Metallic MOS: Difference between revisions

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But phi is only the first of an infinite sequence of such metallic means which can be used to generate scales offering interesting musical possibilities. And while some attention has been given to silver scales, what we seek to do here is centralize all met-MOS knowledge and generalize principles across all of the metallic means.

MOS concepts are logarithmic, not acoustic. In other words, we are not dealing with frequency ratios here. Frequency ratios related to metallic means, such as "[[acoustic phi]]" (approximately 833.~~09¢~~), have interesting properties too—creating recursive combination tones, for example—but these musical applications of metallic means will not be discussed here.

MOS concepts are logarithmic, not acoustic. In other words, we are not dealing with frequency ratios here. Frequency ratios related to metallic means, such as "[[acoustic phi]]" (approximately 833.09{{c}}), have interesting properties too—creating recursive combination tones, for example—but these musical applications of metallic means will not be discussed here.

The met-MOS concepts discussed here may be used to define generators as fractions of an octave, as is most common. But these concepts are more abstract than that, and may be used to define generators as fractions of ''any'' period. As such, they only depend on the ratio between the generator and the period, and so for convenience we can lock one of these two values to 1 and only vary the value of the other. We'll be conforming here with the convention of choosing the period as the interval to lock to 1.

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As for why we pick the lower half of the period rather than the upper half, this is somewhat arbitrary, but it seems objectively simpler to keep our lower bound at 0.

Sure, depending on the context, the generator complement greater than 0.5 may be the one we want to describe our scale in terms of. For example, we may be thinking of the generator as the perfect fifth instead of the perfect fourth. Or we may want to use 0.618034 instead of its complement 0.381966 (we've been using the latter and calling it the golden generator, but some readers may be more familiar with the former, known as "[[logarithmic phi]]", which is 741.~~64¢~~ when the period is an octave). But for purposes of cataloging we prefer the smaller, or ''reduced'' of the two complements.

Sure, depending on the context, the generator complement greater than 0.5 may be the one we want to describe our scale in terms of. For example, we may be thinking of the generator as the perfect fifth instead of the perfect fourth. Or we may want to use 0.618034 instead of its complement 0.381966 (we've been using the latter and calling it the golden generator, but some readers may be more familiar with the former, known as "[[logarithmic phi]]", which is 741.64{{c}} when the period is an octave). But for purposes of cataloging we prefer the smaller, or ''reduced'' of the two complements.

And this is a subtle point, but it's another reason to prefer leaning intervals parentward. We have a potential problem: we don't want to find generators <math>> 0.5</math>. Almost every interval we include does not even allow for that possibility, but one interval does threaten this: the interval {{sfrac|0|1}} to {{sfrac|1|1}}. We include this interval because it occupies space between {{sfrac|0|1}} and {{sfrac|1|2}}—so has potential to find useful generators—but we have to be careful with it to avoid finding generators <math>> 0.5</math>. The method for this is simple. First, note that the unweighted mediant in the interval {{sfrac|0|1}} to {{sfrac|1|1}} is {{sfrac|1|2}}, or exactly 0.5. So if we want to avoid generators <math>> 0.5</math>, all we must do is make sure to weight more toward {{sfrac|0|1}}. Since of these two ratios {{sfrac|0|1}} and {{sfrac|1|1}}, the parent ratio is {{sfrac|0|1}}, weighting parentward is the solution.

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[[File:MOS iteration rules for L and s.png|452x452px]]

We are reasoning about MOS concepts in the abstract here. These truths about large and small steps are true whether they are ~~100¢~~ or 4516.8¢, and all we really care about are their ratios. So if we treat our small steps' size as 1 then we can treat our large steps' size as equal to the ''L'':''s'' ratio.

We are reasoning about MOS concepts in the abstract here. These truths about large and small steps are true whether they are 100{{c}} or 4516.8{{c}}, and all we really care about are their ratios. So if we treat our small steps' size as 1 then we can treat our large steps' size as equal to the ''L'':''s'' ratio.

So the ''L'':''s'' ratio decreases by 1 because if an ''s''-sized chunk has been sliced off ''L'', and ''s''{{`s}} size is 1, then 1 should be subtracted from ''L''.

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=== Why they cycle ===

That is true of scale iterations where {{nowrap|''L'' − ''s'' > s}}. For the other type of scale iteration, where {{nowrap|''L'' − ''s'' < s}}, the result is simply reciprocated:

That is true of scale iterations where {{nowrap|''L'' − ''s'' > ''s''}}. For the other type of scale iteration, where {{nowrap|''L'' − ''s'' < ''s''}}, the result is simply reciprocated:

<math>

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Find any ''s'' in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, ''s'' will be replaced with ''L''. After two iterations, the original ''s'' interval is now represented by an ''L'' and an ''s''. And so forth.

Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier ''L''{{'s}} and s's that remain—only now spanning many ''L''{{'s}} and ''s''{{`s}} each—are precisely the larger intervals in the scale that also exhibit the φ ratio to each other.

Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier ''L''{{'s}} and ''s''{{`s}} that remain—only now spanning many ''L''{{'s}} and ''s''{{`s}} each—are precisely the larger intervals in the scale that also exhibit the φ ratio to each other.

=== Beyond golden cases ===

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</math>

[[File:Silver horogram.png|alt=horogram for g = 0.292893, 7 iterations|horogram for g = 0.292893, 7 iterations|none|frame]]

[[File:Silver horogram.png|alt=horogram for ''g'' = 0.292893, 7 iterations|horogram for {{nowrap|''g'' {{=}} 0.292893}}, 7 iterations|none|frame]]

We can use the horogram for the silver generator to see how its interval pattern cycle is length 2, i.e. that it alternates between two different interval patterns. If we want to understand the interval pattern for δ<sub>''s''</sub>, we'll look at the right and left sides separately, as we did with the golden:

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</math>

As expected, {{nowrap|''L'':''s'' {{=}} (3''L'' + s):L}} is only true of every ''third'' scale the bronze generator generates. The remaining interval relationships are left as an exercise for the reader.

As expected, {{nowrap|''L'':''s'' {{=}} (3''L'' + s):''L''}} is only true of every ''third'' scale the bronze generator generates. The remaining interval relationships are left as an exercise for the reader.

= Gallery of generators =

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== Non-Octave Golden ==

Golden lemba is a Multi-MOS, fitting two periods into an octave (~~600¢~~ period, 229.~~179¢~~ generator).

Golden lemba is a Multi-MOS, fitting two periods into an octave (600{{c}} period, 229.179{{c}} generator).

Golden triforce similarly fits three periods into an octave (~~400¢~~ period, 152.~~786¢~~ generator).

Golden triforce similarly fits three periods into an octave (400{{c}} period, 152.786{{c}} generator).

== Argent Temperament ==

[[File:Argent horogram.png|alt=horogram for g = 0.414214, 7 iterations|right|388x388px]]

[[File:Argent horogram.png|alt=horogram for {{nowrap|''g'' {{=}} 0.414214}}, 7 iterations|right|388x388px]]

Scales based on the bronze mean and metallic means beyond it have not been extensively explored. However, the silver mean has gotten some attention.

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Which we can see is the weight multiplied by the bounding ratios' denominators:

<math> 1.078674 = \phi ~~·~~ \frac 23

<math> 1.078674 = \phi \cdot \frac{2}{3}

</math>

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! 4th Metallic mean or isotope

|-

~~| '''~~1~~'''~~

! 1

| 1

Line 1,099:

| 1

|-

~~| '''~~2~~'''~~

! 2

| 1*

| 2

Line 1,105:

| 2

|-

~~| '''~~3~~'''~~

! 3

| 2

| 3*

Line 1,111:

| 4

|-

~~| '''~~4~~'''~~

! 4

| 4

| 6

Line 1,117:

| 8

|-

~~| '''~~5~~'''~~

! 5

| 8

| 12

Line 1,123:

| 15*

|-

~~| '''~~6~~'''~~

! 6

| 16

| 24

Line 1,129:

| 30

|-

~~| '''~~7~~'''~~

! 7

| 32

| 48

Line 1,180:

| 32

|-

~~| '''~~6~~'''~~

! 6

| 16

| 32

Line 1,186:

| 64

|-

~~| '''~~7~~'''~~

! 7

| 32

| 64

@@ Line 13: / Line 13: @@
 But phi is only the first of an infinite sequence of such metallic means which can be used to generate scales offering interesting musical possibilities. And while some attention has been given to silver scales, what we seek to do here is centralize all met-MOS knowledge and generalize principles across all of the metallic means.
-MOS concepts are logarithmic, not acoustic. In other words, we are not dealing with frequency ratios here. Frequency ratios related to metallic means, such as "[[acoustic phi]]" (approximately 833.09¢), have interesting properties too—creating recursive combination tones, for example—but these musical applications of metallic means will not be discussed here.
+MOS concepts are logarithmic, not acoustic. In other words, we are not dealing with frequency ratios here. Frequency ratios related to metallic means, such as "[[acoustic phi]]" (approximately 833.09{{c}}), have interesting properties too—creating recursive combination tones, for example—but these musical applications of metallic means will not be discussed here.
 The met-MOS concepts discussed here may be used to define generators as fractions of an octave, as is most common. But these concepts are more abstract than that, and may be used to define generators as fractions of ''any'' period. As such, they only depend on the ratio between the generator and the period, and so for convenience we can lock one of these two values to 1 and only vary the value of the other. We'll be conforming here with the convention of choosing the period as the interval to lock to 1.
@@ Line 402: / Line 402: @@
 As for why we pick the lower half of the period rather than the upper half, this is somewhat arbitrary, but it seems objectively simpler to keep our lower bound at 0.
-Sure, depending on the context, the generator complement greater than 0.5 may be the one we want to describe our scale in terms of. For example, we may be thinking of the generator as the perfect fifth instead of the perfect fourth. Or we may want to use 0.618034 instead of its complement 0.381966 (we've been using the latter and calling it the golden generator, but some readers may be more familiar with the former, known as "[[logarithmic phi]]", which is 741.64¢ when the period is an octave). But for purposes of cataloging we prefer the smaller, or ''reduced'' of the two complements.
+Sure, depending on the context, the generator complement greater than 0.5 may be the one we want to describe our scale in terms of. For example, we may be thinking of the generator as the perfect fifth instead of the perfect fourth. Or we may want to use 0.618034 instead of its complement 0.381966 (we've been using the latter and calling it the golden generator, but some readers may be more familiar with the former, known as "[[logarithmic phi]]", which is 741.64{{c}} when the period is an octave). But for purposes of cataloging we prefer the smaller, or ''reduced'' of the two complements.
 And this is a subtle point, but it's another reason to prefer leaning intervals parentward. We have a potential problem: we don't want to find generators <math>> 0.5</math>. Almost every interval we include does not even allow for that possibility, but one interval does threaten this: the interval {{sfrac|0|1}} to {{sfrac|1|1}}. We include this interval because it occupies space between {{sfrac|0|1}} and {{sfrac|1|2}}—so has potential to find useful generators—but we have to be careful with it to avoid finding generators <math>> 0.5</math>. The method for this is simple. First, note that the unweighted mediant in the interval {{sfrac|0|1}} to {{sfrac|1|1}} is {{sfrac|1|2}}, or exactly 0.5. So if we want to avoid generators <math>> 0.5</math>, all we must do is make sure to weight more toward {{sfrac|0|1}}. Since of these two ratios {{sfrac|0|1}} and {{sfrac|1|1}}, the parent ratio is {{sfrac|0|1}}, weighting parentward is the solution.
@@ Line 416: / Line 416: @@
 [[File:MOS iteration rules for L and s.png|452x452px]]
-We are reasoning about MOS concepts in the abstract here. These truths about large and small steps are true whether they are 100¢ or 4516.8¢, and all we really care about are their ratios. So if we treat our small steps' size as 1 then we can treat our large steps' size as equal to the ''L'':''s'' ratio.
+We are reasoning about MOS concepts in the abstract here. These truths about large and small steps are true whether they are 100{{c}} or 4516.8{{c}}, and all we really care about are their ratios. So if we treat our small steps' size as 1 then we can treat our large steps' size as equal to the ''L'':''s'' ratio.
 So the ''L'':''s'' ratio decreases by 1 because if an ''s''-sized chunk has been sliced off ''L'', and ''s''{{`s}} size is 1, then 1 should be subtracted from ''L''.
@@ Line 429: / Line 429: @@
 === Why they cycle ===
-That is true of scale iterations where {{nowrap|''L'' − ''s'' &gt; s}}. For the other type of scale iteration, where {{nowrap|''L'' − ''s'' &lt; s}}, the result is simply reciprocated:
+That is true of scale iterations where {{nowrap|''L'' − ''s'' &gt; ''s''}}. For the other type of scale iteration, where {{nowrap|''L'' − ''s'' &lt; ''s''}}, the result is simply reciprocated:
 <math>
@@ Line 758: / Line 758: @@
 Find any ''s'' in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, ''s'' will be replaced with ''L''. After two iterations, the original ''s'' interval is now represented by an ''L'' and an ''s''. And so forth.
-Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier ''L''{{'s}} and s's that remain—only now spanning many ''L''{{'s}} and ''s''{{`s}} each—are precisely the larger intervals in the scale that also exhibit the φ ratio to each other.
+Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier ''L''{{'s}} and ''s''{{`s}} that remain—only now spanning many ''L''{{'s}} and ''s''{{`s}} each—are precisely the larger intervals in the scale that also exhibit the φ ratio to each other.
 === Beyond golden cases ===
@@ Line 804: / Line 804: @@
 </math>
-[[File:Silver horogram.png|alt=horogram for g = 0.292893, 7 iterations|horogram for g = 0.292893, 7 iterations|none|frame]]
+[[File:Silver horogram.png|alt=horogram for ''g'' = 0.292893, 7 iterations|horogram for {{nowrap|''g'' {{=}} 0.292893}}, 7 iterations|none|frame]]
 We can use the horogram for the silver generator to see how its interval pattern cycle is length 2, i.e. that it alternates between two different interval patterns. If we want to understand the interval pattern for δ<sub>''s''</sub>, we'll look at the right and left sides separately, as we did with the golden:
@@ Line 873: / Line 873: @@
 </math>
-As expected, {{nowrap|''L'':''s'' {{=}} (3''L'' + s):L}} is only true of every ''third'' scale the bronze generator generates. The remaining interval relationships are left as an exercise for the reader.
+As expected, {{nowrap|''L'':''s'' {{=}} (3''L'' + s):''L''}} is only true of every ''third'' scale the bronze generator generates. The remaining interval relationships are left as an exercise for the reader.
 = Gallery of generators =
@@ Line 932: / Line 932: @@
 == Non-Octave Golden ==
-Golden lemba is a Multi-MOS, fitting two periods into an octave (600¢ period, 229.179¢ generator).
+Golden lemba is a Multi-MOS, fitting two periods into an octave (600{{c}} period, 229.179{{c}} generator).
-Golden triforce similarly fits three periods into an octave (400¢ period, 152.786¢ generator).
+Golden triforce similarly fits three periods into an octave (400{{c}} period, 152.786{{c}} generator).
 == Argent Temperament ==
-[[File:Argent horogram.png|alt=horogram for g = 0.414214, 7 iterations|right|388x388px]]
+[[File:Argent horogram.png|alt=horogram for {{nowrap|''g'' {{=}} 0.414214}}, 7 iterations|right|388x388px]]
 Scales based on the bronze mean and metallic means beyond it have not been extensively explored. However, the silver mean has gotten some attention.
@@ Line 1,031: / Line 1,031: @@
 Which we can see is the weight multiplied by the bounding ratios' denominators:
-<math> 1.078674 = \phi &#xB7; \frac 23
+<math> 1.078674 = \phi \cdot \frac{2}{3}
 </math>
@@ Line 1,093: / Line 1,093: @@
 ! 4th Metallic mean or isotope
 |-
-| '''1'''
+! 1
 | 1
 | 1
@@ Line 1,099: / Line 1,099: @@
 | 1
 |-
-| '''2'''
+! 2
 | 1*
 | 2
@@ Line 1,105: / Line 1,105: @@
 | 2
 |-
-| '''3'''
+! 3
 | 2
 | 3*
@@ Line 1,111: / Line 1,111: @@
 | 4
 |-
-| '''4'''
+! 4
 | 4
 | 6
@@ Line 1,117: / Line 1,117: @@
 | 8
 |-
-| '''5'''
+! 5
 | 8
 | 12
@@ Line 1,123: / Line 1,123: @@
 | 15*
 |-
-| '''6'''
+! 6
 | 16
 | 24
@@ Line 1,129: / Line 1,129: @@
 | 30
 |-
-| '''7'''
+! 7
 | 32
 | 48
@@ Line 1,180: / Line 1,180: @@
 | 32
 |-
-| '''6'''
+! 6
 | 16
 | 32
@@ Line 1,186: / Line 1,186: @@
 | 64
 |-
-| '''7'''
+! 7
 | 32
 | 64