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.

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.

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

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]].

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

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...

~~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~~)

And OS(1/p) could be read as "every harmonic but over p" (again, always starting with harmonic 1). For example, OS(1/5) gives <math>\frac 55, \frac 65, \frac 75, \frac 85, etc.</math>

~~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...)~~

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.

~~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]].

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== Derivation ==

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.

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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.
+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.
-(n-)OSp: (n pitches of an) otonal sequence adding by p
+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]].
-OS(1/n) as "1 out of every n harmonics of the harmonic series" (starting with harmonic 1).
+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...
-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)
+And OS(1/p) could be read as "every harmonic but over p" (again, always starting with harmonic 1). For example, OS(1/5) gives <span><math>\frac 55, \frac 65, \frac 75, \frac 85, etc.</math></span>
-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...)
+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.
-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]].
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 == Derivation ==
 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.