Arithmetic tuning

An arithmetic tuning is one which has equal step sizes of any kind of quantity, whether that be pitch, frequency, or length (of the resonating entity producing the sound).

This sense of the word arithmetic comes from the mathematical concept of an arithmetic sequence, which means a sequence with a constant difference between successive terms.

All arithmetic tunings are harmonotonic tunings.

Types

Equal frequency steps:

Equal pitch steps:

Equal length steps:

Basic examples

Basic examples of arithmetic tunings:

1. the overtone series has equal steps of frequency (1, 2, 3, 4, etc.; adding 1 each step)
2. any edo has equal steps of pitch (12edo goes 0\12, 1\12, 2\12, 3\12, etc.; adding 1\12 each step)
3. the undertone series has equal steps of length (to play the first four steps of the undertone series you would pluck the whole length of a string, then 3/4 the string, then 2/4, then 1/4; adding -1/4 length each step)

Sequences

Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning $1, 1\frac 34, 2\frac 24, 3\frac14$, which is equivalent to $\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}$, 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.

If the new step size is irrational, the tuning is no longer JI, so we use a different acronym to distinguish it: AFS, for arithmetic frequency sequence. For example, if we wanted to move by steps of φ, like this: $1, 1+φ, 1+2φ, 1+3φ...$ etc. we could have the AFSφ.

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 of OS.

The same principles that were just described for frequency are also possible for length: by varying the undertone series step size to some rational number you can produce a utonal sequence (US), and varying it to an irrational number you can produce an arithmetic length sequence (ALS). In other words, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but their relationship in pitch changes.

Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.

Divisions

So far we have looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having equal step sizes of some kind of quantity (frequency, pitch, or length), then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind of quantity.

The most common example of this type of tuning is 12edo, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12epdo, for 12 equal pitch divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12edo is the better name).

But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by frequency, or length. In the former case, you will have 12efdo, and in the latter case, you will have 12eldo. However, that is not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD and ELD are typically reserved for irrational tunings, such as 12efdφ. So it would be more appropriate to name these two tunings 12odo and 12udo, for otonal divisions of the octave and utonal divisions of the octave, respectively.

Comparing arithmetic tunings

We can state a few helpful analogies:

(n-)AQSp : n-EQDp

An arithmetic sequence of some kind of quantity Q is analogous to an equal division. Both require an interval p to be specified. The key difference is that a sequence is potentially open-ended — proceeding forever without repeating (such as the overtone series) — so its parameter n, for the total number of pitches, is optional.

OS : AFS :: OD : EFD

AS : APS :: __ : EPD

US : ALS :: UD : ELD

Each of these rows has the form rational sequence : irrational division :: rational division : irrational division. The first row is for frequency, the second for pitch, and the third for length.

We haven't looked in detail at the middle row, for pitch. EPD, again, is long for simply ED. AS stands for ambitonal sequence; these are sequences which are rational but ambiguous between otonality and utonality, such as a chain of the same JI pitch. There is one blank space in the system of analogies for rational divisions of pitch; these are theoretically impossible.

Every rational arithmetic tuning is a subtype of its corresponding irrational arithmetic tuning:

• An OS is a specific (rational) type of AFS.
• An OD is a specific (rational) type of EFD.
• An AS is a specific (rational) type of APS.
• A US is a specific (rational) type of ALS.
• A UD is a specific (rational) type of ELD.

It is convenient that the three basic types of divisions — OD, ED, and UD — all begin with vowels. As do the three rational types of sequences — OS, AS, and US.