# Equal-step tuning

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The equal-step tuning or equal tuning is the 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 temperament) or as a fraction of a larger interval (e.g. 13 equal tones per 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 edn can be used, where n 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 infinite intervals, there are infinite 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.

## 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

• EDF (… of the perfect fifth, 3/2)
• most famously Carlos Alpha, Beta and Gamma, but lots of others, too
• Ed5/3 (… of the classic major sixth)
• Ed7/4 (… of the septimal minor seventh)
• Ed9/5 (… of the classic minor seventh)
• EDO (… of the octave, 2/1)
• 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)
• EDT (… of the tritave/twelfth, 3/1)
• most famously the BP scale, but lots of others, too
• Ed10/3 (… of the classic major thirteenth)
• Ed7/2 (… of the septimal minor fourteenth)
• Ed4 (… of the double octave)
• Ed5 (… of the 5th harmonic)
• 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)

### 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¢.

… of a given cents value
… of a rational interval