Mathematics of MOS: Difference between revisions

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# 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{{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")

# 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.

# 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 ~~[[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 {{w|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.

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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].

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{{c}}. 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].

[[File:map_of_sensi-8-.png|alt=Map of Sensi[8].png]]

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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{{c}}). 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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== 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 {{w|maple (software)|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>.

<pre<includeonly />>

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

# logarithm base 2

<nowiki /># logarithm base 2

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

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

# 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;

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

# 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;

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

# Farey pair with q as its mediant

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

local n;

n := denom(q);

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

# 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)

# convergent list for z

<nowiki /># convergent list for z

local q;

convert(z,confrac,'q');

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

# 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;

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

# semiconvergent list for z

<nowiki /># semiconvergent list for z

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

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

# penultimate convergent to q

<nowiki /># penultimate convergent to q

local i, u;

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

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

# large-small steps from mediant q

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

local u;

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

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

# 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;

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

# 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)

# reverse of list

<nowiki /># reverse of list

local i, v, e;

e := nops(l);

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

# inverse continued fraction

<nowiki /># inverse continued fraction

local d, i, h, k;

h[-2] := 0;

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

# Minkowski ? function

<nowiki /># Minkowski ? function

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

l := convert(x, confrac);

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

# inverse ? function

<nowiki /># inverse ? function

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

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

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w := n[nops(n)];

if type(x, rational) and modp(denom(x), 2)=0 then RETURN(w) fi;

evalf(w) end:</pre>

evalf(w) end:

</pre>

== Proofs ==

@@ Line 6: / Line 6: @@
 # 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{{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")
 # 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.
+# 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 [[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 {{w|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 46: / Line 46: @@
 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].
+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{{c}}. 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].
 [[File:map_of_sensi-8-.png|alt=Map of Sensi[8].png]]
@@ 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—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{{c}}). 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 85: / Line 85: @@
 == 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 {{w|maple (software)|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,&nbsp;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>.
+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 />>
 '''log2''' := proc(x)
-# logarithm base 2
+<nowiki /># logarithm base 2
 evalf(ln(x)/ln(2)) end:
 '''nextfarey''' := proc(q, n)
-# 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)
-# 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)
-# 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)
-# 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)
-# convergent list for z
+<nowiki /># convergent list for z
 local q;
 convert(z,confrac,'q');
@@ Line 132: / Line 132: @@
 '''exlist''' := proc(l)
-# 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)
-# semiconvergent list for z
+<nowiki /># semiconvergent list for z
 '''exlist'''('''convergents'''(z)) end:
 '''penult''' := proc(q)
-# penultimate convergent to q
+<nowiki /># penultimate convergent to q
 local i, u;
 u := '''convergents'''(q);
@@ Line 159: / Line 159: @@
 '''Ls''' := proc(q)
-# 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)
-# 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)
-# 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)
-# reverse of list
+<nowiki /># reverse of list
 local i, v, e;
 e := nops(l);
@@ Line 193: / Line 193: @@
 '''invcon''' := proc(l)
-# inverse continued fraction
+<nowiki /># inverse continued fraction
 local d, i, h, k;
 h[-2] := 0;
@@ Line 206: / Line 206: @@
 '''quest''' := proc(x)
-# Minkowski ? function
+<nowiki /># Minkowski ? function
 local i, j, d, l, s, t;
 l := convert(x, confrac);
@@ Line 220: / Line 220: @@
 '''Box''' := proc(x)
-# inverse ? function
+<nowiki /># inverse ? function
 local d, e, i, n, w, y;
 if type(x, integer) then RETURN(x) fi;
@@ Line 245: / Line 245: @@
 w := n[nops(n)];
 if type(x, rational) and modp(denom(x), 2)=0 then RETURN(w) fi;
-evalf(w) end:</pre>
+evalf(w) end:
+</pre>
 == Proofs ==