Maximal evenness
- Not to be confused with Distributional evenness.
A periodic binary scale is maximally even (ME) with respect to an equal-step tuning if it is the result of rounding a smaller equal tuning to the nearest notes of the parent equal tuning with the same equave. Equivalently, a scale is maximally even if its two step sizes are 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 scales, 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. An m-note maximally even scale in n-edo is the closest n-edo can get to representing m-edo.
Mathematics
Definition
Mathematically, if 0 < n < m, a maximally even (sub)set of size n in Z/mZ is any translate of the set
[math]\operatorname{ME}(n, m) = \{ m\mathbb{Z} + \lceil \frac{im}{n} \rceil : i \in \{0, ..., n-1\} \} \subseteq \mathbb{Z}/m\mathbb{Z},[/math]
where the ceiling function fixes integers and rounds up non-integers to the next higher integer. It can be proven that when n does not divide m, ME(n, m) is a MOS subset of Z/mZ where the two step sizes differ by exactly 1, and that the set of degrees where each step size occurs is itself maximally even in Z/nZ, satisfying the informal definition above. ME(n, m) is the lexicographically first mode among its rotations, and combined with the fact that it is a MOS, this implies that ME(n, m) is the brightest mode in the MOS sense.
It is easy to show that replacing ceil() with round() (rounding half-integers up) gives an equivalent definition; floor() does too, since ME(n, m) is a MOS and thus achiral.
Complement of a maximally even subset is maximally even
Proof sketch: We may assume that gcd(n, m) = 1; there are two cases.
Case 1: ME(n, m) where n < m/2. This is a maximally even subset of Z/mS with step sizes L > s > 1, which determines the locations of step sizes of 2 in the complement. The rest of the complement's step sizes are 1. The sizes of the chunks of 1 are L - 2 and s - 2 (0 is a valid chunk size), and the sizes form a maximally even MOS.
Case 2: ME(n, m) where n > m/2. This has step sizes 1 and 2. The chunks of 1 (of nonzero size since n > m/2) occupy a maximally even subset of the slots of ME(n, m) (*). Now replace each 1 with "|" and each 2 with "$|".
(e.g.) 2112111 -> $|||$||||
Consider the resulting binary word of "|" and "$". The "|"s form chunks of sizes that differ by 1 and are distributed in a MOS way by (*). The desired complement, occupied by the "$"'s, thus forms a maximally even subset.
Concoctic scales
The generator of a maximally even scale is a modular multiplicative inverse of it's number of notes and the EDO size. A maximal even 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 diatonic scale.
- The maximally even heptatonic sets of 17edo and 24edo, in contrary, are mosh scales (Neutrominant[7]).
- The maximally even heptatonic set of 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 temperaments with the right timbre.
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, 2 2 1 2 2 2 1, 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.
Quasi-equal scales
Note that "maximally even" is equivalent to "quasi-equal-interval-symmetrical" in Joel Mandelbaum's 1961 thesis 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.
Examples of quasi-equal scales include equipentatonic and equiheptatonic scales among others.
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 Hebrew calendar.
- Maximally even octatonic scale of 33edo is a leap year arrangement of the Dee calendar and the tabular, evened version of the Persian calendar.