OS: Difference between revisions

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

Its full specification is (n-)OSp: (n pitches of an) otonal sequence adding by rational interval p. An OS is a specific (rational) type of [[AFS]]; the only difference is that the p for an n-AFSp is irrational.

== Specification ==

The "n" is optional. If unspecified, you describe an open-ended sequence~~. By specifying n, your sequence will be equivalent to some [[OD|OD (otonal division)]]. E.g. 8-OS3/4 = 8-OD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7~~.

Its full specification is (n-)OSp: (n pitches of an) otonal sequence adding by rational interval p. The "n" is optional. If unspecified, you describe an open-ended sequence.

~~The analogous undertone equivalent of an OS is a [[US]].~~

== Formula ==

The formula for step <math>k</math> of an OSp is:

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</math>

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 [[OS#Derivation|derivation]].

== Tips ==

The OSp could be read as "1 out of every p harmonics of the harmonic series" (starting with harmonic 1). So OS2 would give the odd harmonics: 1, 3, 5, 7...

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For an example combining specifying the numerator and denominator: if you say OS3/4, in other words 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.

== Relationship to other tunings ==

=== vs. AFS ===

An OS is a specific (rational) type of [[AFS]]; the only difference is that the p for an n-AFSp is irrational.

=== As shifted overtone series ===

Both OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency (for OS it is rational, for AFS it is irrational). 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 [[OS#Derivation|derivation]].

=== vs. OS ===

By specifying n, your OS will be equivalent to some [[OD|OD (otonal division)]]. E.g. 8-OS3/4 = 8-OD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.

=== vs. US ===

The analogous undertone equivalent of an OS is a [[US]].

== Examples ==

{| class="wikitable"

@@ Line 1: / Line 1: @@
 An '''OS''', or '''otonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-Its full specification is (n-)OSp: (n pitches of an) otonal sequence adding by rational interval p. An OS is a specific (rational) type of [[AFS]]; the only difference is that the p for an n-AFSp is irrational.
+== Specification ==
-The "n" is optional. If unspecified, you describe an open-ended sequence. By specifying n, your sequence will be equivalent to some [[OD|OD (otonal division)]]. E.g. 8-OS3/4 = 8-OD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.
+Its full specification is (n-)OSp: (n pitches of an) otonal sequence adding by rational interval p. The "n" is optional. If unspecified, you describe an open-ended sequence.
-The analogous undertone equivalent of an OS is a [[US]].
+== Formula ==
 The formula for step <span><math>k</math></span> of an OSp is:
@@ Line 13: / Line 13: @@
 </math>
-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 [[OS#Derivation|derivation]].
+== Tips ==
 The OSp could be read as "1 out of every p harmonics of the harmonic series" (starting with harmonic 1). So OS2 would give the odd harmonics: 1, 3, 5, 7...
@@ Line 20: / Line 20: @@
 For an example combining specifying the numerator and denominator: if you say OS3/4, in other words 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.
+== Relationship to other tunings ==
+=== vs. AFS ===
+An OS is a specific (rational) type of [[AFS]]; the only difference is that the p for an n-AFSp is irrational.
+=== As shifted overtone series ===
+Both OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency (for OS it is rational, for AFS it is irrational). 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 [[OS#Derivation|derivation]].
+=== vs. OS ===
+By specifying n, your OS will be equivalent to some [[OD|OD (otonal division)]]. E.g. 8-OS3/4 = 8-OD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.
+=== vs. US ===
+The analogous undertone equivalent of an OS is a [[US]].
+== Examples ==
 {| class="wikitable"