Irvian mode

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Irvian mode is the mode of the maximal evenness scale where the notes are symmetrically arranged.

The term was proposed in 2021 by Eliora, and it is named after a calendar designer Irvin Bromberg, whose Sym454 calendar's leap year patterns are arranged in the same way.


In 2004, Dr. Irvin Bromberg of University of Toronto developed a calendar called Symmetry454, 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 unsmooth distribution of days, weeks, months, and leap years. 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 mode of a maximally even scale, which is named an 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. In this paradigm, years correspond to individual steps of the scale, and leap years correspond to steps that are part of the mode. The length of the cycle is the size of an EDO.

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

In addition, there's two more numbers in this math - U and the accumulator. The U number corresponds to the generator in MOS theory. Bromberg does not provide a name for it, and the reason for this is unknown. Accumulator is the amount of generators for a given note, counting from the tonic.

The current, "canonical" usage of the cycle is that of 52 leap week years in 293 years - year is leap if the remainder of (52 x Year + 146)/293 is less than 52. Musically, this would correspond to a 33L 19s MOS scale of 293edo. 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. The U of this cycle is equal to 62, and it represends a mode of a maximal evenness 62\293 scale.

Even-length Irvian modes with odd number of years per cycle (that is notes) 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. Thus it is called almost symmetrical in the calendar lore. That being said, they fulfill their function just like odd cycles do, and therefore belong in Irvian modes.

Relationship to the standard Western music theory

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.

Owing to the definition of the "accumulator" as prescribed on the Sym454's "Solar Calendar Leap Rules" page, the accumulator of the first year is always equal to the number K, in this case 6. In this case, the generator U in question is equal to 7, coinciding with the note amount in the scale. The note whose accumulator is equal to 0 in this case is the 6th note, which is F on a piano. Counting 7 notes forward from F makes F, C, G, D, A, E, B, which goes through all white keys once and when octave-sorted is just the C-major. Thus, the way keys are arranged on a 12edo piano is the Irvian mode of the diatonic scale.

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. Choosing 5 instead of 6 for the K would produce a Lydian scale on C, parralel to an F major scale - patterns of keys are reversed. Having different K choices will go through all 7 modes of the diatonic scale such as Phrygian, Mixolydian, etc., however they aren't Irvian in their conception.

Other examples


3L 4s:

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


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.


L L s L L L s L L s


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


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

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


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.

See also

External links