OS: Difference between revisions

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An '''OS''', or '''otonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] tuning.

An OS is a specific (rational) type of AFS.

(n-)OSp: (n pitches of an) otonal sequence adding by p

OS(1/n) as "1 out of every n harmonics of the harmonic series" (starting with harmonic 1).

That way I could just say AFS4 and that gives me harmonics 1 5 9 13 ... (though since that ones rational we'd prefer IOS4 for iso-otonal sequence)

If you wanted the iso-otonal sequence going by 4's but starting on 3 instead, you'd just need AFS(4/3). (Technically that'd give you 3/3 7/3 11/3 15/3...)

Consider AFS(3/4). That means start on 4/3 and move by 1's, so 4/3, 1+(4/3), 2+(4/3), 3+(4/3), etc. which becomes 4/4, 7/4, 10/4, 13/4... In other words, move by 3's, start on 4.

thats the way it works for 7OD3

1+(1/7*2)=~1.28571

1+(2/7*2)=~1.57143

For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <math>1, 1\frac 34, 2\frac 24, 3\frac14</math>, which is equivalent to <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math>, or in other words, a class iii [[isoharmonic_chords|isoharmonic]] tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4.

OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[Monotonic tunings#Derivation of OS|derivation of OS]].

== Derivation of OS ==

The tuning OS3/4 is the sequence <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}...</math> and so on. Any OS is equivalent to shifting the overtone series by a constant amount of frequency. In the case of OS3/4, it is a shift by <math>\frac 13</math>. Let's show how.

Begin with the overtone series:

Shift it by <math>\frac 13</math>:

<math>

1\frac 13, 2\frac 13, 3\frac 13, 4\frac 13... \\

</math>

Convert to improper fractions by first expanding the whole number:

<math>

\frac 33 + \frac 13, \frac 63 + \frac 13, \frac 93 + \frac 13, \frac {12}{3} + \frac 13... \\

</math>

...then consolidating numerators:

<math>

\frac 43, \frac 73, \frac{10}{3}, \frac{13}{3}...

</math>

Resize to start at <math>\frac 11</math> by multiplying every term by the reciprocal of the first term, <math>\frac 43</math>, which is <math>\frac 34</math>:

<math>

\frac 43 \cdot \frac 34, \frac 73 \cdot \frac 34, \frac{10}{3} \cdot \frac 34, \frac{13}{3} \cdot \frac 34...

</math>

Cancel out:

<math>

\frac{4}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{7}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{10}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{13}{\cancel{3}} \cdot \frac{\cancel{3}}{4}...

</math>

And we've arrived:

<math>

\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}...

</math>

So we can see that <math>\frac 13</math> was the right amount to shift by because it is the delta from the starting position <math>1</math> to <math>\frac 43</math>, the latter of which is the reciprocal of the target step size <math>\frac 34</math> and therefore the value that we need the starting position to equal in order to be sent ''back'' to <math>1</math> when we resize all steps from 1 to the target step size by multiplying everything by it.

@@ Line 1: / Line 1: @@
 An '''OS''', or '''otonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] tuning.
+An OS is a specific (rational) type of AFS.
+(n-)OSp: (n pitches of an) otonal sequence adding by p
+OS(1/n) as "1 out of every n harmonics of the harmonic series" (starting with harmonic 1).
+That way I could just say AFS4 and that gives me harmonics 1 5 9 13 ... (though since that ones rational we'd prefer IOS4 for iso-otonal sequence)
+If you wanted the iso-otonal sequence going by 4's but starting on 3 instead, you'd just need AFS(4/3). (Technically that'd give you 3/3 7/3 11/3 15/3...)
+Consider AFS(3/4). That means start on 4/3 and move by 1's, so 4/3, 1+(4/3), 2+(4/3), 3+(4/3), etc. which becomes 4/4, 7/4, 10/4, 13/4... In other words, move by 3's, start on 4.
+thats the way it works for 7OD3
++(1/7*2)=~1.28571
++(2/7*2)=~1.57143
+For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <span><math>1, 1\frac 34, 2\frac 24, 3\frac14</math><span>, which is equivalent to <span><math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math></span>, or in other words, a class iii [[isoharmonic_chords|isoharmonic]] tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4.
+OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[Monotonic tunings#Derivation of OS|derivation of OS]].
+== Derivation of OS ==
+The tuning OS3/4 is the sequence <span><math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}...</math></span> and so on. Any OS is equivalent to shifting the overtone series by a constant amount of frequency. In the case of OS3/4, it is a shift by <span><math>\frac 13</math></span>. Let's show how.
+Begin with the overtone series:
+<math>1, 2, 3, 4...</math>
+Shift it by <span><math>\frac 13</math></span>:
+<math>
+\frac 13, 2\frac 13, 3\frac 13, 4\frac 13... \\
+</math>
+Convert to improper fractions by first expanding the whole number:
+<math>
+\frac 33 + \frac 13, \frac 63 + \frac 13, \frac 93 + \frac 13, \frac {12}{3} + \frac 13... \\
+</math>
+...then consolidating numerators:
+<math>
+\frac 43, \frac 73, \frac{10}{3}, \frac{13}{3}...
+</math>
+Resize to start at <span><math>\frac 11</math></span> by multiplying every term by the reciprocal of the first term, <span><math>\frac 43</math><span>, which is <span><math>\frac 34</math><span>:
+<math>
+\frac 43 \cdot \frac 34, \frac 73 \cdot \frac 34, \frac{10}{3} \cdot \frac 34, \frac{13}{3} \cdot \frac 34...
+</math>
+Cancel out:
+<math>
+\frac{4}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{7}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{10}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{13}{\cancel{3}} \cdot \frac{\cancel{3}}{4}...
+</math>
+And we've arrived:
+<math>
+\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}...
+</math>
+So we can see that <span><math>\frac 13</math></span> was the right amount to shift by because it is the delta from the starting position <span><math>1</math></span> to <span><math>\frac 43</math></span>, the latter of which is the reciprocal of the target step size <span><math>\frac 34</math></span> and therefore the value that we need the starting position to equal in order to be sent ''back'' to <span><math>1</math></span> when we resize all steps from 1 to the target step size by multiplying everything by it.