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{{Distinguish|Distributional evenness}}
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A [[periodic scale|periodic]] [[binary scale]] is '''maximally even''' ('''ME''') with respect to an [[equal-step tuning]] if its two [[step]] sizes are [[Distributional evenness|evenly distributed]] within its [[step pattern]] and differ by exactly one step of the parent tuning. In other words, such a scale satisfies the property of '''maximal evenness'''. The first condition implies that ME scales are [[MOS scale]]s, and the second condition implies that the scale's [[step ratio]] is [[superparticular]].

In particular, within every [[edo]], one can specify such a scale for every smaller number of notes. In terms of sub-edo representation, a maximally even scale is the closest the parent edo can get to representing the smaller edo.

== Mathematics ==
Mathematically, if ''n'' < ''m'', a ''maximally even (sub)set of size n'' in '''Z'''/''m'''''Z''' is any translate of the set

<math>\operatorname{ME}(n, m) = \{ \mathbb{Z} + \lfloor \frac{im}{n} \rfloor : i \in \{1, ..., n\} \} \subseteq \mathbb{Z}/m\mathbb{Z},</math>

where the [[Wikipedia:Floor and ceiling functions|floor]] function fixes integers and rounds down non-integers to the next lower integer. It can be proven that ME(''n'', ''m'') is a [[MOS scale]] where the two step sizes differ by exactly 1\''m''⟨''E''⟩ when interpreted as a subset of ''m''-ed''E'', and that the indices for the two step sizes are themselves ME when considered as subsets of ''n''-ed''E'', satisfying the informal definition above.

From the MOS theory standpoint, the generator of the scale is a modular multiplicative inverse of it's number of notes and the EDO size. Maximal evenness scale whose generator is equal to it's note amount is called [[concoctic]]. Major and minor scales in standard Western music are such - the generator is a perfect fifth of 7 semitones, as inferred through Pythagorean tuning, and the scale has 7 notes in it.

== Sound perception ==
The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31st of an octave instead of one 13th).

The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo.

Maximally even sets tend to be familiar and musically relevant scale collections. Examples:

* The maximally even heptatonic set of [[19edo]] is, like the one in [[12edo]], a [[5L 2s|diatonic scale]].
* The maximally even heptatonic sets of [[17edo|17edo]] and [[24edo|24edo]], in contrary, are [[4L 3s|mosh scales]] (Neutrominant[7]).
* The maximally even heptatonic set of [[22edo|22edo]] is Porcupine[7] (the superpythagorean diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is the octatonic scale of Hedgehog, the maximally even nonatonic set of 22edo is Orwell[9], (as well as 13-tonic being an Orwell[13]), while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara.
* The maximally even 13-element set in 24edo is Ivan Wyschnegradsky's diatonicized chromatic scale.
* The maximally even sets in edos 40 and higher have step sizes so close together that they can sound like [[circulating temperament]]s with the right timbre.

Note that "maximally even" is equivalent to "quasi-equal-interval-symmetrical" in [[Joel Mandelbaum]]'s 1961 thesis [http://www.anaphoria.com/mandelbaum.html Multiple Divisions of the Octave and the Tonal Resources of 19-Tone Temperament]. Previous versions of this article have conflated "quasi-equal" with "quasi-equal-interval symmetrical". In fact, "quasi-equal" scales, according to Mandelbaum, meet the first criterion listed above, but not necessarily the second.

[[Irvian mode]] is a specific mode of the scale, where the notes are also symmetrically arranged. For example, the major mode of the basic diatonic scale from 12edo, <code>2 2 1 2 2 2 1</code>, is not only a maximally even scale, but also the Irvian mode of such scale. Such a mode is best shown in odd EDOs, which truly have a "middle" note owing to being odd, and therefore allowing for true symmetric arrangements of notes.

== Discovery of temperaments with a given generator ==
Maximum evenness scales' generator and amount of notes follow the formula LU mod N = 1, where L is the note amount per period, U is the generator, and N is the EDO's cardinality. Note: L and U have to be coprime for the period to be 1 octave. 

As such, it's possible to discover a temperament with a given generator in a given EDO simply by [[temperament merging]] the amount of notes with the EDO's cardinality.

=== Example 1: 12edo's diatonic ===
Generator of 12edo's diatonic is 7\12, as is the amount of notes. As such, we simply carry out 7 & 12 to find the desired temperament. In 5-limit, that's meantone, tempering out 81/80, and consistent with world musical practices today.

=== Example 2: 37edo's 11/8 ===
Let's say we want to see what would repeatedly stacking 11th harmonic do well in all of 11-limit, in an EDO that presents it well.

11/8 amounts to 17 steps of 37edo, and the solution to the problem 17*x mod 1 = 37 is 24, meaning if the generator is 11/8, we are dealing with a 24 tone maximally even scale. As such, the temperament we are looking for is 24 & 37, which can be interpreted as [[freivald]] or [[emka]].

=== Example 3: On-request maximum evenness scales ===
Let's say we want to see what rank two temperament does Sym454 leap rule represent, 62\293 generator with 52/293 note count. 

We simply merge 52 & 293 in a selected limit to get our answer. Let's say 17 limit, we get 52 & 243c temperament with a comma list 225/224, 715/714, 2880/2873, 22750/22627 and 60112/60025.

Let's see what temperament does the Tabular Persian or Dee calendar offer (29 & 33). In the 5-limit, we get a contorted Lala-Quinyo (553584375:536870912). 

== Sonifications ==
* Maximally even heptatonic scale of [[19edo]] is the leap year arrangement of the [[Wikipedia:Hebrew calendar|Hebrew calendar]].
* Maximally even octatonic scale of [[33edo]] is a leap year arrangement of the Dee calendar and the tabular, evened version of the [[Wikipedia:Iranian calendars|Persian calendar]].

[[Category:Scale]]
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