Irvic scale
Irvic scale, or a smoothly spread symmetrical scale, is a type of MOS scale where intervals are arranged in a specific smoothly spread symmetrical order.
Irvic scales can be interpreted as a way to arrive at maximal evenness through an external concept.
Origin
In 2004, Dr. Irvin Bromberg developed a calendar called Sym454, and a leap year pattern that is symmetrical and as smoothly spread as possible. The goal of the initial pattern was to minimize divergence of calendar days from dates such as equinoxes and solstices, however the pattern also has an interpretation in terms of MOS scale making and keyboard mapping.
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 Irvic scale is a MOS scale, but not every MOS scale can be made into an Irvic scale.
Example on a standard 12edo piano
The 12edo piano key layout, which is predominantly use in the world today, is an example of an Irvic scale 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.
Other examples
Irvic scale also encompasses other previously discovered scales in different temperaments.
17edo
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 also Irvic.
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. However, 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
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 Irvic mapping as directly taken from the formula.
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 Irvic mappings 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 |