Maximal evenness

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A maximally even (ME) scale is a scale inscribed in an equal-step tuning which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent tuning system), and whose steps are distributed as evenly as possible. In other words, such a scale satisfies the property of maximal evenness. These conditions infer that an ME scale is necessarily an MOS scale.

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. Mathematically, ME scales of n notes in m edo are any mode of the sequence ME(n, m) = [floor(i*m/n) | i = 1…n], where the floor function rounds down to the nearest integer.

A special case of the maximally even scale is the Irvian mode, which originates from a calendar reform to smoothly spread inaccuracies arising from the uneven number of days or weeks per year. 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. Every mode of any diatonic scale is maximally even, but not necessarily Irvian.

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 Maqamic[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.

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.

Real life counterparts

  • 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.