Maximal evenness

Within every edo one can specify a maximally even (ME) scale for every smaller edo. The 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.

The maximally even scale will be one:

a. which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent edo).

b. whose steps are distributed as evenly as possible.

(a) and (b) above imply that the ME scale will be a moment of symmetry scale.

The probably most popular heptatonic ME scale is the major scale of 12edo: 2 2 1 2 2 2 1, but also every diatonic scale of 12edo is maximally even. Some more detailed examples follow.

A variant of the maximal evenness scale is the Irvian mode, which originates from leap year arrangements of Sym454 calendar (see below).

Irvian mode and a relation to a proposed calendar reform

In 2004, Dr. Irvin Bromberg of University of Toronto developed a calendar called Sym454, and a leap year pattern for the calendar that is symmetrical and as smoothly spread as possible. The calendar is proposed as a variant to replace Gregorian calendar's uneven distribution of dates. The goal of the initial pattern was to minimize divergence of calendar days from cardinal dates such as equinoxes, solstices, and "new year moments", however the pattern also has an interpretation in terms of MOS scale making and keyboard mapping.

Such a pattern produces a specific maximum evenness mode, which is named Irvian mode. A stand-alone leap week at the end of year in Sym454 lore is called Irvember, and therefore the constructed name of the mode would be Irvian.

The pattern is defined by the following:

Year is leap if the remainder of (L x Y + K)/ C is less than L.

L = number of leap years per cycle,

Y = number of the year

C = number of years per cycle

K = (C-1)/2 if odd, can choose between (C-1)/2 and C/2 if even

The current, "canonical" usage of the cycle is that of 52 leap week years in 293 years - year is leap if (52 x Year + 146)/293 is less than 52. Musically, this would correspond to a 33L 19s MOS scale. In addition, if the remainder of the leap year is less than the count of long intervals in the MOS, the next year will be in a long interval, otherwise in a short interval. For example here, this means if remainder is less than 33, next leap year (or key) will be 6 years later (6 steps above), otherwise 5 years later.

Even-length symmetrical cycles with an irreducible (that is odd) number of years per cycle have a feature where they aren't 100% symmetrical - two middle years follow a pattern of non-leap - leap. If the K is chosen as (C-1)/2 instead of C/2, the sequence will be leap, nonleap.

Every Irvian mode is a maximal evenness scale, but not every maximal evenness scale is Irvian.

Example on a standard 12edo piano

The 12edo piano key layout, which is predominantly use in the world today, is an example of an Irvian mode that is subject to even-length leap rule modification.

Year is leap if the remainder of (7 x Year + 6) / 12 is less than 7.

Such a pattern generates keys number 1-3-5-6-8-10-12-1 to be the keys on the scale, which is a 5L 2s scale in a pattern of LLsLLLs. White keys are leap years, and black keys are common years.

Years 1,3,6,8,10, that is notes C, D, F, G, A have a long interval - a tone - after them, while E and B, with remainder of 6, have a semitone. When started on C turns out to be plain C major. In this case, the accumulator K is taken to be C/2 instead of (C-1)/2 as with odd cycles, therefore middle of the cycle is nonleap-leap, that is F and F#. Choosing 5 instead of 6 for the K would produce a Lydian scale on C, or a F major scale - patterns of keys are reversed.

17edo

3L 4s:

Year is leap if the remainder of (7 x Year + 8) / 17 is less than 7

1-3-6-8-10-13-15

s L s s L s L.

Starting from the other key, it's bayati 3232322. 17edo is the only temperament where bayati is parallel to the Irvian mode.

7L 3s:

Year is leap if the remainder of (10 x Year + 8) / 17 is less than 10.

0-2-4-5-7-9-11-12-14-16-17

L L s L L L s L L s

Maqamic alternative as listed on the 17edo page:

0-2-4-6-7-9-11-12-14-16-17

L L L s L L s L L s

Such a scale ends up skipping the perfect fifth. Starting on a different note, the scale can be made to have a perfect fifth, for example:

0-1-3-5-7-8-10-12-13-15-17

s L L L s L L s L L

However, such note arrangements are not Irvian, although they are maximal evenness.

22edo

Year is leap if the remainder of (13 x Year + 11) / 22 is less than 13.

Orwell[13]:

0-2-4-5-7-9-10-12-14-16-17-19-21-0, proper Irvian mapping as directly taken from the formula.

Following mappings are ME but not Irvian:

0-2-3-5-7-8-10-12-14-15-17-19-20-22, as mentioned on the 22edo page.

Alternatives that do not skip the perfect fifth:

0-2-3-5-7-8-10-12-13-15-17-19-20-22

0-1-3-5-6-8-10-12-13-15-17-18-20-22

As it is tenuous to write out all the notes, this is a table of a few possible Irvian modes of 22edo:

Name	Formula core
Porcupine[15]	(15 x Year + 11) / 22
Superpyth[5]	(5 x Year + 11) / 22
Porcupine[7]	(7 x Year + 11) / 22

31edo

31 edo contains the following Irvian modes, derived from ME 31edo MOS scales:

31edo Irvian modes

Name	Formula core	Key layout
Würschmidt[3]	(3 x Year + 15) / 31	6-16-26
Myna[4]	(4 x Year + 15) / 31	4-12-20-28
Mothra[5]	(5 x Year + 15) / 31	4-10-16-22-28
Hemithirds[6]	(6 x Year + 15) / 31	3-8-13-19-24-29
Mohajira[7]	(7 x Year + 15) / 31	3-7-12-16-20-25-29
Nusecond[8]	(8 x Year + 15) / 31	2-6-10-14-18-22-26-30
Orwell[9]	(9 x Year + 15) / 31	2-6-9-13-16-19-23-26-30
Miracle[10]	(10 x Year + 15) / 31	2-5-8-11-14-18-21-24-27-30

and so on.

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], 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.

Trivia

• 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.

External links

• 52/293 Symmetry454 Leap Years

• Solar Calendar Leap Rules - subsection Symmetrical Leap Cycles