Equal-step tuning
An equal-step tuning, equal tuning, or equal division (ED) is a periodic tuning system where the distance between adjacent steps is of constant size. The size of this single step is given explicitly (e.g. 88-cent equal tuning) or as a fraction of a larger interval (e.g. 13 equal divisions of the octave). Any interval, rational/just, or irrational, may be used as the basis for an equal tuning, although divisions of the octave are most common, leading to edo systems. When a just interval is equally divided, it is assumed none of the resulting intervals are just, because if the interval has a rational root it is seen as a division of that root.
When a tuning is called n-tone equal temperament (abbreviated n-tet or n-et), this usually means "n divisions of 2/1, the octave, or some approximation thereof", but it also implies a mindset of temperament – that is, of a JI-approximation-based understanding of the scale. If you are wondering how equal divisions of the octave can become associated with temperaments, the page EDOs to ETs may help clarify.
There are many reasons why one might choose to not consider JI approximations when dealing with equal tunings, and thus not treat equal tunings as temperaments. In such case, the less theory-laden term edo (occasionally written ed2), meaning equal divisions of the octave (or equal divisions of 2/1), leaves comparison to JI out of the picture, aside from the octave itself (which is assumed to be just). There are other less standard terms, many in the Tonalsoft Encyclopedia. More generally, the term ed-p can be used, where p is any frequency ratio. For example, the equal-tempered Bohlen-Pierce scale may also be referred to as 13ed3, for 13 equal divisions of 3/1 (the 3rd harmonic).
As the steps are tuned to be equal, equal scales may be taken to close anywhere composers wish them to. Barring the convention of closing equal divisions of particular just intervals at those stated just intervals, there are infinite synonymous names for each equal scale. Barring further the large number of names which would be avoided in discourses on comparative modality and tonality, there is still a a great width to the universe of modes and keys which modal and tonal compositional art can access.
As there are infinitely many intervals, there are infinitely many equal scales. Barring technicalities, there are large quantities of perceivably different equal scales. Seeing such a diverse menagerie at their disposal, some composers choose to combine multiple equal tunings sequentially or simultaneously.
An equal-step tuning is an arithmetic and harmonotonic tuning. In terms of what musical resource is divided, it divides pitch, so it is an equal pitch division (EPD). Because pitch is the overwhelmingly most common musical resource to divide equally, this may be abbreviated to ED, or equal division.
Formula
To find the step size of n-ed-p in terms of cents, divide the cents of p by n. The size s of k steps of n-ed-p (k\n <p>) is
[math]\displaystyle s = 1200 \log_2 (p) \cdot k/n[/math]
To find the step size of n-edo in terms of frequency ratio, take the n-th root of p. For example, the step of 12edo is 21/12 (≈ 1.059). So the ratio c of k steps of n-ed-p is
[math]\displaystyle c = p^{k/n}[/math]
In particular, when k is 0, c is simply 1, because any number to the 0th power is 1. And when k = n, c is simply p, because any number to the 1st power is itself.
Simultaneous equal divisions
What do 12ed2, 19ed3, and 28ed5 all have in common? They are all approximately the same scale. This happens because 12ed2 is an accurate temperament (for its size) that contains relatively close approximations of 3/1 and 5/1. In contrast, 11ed2 does not correspond closely to any equal division of 3/1 or 5/1.
The following plot shows equal divisions of 2/1, 3/1, 5/1, and 7/1, and points out some instances when three or more of them happen to be close together. Note that any equal division of 2/1 is automatically an equal division of 4/1; and if something is simultaneously a good equal division of both 2/1 and 3/1, then it is a good equal division of 6/1 as well.
(Unlimited resolution version: equal.svg)
For the mathematically inclined, this kind of diagram is closely related to the Riemann zeta function.
Catalog of equal-step tunings
Equal divisions
- Ed343/338 (“sooty fox scales” [idiosyncratic term ])
- Ed16/15 (… of the classic diatonic semitone)
- 8 – Delta scale (8ed16/15)
- Ed13/11 (… of the tridecimal minor third)
- Ed5/4 (… of the classic major third)
- Ed666c (“unlucky scales” [idiosyncratic term ])
- EDF (… of the perfect fifth, 3/2)
- most famously Carlos Alpha, Beta and Gamma, but lots of others, too
- Ed777c (“lucky scales” [idiosyncratic term ])
- Ed11/7 (… of the undecimal minor sixth)
- Ed8/5 (… of the classic minor sixth)
- Edφ (… of acoustic phi)
- Ed5/3 (… of the classic major sixth)
- Ed7/4 (… of the septimal minor seventh)
- Ed16/9 (… of the Pythagorean minor seventh)
- Ed9/5 (… of the classic minor seventh)
- Ed15/8 (… of the classic major seventh)
- Ed255/128 (… of the reduced harmonic 255)
- Ed257/128 (… of the reduced harmonic 257)
- EDφφ (… of phi to the phi)
- Ed11/5 (… of the neutral ninth)
- Ed9/4 (… of the major ninth)
- Ed7/3 (… of the septimal minor tenth)
- Ed5/2 (… of the classic major tenth)
- Ed8/3 (… of the perfect eleventh)
- EDN (… of the natave, e/1)
- Ed11/4 (… of the undecimal eleventh)
- EDT (… of the tritave/twelfth, 3/1)
- most famously the Bohlen-Pierce scale, but lots of others, too
- Edπ (… of acoustic pi)
- Ed10/3 (… of the classic major thirteenth)
- Ed7/2 (… of the septimal minor fourteenth)
- Ed11/3 (… of the undecimal neutral fourteenth)
- Ed4 (… of the double octave)
- Ed9/2 (… of the major sixteenth)
- Ed5 (… of the 5th harmonic)
- Ed11/2 (… of the undecimal eighteenth)
- Ed6 (… of the 6th harmonic)
- Ed7 (… of the 7th harmonic)
- Ed8 (… of the 8th harmonic)
- Ed9 (… of the 9th harmonic)
- Ed10 (… of the 10th harmonic)
- Ed11 (… of the 11th harmonic)
- Ed12 (… of the 12th harmonic)
- Ed13 (… of the 13th harmonic)
- Ed100 (… of the 100th harmonic)
Equal multiplications
An equal multiplication of a rational interval can also be called an ambitonal sequence (AS). For example, the 25/24 equal-step tuning could also be written AS25/24. An equal multiplication of an irrational interval can also be called an arithmetic pitch sequence (APS). For example, the 65cET could also be written APS65c. The union of both is equivalent to the unity division of a target interval. For example, AS25/24 is 1ed25/24, and 65cET is 1ed65¢.