Mathematics of MOS: Difference between revisions

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

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.

# Mode of a Christoffel word. (If gcd(''p'', ''q'') = 1, a ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line {{nowrap|''y'' {{=}} (''p''/''q'')''x''}} that stays as close to the line as possible without crossing it.)

# [[Maximum variety]] n

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.

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}})}}.

<pre<includeonly />>

<nowiki /># logarithm base 2

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

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

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

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

<nowiki /># convergent list for z

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

<nowiki /># semiconvergent list for z

<nowiki /># penultimate convergent to q

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

<nowiki /># mediant from Large-small steps

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

<nowiki /># reverse of list

<nowiki /># inverse continued fraction

<nowiki /># Minkowski ? function

<nowiki /># inverse ? function

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

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

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

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

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'' − 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'' {{=}} −''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.