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An '''OS''', or '''otonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | An '''OS''', or '''otonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | ||
== Specification == | |||
The "n" is optional. If unspecified, you describe an open-ended sequence. | 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. | ||
== Formula == | |||
The formula for step <span><math>k</math></span> of an OSp is: | |||
<math> | |||
f(k) = 1 + k⋅p | |||
</math> | |||
== 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... | 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 <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. | 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-OSp must be rational. | |||
=== 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 probably 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]]. | |||
Yet another term for this structure is an [[isoharmonic series]]. | |||
=== Vs. OD === | |||
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" | {| class="wikitable" | ||
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! 8 | ! 8 | ||
|- | |- | ||
! frequency (f) | ! frequency (''f'', ratio) | ||
|(4/4) | |(4/4) | ||
|7/4 | |7/4 | ||
| Line 38: | Line 68: | ||
|28/4 | |28/4 | ||
|- | |- | ||
! pitch ( | ! pitch (log₂''f'', octaves) | ||
|(0) | |(0) | ||
|0.81 | |0.81 | ||
| Line 49: | Line 79: | ||
|2.81 | |2.81 | ||
|- | |- | ||
! length (1/f) | ! length (1/''f'', ratio) | ||
|(1/1) | |(1/1) | ||
|4/7 | |4/7 | ||
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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. | 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. | ||
[[Category:Otonality]] | [[Category:Otonality]] | ||
[[Category:Harmonic]] | [[Category:Harmonic]] | ||
[[Category:Harmonic series]] | [[Category:Harmonic series]] | ||
[[Category:Xenharmonic series]] | |||