OS

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

Specification

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 [math]\displaystyle{ k }[/math] of an OSp is:

[math]\displaystyle{ 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...

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.

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 derivation.

Yet another term for this structure is an isoharmonic series.

Vs. OD

By specifying n, your OS 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.

Vs. US

The analogous undertone equivalent of an OS is a US.

Examples

example: 8-OS(3/4)

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