OS

An OS, or otonal sequence, is a kind of arithmetic and 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.

The "n" is optional. If unspecified, you describe an open-ended sequence. By specifying n, your sequence will be equivalent to some 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.

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 derivation.

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...

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]\displaystyle{ \frac 55, \frac 65, \frac 75, \frac 85, etc. }[/math]

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]\displaystyle{ 1, 1\frac 34, 2\frac 24, 3\frac14 }[/math], which is equivalent to [math]\displaystyle{ \frac 44, \frac 74, \frac{10}{4}, \frac{13}{4} }[/math], or in other words, a class iii isoharmonic tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4.

example: 8-OS(3/4)

quantity	(0)	1	2	3	4	5	6	7	8
frequency (f)	(4/4)	7/4	10/4	13/4	16/4	19/4	22/4	25/4	28/4
pitch (log₂f)	(0)	0.81	1.32	1.70	2.00	2.25	2.46	2.64	2.81
length (1/f)	(1/1)	4/7	2/5	4/13	1/4	4/19	2/11	4/25	1/7

Derivation

The tuning OS3/4 is the sequence [math]\displaystyle{ \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]\displaystyle{ \frac 13 }[/math]. Let's show how.

Begin with the overtone series:

[math]\displaystyle{ 1, 2, 3, 4... }[/math]

Shift it by [math]\displaystyle{ \frac 13 }[/math]:

[math]\displaystyle{ 1\frac 13, 2\frac 13, 3\frac 13, 4\frac 13... \\ }[/math]

Convert to improper fractions by first expanding the whole number:

[math]\displaystyle{ \frac 33 + \frac 13, \frac 63 + \frac 13, \frac 93 + \frac 13, \frac {12}{3} + \frac 13... \\ }[/math]

...then consolidating numerators:

[math]\displaystyle{ \frac 43, \frac 73, \frac{10}{3}, \frac{13}{3}... }[/math]

Resize to start at [math]\displaystyle{ \frac 11 }[/math] by multiplying every term by the reciprocal of the first term, [math]\displaystyle{ \frac 43 }[/math], which is [math]\displaystyle{ \frac 34 }[/math]:

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}... }[/math]

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