representing the large and small steps of sensi[8].

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** an equal scale (when {{nowrap|L:s {{=}} 2:1}}).

Below is a diagram showing four MOS scales in logarithmic pitch space generated by 7\[[37edo]] (approx. 227¢). Each contains the ones above it and is contained by the ones below it. There are additional MOS scales that would appear above the first line with 1, 2, 3, 4, and 5 tones, but they have been omitted. The bottom line is a case where {{nowrap|L:s {{=}} 2:1}}, which means that there are no more MOS scales to be ~~had—the~~ next stopping point is at a complete chromatic scale of [[37edo]].

Below is a diagram showing four MOS scales in logarithmic pitch space generated by 7\[[37edo]] (approx. 227¢). Each contains the ones above it and is contained by the ones below it. There are additional MOS scales that would appear above the first line with 1, 2, 3, 4, and 5 tones, but they have been omitted. The bottom line is a case where {{nowrap|L:s {{=}} 2:1}}, which means that there are no more MOS scales to be had—the next stopping point is at a complete chromatic scale of [[37edo]].

[[File:37edo_mos_07.png]]

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== Classification of MOS scales ==

Since MOS scales always consist of some number of large steps and some number of small steps, they can be classified simply by the number of large steps and the number of small steps, in the form #L #~~s—e~~.g., the diatonic scale can be described as 5L 2s (5 large steps and 2 small steps) or simply [5, 2]. It is typical to ignore the period when specifying MOS scales and instead use the number of large and small steps that make up the interval of equivalence (typically assumed to be the ~~octave—a~~ frequency ratio of 2/~~1—unless~~ otherwise specified). For instance, the diminished scale in 12-TET is typically classified as 4L4s rather than 1L1s, since there are 4 large and 4 small steps that make up an octave.

Since MOS scales always consist of some number of large steps and some number of small steps, they can be classified simply by the number of large steps and the number of small steps, in the form #L #s—e.g., the diatonic scale can be described as 5L 2s (5 large steps and 2 small steps) or simply [5, 2]. It is typical to ignore the period when specifying MOS scales and instead use the number of large and small steps that make up the interval of equivalence (typically assumed to be the octave—a frequency ratio of 2/1—unless otherwise specified). For instance, the diminished scale in 12-TET is typically classified as 4L4s rather than 1L1s, since there are 4 large and 4 small steps that make up an octave.

Alternatively, we could give a mediant for a Farey pair associated to the MOS, where this mediant is less than any generator for the MOS. In other words, we use the right hand part of the Farey pair interval, which means we must replace ''g'' with {{nowrap|1 ~~−~~ ''g''}} and use the complementary pair if ''g'' is in the left hand side. This method is rarely used in discussions, however.

Alternatively, we could give a mediant for a Farey pair associated to the MOS, where this mediant is less than any generator for the MOS. In other words, we use the right hand part of the Farey pair interval, which means we must replace ''g'' with {{nowrap|1 − ''g''}} and use the complementary pair if ''g'' is in the left hand side. This method is rarely used in discussions, however.

The two systems are equivalent; in the Algorithms section you will find code for routines starting from the mediant and going to the Ls pair (the "Ls" routine) and for starting from an Ls pair and going to the mediant (the "medi" routine.) The Ls routine uses {{w|modular multiplicative inverse}}s, whereas the medi routine uses continued fractions.

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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''n''}} 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~~−~~''n'' ~~−~~ 1)}}. 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~~−~~''n'')}}, and the proper generators will be {{nowrap|Box(''r'') < ''g'' < Box(''r'' + 2~~−~~''n'' ~~−~~ 1)}}. 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−''n'' − 1)}}. 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−''n'')}}, and the proper generators will be {{nowrap|Box(''r'') < ''g'' < Box(''r'' + 2−''n'' − 1)}}. 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}})}}.

== MOS in equal tunings ==

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== Blackwood ''R'' constant ==

In the context of the "recognizable diatonic" scales deriving from the Farey pair [{{frac|1|2}}, {{frac|3|5}}] {{w|Easley Blackwood Jr.}} defined a characterizing constant ''R'' which we may generalize to any MOS as follows: If {{nowrap|{{frac|''a''|''b''}} < ''g'' < {{frac|''c''|''d''}}}} is a generator with the given Farey pair, take the ratio of relative errors {{nowrap|''R'' {{=}} {{sfrac|''bg'' ~~−~~ ''a''|''c'' ~~−~~ ''dg''}}}}. Since this is a ratio of positive numbers, it is positive. As ''g'' tends towards {{frac|''a''|''b''}}, ''R'' tends to zero, and as ''g'' goes to {{frac|''c''|''d''}}, ''R'' goes to infinity. When {{nowrap|''g'' {{=}} {{sfrac|''a'' + ''c''|''b'' + ''d''}}}} it takes the value of 1, and the range of propriety is {{nowrap|{{frac|1|2}} ≤ ''R'' ≤ 2}}.

In the context of the "recognizable diatonic" scales deriving from the Farey pair [{{frac|1|2}}, {{frac|3|5}}] {{w|Easley Blackwood Jr.}} defined a characterizing constant ''R'' which we may generalize to any MOS as follows: If {{nowrap|{{frac|''a''|''b''}} < ''g'' < {{frac|''c''|''d''}}}} is a generator with the given Farey pair, take the ratio of relative errors {{nowrap|''R'' {{=}} {{sfrac|''bg'' − ''a''|''c'' − ''dg''}}}}. Since this is a ratio of positive numbers, it is positive. As ''g'' tends towards {{frac|''a''|''b''}}, ''R'' tends to zero, and as ''g'' goes to {{frac|''c''|''d''}}, ''R'' goes to infinity. When {{nowrap|''g'' {{=}} {{sfrac|''a'' + ''c''|''b'' + ''d''}}}} it takes the value of 1, and the range of propriety is {{nowrap|{{frac|1|2}} ≤ ''R'' ≤ 2}}.

When {{nowrap|''R'' < 1}}, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing ''g'' with {{nowrap|1 ~~−~~ ''g''}} if necessary, we can reduce always to the case where {{nowrap|''R'' > 1}} (or {{nowrap|''R'' < 1}} if we prefer.)

When {{nowrap|''R'' < 1}}, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing ''g'' with {{nowrap|1 − ''g''}} if necessary, we can reduce always to the case where {{nowrap|''R'' > 1}} (or {{nowrap|''R'' < 1}} if we prefer.)

== Algorithms ==

Below is some Maple code for various mathematical routines having to do with MOS. If you have access to Maple, you can of course copy and run these programs. Even if you do not, Maple code makes better pseudocode than most languages or computer algebra packages afford, so it can be used as pseudocode. For that purpose, it will be helpful to know that <code>modp(x, n)</code> means reducing ''x'' mod the integer ''n'' to 0, 1, …, {{nowrap|''n'' ~~−~~ 1}} not only when ''x'' is an integer, but also when it is a rational number with denominator prime to ''n''. In that case, {{nowrap|{{frac|''p''|''q''}} (mod ''n'') {{=}} ''r''}} means {{nowrap|''p'' {{=}} ''qr'' (mod ''n'')}}.

Below is some Maple code for various mathematical routines having to do with MOS. If you have access to Maple, you can of course copy and run these programs. Even if you do not, Maple code makes better pseudocode than most languages or computer algebra packages afford, so it can be used as pseudocode. For that purpose, it will be helpful to know that <code>modp(x, n)</code> means reducing ''x'' mod the integer ''n'' to 0, 1, …, {{nowrap|''n'' − 1}} not only when ''x'' is an integer, but also when it is a rational number with denominator prime to ''n''. In that case, {{nowrap|{{frac|''p''|''q''}} (mod ''n'') {{=}} ''r''}} means {{nowrap|''p'' {{=}} ''qr'' (mod ''n'')}}.

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

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=== 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|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

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

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Steps of ME(''m'', ''n'') have exactly two sizes because if it were one size, we would have {{nowrap|''m'' {{!}} {{frac|''n''|''d''}}}}, which is a contradiction.

This maximally even MOS has {{nowrap|''n'' % ''m''}} large steps and {{nowrap|''m'' ~~−~~ (''n'' % ''m'')}} small steps.

This maximally even MOS has {{nowrap|''n'' % ''m''}} large steps and {{nowrap|''m'' − (''n'' % ''m'')}} small steps.

=== Binary generated scales with independent period and generator are MOS ===

This proof justifies the common description of "stack until binary" for MOS building and Erv Wilson's terminology ''moment of symmetry'' where MOS sizes are "moments" in time (when stacking) where the "symmetry" of binarity and MV2 holds.

By ''generatedness'', we mean that every interval in the scale is of the form {{nowrap|''jg'' + ''kp''}} where ''g'' is a generator, ''p'' is the period, and {{nowrap|''j'', ''k'' ∈ ℤ}}, and that either ''g'' or ~~−~~''g'' occurs on every note. We claim that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes in any scale satisfying the antecedent.

By ''generatedness'', we mean that every interval in the scale is of the form {{nowrap|''jg'' + ''kp''}} where ''g'' is a generator, ''p'' is the period, and {{nowrap|''j'', ''k'' ∈ ℤ}}, and that either ''g'' or −''g'' occurs on every note. We claim that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes in any scale satisfying the antecedent.

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:

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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 20: / Line 20: @@
 # [[Binary]] and has a generator
 # Binary and [[distributionally even]]
-# Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or {{nowrap|L &minus; 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,&nbsp;0) and (''p'',&nbsp;''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line {{nowrap|''y'' {{=}} ''p''/''q''&#x200A;''x''}} that stays as close to the line without crossing it.)
@@ Line 33: / Line 33: @@
 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'')&#x200A;{{frac|''a''|''b''}} + (''t'')&#x200A;{{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.
+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'' − ''ad'' {{=}} 1}}. If {{nowrap|''g'' {{=}} (1 − ''t'')&#x200A;{{frac|''a''|''b''}} + (''t'')&#x200A;{{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.
 While all the scales constructed by generators ''g'' with {{nowrap|{{frac|''a''|''b''}} &lt; ''g'' &lt; {{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''}} &le; ''g''}} &le;&nbsp;{{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''}} &lt; ''g''}} &lt;&nbsp;{{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}},&nbsp;{{frac|3|5}}) we find the range of propriety for these seven-note MOS to be [{{frac|5|9}},&nbsp;{{frac|7|12}}].
@@ Line 44: / Line 44: @@
 As MOS Scales are generated by repeated iterations of a single interval, the generator, it is useful to visualize a contiguous "generator chain" that organizes the scale. For example, if the generator is some kind of perfect fifth, then the generator chain is a chain of fifths: {{dash|F, C, G, D, A, E, B|med}}. We know that adjacent tones in the chain are a perfect fifth apart (possibly with octave-displacement), such as F to C, and that tones two spaces away are some kind of second or ninth apart, such as F to G. It is clear from the chain that B to F is *not* a perfect fifth, but must be something else (unless the chain closes to form a circle, as would be the case in [[7edo]]). The generator chain shows us that every interval of the MOS scale represents a move on the generator chain some number of generators up or down.
-Another common way to view the tones of an MOS scale is as points in logarithmic pitch space, with larger gaps between points representing larger intervals and smaller gaps between points representing smaller intervals. Then we see that our scale has large steps and small steps and intervals that are composed of some stackings of large and small steps. It is not obvious, looking at the generator chain or looking at the tones in pitch space, what the relationship is. Indeed, it is different for different MOS scales&mdash;an "L" will not always represent the same number of generators up or down when we move to a different scale.
+Another common way to view the tones of an MOS scale is as points in logarithmic pitch space, with larger gaps between points representing larger intervals and smaller gaps between points representing smaller intervals. Then we see that our scale has large steps and small steps and intervals that are composed of some stackings of large and small steps. It is not obvious, looking at the generator chain or looking at the tones in pitch space, what the relationship is. Indeed, it is different for different MOS scales—an "L" will not always represent the same number of generators up or down when we move to a different scale.
 Since the generator chain and logarithmic pitch space are both 1-dimensional, it may be helpful to graph them together in 2 dimensions. Here is a diagram for sensi[8], an octatonic [[3L&nbsp;5s]] MOS scale with a generator of between 442 and 445{{nbhsp}}¢. The x-axis shows the generator chain and the y-axis shows the nine tones (eight plus octave) in logarithmic pitch space. You can see that the vertical lines are evenly-spaced (since every generator is the same), while the horizontal lines have large and small gaps, representing the large and small steps of sensi[8].
@@ Line 56: / Line 56: @@
 ** an equal scale (when {{nowrap|L:s {{=}} 2:1}}).
-Below is a diagram showing four MOS scales in logarithmic pitch space generated by 7\[[37edo]] (approx. 227¢). Each contains the ones above it and is contained by the ones below it. There are additional MOS scales that would appear above the first line with 1, 2, 3, 4, and 5 tones, but they have been omitted. The bottom line is a case where {{nowrap|L:s {{=}} 2:1}}, which means that there are no more MOS scales to be had&mdash;the next stopping point is at a complete chromatic scale of [[37edo]].
+Below is a diagram showing four MOS scales in logarithmic pitch space generated by 7\[[37edo]] (approx. 227¢). Each contains the ones above it and is contained by the ones below it. There are additional MOS scales that would appear above the first line with 1, 2, 3, 4, and 5 tones, but they have been omitted. The bottom line is a case where {{nowrap|L:s {{=}} 2:1}}, which means that there are no more MOS scales to be had—the next stopping point is at a complete chromatic scale of [[37edo]].
 [[File:37edo_mos_07.png]]
@@ Line 63: / Line 63: @@
 == Classification of MOS scales ==
-Since MOS scales always consist of some number of large steps and some number of small steps, they can be classified simply by the number of large steps and the number of small steps, in the form #L&nbsp;#s&mdash;e.g., the diatonic scale can be described as 5L&nbsp;2s (5 large steps and 2 small steps) or simply [5,&nbsp;2]. It is typical to ignore the period when specifying MOS scales and instead use the number of large and small steps that make up the interval of equivalence (typically assumed to be the octave&mdash;a frequency ratio of 2/1&mdash;unless otherwise specified). For instance, the diminished scale in 12-TET is typically classified as 4L4s rather than 1L1s, since there are 4 large and 4 small steps that make up an octave.
+Since MOS scales always consist of some number of large steps and some number of small steps, they can be classified simply by the number of large steps and the number of small steps, in the form #L&nbsp;#s—e.g., the diatonic scale can be described as 5L&nbsp;2s (5 large steps and 2 small steps) or simply [5,&nbsp;2]. It is typical to ignore the period when specifying MOS scales and instead use the number of large and small steps that make up the interval of equivalence (typically assumed to be the octave—a frequency ratio of 2/1—unless otherwise specified). For instance, the diminished scale in 12-TET is typically classified as 4L4s rather than 1L1s, since there are 4 large and 4 small steps that make up an octave.
-Alternatively, we could give a mediant for a Farey pair associated to the MOS, where this mediant is less than any generator for the MOS. In other words, we use the right hand part of the Farey pair interval, which means we must replace ''g'' with {{nowrap|1 &minus; ''g''}} and use the complementary pair if ''g'' is in the left hand side. This method is rarely used in discussions, however.
+Alternatively, we could give a mediant for a Farey pair associated to the MOS, where this mediant is less than any generator for the MOS. In other words, we use the right hand part of the Farey pair interval, which means we must replace ''g'' with {{nowrap|1 − ''g''}} and use the complementary pair if ''g'' is in the left hand side. This method is rarely used in discussions, however.
 The two systems are equivalent; in the Algorithms section you will find code for routines starting from the mediant and going to the Ls pair (the "Ls" routine) and for starting from an Ls pair and going to the mediant (the "medi" routine.) The Ls routine uses {{w|modular multiplicative inverse}}s, whereas the medi routine uses continued fractions.
@@ Line 74: / Line 74: @@
 Yet another way of classifying MOS is via {{w|Minkowski's question-mark function}}. Here ?(''x''), {{nowrap|? : ℝ &rarr; 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 &lt; ''q'' &lt; 1}} in use in the mediant system of classifying MOS scales, {{nowrap|''r'' {{=}} ?(''q'')}} =&nbsp;{{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&ndash;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'' &#177; 2<sup>&minus;''n'' &minus; 1</sup>)}}. If ''r'' represents a MOS, the range of possible values for a generator of the MOS will be {{nowrap|Box(''r'') &lt; ''g''}} {{nowrap|&lt; Box(''r'' + 2<sup>&minus;''n''</sup>)}}, and the proper generators will be {{nowrap|Box(''r'') &lt; ''g'' &lt; Box(''r'' + 2<sup>&minus;''n'' &minus; 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}} &lt; ''g'' &lt; {{frac|1|10}}}}, and will be proper if {{nowrap|{{frac|2|21}} &lt; ''g''}} &lt;&nbsp;{{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}} &gt; {{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'' &#177; 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'') &lt; ''g''}} {{nowrap|&lt; Box(''r'' + 2<sup>−''n''</sup>)}}, and the proper generators will be {{nowrap|Box(''r'') &lt; ''g'' &lt; 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}} &lt; ''g'' &lt; {{frac|1|10}}}}, and will be proper if {{nowrap|{{frac|2|21}} &lt; ''g''}} &lt;&nbsp;{{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}} &gt; {{frac|3|31}}}} {{nowrap|{{=}} Box({{frac|3|2048}} + {{frac|1|4096}})}}.
 == MOS in equal tunings ==
@@ Line 80: / Line 80: @@
 == Blackwood ''R'' constant ==
-In the context of the "recognizable diatonic" scales deriving from the Farey pair [{{frac|1|2}},&nbsp;{{frac|3|5}}] {{w|Easley Blackwood Jr.}} defined a characterizing constant ''R'' which we may generalize to any MOS as follows: If {{nowrap|{{frac|''a''|''b''}} &lt; ''g'' &lt; {{frac|''c''|''d''}}}} is a generator with the given Farey pair, take the ratio of relative errors {{nowrap|''R'' {{=}} {{sfrac|''bg'' &minus; ''a''|''c'' &minus; ''dg''}}}}. Since this is a ratio of positive numbers, it is positive. As ''g'' tends towards {{frac|''a''|''b''}}, ''R'' tends to zero, and as ''g'' goes to {{frac|''c''|''d''}}, ''R'' goes to infinity. When {{nowrap|''g'' {{=}} {{sfrac|''a'' + ''c''|''b'' + ''d''}}}} it takes the value of 1, and the range of propriety is {{nowrap|{{frac|1|2}} &le; ''R'' &le; 2}}.
+In the context of the "recognizable diatonic" scales deriving from the Farey pair [{{frac|1|2}},&nbsp;{{frac|3|5}}] {{w|Easley Blackwood Jr.}} defined a characterizing constant ''R'' which we may generalize to any MOS as follows: If {{nowrap|{{frac|''a''|''b''}} &lt; ''g'' &lt; {{frac|''c''|''d''}}}} is a generator with the given Farey pair, take the ratio of relative errors {{nowrap|''R'' {{=}} {{sfrac|''bg'' − ''a''|''c'' − ''dg''}}}}. Since this is a ratio of positive numbers, it is positive. As ''g'' tends towards {{frac|''a''|''b''}}, ''R'' tends to zero, and as ''g'' goes to {{frac|''c''|''d''}}, ''R'' goes to infinity. When {{nowrap|''g'' {{=}} {{sfrac|''a'' + ''c''|''b'' + ''d''}}}} it takes the value of 1, and the range of propriety is {{nowrap|{{frac|1|2}} &le; ''R'' &le; 2}}.
-When {{nowrap|''R'' &lt; 1}}, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing ''g'' with {{nowrap|1 &minus; ''g''}} if necessary, we can reduce always to the case where {{nowrap|''R'' &gt; 1}} (or {{nowrap|''R'' &lt; 1}} if we prefer.)
+When {{nowrap|''R'' &lt; 1}}, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing ''g'' with {{nowrap|1 − ''g''}} if necessary, we can reduce always to the case where {{nowrap|''R'' &gt; 1}} (or {{nowrap|''R'' &lt; 1}} if we prefer.)
 == Algorithms ==
-Below is some Maple code for various mathematical routines having to do with MOS. If you have access to Maple, you can of course copy and run these programs. Even if you do not, Maple code makes better pseudocode than most languages or computer algebra packages afford, so it can be used as pseudocode. For that purpose, it will be helpful to know that <code>modp(x, n)</code> means reducing ''x'' mod the integer ''n'' to 0, 1, …, {{nowrap|''n'' &minus; 1}} not only when ''x'' is an integer, but also when it is a rational number with denominator prime to ''n''. In that case, {{nowrap|{{frac|''p''|''q''}} (mod ''n'') {{=}} ''r''}} means {{nowrap|''p'' {{=}} ''qr'' (mod ''n'')}}.
+Below is some Maple code for various mathematical routines having to do with MOS. If you have access to Maple, you can of course copy and run these programs. Even if you do not, Maple code makes better pseudocode than most languages or computer algebra packages afford, so it can be used as pseudocode. For that purpose, it will be helpful to know that <code>modp(x, n)</code> means reducing ''x'' mod the integer ''n'' to 0, 1, …, {{nowrap|''n'' − 1}} not only when ''x'' is an integer, but also when it is a rational number with denominator prime to ''n''. In that case, {{nowrap|{{frac|''p''|''q''}} (mod ''n'') {{=}} ''r''}} means {{nowrap|''p'' {{=}} ''qr'' (mod ''n'')}}.
 Note that some procedures below depend on others.  Special procedure names (i.e. not built into Maple) are shown in <code>'''bold type'''</code>.
@@ Line 258: / Line 258: @@
 === Maximally even scales are MOS ===
-Let {{nowrap|1 &gt; ''m'' &gt; ''n''|gcd(''m'', ''n'') {{=}} d}}, and ME(''m'',&nbsp;''n'') be as in [[Maximal evenness]]. An arbitrary ''k''-step in ME(''m'',&nbsp;''n'') has size {{nowrap|''u'' {{=}} &lfloor;(''i'' + ''k'')&#x200A;{{frac|''n''|''m''}}&rfloor; &minus; &lfloor;{{frac|''in''|''m''}}&rfloor;}}, and
+Let {{nowrap|1 &gt; ''m'' &gt; ''n''|gcd(''m'', ''n'') {{=}} d}}, and ME(''m'',&nbsp;''n'') be as in [[Maximal evenness]]. An arbitrary ''k''-step in ME(''m'',&nbsp;''n'') has size {{nowrap|''u'' {{=}} &lfloor;(''i'' + ''k'')&#x200A;{{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
@@ Line 266: / Line 266: @@
 Steps of ME(''m'',&nbsp;''n'') have exactly two sizes because if it were one size, we would have {{nowrap|''m'' {{!}} {{frac|''n''|''d''}}}}, which is a contradiction.
-This maximally even MOS has {{nowrap|''n'' % ''m''}} large steps and {{nowrap|''m'' &minus; (''n'' % ''m'')}} small steps.
+This maximally even MOS has {{nowrap|''n'' % ''m''}} large steps and {{nowrap|''m'' − (''n'' % ''m'')}} small steps.
 === Binary generated scales with independent period and generator are MOS ===
 This proof justifies the common description of "stack until binary" for MOS building and Erv Wilson's terminology ''moment of symmetry'' where MOS sizes are "moments" in time (when stacking) where the "symmetry" of binarity and MV2 holds.
-By ''generatedness'', we mean that every interval in the scale is of the form {{nowrap|''jg'' + ''kp''}} where ''g'' is a generator, ''p'' is the period, and {{nowrap|''j'', ''k'' ∈ ℤ}}, and that either ''g'' or &minus;''g'' occurs on every note. We claim that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes in any scale satisfying the antecedent.
+By ''generatedness'', we mean that every interval in the scale is of the form {{nowrap|''jg'' + ''kp''}} where ''g'' is a generator, ''p'' is the period, and {{nowrap|''j'', ''k'' ∈ ℤ}}, and that either ''g'' or −''g'' occurs on every note. We claim that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes in any scale satisfying the antecedent.
 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|(−''n'' + 1)''g''}}, …, −''g'', 0, ''g'', …, {{nowrap|(''n'' − 1)''g''}}{{)}},
-and all sizes {0, ''g'', …, {{nowrap|(''n'' &minus; 1)''g''}}{{)}} are distinct.
+and all sizes {0, ''g'', …, {{nowrap|(''n'' − 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'' − 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'' {{=}} −''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.
 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.