Metallic MOS: Difference between revisions

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But 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.

The met-MOS concepts discussed here are abstract — unrelated to the octave, or neutral thirds, or any other size to which we might assign the period or generator. Being abstract, they only depend on the ratio between the period and the generator. Therefore we can set one of these two values to 1, for convenience, so that we only have to worry about the value of the other. We’ll be conforming here with the convention of setting the period to 1.~~<math>φ</math>~~Since no other types of scales besides MOS scales will be discussed here, we can assume MOS scale whenever we write “scale”.

The met-MOS concepts discussed here are abstract — unrelated to the octave, or neutral thirds, or any other size to which we might assign the period or generator. Being abstract, they only depend on the ratio between the period and the generator. Therefore we can set one of these two values to 1, for convenience, so that we only have to worry about the value of the other. We’ll be conforming here with the convention of setting the period to 1. Since no other types of scales besides MOS scales will be discussed here, we can assume MOS scale whenever we write “scale”.

MOS concepts are logarithmic, not acoustic. 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¢), have interesting properties too — creating recursive combination tones, for example — but these musical applications of metallic means will not be discussed here.

= Behavior =

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=== Noble cases ===

Another way to think about the period is the interval from <math>\frac 01</math> to <math>\frac 11</math>. We can find slightly more complex metallic generators by choosing an interval other than the entire period to split into two segments related by <math>φ</math>. For example, we could pick <math>\frac 13</math> to <math>\frac 12</math>, giving us approximately <math>0.419821</math>:

Another way to think about the period is the interval from <math>\frac 01</math> to <math>\frac 11</math> (these are not frequency ratios, but just another way of writing 0 and 1, the motivation for which will become clear soon). We can find slightly more complex metallic generators by choosing an interval other than the entire period to split into two segments related by <math>φ</math>. For example, we could pick <math>\frac 13</math> to <math>\frac 12</math>, giving us approximately <math>0.419821</math>:

[[File:Noble generator.png|618x618px]]

@@ Line 12: / Line 12: @@
 But  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.
-The met-MOS concepts discussed here are abstract — unrelated to the octave, or neutral thirds, or any other size to which we might assign the period or generator. Being abstract, they only depend on the ratio between the period and the generator. Therefore we can set one of these two values to 1, for convenience, so that we only have to worry about the value of the other. We’ll be conforming here with the convention of setting the period to 1.<span><math>φ</math></span>Since no other types of scales besides MOS scales will be discussed here, we can assume MOS scale whenever we write “scale”.
+The met-MOS concepts discussed here are abstract — unrelated to the octave, or neutral thirds, or any other size to which we might assign the period or generator. Being abstract, they only depend on the ratio between the period and the generator. Therefore we can set one of these two values to 1, for convenience, so that we only have to worry about the value of the other. We’ll be conforming here with the convention of setting the period to 1. Since no other types of scales besides MOS scales will be discussed here, we can assume MOS scale whenever we write “scale”.
-MOS concepts are logarithmic, not acoustic. 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¢), have interesting properties too — creating recursive combination tones, for example — but these musical applications of metallic means will not be discussed here.
 = Behavior =
@@ Line 32: / Line 32: @@
 === Noble cases ===
-Another way to think about the period is the interval from <span><math>\frac 01</math></span> to <span><math>\frac 11</math></span>. We can find slightly more complex metallic generators by choosing an interval other than the entire period to split into two segments related by <span><math>φ</math></span>. For example, we could pick <span><math>\frac 13</math></span> to <span><math>\frac 12</math></span>, giving us approximately <span><math>0.419821</math></span>:
+Another way to think about the period is the interval from <span><math>\frac 01</math></span> to <span><math>\frac 11</math></span> (these are not frequency ratios, but just another way of writing 0 and 1, the motivation for which will become clear soon). We can find slightly more complex metallic generators by choosing an interval other than the entire period to split into two segments related by <span><math>φ</math></span>. For example, we could pick <span><math>\frac 13</math></span> to <span><math>\frac 12</math></span>, giving us approximately <span><math>0.419821</math></span>:
 [[File:Noble generator.png|618x618px]]